B.E.S.T.

ALGEBRA 2

Teacher’s Edition
Program Overview
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TEACHER’S EDITION PROGRAM OVERVIEW
CONTENTS

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

OVERVIEW of B.E.S.T. A|G|A

Algebra 2 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Algebra 2 Pacing Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Algebra 2 Honors Pacing Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
From the Authors: Program Goal and Organization . . . . . . . . . . . . . . . . . . . . . . 18
From the Authors: Instructional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Focused Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Connected Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Balanced Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Mathematical Thinking and Reasoning Standards . . . . . . . . . . . . . . . . . . . . . . . 25

USER’S GUIDE
Welcome to the enVision Florida B.E.S.T. A|G|A User’s Guide . . . . . . . . . 28
Program Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Using a Lesson: Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Using a Lesson: Explore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Using a Lesson: Understand & Apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Using a Lesson: Practice & Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Using a Lesson: Assess and Differentiate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Preparing for a Topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Mathematical Thinking and Reasoning Standards in This Program . . . . . . . . . . . . 52
Effective Mathematics Teaching Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
enVision STEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Building Literacy in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Supporting English Language Learners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Preparing Students for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Differentiated Instruction and Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Assessment Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Professional Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

CORRELATION
enVision Florida B.E.S.T. Algebra 2 Correlation . . . . . . . . . . . . . . . . . . . 72
enVision Florida B.E.S.T. A|G|A Correlation . . . . . . . . . . . . . . . . . . . . . 89

PROFESSIONAL DEVELOPMENT
Teaching for Understanding by Eric Milou . . . . . . . . . . . . . . . . . . . . . . 122
Facilitating Mathematical Discourse by Christine Thomas . . . . . . . . . . . 125
Developing Mathematical Modelers by Rose Zbiek . . . . . . . . . . . . . . . 128
Mathematical Thinking and Reasoning by Al Cuoco . . . . . . . . . . . . . 136
 About the Authors

Authors
Dan Kennedy, Ph.D
• Classroom teacher and the Lupton Distinguished Professor of
Mathematics at the Baylor School in Chattanooga, TN
• Co-author of textbooks Precalculus: Graphical, Numerical, Algebraic
and Calculus: Graphical, Numerical, Algebraic, AP Edition
• Past chair of the College Board’s AP Calculus Development Committee
• Previous Tandy Technology Scholar and Presidential Award winner

Eric Milou, Ed.D

• Professor of Mathematics, Rowan University, Glassboro, NJ
• Member of the author team for Savvas’ enVisionmath2.0 6-8
• Past President, Association of Mathematics Teachers of NJ
• Author of Teaching Mathematics to Middle School Students
• 2009 Max Sobel Outstanding Mathematics Educator Award

Christine D. Thomas, Ph.D

• Professor of Mathematics Education at Georgia State University, Atlanta, GA
• Past–President of the Association of Mathematics Teacher Educators (AMTE)
• Past NCTM Board of Directors Member
• Past member of the editorial panel of the NCTM journal Mathematics
Teacher
• Past co-chair of the steering committee of the North American chapter
of the International Group of the Psychology of Mathematics Education

Rose Mary Zbiek, Ph.D

• Professor of Mathematics Education, Pennsylvania State University,
College Park, PA
• Series editor for the NCTM Essential Understanding project

Contributing Author
Al Cuoco, Ph.D
• Lead author of CME Project, a National Science Foundation
(NSF)-funded high school curriculum
• Team member to revise the Conference Board of the Mathematical
Sciences (CBMS) recommendations for teacher preparation and
professional development
• Co-author of several books published by the Mathematical Association
of America and the American Mathematical Society
• Consultant to the writers of the PARCC Content Frameworks for high
school mathematics

4
OVERVIEW of B.E.S.T. A|G|A

Algebra 2 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Algebra 2 Pacing Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Algebra 2 Honors Pacing Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
From the Authors: Program Goal and Organization . . . . . . . . . . . . . . . . . . . . . . 18
From the Authors: Instructional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Focused Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Connected Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Balanced Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Mathematical Thinking and Reasoning Standards . . . . . . . . . . . . . . . . . . . . . . . 25
 Contents in Brief

B.E.S.T. ALGEBRA 2

1 Functions, Inequalities, and Systems................................................... 2A

2 Quadratic Functions and Equations................................................... 56A

3 Polynomial Functions.............................................................................. 122A

4 Rational Functions.................................................................................... 182A

5 Rational Exponents and Radical Functions.................................. 228A

6 Exponential and Logarithmic Functions........................................ 286A

7 Matrices.......................................................................................................... 346A

8 Probability..................................................................................................... 384A

Glossary............................................................................................................... G2

Additional Answers.................................................................................... AA1

Index..................................................................................................................... N2

Acknowledgments ........................................................................................ A1

6
TOPIC Functions, Inequalities, and Systems

1
Topic Overview.......................................................................................................... 2A

Topic Readiness Assessment................................................................................... 2H

Topic Opener.............................................................................................................2–3

............................................................................................................... 4

1-1 Key Features of Functions........................................................................ 5A

AR.3.8, AR.4.4, F.1.1, MTR.4.1, MTR.5.1, MTR.7.1

1-2 Transformations of Functions............................................................... 13A

AR.4.4, F.1.1, F.1.7, F.2.2, F.2.3, F.2.5, MTR.2.1, MTR.5.1, MTR.6.1

11-6
1-3 Honors Piecewise-Defined Functions.................................................. 23A
AR.9.10♦, MTR.1.1, MTR.5.1, MTR.7.1

11-6
1-4 Honors Arithmetic Sequences............................................................... 31A
AR.10.1♦, MTR.3.1, MTR.5.1, MTR.7.1

1-5 Quadratic and Absolute Value Inequalities........................................ 38A

AR.3.3, AR.3.8, AR.4.2, MTR.2.1, MTR.3.1, MTR.5.1

1-6 Linear Systems......................................................................................... 45A

AR.9.2, AR.9.3, AR.9.5, AR.9.7, MTR.2.1, MTR.3.1, MTR.5.1

Mathematical Modeling in 3 Acts:

Current Events.............................................................................................51
AR.9.3, AR.9.7, MTR.7.1

Topic Review ..............................................................................................................52

Topic Assessment.................................................................................................... 55A

Topic Performance Task......................................................................................... 55C

♦ Standard required for Algebra 2 Honors course only.

Honors Lessons address honors benchmark expectations. See each Lesson Overview for details.

7
 TOPIC Quadratic Functions and Equations

2
Topic Overview........................................................................................................ 56A

Topic Readiness Assessment................................................................................. 56H

Topic Opener........................................................................................................ 56–57

.............................................................................................................58

2-1 Vertex Form of a Quadratic Function.................................................. 59A

AR.3.4, AR.3.8, F.1.1, F.1.7, F.2.2, F.2.3, F.2.7, MTR.1.1, MTR.4.1, MTR.5.1

2-2 Standard Form of a Quadratic Function.............................................. 66A

DP.2.8, AR.3.4, AR.3.8, F.1.1, MTR.1.1, MTR.4.1, MTR.5.1

2-3 Factored Form of a Quadratic Function............................................... 74A

AR.1.1, AR.3.2, AR.3.4, AR.3.8, MTR.1.1, MTR.2.1, MTR.5.1

2-4 Complex Numbers and Operations...................................................... 81A

AR.3.2, NSO.2.1, MTR.6.1, MTR.7.1

Mathematical Modeling in 3 Acts:

Swift Kick.....................................................................................................88
AR.3.4, AR.3.8, MTR.7.1

2-5 Completing the Square........................................................................... 89A

AR.3.2, AR.3.4, AR.3.8, MTR.1.1, MTR.2.1, MTR.6.1

2-6 The Quadratic Formula........................................................................... 96A

AR.3.2, AR.3.8, MTR.3.1, MTR.4.1

2-7 Quadratic Inequalities.......................................................................... 103A

AR.3.9, AR.3.10, MTR.2.1, MTR.6.1, MTR.7.1

2-8 Systems Involving Quadratic Equations and Inequalities............. 109A

AR.9.2, AR.9.3, AR.9.5, AR.9.7, MTR.1.1, MTR.2.1, MTR.4.1

Topic Review ........................................................................................................... 116

Topic Assessment.................................................................................................. 121A

Topic Performance Task....................................................................................... 121C

Cumulative Assessment....................................................................................... 121E

8
TOPIC Polynomial Functions

3
Topic Overview...................................................................................................... 122A

Topic Readiness Assessment............................................................................... 122H

Topic Opener.................................................................................................... 122–123

.......................................................................................................... 124

3-1 Graphing Polynomial Functions.......................................................... 125A

AR.1.1, AR.6.5, MTR.1.1, MTR.2.1, MTR.5.1

3-2 Adding, Subtracting, and Multiplying Polynomials........................... 133A

AR.1.3, AR.1.6, F.1.7, F.3.2, MTR.4.1, MTR.6.1, MTR.7.1

11-6
3-3 Honors Polynomial Identities............................................................. 140A
AR.1.3, AR1.8, AR.1.11♦, MTR.1.1, MTR.2.1, MTR.5.1

11-6
3-4 Honors Dividing Polynomials............................................................. 148A
AR.1.5, AR.1.6, AR.1.8, AR.6.2♦, MTR.4.1, MTR.5.1, MTR.6.1

3-5 Zeros of Polynomial Functions............................................................ 156A

AR.1.8, AR.6.1, AR.6.5, F.1.1, MTR.1.1, MTR.5.1, MTR.7.1

Mathematical Modeling in 3 Acts:

What Are the Rules................................................................................. 164
AR.1.3, AR.1.6, AR.6.1, MTR.2.1, MTR.6.1, MTR.7.1

3-6 Roots of Polynomial Equations........................................................... 165A

AR.1.8, AR.6.1, MTR.1.1, MTR.4.1, MTR.5.1

3-7 Transformations of Polynomial Functions........................................ 171A

F.1.9, F.2.2, F.2.3, F.2.5, MTR.3.1, MTR.4.1, MTR.5.1

Topic Review ........................................................................................................... 179

Topic Assessment.................................................................................................. 181A

Topic Performance Task....................................................................................... 181C

♦ Standard required for Algebra 2 Honors course only.

Honors Lessons address honors benchmark expectations. See each Lesson Overview for details.

9
 TOPIC Rational Functions

4
Topic Overview...................................................................................................... 182A

Topic Readiness Assessment............................................................................... 182H

Topic Opener.................................................................................................... 182–183

.......................................................................................................... 184

4-1 Inverse Variation and the Reciprocal Function................................ 185A

AR.8.2, AR.8.3, F.1.1, F.2.2, MTR.1.1, MTR.5.1, MTR.7.1

4-2 Graphing Rational Functions............................................................... 193A

AR.8.2, AR.8.3, MTR.2.1, MTR.4.1, MTR.6.1

4-3 Multiplying and Dividing Rational Expressions............................... 202A

AR.1.9, MTR.2.1, MTR.4.1, MTR.6.1

4-4 Adding and Subtracting Rational Expressions................................. 209A

AR.1.9, MTR.3.1, MTR.5.1, MTR.7.1

4-5 Solving Rational Equations.................................................................. 216A

AR.8.1, AR.8.3, MTR.1.1, MTR.3.1, MTR.5.1

Mathematical Modeling in 3 Acts:

Real Cool Waters...................................................................................... 224
AR.1.9, AR.8.3, MTR.7.1

Topic Review ........................................................................................................... 225

Topic Assessment.................................................................................................. 227A

Topic Performance Task....................................................................................... 227C

Cumulative Assessment....................................................................................... 227E

10
TOPIC Rational Exponents and Radical Functions

5
Topic Overview...................................................................................................... 228A

Topic Readiness Assessment............................................................................... 228H

Topic Opener.................................................................................................... 228–229

.......................................................................................................... 230

5-1 nth Roots, Radicals, and Rational Exponents................................... 231A

NSO.1.3, MTR.3.1, MTR.4.1, MTR.5.1

5-2 Properties of Exponents and Radicals............................................... 239A

NSO.1.3, NSO.1.5, MTR.3.1, MTR.5.1, MTR.6.1

5-3 Graphing Radical Functions................................................................. 247A

AR.7.2, AR.7.3, F.1.7, F.2.2, F.2.3, F.2.5, MTR.2.1, MTR.5.1, MTR.7.1

5-4 Solving Radical Equations.................................................................... 255A

AR.7.1, AR.7.3, MTR.1.1, MTR.4.1, MTR.7.1

Mathematical Modeling in 3 Acts:

The Snack Shack....................................................................................... 264
AR.7.1, MTR.7.1

5-5 Function Operations............................................................................. 265A

F.3.2, F.3.4, MTR.1.1, MTR.2.1, MTR.6.1

5-6 Inverse Relations and Functions......................................................... 273A

F.3.4, F.3.6, F.3.7, MTR.1.1, MTR.2.1, MTR.4.1

Topic Review ........................................................................................................... 282

Topic Assessment.................................................................................................. 285A

Topic Performance Task....................................................................................... 285C

11
 TOPIC Exponential and Logarithmic Functions

6
Topic Overview...................................................................................................... 286A

Topic Readiness Assessment............................................................................... 286H

Topic Opener.................................................................................................... 286–287

.......................................................................................................... 288

6-1 Key Features of Exponential Functions............................................. 289A

AR.5.4, AR.5.5, AR.5.7, F.1.1, F.1.7, MTR.1.1, MTR.2.1, MTR.7.1

6-2 Exponential Models................................................................................ 297A

AR.5.4, AR.5.5, AR.5.7, F.1.1, FL.3.1, FL.3.2, FL.3.4, MTR.2.1, MTR.3.1, MTR.6.1

Mathematical Modeling in 3 Acts:

The Crazy Conditioning.......................................................................... 305
AR.5.4, AR.5.5, MTR.7.1

6-3 Logarithms.............................................................................................. 306A

NSO.1.6, AR.5.2, AR.5.7, F.3.7, MTR.2.1, MTR.4.1, MTR.5.1

6-4 Logarithmic Functions ......................................................................... 313A

AR.5.7, AR.5.8, AR.5.9, F.1.7, F.2.2, F.2.3, F.2.5, F.3.7, DP.2.9, MTR.1.1, MTR.5.1, MTR.7.1

6-5 Properties of Logarithms...................................................................... 321A

NSO.1.6, NSO.1.7, AR.5.2, MTR.4.1, MTR.5.1, MTR.6.1

6-6 Exponential and Logarithmic Equations........................................... 327A

AR.5.2, MTR.3.1, MTR.4.1, MTR.7.1

11-6
6-7 Honors Geometric Sequences............................................................. 334A
AR.10.2♦, MTR.4.1, MTR.6.1, MTR.7.1

Topic Review ........................................................................................................... 341

Topic Assessment.................................................................................................. 345A

Topic Performance Task....................................................................................... 345C

Cumulative Assessment....................................................................................... 345E

♦ Standard required for Algebra 2 Honors course only.

Honors Lessons address honors benchmark expectations. See each Lesson Overview for details.

12
TOPIC Matrices

7
Topic Overview............................................................................................. 346A

Topic Readiness Assessment........................................................................ 346H

Topic Opener............................................................................................346–347

.................................................................................................. 348

11-6
7-1 Honors Operations With Matrices................................................ 349A
NSO.4.1♦, NSO.4.3♦, MTR.2.1, MTR.3.1, MTR.5.1

11-6
7-2 Honors Matrix Multiplication....................................................... 357A
NSO.4.1♦, NSO.4.3♦, MTR.1.1, MTR.2.1, MTR.6.1

11-6
7-3 Honors Inverses and Determinants.............................................. 364A
NSO.4.4, MTR.1.1, MTR.3.1, MTR.4.1

11-6
7-4 Honors Inverse Matrices and Systems of Equations.................... 373A
NSO.4.2♦, NSO.4.4♦, MTR.4.1, MTR.5.1, MTR.7.1

Mathematical Modeling in 3 Acts:

The Big Burger.................................................................................. 380
NSO.4.2♦, NSO.4.4♦, MTR.7.1

Topic Review .................................................................................................. 381

Topic Assessment.......................................................................................... 383A

Topic Performance Task................................................................................ 383C

♦ Standard required for Algebra 2 Honors course only.

Honors Lessons address honors benchmark expectations. See each Lesson Overview for details.

13
 TOPIC Probability

8
Topic Overview...................................................................................................... 384A

Topic Readiness Assessment............................................................................... 384H

Topic Opener.................................................................................................... 384–385

Project............................................................................................. 386

11-6
8-1 Honors Probability Events................................................................... 387A
DP.4.1♦, DP.4.2♦, MTR.5.1, MTR.6.1, MTR.7.1

11-6
8-2 Honors Conditional Probability.......................................................... 396A
DP.4.3♦, DP.4.4♦, MTR.1.1, MTR.4.1, MTR.5.1

Mathematical Modeling in 3 Acts:

Place Your Guess...................................................................................... 403
DP.4.1♦, DP.4.9♦, MTR.7.1

11-6
8-3 Honors Permutations and Combinations.......................................... 404A
DP.4.9♦, DP.4.10♦, AR.1.11♦, MTR.1.1, MTR.2.1, MTR.3.1

Topic Review ........................................................................................................... 412

Topic Assessment.................................................................................................. 413A

Topic Performance Task....................................................................................... 413C

Cumulative Assessment....................................................................................... 413E

♦ Standard required for Algebra 2 Honors course only.

Honors Lessons address honors benchmark expectations. See each Lesson Overview for details.

14
 PACING GUIDE
B.E.S.T. ALGEBRA 2

enVision Florida B.E.S.T. Algebra 2 was designed to provide students rich opportunities to build understanding
of important new mathematical concepts and develop fluency with key skills as described in Florida’s B.E.S.T.
Standards and Benchmarks, and to develop expertise with the skills described in the Mathematical Thinking and
Reasoning Standards. To achieve these goals, the program includes content-focused core lessons, Mathematical
Modeling in 3 Acts lessons, and enVision STEM projects. All of these instructional activities are integral to helping
students achieve success, and the pacing of the program reflects this.
Teachers are encouraged to spend 3 days on each core lesson, giving students time to build deep understanding of
the concepts presented, 1 to 2 days for the Mathematical Modeling in 3 Acts lesson and 1 day for the STEM project.
This pacing allows for 1 to 2 days for each Topic Review and Topic Assessment, plus additional time of 2 to 3 days
per Topic to be spent on remediation, differentiation, and other assessment.
This pacing guide shows recommended pacing for both a 45-minute (Traditional) and a 90-minute (Block) math class.

Traditional Block

Core Core
Lessons Total Lessons Total

TOPIC 1 Functions, Inequalities, and Systems 12 days 16–17 days 6 days 9 days

TOPIC 2 Quadratic Functions and Equations 24 days 28–29 days 12 days 15 days

TOPIC 3 Polynomial Functions 21 days 25–26 days 11 days 14 days

TOPIC 4 Rational Functions 15 days 19–20 days 8 days 11 days

TOPIC 5 Rational Exponents and Radical Functions 18 days 22–23 days 9 days 12 days

TOPIC 6 Exponential and Logarithmic Functions 18 days 22–23 days 9 days 12 days

Total 108 days 132–138 days 55 days 73 days

15
 PACING GUIDE (continued)
B.E.S.T. ALGEBRA 2

Honors Traditional Block

Core Core
Lessons Total Lessons Total

TOPIC 1 Functions, Inequalities, and Systems 12 days 16–17 days 6 days 9 days

TOPIC 2 Quadratic Functions and Equations 16 days 20–21 days 8 days 11 days

TOPIC 3 Polynomial Functions 21 days 25–26 days 11 days 14 days

TOPIC 4 Rational Functions 15 days 19–20 days 8 days 11 days

TOPIC 5 Rational Exponents and Radical Functions 18 days 22–23 days 9 days 12 days

TOPIC 6 Exponential and Logarithmic Functions 21 days 25–26 days 11 days 14 days

TOPIC 7 Matrices 12 days 16–17 days 6 days 9 days

TOPIC 8 Probability 9 days 13–14 days 5 days 8 days

Total 124 days 146–154 days 64 days 88 days

16
NOTES
FROM THE AUTHORS A program organization that
PROGRAM GOAL AND ORGANIZATION promotes understanding!

VISION

Welcome to enVision Florida B.E.S.T. A|G|A. In this introduction to the Algebra 1,

Geometry, and Algebra 2 series, we share our major goals for enVision Florida B.E.S.T.
A|G|A, and explain how those goals were translated into the instructional model of the program.

PROGRAM GOAL

What were the major goals in developing enVision Florida B.E.S.T. A|G|A?
One major goal of enVision Florida B.E.S.T. A|G|A was to create a high school

“
mathematics program that reflects the latest research in mathematics education and learning
theory and supports all learners on their pathway to college- and career-readiness. To
One of the major achieve this goal, we developed a brand new program built from the ground up around three
goals in developing foundational principles:

enVision Florida • A balanced pedagogy. Research has shown that teaching for understanding requires
equal attention to helping students develop deep understanding of concepts, fluency with
B.E.S.T. A|G|A important processes and skills, and the ability to apply these concepts and skills to solve
real-world and mathematical problems. The organizing structure and the lesson features
was to create a of enVision Florida B.E.S.T. A|G|A provide students with appropriate and frequent
opportunities to develop understanding and proficiency, and apply these new concepts and
high school math skills to solve problems.

program that… • A focus on visual learning. Recent research (Park & Brannon, 2013) has found that
powerful learning occurs when students use different areas of the brain, specifically the
supports all learners area that governs symbolic thinking and the area that focuses on visual thinking. enVision
Florida B.E.S.T. A|G|A makes extensive use of visuals to encourage students to use these
on their pathway different parts of the brain, resulting in powerful learning.
to college- and • A focus on effective teaching and learning. Recent research has also shown that

”
career-readiness. students make significant academic gains when they explore “worthwhile tasks” and
engage in meaningful mathematical discourse using mathematical language. Research also
suggests that teachers need to create learning environments that facilitate and encourage
this meaningful discourse. Every lesson in enVision Florida B.E.S.T. A|G|A opens with
a worthwhile task, a student-centered activity that requires students to think critically and
construct sound mathematical arguments to defend their reasoning and their solutions. The
teacher support was created using the NCTM’s Guiding Principles for School Mathematics,
in particular Teaching and Learning. To ensure that teachers have the necessary tools to
facilitate meaningful mathematical discourse in the classroom, the program used the Effective
Mathematics Teaching Practices as a framework within which probing questions were
developed.

Figure 1: Algebra 1, Geometry, and Algebra 2 of enVision Florida B.E.S.T.

A|G|A available at SavvasRealize.com

18
 Digital

PROGRAM ORGANIZATION

In recent years, the mathematics education community has placed increasing emphasis on
having students be able to create mathematical models to explain real-world phenomena. This
process, teased out in detail in the GAIMME (Guidelines for Assessment and Instruction in
Mathematical Modeling Education) Report, is an iterative process (shown in Figure 2) in which
students think through a mathematical model for a given real-world phenomenon, apply the
model, analyze and assess the solution, and then iterate on the model as needed.

IDENTIFY AND MAKE

SPECIFY THE ASSUMPTIONS AND DO THE MATH:
PROBLEM TO BE DEFINE ESSENTIAL GET A SOLUTION
SOLVED VARIABLES

IMPLEMENT THE ITERATE AS NEEDED ANALYZE AND

MODEL AND REPORT TO REFINE AND ASSESS THE MODEL
THE RESULTS EXTEND THE MODEL AND THE SOLUTIONS

Figure 2

enVision Florida B.E.S.T. A|G|A was designed with a particular focus on mathematical
modeling and the mathematical modeling process. This focus can be seen in two key features,
the Mathematical Modeling in 3 Acts activities found in each topic of the program and the
Model & Discuss tasks that are found at the start of many lessons.

The Mathematical Modeling in 3 Acts tasks offer students real-world scenarios with
interesting problems. Students are given little information to start and are asked to come up
with solution conjectures to answer the main question. As students test out their conjectures by
coming up with a mathematical model and using that model to solve the problem, they are
working through the steps of the modeling process shown in Figure 2.

The Model & Discuss tasks focus on parts of the modeling process, helping students become
better modelers and build proficiency with the modeling process.

19
FROM THE AUTHORS A carefully designed instructional
INSTRUCTIONAL MODEL model to build understanding

INSTRUCTIONAL MODEL

What is the core instructional model for enVision Florida B.E.S.T. A|G|A?
Over the past twenty years, leaders in mathematics education have proposed a range of
reforms to bring about higher student achievement in mathematics. These reforms are both
curricular and pedagogical in nature. In addition, recent research in mathematics instruction
highlight the importance of having students actively engage in worthwhile, meaningful tasks.

The instructional model for enVision Florida B.E.S.T. A|G|A is grounded in these two
research foci.

“
An integral part of the instructional model is a focus on the thinking and reasoning skills that
help students’ deeper understanding of mathematics. Throughout every lesson are multiple
Research shows opportunities to help students develop expertise in these important skills.
that conceptual In addition, each topic features a lesson called Mathematical Modeling in 3 Acts that is
designed to engage students in the mathematical modeling process.
understanding is
STEP 1: EXPLORE STEP 2: UNDERSTAND & APPLY
developed when new Introduce concepts and procedures Make the important mathematics
mathematics with “worthwhile tasks.” explicit with enhanced direct
instruction connected to Step 1.
Research shows that conceptual
is introduced in understanding is developed when new The second step, Understand and Apply,
the context of solving mathematics is introduced in the context of
solving a real problem in which ideas related
is designed to connect students’ thinking
about the opening activity to the new ideas
a real problem in to the new content are embedded (Kapur, of the lesson. These concepts are presented
2010; Lester and Charles, 2003; Scott, through a series of visually rich example
which ideas related 2014). Conceptual understanding results types purposefully designed to promote
because the process of solving a problem understanding.
to the new content that involves a new concept or procedure

” requires students to make connections Conceptual Understanding examples

are embedded. between existing knowledge and the new present a key mathematical concept in
concept or procedure. The process of the lesson to help students develop deep
making connections between ideas builds understanding of the mathematical content.
understanding. In enVision Florida Skill examples focus on helping students
B.E.S.T. A|G|A, these problem-solving build fluency with skills. Finally, Application
experiences open every lesson. examples show students how mathematics
can be used to solve real-world problems.
Teacher support for each example includes
probing questions that help students make the
connections between their thinking from the
opening activity and the mathematics in the
examples. Further enhancing the learning for
students is the robust and engaging online
student and teacher experience available for
each lesson.

STEP 1: EXPLORE STEP 2: UNDERSTAND & APPLY

Introduce concepts and procedures Make the important mathematics explicit with
informally with a problem-solving enhanced direct instruction that connects
experience. back to the problem in Step 1.

20
 Digital

STEP 3: PRACTICE & PROBLEM SOLVING: STEP 4: ASSESS & DIFFERENTIATE:

Offer robust and balanced practice Check for understanding and provide
to solidify understanding. remediation and enrichment.
In Step 3, students embark on a series of enVision Florida B.E.S.T. A|G|A
carefully sequenced and crafted exercises to provides quality assessment and
apply what they just learned and to practice differentiation support. enVision Florida
toward mastery. The design of the Practice B.E.S.T. A|G|A offers diagnostic, formative,
& Problem Solving section is intentionally and summative assessments in print and
sequenced into four parts: Understand, digital formats. The digital assessments offer
Practice, Apply, and Assessment a wide range of item types that students
Practice. may encounter in their state-mandated
assessment, from open response questions,

“
First, with the goal of ensuring students’ multiple-response multiple-choice items to rich,
understanding of new concepts, students multi-part performance tasks. The inclusion The instructional
work through exercises focused on conceptual of performance tasks in the assessment
understanding in the Understand section. resources provides a deeper level of probing model for
Students are asked to show conceptual
understanding by explaining their reasoning,
into student learning, giving teachers a view
into student thinking around problem solving enVision Florida
constructing arguments, and analyzing
errors, among other tasks. Next, students
and conceptual understanding.
B.E.S.T. A|G|A
practice their newly learned skills and enVision Florida B.E.S.T. A|G|A
concepts with Practice exercises. Students includes a differentiated learning library
begins with a
apply their knowledge by working through
real-world examples in the Apply set
with an array of support to meet the
needs of all learners. The library includes
problem-based
of exercises. Finally, the Assessment Mathematical Literacy & Vocabulary, Reteach activity that connects
Practice section offers students practice for Understanding, Additional Practice, and
with high-stake assessment items, including Enrichment. enVision Florida B.E.S.T. students’ informal
a performance task. A|G|A also offers personalized study plans
to provide targeted remediation for students. mathematical
Some Final Thoughts understandings to
enVision Florida B.E.S.T. A|G|A has formal mathematical

”
connects students’ informal mathematical
been developed around an instructional
model with proven efficacy designed to
understandings to formal mathematical
concepts. The promise for enVision Florida
concepts.
• Prepare students for college and career B.E.S.T. A|G|A grows from the core
instructional model. However, there are many
• Build mathematicians skilled in thinking other elements that contribute to making
and reasoning enVision Florida B.E.S.T. A|G|A a
high-quality curriculum. The remainder of this
• Support the learning needs of all students Teacher’s Edition Program Overview shares
many of these.
This instructional model begins with a
problem-based learning activity that

STEP 3: PRACTICE & PROBLEM SOLVING STEP 4: ASSESS & DIFFERENTIATE

Offer robust and balanced practice Check for understanding and
to solidify understanding. provide remediation.

21
FOCUSED MATHEMATICAL
CURRICULUM FOCUS CONNECT BALANCE THINKING AND
REASONING

On a Pathway to College- and Career-Readiness…

High school graduates who are college- and career-ready have the B.E.S.T. A|G|A
math knowledge and skills to be successful either in their college
The program is organized specifically to provide students robust
studies or in any postsecondary job training. To be college- and
coverage of all of Florida’s B.E.S.T. Standards to promote
career-ready, students need to have studied a rigorous curriculum.
success in postsecondary programs and careers.
A rigorous mathematics curriculum is focused on key concepts that
See the Math Background sections of the Topic Overviews, and
students study in depth.
Lesson Overviews in the Teacher’s Edition.

Focus Within a Topic Focus Within a Lesson

• The Focus of a Topic At the start of a topic, the Essential • The Focus of a Lesson Each lesson includes an
Question focuses students on key ideas in the topic. Essential Question that focuses students’ thinking on the key
concepts of the lesson.

6-2
Exponential
EXPLORE & REASON
Juan is studying exponential growth of bacteria cultures. Each culture is
TOPIC Exponential and carefully controlled to maintain a specific growth rate. Copy and complete

6
Models the table to find the number of bacteria cells in each culture.
Logarithmic Functions Initial Number Growth Rate Time Final Number
TOPIC ESSENTIAL QUESTION
Culture of Bacteria per Day (days) of Bacteria

How do you use exponential and logarithmic functions to I CAN… write exponential A 10,000 8% 1
model situations and solve problems? models in different ways to B 10,000 4% 2
solve problems.
C 10,000 2% 4
VOCABULARY D 10,000 1% 8
Topic Overview Topic Vocabulary
enVision® STEM Project: • Change of Base Formula
• compound interest formula
Analyze Elections • common logarithm • continuously compounded A. What is the relationship between the daily growth rate and the time in
6-1 Key Features of Exponential Functions • compound interest interest formula days for each culture?
AR.5.4, AR.5.5, AR.5.7, F.1.1, F.1.7, MTR.1.1, MTR.2.1, MTR.7.1
• continuously compounded • natural base e
6-2 Exponential Models interest
B. Generalize Would you expect a culture with a growth rate of __12 % and a
AR.5.4, AR.5.5, AR.5.7, F.1.1, FL.3.1, FL.3.2, FL.3.4, DP.2.9,
MTR.2.1, MTR.3.1, MTR.6.1 • decay factor time of 16 days to have more or fewer cells than the others in the table?
Mathematical Modeling in 3 Acts: The Crazy Conditioning • exponential equation MA.912.FL.3.2–Solve real- Explain.
world problems involving simple,
AR.5.4, AR.5.5, MTR.7.1 • exponential function
compound and continuously
6-3 Logarithms • exponential decay function compounded interest. Also AR.5.4,
NSO.1.6, AR.5.2, AR.5.7, F.3.7, MTR.2.1, MTR.4.1, MTR.5.1
• exponential growth AR.5.5, AR.5.7, F.1.1, FL.3.1, How can you develop exponential models to
6-4 Logarithmic Functions function FL.3.4, DP.2.9 ESSENTIAL QUESTION
AR.5.7, AR.5.8, AR.5.9, F.1.7, F.2.2, F.2.3, F.2.5, F.3.7, MTR.1.1, represent and interpret situations?
MTR.5.1, MTR.7.1 • growth factor MA.K12.MTR.2.1, MTR.3.1,
• logarithm MTR.6.1
6-5 Properties of Logarithms
NSO.1.6, NSO.1.7, AR.5.2, MTR.4.1, MTR.5.1, MTR.6.1 • logarithmic equation EXAMPLE 1 Rewrite an Exponential Function to Identify a Rate
CONCEPT SUMMARY Writing Exponential Models
6-6 Exponential and Logarithmic Equations • logarithmic function
AR.5.2, MTR.3.1, MTR.4.1, MTR.7.1
• natural base e In 2015, the population of a small town was 8,000. The population is
6-7 Geometric Sequences • natural logarithm General
increasing at a rate Continuously
of 2.5% per year. Rewrite an exponential growth
AR.10.2, MTR.4.1, MTR.6.1, MTR.7.1 Exponential Model Compound Interest Compounded Interest
function to find the monthly growth rate.
ALGEBRA y=a∙b xA = P(1 + n ) A = Per nt
__ rt
Write an exponential growth function using the annual rate to model the
town’s population y, in
A necklace costs t years after
$250 2015.of $3,000 is
A principal A principal of
Digital Experience NUMBERS and increases in value by invested at 5% annual annual $3,000
growthisrate
invested
2% population
initial per year. interest, compounded at 5% continuously

• Students revisit the Essential Question at the end of the • Students answer the Essential Question years atafterthe
2015 end of the
INTERACTIVE STUDENT EDITION monthly, for 4 years.t compounded interest
Access online or offline. a = initial amount $250 y = 8,000(1 + 0.025)
P = 3,000 for 4 years.
FAMILY ENGAGEMENT b = growth factor 1.02 t
y = 8,000(1.025) P = 3,000

topic as they complete the Topic Review. lesson in the Do You Understand section.
Involve family in your learning. x = number of years r = 5%
ACTIVITIES Complete Explore & Reason, n = 12 compounding periods r = 5%
y = 250(1.02)x
Model & Discuss, and Critique & Explain To identify the monthly growth rate, you need the exponent to be the per year t = 4 years
activities. Interact with Examples and Try Its.
number of months in t years, or 12t.
t = 4 years A = 3000e (0.05)(4)
ANIMATION View and interact with (12)(4)
COMMON ERROR A = 3000(1 + ____
12 )
0.05
12 so that 12t
real-world applications.
y = 8,000(1.025) 12
___
12t
Multiply the exponent by __
PRACTICE Practice what Dividing the annual growth 12
Go online | SavvasRealize.com represents the number of months.
you’ve learned. rate by 12 does not give the __
1
exact monthly growth rate. This y = 8,000(1.025 12 )12t
TOPIC
Topic Review Example shows how to find an

6
286 TOPIC 6 Exponential and Logarithmic Functions Do You UNDERSTAND? Do You
Applying the Power of a KNOW HOW?
Power rule helps to
expression for the exact monthly y ≈ 8,000(1.00206
1.
) 12t
ESSENTIAL QUESTION Why doreveal
you the monthly growth ratefunction
The exponential by producing
models the annual
__
1

TOPIC ESSENTIAL QUESTION

rate: 1.025 12 − 1. an expression with
develop exponential models to represent rate oftheincrease.
exponent 12t.
Find the monthly and
HSM23_SEA2_FL_T06_TO.indd Page 286 13/02/21 2:00 PM eteam /154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ... and interpret situations? quarterly rates.
1. How do you use exponential and logarithmic functions to model t
situations and solve problems? The monthly growth rate is about 1.00206 −
2. Error Analysis The exponential model
1 ==0.00206
5. f(t) 2,000(1.03. )The population is
increasing about
y = 5,000(1.05 0.206%
) t represents theper month.
amount 6. f(t) = 500(1.055) t
Yori earns in an account after t years when
Vocabulary Review $5,000 is invested. Yori said the monthly Find the total amount of money in an account
interest rate of the exponential model is at the end of the given time period.
Choose the correct term to complete each sentence. Try It! 1. The population in a small town
5%. Explain Yori’s error.
is increasing annually by 1.8%.
• decay factor 7. compounded monthly, P = $2,000,
2. A(n) has base e. What is the quarterly rate of population increase?
• exponential function 3. Vocabulary Explain the similarities and r = 3%, t = 5 years
3. A(n) has the form f(x) = a ∙ b x.
• logarithmic function differences between compound interest and 8. continuously compounded, P = $1,500,
4. In an exponential function, when 0 < b < 1, b is a(n) .
• growth factor continuously compounded interest.
5. The is helpful for evaluating logarithms with a base r = 1.5%, t = 6 years
other than 10 or e. • common logarithm
4. Represent and Connect Write a math story
• natural logarithm LESSON
Write an6-2 Exponential
exponential model Models 297
given two points.
6. A(n) has base 10. that would use the general exponential
7. The inverse of an exponential function is a(n) . • Change of Base model to solve. Explain the meanings of the 9. (3, 55) and (4, 70)
Formula variables and parameters in your model.
10. (7, 12) and (8, 25)

11. Paul invests $6,450 in an account that earns

Concepts & Skills Review HSM23_SEA2_FL_T06_L02.indd 297
continuously compounded interest at an
1/11/21 7:40 PM
TOPIC 6 REVIEW

annual rate of 2.8%. What is the value of the

LESSON 6-1 Key Features of Exponential Functions account after 8 years?

Quick Review Practice & Problem Solving

An exponential function has the form Identify the domain, range, intercept, and
f(x) = a ∙ b x. When a > 0 and b > 1, the asymptote of each exponential function. Then
function is an exponential growth function. describe the end behavior.
8. f(x) = 400 ∙ (__1 )
When a > 0 and 0 < b < 1, the function is an x
exponential decay function. 2
9. f(x) = 2 ∙ (3) x
302 TOPIC 6 Exponential and Logarithmic Functions
Example 10. Represent and Connect Seth invests $1,400
Paul invests $4,000 in an account that pays 2.5% at 1.8% annual compound interest for
interest annually. How much money will be in 6 years. How much will Seth have at the
the account after 5 years? end of the sixth year?
HSM23_SEA2_FL_T06_L02.indd 302 1/11/21 7:40 PM
Write and use the exponential growth function 11. Apply Math Models Bailey buys a car for
model. $25,000. The car depreciates in value 18%
per year. How much will the car be worth
A(t) = a(1 + r) n after 3 years?
A(5) = 4,000(1 + 0.025) 5
12. Identify the domain, y
A(5) = 4,000(1.025) 5 range, intercept, and
4
A(5) = 4,525.63 asymptote.
2
There will be about $4,525.63 in Paul’s account
after 5 years. x
−4 −2 O 2 4

TOPIC 6 Topic Review 341

22 HSM23_SEA2_FL_T06_TR.indd 341 15/01/21 1:04 AM

CONNECTED MATHEMATICAL
CURRICULUM FOCUS CONNECT BALANCE THINKING AND
REASONING

On a Pathway to College- and Career-Readiness…

A rigorous mathematics curriculum is also a coherent one. B.E.S.T. A|G|A
Students see connections between and among concepts both
The program was also designed around a logical and coherent
within a course and across courses.
progression of concepts from course to course and clear
connections among concepts within each course.
To achieve coherence across and within courses, new content
is presented as an extension of prior learning. For example, the
opening activity found at the start of each lesson engages students
in a problem-based learning experience that helps them connect
their prior knowledge to the new concepts they will encounter in
Step 2 of the lesson: Understand & Apply.

Connections Across Grades Connections Across Topics and Lessons

• Looking Back and Looking Ahead Connections to • Connections Across Topics and Lessons The Lesson
content in previous courses or grades and in future grades Overview in the Teacher’s Edition describes connections
are highlighted in the Math Background pages of the Topic between lessons in the topic and to other lessons that
Overview in the Teacher’s Edition. precede or follow the lesson.

TOPIC 6 Honors LESSON 6-7

Exponential and Logarithmic Functions Geometric Sequences Glossary

MATH BACKGROUND Lesson Overview Vocabulary Builder

Objective REVIEW VOCABULARY English | Spanish
Students learn best when concepts are connected through the curriculum. This coherence is • arithmetic sequence | secuencia aritmética
achieved within topics, across topics, across strands, and across grade levels. Students will be able to:
✔ Construct a geometric sequence given a graph, table, or NEW VOCABULARY
description of a relationship. • common ratio | razón común
IN THIS TOPIC ✔ Translate between geometric sequences written in recursive • geometric sequence | secuencia geométrica
and explicit forms.
MAKING MATHEMATICAL CONNECTIONS How is content connected within Topic 6? MAKING MATHEMATICAL CONNECTIONS
VOCABULARY ACTIVITY
Looking Back • Exponential and Logarithmic Functions Looking Ahead Essential Understanding
Help students recall the relationship between arithmetic
How does Topic 6 connect to what Students use what they learn about the How does Topic 6 connect to what A geometric sequence is a sequence of numbers in which terms sequences and common differences. Explain that for geometric
students learned earlier? key features of exponential functions students will learn later? are related to the previous term by a common ratio, r. sequences, rather than the terms being separated by a common
in Lesson 6-1 to identify rates of difference, consecutive terms will have a common ratio. Have
ALGEBRA 1 growth of exponential functions and Pre-Calculus Previously in this course, students: students use the sentence frames to describe these relationships.
• Properties of Exponents In Topic 5 of solve problems involving compound • Exponential and Logarithmic Functions • Wrote arithmetic sequences recursively and then translated
Algebra 1, students used the Product, interest in Lesson 6-2. They also apply In Topic 6, students study the behavior 1, 3, 9, 27,… is a ____ because the terms are separated by a ____.
from a recursive formula to an explicit formula.
Quotient, and Power Properties of their knowledge from Lesson 6-1, along of exponential and logarithmic 2, 4, 6, 8,… is a ____ because the terms are separated by a ____.
with the new concept of exponential • Solved exponential equations.
Exponents to simplify exponential functions. In pre-calculus, they apply
expressions. In Topic 6 of Algebra 2, and logarithmic functions as inverses, their understanding of these effects to In this lesson, students:
students will recognize that the to identify key features of the graphs the graphs of trigonometric functions.
• Write geometric sequences recursively and with an explicit
Product, Quotient, and Power of logarithmic functions in Lesson 6-4. They also learn how the value of b
formula and translate between the two forms. Student Companion
Properties of Logarithms are very affects the period of the graph of a
similar to the Properties of Exponents, trigonometric function. • Determine the number of terms in a geometric sequence. Students can do their work for the lesson in their Student
• Exponential and Logarithmic Equations Companion or in Savvas Realize.
and will use the new properties to
solve logarithmic equations.
In Lesson 6-3, students learn how to
CALCULUS In later courses, students:
convert between exponential and • Will apply the concept of sequences and series to convergent
logarthmic forms of an expression and • Differentiating Logarithmic Functions
• Transformations of Functions In and divergent series.
how to evaluate logarithms. In In Lesson 6-4, students learn to
Algebra 1, Lessons 7-1 and 7-2, students
Lesson 6-5, they learn the Properties calculate the rate of change between
learned how a, h, and k affect the
of Logarithms. These two lessons two points of a logarithmic function. Honors Lesson 6-7 is required for Algebra 2 Honors course
graph of a function in the form
provide the background knowledge In Calculus, students will extend their only.
f(x) = a(x − h) 2 + k. In Topic 6, students
students need to solve exponential understanding of rate of change.
will apply that knowledge to graph Skills Students use the common ratio in a geometric sequence
and logarithmic equations in Students will use differentiation to
functions such as g(x) = a ∙ b (x − h) + k and use that to write a geometric series recursively. They also
Lesson 6-6. Students solve exponential calculate the rate of change of a
and j(x) = a ∙ log(x − h) + k. use a pattern to write an explicit definition for a sequence.
equations by rewriting expressions with logarithmic function at a single point.
a common base and converting them Applications Students apply geometric sequences to solve
to logarithmic form. They also solve real-world problems.
TOPIC 2 equations containing exponential and
• Linear-Quadratic Systems Students logarithmic expressions by graphing.
learned to solve systems of linear and
quadratic equations. In Topic 6,
• Exponential and Logarithmic Models
students will solve systems of linear
equations and exponential or
Throughout the topic, students model Florida’s B.E.S.T. Standards and Benchmarks Honors
real-life situations with exponential
logarithmic equations.
and logarithmic functions.
In this lesson, students focus on this benchmark: These standards is highlighted in this lesson:
MA.912.AR.10.2♦ Given a mathematical or real-world context, MA.K12.MTR.4.1 Engage in discussions that reflect on the
Painting Value
80 (5, 64.4)
write and solve problems involving geometric sequences. mathematical thinking of self and others.
Value (thousands of dollars)

60 (0, 40) Students justify whether a sequence is a geometric sequence by

40
identifying the common ratio of the sequence.
20
0
0 1 2 3
Years
4 5
MTR.7.1 Apply mathematics to real-world contexts.
Sculpture Value
Students use geometric sequences to model the number of people
f(x) = 50(1.075)x
(thousands of dollars in x years)
contacted through a phone tree.

♦ Benchmark required for Algebra 2 Honors course only.

TOPIC 6 286B TOPIC OVERVIEW TOPIC 6 334A LESSON 7

HSM23_TEA2_FL_T06_OV.indd 2 12/04/21 2:30 AM HSM23_TEA2_FL_T06_L07.indd 1 12/04/21 3:24 AM

23
 BALANCED MATHEMATICAL
CURRICULUM FOCUS CONNECT BALANCE THINKING AND
REASONING

On a Pathway to College- and Career-Readiness…

A rigorous mathematics curriculum focuses not just on skills B.E.S.T. A|G|A
and procedures; but equally on the concepts that underlie these
The instructional model is built on the three elements of rigor.
skills and the applicability of these concepts and skills.
The three example types -- Conceptual Understanding, Skill, and
Application -- were specifically designed to address these three
elements of rigor. The exericse set also follows this structure.
See the Math Background sections of the Topic Overviews, and
Lesson Overviews in the Teacher’s Edition.

Conceptual Understanding Procedural Fluency VOCABULARY EXAMPLE 5 Solve Logarithmic Equations

A logarithmic equation

• Problem-Based Learning The opening activity of the • Developing Procedural Fluency Through
contains one or more What is the solution to ln (x 2 – 16) = ln (6x)?
logarithms of variable
expressions. ln (x 2 − 16) = ln (6x) Write the original equation.

lesson helps students connect what they know to new Understanding Students develop procedural fluency
x 2 − 16 = 6x Property of Equality for Logarithmic Equations
x 2 − 6x − 16 = 0 Set quadratic equation equal to 0.

ideas embedded in the activity. When students make when the procedures make sense to them. In Step 2
(x − 8)(x + 2) = 0 Factor.
x = 8 or −2 Apply the Zero Product Property.

these connections, conceptual understanding takes seed. are examples designed to help students understand
Check Substitute each value into the original equation.
x=8 x = −2

procedures. In the Do you Know How? section of the

ln (8 2 − 16) = ln (6 ∙ 8) ln ((−2) 2 – 16) = ln (6 ∙ (−2))
ln (48) = ln (48) ✔ ln (−12) = ln (−12) ✘

lesson, students practice these procedures. Because logarithms are not defined for negative values, only x = 8 is a
solution. The value x = −2 is an extraneous solution.

Try It! 5. Solve each equation.

a. log 5 (x 2 − 45) = log 5 (4x) b. ln (−4x − 1) = ln (4x 2)

6-6
Exponential and
MODEL & DISCUSS
A store introduces two new models of fitness trackers to its product line. CHOOSE EFFICIENT
EXAMPLE 6 Solve Logarithmic and Exponential
Equations by Graphing
What is the solution to log (2x + 1) 5 = x − 2?
Powered By

A glance at the data is enough to see that sales of both types of fitness METHODS
Let y 1 = 5 log (2x + 1) and y 2 = x − 2.
Logarithmic trackers are increasing. Unfortunately, the store has limited space for the When typing this equation into
a calculator, it is helpful to write
Graph both equations.
Equations merchandise. The manager decides that the store will sell both models until the equation using the Power
sales of TrackSmart exceed those of FitTracker. Property of Logarithms rather Use the INTERSECT feature to find the
than risking incorrect input of the point(s) of intersection.
exponent.
The points of intersection, to the nearest
thousandth, are (−0.329, −2.329) and
I CAN… solve exponential FitTracker TrackSmart (8.204, 6.204). x scale: 1 y scale: 1
and logarithmic equations. Number Sold Number Sold
Check
VOCABULARY Week 4 228 130 log (2(−0.329) + 1) 5 = −0.329 − 2 log (2(8.204) + 1) 5 = 8.204 − 2
• exponential equation Week 3 112 44 log (0.342) 5 = −2.329 log (17.408) 5 = 6.204
• logarithmic equation Week 2 54 17 −2.329 = −2.329 ✔ 6.204 = 6.204 ✔
Week 1 28 5 The solutions are x ≈ −0.329 and x ≈ 8.204.
MA.912.AR.5.2–Solve one-
variable equations involving
logarithms or exponential
A. Apply Math Models Find an equation of an exponential function that Try It! 6. Solve each equation by graphing. Round to the nearest
expressions. Interpret solutions as models the sales for each fitness tracker. Describe your method. thousandth.
viable in terms of the context and
identify any extraneous solutions. B. Based on the equations that you wrote, determine when the store will a. 3(2) x+2 – 1 = 3 – x b. ln (3x – 1) = x – 5
MA.K12.MTR.3.1, MTR.4.1, stop selling FitTracker.
MTR.7.1

How do properties of exponents and

ESSENTIAL QUESTION 330 TOPIC 6 Exponential and Logarithmic
EXAMPLE 5 Functions
Solve Equations With Logarithms
logarithms help you solve equations?
What is the solution to each equation? Round to the nearest thousandth.
COMMON ERROR
A. 25 = 10 x−1
CONCEPT Property of Equality for Exponential Equations Remember that 10 is not 330
HSM23_SEA2_FL_T06_L06.indd a 10/01/21 5:14 AM
coefficient, but a base. You cannot 25 = 10 x−1

Applications
divide both sides by 10 and then
VOCABULARY Symbols Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y. add 1 to solve for x. log 25 = x − 1 Convert to logarithmic form.
An exponential equation is an 1 + log 25 = x Addition Property
Words If two powers of the same base are equal, then their exponents
equation that contains variables
are equal; if two exponents are equal, then the powers with 2.398 ≈ x Use calculator to evaluate.
in the exponents.
the same base are equal.

• Conceptual Understanding Examples In Step 2 • Application Examples Students encounter

B. ln (2x + 3) = 4
ln (2x + 3) = 4
2x + 3 = e 4

of the instructional model are designed to help students application examples that help them learn how to apply
Convert to exponential form.
EXAMPLE 1 Solve Exponential Equations Using a Common Base
2x + 3 ≈ 54.598 Use calculator to evaluate.
1 x+7
(2)
What is the solution to =4 ? __ 3x

develop deep =understanding Write the original of important math concepts. the newly learned mathematics to real-world contexts.
2x ≈ 51.598 Addition Property
1 x+7
(2)
4 __ 3x equation. x ≈ 25.799 Multiplication Property
x+7
(2 −1) = (2 2) 3x Rewrite each side with a common base.
Try It! 5. Solve each equation. Round to the nearest thousandth.
2 −x−7 = 2 6x Power of a Power Property
a. log (3x − 2) = 2 b. e x+2 = 8
−x − 7 = 6x Property of Equality for Exponential Equations
CONCEPTUAL
−7 = 7x
UNDERSTANDING Add
EXAMPLE 3 x to each with
Model side. Exponential Functions APPLICATION EXAMPLE 6 Use Logarithms to Solve Problems
−1 = x The population
Divideofeach
a large
sidecity
by was
7. about 4.6 million in the year 2010 and The seismic energy, x, in joules can be
grew at a rate of 1.3% for the next four years. Earthquake Magnitude Scale
estimated based on the magnitude, m, of an
A. What exponential function models the population of the city over that earthquake by the formula x = 10 1.5m+12. 2.5 or less
4-year period? What is the magnitude of an earthquake with
Try It! 1. Solve each equation using a common base. a seismic energy of 4.2 × 10 20 joules?
Compute the population for the first few years to look for a pattern.
2.5 to 5.4
USE PATTERNS AND
STRUCTURE
a. 25 3x = 125 x+2 b. 0.001
t (years since 2010) = 10 6x
p (population in millions) Formulate Substitute 4.2 × 10 20 for x in the formula.
Exponential functions of 4.2 × 10 20 = 10 1.5m+12
the form y = a ∙ b x involve 5.5 to 6.0
In 2011, there are 4.6 million 0 4.6
repeated multiplication by the people, and another 1.3% of Compute Solve the equation for m.
factor b. To understand how 4.6 million are added. 1 4.6 + 4.6(0.013) = 4.6(1.013) 6.1 to 6.9
to model population with an 4.2 × 10 20 = 10 1.5m+12 Write the original
exponential function, look for equation.
repeated multiplication in your
LESSON 6-6repeats
The process Exponential
2 and Logarithmic Equations
4.6(1.013)(1.013) 3272
= 4.6(1.013)
computations. each year. The 1 in log (4.2 × 10 20) = 1.5m + 12 Write the equation 7.0 to 7.9
1.013 represents the 3 4.6(1.013)3 in logarithmic form.
current population
and the 0.013 4 4.6(1.013) 4 20.6 ≈ 1.5m + 12 Evaluate the 8.0 or greater
represents the The exponent
logarithm.
HSM23_SEA2_FL_T06_L06.indd 327 yearly increase. is 10/01/21
the number
5:14 AM
of years since 5.75 ≈ m Solve for m.
2010.
Interpret The magnitude of the earthquake is about 5.75.
The population can be modeled by the exponential function: Verify the answer: 10 1.5(5.75)+12 ≈ 4.2 × 10 20

population after t years population in 2010 growth factor years since 2010
Try It! 6. What is the magnitude of an earthquake with a seismic energy
P = 4.6(1.013) t of 1.8 × 10 23 joules?

B. If the population continues to grow at the same rate, what will the
population be in 2040?
LESSON 6-3 Logarithms 309
To find the population in 2040, solve the equation for t = 30:
P = 4.6(1.013) 30 ≈ 6.78.
In 2040, the population will be about 6.78 million.
HSM23_SEA2_FL_T06_L03.indd 309 10/01/21 5:14 AM

Try It! 3. A factory purchased a 3D printer in 2012 for $15,000. The value
of the printer is modeled by the function f(x) = 15,000(0.93) x,
where x is the number of years since 2012.
a. What is the value of the printer after 3 years?
b. Does the printer lose more of its value in the first 3 years or
in the second 3 years after it was purchased?

24
LESSON 6-1 291
 MATHEMATICAL MATHEMATICAL
THINKING AND FOCUS CONNECT BALANCE
THINKING AND
REASONING STANDARDS REASONING

On a Pathway to College- and Career-Readiness…

A rigorous mathematics curriculum helps students deepen B.E.S.T. A|G|A
their understanding of important math concepts while also
Throughout the program, students encounter multiple opportunities
strengthening and refining the thinking and reasoning skills of
to help them build expertise with the Mathematical Thinking
mathematicians.
and Reasoning (MTR) Standards. In each lesson, students are
Developing these skills is integral to helping students’ learning reminded to think about different approaches and strategies
and understanding of mathematics. as they are guided through the examples or complete the
Try Its. In addition, teachers are provided with robust support
to facilitate mathematical discourse and ask Thinking and
Reasoning questions at appropriate moments throughout the
lesson. Mathematical Modeling in 3 Acts lessons found in
each topic, and the Model & Discuss, Explore & Reason,
and Critique & Explain tasks that open lessons provide
opportunities to focus on specific MTR Standards.

MTR Standards in Instruction MTR Standards in Practice

CONCEPT SUMMARY Writing Exponential Models

General Continuously
Exponential Model Compound Interest Compounded Interest

• Core Instruction Driven by a Marriage of Content • Do You Understand? and Do You Know How?
ALGEBRA y = a ∙ bx A = P(1 + __
n)
r nt
A = Pe rt

A necklace costs $250 A principal of $3,000 is A principal of

and MTR Standards MTR Standards are infused and offer many exercises that have students apply the MTR
NUMBERS and increases in value by invested at 5% annual $3,000 is invested
2% per year. interest, compounded at 5% continuously

explicitly highlighted in lesson instruction. Look for MTR

monthly, for 4 years. compounded interest

Standards as they solve problems

a = initial amount $250
P = 3,000 for 4 years.
b = growth factor 1.02

2 r = 5% P = 3,000
x = number of years

boxes throughout the lesson that model the thinking that

STEP Understand & Apply
n = 12 compounding periods r = 5%
Activity Assess
y = 250(1.02)x
per year t = 4 years
Graph Transformations of

these standards embody. (See also the Mathematical

EXAMPLE 2 t = 4 years A = 3000e (0.05)(4)
Logarithmic Functions, AVAILABLE ONLINE (12)(4)
f(x) = alogb(x − h) + k A = 3000(1 + ____
12 )
0.05

Thinking and Reasoning in This Program pages.)

EXAMPLE 2 Graph Transformations of Logarithmic Functions,
Use and Connect Mathematical f(x) = alogb (x − h) + k

Representations Graph the function. How do the asymptote and x-intercept of the given

 Notes
function compare to those of the parent function?

Q: How is graphing transformations of logarithmic functions the g(x) = log 2 (x + 3)

same as graphing transformations of exponential functions? COMMON ERROR In g, the value of h is –3, so the
4
y
g
Assess
How is it different?
With exponential functions,
the value of k dictates the shift
of the horizontal asymptote.
logarithmic function is translated
3 units to the left.
O
2

2
x Do You UNDERSTAND? Do You KNOW HOW?
[same: h = −3 implies a shift to the left; different: h causes the With logarithmic functions, the
asymptote is vertical. So the value
f(x) = log 2 x −2 f

Try It! Identify Key Features of Logarithmic Functions 1. Why do you The exponential function models the annual
g(x) = log 2 (x + 3) = f(x −(−3))
shift in the asymptote of a logarithmic function, whereas k of h determines the shift of the ESSENTIAL QUESTION
EXAMPLE 1 causes the shift of the asymptote in an exponential function.]
asymptote. The vertical asymptote and the x-intercept each shift 3 units to the left.

develop exponential models to represent rate of increase. Find the monthly and
Try It! 2. Describe how each graph compares to the graph of f(x) = ln x.
Try It! Answers and interpret situations? quarterly rates.
1. 2.Graph each function and identify the domain and range. List any intercepts
x 0.1 e −1 1 2 e e2
a. g(x) = ln x + 4 b.
h(x) −11.51 −5 0 3.47 5 10
a. translated up 4 units CONCEPTUAL 5. f(t) = 2,000(1.03) t
or asymptotes.
b. vertical stretch of 5 Describe the end behavior. UNDERSTANDING EXAMPLE 3 Inverses of Exponential and Logarithmic Functions 2. Error Analysis The exponential model
y = 5,000(1.05) t represents the amount 6. f(t) = 500(1.055) t
What is the equation of the inverse of the functions?
A. f (x) = 10 x+1

a. y = In x
THINKING AND REASONING
Use with EXAMPLES 1 & 2
b. y = log __1 x
GENERALIZE
f(x) is a translation of the parent
Write the function in y = f(x) form and then interchange x and y.

y = 10 x+1 Write the function in y = f(x) form. y

Yori earns in an account after t years when
Use Patterns and Structure Does the graph of either y = ln x + 4
function y = 10 x one unit left.
f –1(x) is a translation of the 2x = 10 y+1 Interchange x and y. f
$5,000 is invested. Yori said the monthly Find the total amount of money in an account
parent function y = log x one y + 1 = log x
or y = ln (x + 4) have an intercept that is different from the Write in log form. x

intercept of y = ln x? Explain.
unit down. Graphical translations
of a function and its inverse are
directly related, with horizontal
y = log x – 1 Solve for y. −2
−2
O
f−1
interest rate of the exponential model is at the end of the given time period.
The equation of the inverse of
and vertical effects switching f(x) = 10 x+1 is f –1 (x) = log x – 1. 5%. Explain Yori’s error.
[The x-intercept of y = ln x + 4 is different from the x-intercept places.
B. g(x) = log 7 (x + 5)
7. compounded monthly, P = $2,000,
of y = ln x, and y = ln x + 4 does not have a y-intercept. The Write the function in y = g(x) form and then interchange x and y.
3. Vocabulary Explain the similarities and r = 3%, t = 5 years
x-intercept of y = ln(x + 4) is different from the x-intercept of y = log 7 (x + 5) Write the function in y = g(x) form. y
2
y = ln x and y = ln(x + 4) has a y-intercept.] x = log 7 (y + 5) Interchange x and y.
x
differences between compound interest and
y + 5 = 7x Write in exponential form. −6 −4 −2 O 8. continuously compounded, P = $1,500,
y = 7x – 5 Solve for y. −2
continuously compounded interest.
Inverses of Exponential and The equation of the inverse of g(x) = log 7 (x + 5)
is g –1(x) = 7 x – 5.
g −1 −4 r = 1.5%, t = 6 years
EXAMPLE 3 −6
Logarithmic Functions
g
4. Represent and Connect Write a math story
Write an exponential model given two points.
Support Productive Struggle in Learning Try It! 3. Find the inverse of each function.
that would use the general exponential
Mathematics
a. f(x) = 3 x+2 b. g(x) = log 7 x – 2
model to solve. Explain the meanings of the 9. (3, 55) and (4, 70)
Q: Why does a horizontal shift in an exponential equation
314 TOPIC 6 Exponential and Logarithmic Functions
variables and parameters in your model.
become a vertical shift in its inverse.
10. (7, 12) and (8, 25)
[With inverse functions, the x and y values are interchanged. HSM23_SEA2_FL_T06_L04.indd Page 314 13/02/21 3:16 PM eteam /154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...

So a transformation that affected the x values of the original THINKING AND REASONING
Use with EXAMPLE 3 11. Paul invests $6,450 in an account that earns
function would affect the y values of its inverse.] Generalize How would you explain, in your own words, how to continuously compounded interest at an
find the inverse of a logarithmic function? annual rate of 2.8%. What is the value of the
Try It! Answers
[Solve the equation for the other variable by isolating the account after 8 years?
3. a. y = log 3 x − 2 logarithm, converting to exponential form, and then isolating
b. y = 7 x + 2 the desired variable.]

Struggling Students Advanced Students

USE WITH EXAMPLE 2 Some students may have difficulty USE WITH EXAMPLE 3 Have students explore finding the inverse
comparing the graph of the function to the parent function. of functions with two translations.
• Have students practice finding the x-intercept or the asymptote • Have students find the inverse of each function.
of each function.
1. f(x) = 2 x + 3 − 3 [f −1(x) = log 2(x + 3) − 3] 302 TOPIC 6 Exponential and Logarithmic Functions
Q: Find the x-intercept of g(x) = log (x − 5). [6] 2. g(x) = log 4(x − 4) − 1 [g −1(x) = 4 x + 1 + 4]
Q: Find the asymptote of h(x) = log (x + 3). [x = −3] 3. h(x) = −log(x + 1) + 6 [h −1(x) = 10 6 − x − 1]

Q: How does the graph of h(x) = log (x + 3) compare to the Q: How are the graphs of these functions and their inverses

EXAMPLE 2 Try It! Graph Transformations of Logarithmic Functions

parent function f(x) = log x? [The graph of h is shifted related?

• MTR Standards in Exercises Look for the red run-in

3 units to the left of the graph of f.] [The graphs of the inverses are reflections of the original HSM23_SEA2_FL_T06_L02.indd 302 1/11/21 7:40 PM

f(x) = a logb(x – h) + k
function across the line y = x.]

TOPIC 6 314 LESSON 4

2. Describe how each graph compares to the graph of f(x) = ln x.
a. g(x) = ln x + 4 b. x 0.1 e −1 1 2 e e2
heads on exercises. These highlighted exercises lend
themselves to a discussion of a specific MTR Standard.
HSM23_TEA2_FL_T06_L04.indd 314 12/04/21 3:16 AM

h(x) −11.51 −5 0 3.47 5 10

• Thinking and Reasoning Questions Facilitate
discussions with questions that help students develop their
thinking and reasoning skills.
PRACTICE & PROBLEM SOLVING

UNDERSTAND PRACTICE

9. Use Patterns and Structure Are the logarithmic Graph each function and identify the domain and
and exponential functions shown inverses of range. List any intercepts or asymptotes. Describe
each other? Explain. the end behavior. SEE EXAMPLE 1
y 14. y = log 5 x 15. y = log 8 x
4
16. y = log __
3 x 17. y = log 0.1 x
2 10

THINKING AND REASONING x Describe the graph in terms of transformations

of the parent function f(x) = log 6 x. Compare the
−4 −2 O 2 4
Use Patterns and Structure Does the graph of either y = ln x + 4 or y = ln (x + 4) −2
asymptote and x-intercept of the given function to
the parent function. SEE EXAMPLE 2
have an intercept that is different from the intercept of y = ln x? Explain. −4 1 log x
18. g(x) = __ 6 19. g(x) = log 6 (–x)
2
20. Describe how the graph of g(x) = – In(x + 0.5) is
10. Communicate and Justify How is the graph
related to the graph of f(x) = In x. SEE EXAMPLE 2
of the logarithmic function g(x) = log 2 (x – 7)
related to the graph of the function
Find the equation of the inverse of each function.
f(x) = log 2 x? Explain your reasoning.
SEE EXAMPLE 3

11. Error Analysis Describe and correct the error 21. f(x) = 5 x–3 22. f(x) = 6 x+7
a student made in finding the inverse of the
exponential function f(x) = 5 x–6 + 2. 23. f(x) = In (x + 3) – 1 24. f(x) = 4 log 2 (x − 3) + 2

25. The altitude y, in feet, of a plane t minutes

y = 5x–6 + 2 Write in y = f(x) form.
after takeoff is approximated by the function
x = 5y–6 + 2 Interchange x and y. y = 5,000 ln(.05t) + 8,000. Solve for t in terms
x – 2 = 5y–6 Subtract 2 from each side. of y. SEE EXAMPLE 4
log5 x – 2 = y – 6 Rewrite in logarithmic form.
log5 x – 2 + 6 = y Add 6 to each side. For items 26–29, let f(x) = e x, g(x) = In x,
2
log5 x + 4 = y Simplify. h(x) = e −x , and j(x) = In x 2. Find a simplified
f–1(x) = log5 x + 4 Rewrite as an
inverse function. ✗ expression for each composition of functions.
SEE EXAMPLE 5

26. f ∘ g 27. g ∘ f

12. Analyze and Persevere The number of 28. g ∘ h 29. f ∘ j

members m who joined a new workout center
30. Apply Math Models A scientist is conducting
296 TOPIC 6 Exponential and Logarithmic Functions Go Online | SavvasRealize.com w weeks after opening is modeled by the
an experiment with a pesticide. Use a calculator
equation m = 1.6 w+2, where 0 ≤ w ≤ 10. Find
to find an exponential model for the data in
the inverse of the function and explain what
the table. Use the model to determine how
the inverse tells you.

25
much pesticide remains after 180 days. Then
13. Use Patterns and Structure The graph shows a transform the function so it graphs a straight
transformation of the parent graph f(x) = log 3 x. line. SEE EXAMPLE 6
Write an equation for the graph. Day 0 20.00g

y Day 1 14.73g

4 Day 2 11.29g
NOTES
 B.E.S.T. A|G|A USER’S GUIDE
Welcome to the enVision Florida B.E.S.T. A|G|A User’s Guide . . . . . . . . . 28
Program Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Using a Lesson: Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Using a Lesson: Explore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Using a Lesson: Understand & Apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Using a Lesson: Practice & Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Using a Lesson: Assess and Differentiate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Preparing for a Topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Mathematical Thinking and Reasoning Standards in This Program . . . . . . . . . . . . 52
Effective Mathematics Teaching Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
enVision STEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Building Literacy in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Supporting English Language Learners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Preparing Students for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Differentiated Instruction and Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Assessment Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Professional Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
 WELCOME TO THE B.E.S.T. A|G|A The right program for you

USER’S GUIDE and your students!

Just a few highlights …

Instructional model supports student learning in print, digital,

or blended classrooms.

Student Edition provides all instructional content in a four-color, Write-in Student Companion encourages students’ active
visually engaging format. engagement during in-class instruction. Students take ownership of
their learning as they take notes, work out problems on their own,
and strengthen their mathematical thinking and reasoning skills.

B.E.S.T. GEOMETRY B.E.S.T. ALGEBRA 2

B.E.S.T. ALGEBRA 2 TOPIC 1 TOPIC 1
Foundations of Geometry Functions, Inequalities, and Systems
TOPIC 1
Functions, Inequalities, and Systems TOPIC 2 TOPIC 2
Parallel and Perpendicular Lines Quadratic Functions and Equations
TOPIC 2 B.E.S.T. ALGEBRA 1
Quadratic Functions and Equations TOPIC 3 TOPIC 3
B.E.S.T. ALGEBRA 1
Transformations TOPIC 1 Polynomial Functions
TOPIC 3 Solving Equations and Inequalities
TOPIC 1 Polynomial Functions TOPIC 4 TOPIC 4
Solving Equations and Inequalities Triangle Congruence TOPIC 2 Rational Functions
TOPIC 4
Linear Equations
TOPIC 2 Rational Functions
TOPIC 5 TOPIC 5
Linear Equations Relationships in Triangles TOPIC 3
TOPIC 5 Rational Exponents and Radical Functions
Rational Exponents and Radical Functions Linear and Absolute Value Functions
TOPIC 3
TOPIC 6 TOPIC 6
Linear and Absolute Value Functions
TOPIC 6 Quadrilaterals TOPIC 4 Exponential and Logarithmic Functions

B.E.S.T. GEOMETRY

B.E.S.T. ALGEBRA 2
TOPIC 4 Exponential and Logarithmic Functions Systems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities TOPIC 7 TOPIC 7
TOPIC 7 Similarity TOPIC 5
Matrices
Matrices
TOPIC 5 Exponents and Exponential Functions
Exponents and Exponential Functions TOPIC 8
B.E.S.T. GEOMETRY

TOPIC 8
B.E.S.T. ALGEBRA 2

TOPIC 8
Right Triangles and Trigonometry TOPIC 6 Probability
TOPIC 6 Probability
Polynomials and Factoring

B.E.S.T. ALGEBRA 1
Polynomials and Factoring TOPIC 9
Coordinate Geometry TOPIC 7
TOPIC 7
Quadratic Functions
Quadratic Functions
TOPIC 10
STUDENT COMPANION

Circles TOPIC 8

STUDENT COMPANION
B.E.S.T. ALGEBRA 1

TOPIC 8
Solving Quadratic Equations Solving Quadratic Equations
TOPIC 11
TOPIC 9 Two- and Three-Dimensional Models TOPIC 9
Working With Functions Working With Functions

TOPIC 10 TOPIC 10
Analyzing Data

STUDENT COMPANION
Analyzing Data
STUDENT COMPANION STUDENT COMPANION
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28
 Digital

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B.E.S.T. ALGEBRA 2
TE TE B.E.S.T. GEOMETRY
VOL 1
VOL 1
B.E.S.T. GEOMETRY B.E.S.T. ALGEBRA 2

) VOLUME 1 (See Volume 1 for Topics 1–4.)

Topic 1 Functions, Inequalities, and Systems

Topic 2 Quadratic Functions and Equations B.E.S.T. ALGEBRA 1
Topic 3 Polynomial Functions TE en español

VOL 1
Topic 4 Rational Functions
B.E.S.T. ALGEBRA 1

VOLUME 1 (See Volume 1 for Topics 1–5.)

VOLUME 2 (See Volume 2 for Topics 5–8.)
Topic 1 Solving Equations and Inequalities
Topic 5 Rational Exponents and Radical Functions
Topic 2 Linear Equations
) Topic 6 Exponential and Logarithmic Functions

B.E.S.T. ALGEBRA 2
B.E.S.T. GEOMETRY
Topic 3 Linear and Absolute Value Functions
Topic 7 Matrices
Topic 4 Systems of Linear Equations and Inequalities
Topic 8 Probability
Topic 5 Exponents and Exponential Functions

VOLUME 2 (See Volume 2 for Topics 6–10.)

B.E.S.T. ALGEBRA 1
B.E.S.T. ALGEBRA 2

Topic 6 Polynomials and Factoring

B.E.S.T. GEOMETRY

ASSESSMENT SOURCEBOOK
Topic 7 Quadratic Functions

ASSESSMENT SOURCEBOOK
Topic 8 Solving Quadratic Equations
Topic 9 Working With Functions
Topic 10 Analyzing Data
B.E.S.T. ALGEBRA 1

ASSESSMENT SOURCEBOOK
Teacher’s Edition Teacher’s Edition ASSESSMENT SOURCEBOOK
ASSESSMENT SOURCEBOOK
VOLUME 1 VOLUME 1
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ASSESSMENT SOURCEBOOK B.E.S.T. ALGEBRA 2
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B.E.S.T. ALGEBRA 1 en español

Resources to Support All Learners include remediation, SavvasRealize.com offers flexibility in planning, teaching,
practice, and enrichment activities as well as support for and progress monitoring. It’s easy to navigate, assign
English Language Learners and struggling readers. resources, search, customize, plan, assess, and analyze data.

Name B.E.S.T. ALGEBRA 2

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6-7 Mathematical Literacy and Vocabulary

Geometric Sequences

For Items 1–9, match the recursive definition in Column A with the explicit
Name B.E.S.T. ALGEBRA 2
definition in Column B. An example has been given.
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6-7 Enrichment Column A Column B

3, if n = 1
Geometric Sequences Ex. an = a. an = a1r n–1
an–1(6), if n > 1

Find the common ratio between the terms going horizontally and vertically on the a , if n = 1
1. an = a 1 (r), if n > 1 b. an = x(y)n–1
n–1
table. Then complete the table for the missing terms.
25, if n = 1
Name B.E.S.T. ALGEBRA 2
2 1,600,000 2. an = 1 c. an = 2(2.6)n–1
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an–1 5 , if n > 1

6-7 Reteach to Build Understanding 5 800,000

3. an =
2, if n = 1 1
d. an = 25 5 n–1
240 an–1(4), if n > 1
Geometric Sequences
48 250 16,000 4, if n = 1
4. an = e. an = 2(4)n–1
an–1(2), if n > 1
1. Complete the table to find the common ratio of the geometric sequence.
Then, use the table to answer the questions. 2.6, if n = 1
5. an = f. an = 3(6)n–1
an–1(2), if n > 1
1 1
Common Ratio (r) 16 64
x, if n = 1
Term Number Term
term in row
1 1 6. an = g. an = 1 (25)n–1
term in previous row 8 64
an–1(y), if n > 1 5

No previous term.
Name B.E.S.T. ALGEBRA 2

1 a1 = 3 16 2, if n = 1 SavvasRealize.com
7. an = h. an = 4(2)n–1
2 a2 = 6
6 =2
3
32 2 6-7 Additional Practice an–1(2.6), if n > 1

a3 = 12 12 = y, if n = 1
3 6 Geometric Sequences 8. an = i. an = 2.6(2)n–1
an–1(x), if n > 1
4 a4 = 24 24 = 1 25 625
12 1
, if n = 1
Is the sequence geometric? If so, write a9.recursive
an = 5 definition for thej. sequence.
an = y(x)n–1
an–1(25), if n > 1
a. If the common ratio is equal for each term, then the sequence is geometric. 1 1. 3, 9, 27, 81, … 2. 4, 8, 12, 16, ... 3. 1, 0.5, 0.25, 0.125, ...
Is this a geometric sequence? Circle the correct answer. Yes No 9

b. The first term number is n. Circle n in the table. The term is a 1. 1

1
What is the value of a 1? 9

{a n − 1(r), if n > n
a 1, if n = n 24
c. What is the recursive definition if a n = ?
Translate between the recursive and explicit definitions for each sequence.

{a n − 1(
, if n =
an = 2, if n = 1
{a n − 1(4), if n > 1 {a n − 1(__
), if n > 5, if n = 1 2 (7) n − 1
4. a n = 5. a n = __ 6. a n =
4)
3 3
, if n > 1
2. Leo says that the missing number in the geometric sequence 30, ,
120 is 90. Is he correct? What should the number be?

enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources

Write the first 6 terms of each sequence.

3. How many terms are in the finite geometric sequence 4, 12, 36, ..., 972? 7. an = −2(3)n − 1 8. an = 3(2)n − 1
HSM23_ANCA2_FL_T06_L07_ML.indd Page 2 06/04/21 7:49 PM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/T ...

Find the common ratio. ___

36 =
Write an explicit definition for each geometric sequence. Then find what term the
12
Then solve for n. 972 = 4( ) n − 1 last number of the sequence is.
enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources
9. __
1 , __
1 , ___
1 , …, ____
1
= ( )n − 1 3 6 12 192

log = (n − 1)log
single # x4
HSM23_ANCA2_FL_T06_L07_EN.indd Page 1 06/04/21 7:49 PM f-0317a
10. −3, −12, −48, …, −3072
/155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/T ...

_______
log
= (n − 1)
log
=n 11. 6, 30, 150, …, 3750
enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources

12. A geometric sequence can be used to describe the population of rabbits on a

farm. The first spring, the farmer purchased 8 rabbits. Five years later, there are
HSM23_ANCA2_FL_T06_L07_RE.indd Page 1 06/04/21 7:49 PM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/T ...
648 rabbits at the farm. Assuming that none of the rabbits leave the farm, how
many rabbits were on the farm in year 3?

enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources

HSM23_ANCA2_FL_T06_L07_AP.indd Page 2 06/04/21 7:49 PM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/T ...

29
WELCOME TO THE B.E.S.T. A|G|A The right program for you

USER’S GUIDE and your students!

Just a few highlights …

Lesson features provide balance among conceptual understanding,

procedural fluency, and application.
Every lesson opens with an engaging activity designed to help students be ready for new learning.
There are three different opening activities:

Model & Discuss Students are presented with a situation that

requires them to engage with an element of the mathematical 6-6 MODEL & DISCUSS

modeling process.
A store introduces two new models of fitness trackers to its product line.
Exponential and A glance at the data is enough to see that sales of both types of fitness
Logarithmic trackers are increasing. Unfortunately, the store has limited space for the
Equations merchandise. The manager decides that the store will sell both models until
sales of TrackSmart exceed those of FitTracker.

I CAN… solve exponential FitTracker TrackSmart

and logarithmic equations. Number Sold Number Sold

VOCABULARY Week 4 228 130

• exponential equation Week 3 112 44
• logarithmic equation Week 2 54 17
Week 1 28 5
MA.912.AR.5.2–Solve one-
variable equations involving
logarithms or exponential
A. Apply Math Models Find an equation of an exponential function that
expressions. Interpret solutions as models the sales for each fitness tracker. Describe your method.
viable in terms of the context and
identify any extraneous solutions. B. Based on the equations that you wrote, determine when the store will
MA.K12.MTR.3.1, MTR.4.1, stop selling FitTracker.
MTR.7.1

How do properties of exponents and

ESSENTIAL QUESTION
logarithms help you solve equations?

CONCEPT Property of Equality for Exponential Equations

VOCABULARY Symbols Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y.

An exponential equation is an
Words If two powers of the same base are equal, then their exponents
equation that contains variables
are equal; if two exponents are equal, then the powers with
in the exponents.
the same base are equal.

Critique & Explain Students evaluate examples of mathematical

reasoning and critique the reasoning as appropriate. In 6-3 EXAMPLE 1 Solve Exponential Equations Using a Common Base
Powered By

CRITIQUE & EXPLAIN x+7

What is the solution to (__
2)
1 = 4 3x?
all instances, students are asked to construct mathematical Logarithms
Earthquakes
1 x+7 = 4 3x
( 2 ) the
__
through
make seismic waves
ground. The equation
Magnitude, x
Write the original equation. Amplitude, y

0 x) x+7 ) 3x height, or Rewrite each side with a common

arguments.
y =(21−1 = (2 2the
relates 2 base. 100
amplitude, in microns, of a seismic 3 1,000
wave, 2 −x−7
y, and
6x
= 2the Power of a Power Property
power, or magnitude,
? 5,500
x, of−xthe−7 ground-shaking
= 6x it canProperty
cause.of Equality for Exponential Equations
I CAN… evaluate and 4 10,000
−7 = 7x Add x to each side.
simplify logarithms.
Taylor and Chen
−1 = x used differentDivide
methods to find
each side by 7. the magnitude of the
VOCABULARY earthquake with amplitude 5,500.
• common logarithm
• logarithm Try It! 1. Solve each equation using a common base.
• logarithmic function Taylor Chen
a. 25 3x = 125 x+2 b. 0.001 = 10 6x
• natural logarithm 5,500 is halfway between y = 10x
1,000 and 10,000.
103 = 1,000
MA.912.AR.5.2–Solve one-
variable equations involving 3.5 is halfway between 3 104 = 10,000
logarithms or exponential
expressions. Interpret solutions as
and 4. 103.5and
LESSON 6-6 Exponential ≈ 3,162
Logarithmic Equations 327
viable in terms of the context and 103.7 ≈ 5,012
identify any extraneous solutions.
Also NSO.1.6, AR.5.7, F.3.7 The magnitude is about 3.5. 103.8 ≈ 6,310
10 3.74 ≈ 5,500
MA.K12.MTR.2.1, MTR.4.1,
MTR.5.1
HSM23_SEA2_FL_T06_L06.indd 327 The magnitude is about 3.74. 10/01/21 5:14 AM

A. What is the magnitude of an earthquake with amplitude 100,000? How

do you know?
B. Communicate and Justify Critique Taylor’s and Chen’s work. Is each
method valid? Could either method be improved?
C. Describe how to express the exact value of the desired magnitude.

Explore & Reason Students explore a mathematical concept and What are logarithms and how are they

6-5
ESSENTIAL QUESTION
evaluated?

use reasoning to draw conclusions. CONCEPTUAL

UNDERSTANDING
EXPLORE & REASON
EXAMPLE 1 Understand Logarithms
Powered By

Look at the graph of y = log x and the ordered pairs shown.

Properties of Solve the equations 2x = 8 and 2 x = 8.
y
Logarithms 4 y8 = log x
Division is the inverse of multiplication,
You can use inverse 2x = __
___ so you can divide both sides by 2 to
operations to solve the 2 (15,
2 1.176)
2 (5, 0.699) solve the equation.
(3, 0.477) x=4
first equation. x
I CAN… TheO operation
2 4 in 26x = 88 is exponentiation.
10 12 14 To solve this equation, you need
USEproperties
PATTERNSofAND an
use logarithms −2inverse for exponentiation that answers the question, “To what exponent
STRUCTURE
to rewrite expressions. would you raise the base 2 to get 8?”
Creating the notation log 2 x to
VOCABULARY
represent the exponent to which The inverse ofthe
A. Complete applying an exponential
table shown.
•you raise 2of
Change to Base
get x Formula
is similar to_ function is applying a logarithm
creating the radical notation √x x This is read “logarithm base 2 of 8”
function. To solve
3 the5equation 15
to represent one number you can x or “log base 2 of 8.”
2 = 8log x can write log 28 = x.
, you
square to get x.
MA.912.NSO.1.6–Given a
Solving this gives log 28 = 3 because 2 3 = 8.
numerical logarithmic expression,
evaluate and generate equivalent
B. Use Patterns and Structure What is the relationship between
CONTINUED the NEXT PAGE
ON THE
numerical expressions using
the properties of logarithms or numbers 3, 5, and 15? What is the relationship between the logarithms
exponents. Also NSO.1.7, AR.5.2 of 3, 5, and 15?
TOPIC 6 MTR.5.1,
MA.K12.MTR.4.1,
306 Exponential and Logarithmic Functions
MTR.6.1 C. What is your prediction for the value of log 45? log 75? Explain.

How are the properties of logarithms used

HSM23_SEA2_FL_T06_L03.indd 306 ESSENTIAL QUESTION to simplify expressions and solve logarithmic 27/01/21 3:40 PM

equations?

CONCEPT Properties of Logarithms

For positive numbers b, m, and n with b ≠ 1, the following properties hold.

log b mn = log b m + log b n Product Property of Logarithms

m = log m − log n
log b __
n b b Quotient Property of Logarithms

log b m n = n log b m Power Property of Logarithms

30 EXAMPLE 1 Prove a Property of Logarithms

LEARN TOGETHER How can you prove the Product Property of Logarithms?
How do you listen actively as Let x = log b m and y = log b n. Then b x = m and b y = n.
 Digital

Different example types ensure that all aspects of rigor are addressed. EXAMPLE 5 Solve Equations With Logarithms

What is the solution to each equation? Round to the nearest thousandth.

COMMON ERROR
A. 25 = 10 x−1
Remember that 10 is not a
coefficient, but a base. You cannot 25 = 10 x−1
divide both sides by 10 and then
add 1 to solve for x. log 25 = x − 1 Convert to logarithmic form.
1 + log 25 = x Addition Property
2.398 ≈ x Use calculator to evaluate.
B. ln (2x + 3) = 4

Conceptual Understanding Examples develop conceptual Application Examples walk students through applying concepts
ln (2x + 3) = 4
2x + 3 = e 4 Convert to exponential form.

understanding. and skills in real-world scenarios.

2x + 3 ≈ 54.598 Use calculator to evaluate.
2x ≈ 51.598 Addition Property
x ≈ 25.799 Multiplication Property

Try It! 5. Solve each equation. Round to the nearest thousandth.

a. log (3x − 2) = 2 b. e x+2 = 8

CONCEPTUAL
UNDERSTANDING EXAMPLE 3 Model with Exponential Functions APPLICATION EXAMPLE 6 Use Logarithms to Solve Problems

The population of a large city was about 4.6 million in the year 2010 and The seismic energy, x, in joules can be Earthquake Magnitude Scale
grew at a rate of 1.3% for the next four years. estimated based on the magnitude, m, of an
earthquake by the formula x = 10 1.5m+12. 2.5 or less
A. What exponential function models the population of the city over that
4-year period? What is the magnitude of an earthquake with
a seismic energy of 4.2 × 10 20 joules?
Compute the population for the first few years to look for a pattern. 2.5 to 5.4
USE PATTERNS AND Formulate Substitute 4.2 × 10 20 for x in the formula.
STRUCTURE t (years since 2010) p (population in millions)
Exponential functions of 4.2 × 10 20 = 10 1.5m+12 5.5 to 6.0
the form y = a ∙ b x involve In 2011, there are 4.6 million 4.6
0
repeated multiplication by the people, and another 1.3% of Compute Solve the equation for m.
factor b. To understand how 4.6 million are added. 6.1 to 6.9
to model population with an
1 4.6 + 4.6(0.013) = 4.6(1.013) 4.2 × 10 20 = 10 1.5m+12 Write the original
exponential function, look for equation.
repeated multiplication in your The process repeats 2 4.6(1.013)(1.013) = 4.6(1.013)2
computations. each year. The 1 in log (4.2 × 10 20) = 1.5m + 12 Write the equation 7.0 to 7.9
1.013 represents the 3 4.6(1.013)3 in logarithmic form.
current population
and the 0.013 4 4.6(1.013) 4
20.6 ≈ 1.5m + 12 Evaluate the 8.0 or greater
represents the The exponent logarithm.
yearly increase. is the number
of years since 5.75 ≈ m Solve for m.
2010.
Interpret The magnitude of the earthquake is about 5.75.
The population can be modeled by the exponential function: Verify the answer: 10 1.5(5.75)+12 ≈ 4.2 × 10 20

population after t years population in 2010 growth factor years since 2010
Try It! 6. What is the magnitude of an earthquake with a seismic energy
P = 4.6(1.013) t of 1.8 × 10 23 joules?

B. If the population continues to grow at the same rate, what will the
population be in 2040?
To find the population in 2040, solve the equation for t = 30:
LESSON 6-3 Logarithms 309

P = 4.6(1.013) 30 ≈ 6.78.

VOCABULARY In 2040, the population will be about 6.78 million.

EXAMPLE 5 Solve Logarithmic Equations
A logarithmic equation HSM23_SEA2_FL_T06_L03.indd 309 10/01/21 5:14 AM
contains one or more What is the solution to ln (x 2 – 16) = ln (6x)?
logarithms of variable Try It! 3. A factory purchased a 3D printer in 2012 for $15,000. The value THEOREM 5-1 Perpendicular Bisector Theorem
expressions. ln (x 2 − 16) = lnprinter
of the (6x) is modeled byWrite
thethefunction f(x) = 15,000(0.93) x,
original equation.
where x is the number of years since 2012. If a point is on the perpendicular If... ℓ
x 2 − 16 = 6x Property of Equality for Logarithmic Equations
a. What is the value of the printer after 3 years? bisector of a segment, then it is P
x 2 − 6x − 16 = 0 Set quadratic equation equal to 0. equidistant from the endpoints of
b. Does the printer lose more of its value in the first 3 years or the segment.
(x − 8)(x + 2) = 0 Factor.
in the second 3 years after it was purchased? X Y
x = 8 or −2 Apply the Zero Product Property. Q

Skill Examples help students develop skill fluency. Check Substitute each value into the original equation.
x=8 x = −2 Proof Examples show students how to prove key theorems. PROOF: SEE EXAMPLE 2. Then... PX = PY

ln (8 2 − 16) = ln (6 ∙ 8)
ln (48) = ln (48) ✔
ln ((−2) 2 – 16) = ln (6 ∙ (−2))
ln (−12) = ln (−12) ✘
(in Geometry only) THEOREM 5-2 Converse of the Perpendicular Bisector Theorem

Because logarithms are not defined for negative values, only x = 8 is a If a point is equidistant from the If... ℓ
solution. The value x = −2 is an extraneous solution. endpoints of a segment, then it is P
on the perpendicular bisector of
LESSON 6-1 Key Features of Exponential Functions 291
the segment.
Try It! 5. Solve each equation. X
Q
Y
a. log 5 (x 2 − 45) = log 5 (4x) b. ln (−4x − 1) = ln (4x 2)
HSM23_SEA2_FL_T06_L01.indd 291 10/01/21 5:14 AM
PROOF: SEE EXAMPLE 2 TRY IT. Then... XQ = YQ

Powered By
EXAMPLE 6 Solve Logarithmic and Exponential
Equations by Graphing PROOF EXAMPLE 2 Prove the Perpendicular Bisector Theorem
CHOOSE EFFICIENT What is the solution to log (2x + 1) 5 = x − 2? Prove the Perpendicular Bisector Theorem. ℓ
METHODS
P
When typing this equation into Let y 1 = 5 log (2x + 1) and y 2 = x − 2. Given: ℓ is the perpendicular bisector of ¯
XY.
a calculator, it is helpful to write
Graph both equations. Prove: PX = PY
the equation using the Power
Property of Logarithms rather Use the INTERSECT feature to find the X Y
Q
than risking incorrect input of the point(s) of intersection. Proof:
exponent.
The points of intersection, to the nearest All right angles are
thousandth, are (−0.329, −2.329) and congruent, so
x scale: 1 y scale: 1 ℓ
(8.204, 6.204). STUDY TIP ∠XQP ≅ ∠YQP. P By the Reflexive
Remember that if a line is a Property, PQ ≅ PQ.
Check perpendicular bisector of a
log (2(−0.329) + 1) 5 = −0.329 − 2 log (2(8.204) + 1) 5 = 8.204 − 2 segment, you can conclude two
X Y
things: the line is perpendicular Q
log (0.342) 5 = −2.329 log (17.408) 5 = 6.204 to the segment, and it bisects
−2.329 = −2.329 ✔ 6.204 = 6.204 ✔ the segment.
Since ℓ is the perpendicular bisector of XY,
The solutions are x ≈ −0.329 and x ≈ 8.204. Q is the midpoint of XY, and XQ ≅ YQ.

By SAS, △XQP ≅ △YQP. Therefore, ¯

PX ≅ ¯
PY by CPCTC, so PX = PY.
Try It! 6. Solve each equation by graphing. Round to the nearest
thousandth.
a. 3(2) x+2 – 1 = 3 – x b. ln (3x – 1) = x – 5 Try It! 2. Prove the Converse of the Perpendicular Bisector Theorem.

330 TOPIC 6 Exponential and Logarithmic Functions

HSM23_SEA2_FL_T06_L06.indd 330 10/01/21 5:14 AM

198 TOPIC 5 Relationships in Triangles

HSM23_SEGM_FL_T05_L01.indd 198 08/01/21 4:51 AM

31
 PROGRAM COMPONENTS Great resources for
students and teachers!

B.E.S.T. ALGEBRA 2 B.E.S.T. GEOMETRY B.E.S.T. ALGEBRA 2

TOPIC 1 TOPIC 1
TOPIC 1 Foundations of Geometry Functions, Inequalities, and Systems
Functions, Inequalities, and Systems
TOPIC 2 TOPIC 2
TOPIC 2 Parallel and Perpendicular Lines Quadratic Functions and Equations
Quadratic Functions and Equations
B.E.S.T. ALGEBRA 1 B.E.S.T. ALGEBRA 1
TOPIC 3 TOPIC 3
TOPIC 3
Transformations TOPIC 1 Polynomial Functions
Polynomial Functions
TOPIC 1 Solving Equations and Inequalities
Solving Equations and Inequalities TOPIC 4 TOPIC 4
TOPIC 4
Rational Functions Triangle Congruence TOPIC 2 Rational Functions
TOPIC 2
Linear Equations
Linear Equations TOPIC 5 TOPIC 5
TOPIC 5
Rational Exponents and Radical Functions Relationships in Triangles TOPIC 3 Rational Exponents and Radical Functions
TOPIC 3
Linear and Absolute Value Functions Linear and Absolute Value Functions
TOPIC 6 TOPIC 6 TOPIC 6
Exponential and Logarithmic Functions Quadrilaterals TOPIC 4 Exponential and Logarithmic Functions

B.E.S.T. GEOMETRY
TOPIC 4

B.E.S.T. ALGEBRA 2
Systems of Linear Equations and Inequalities Systems of Linear Equations and Inequalities
TOPIC 7 TOPIC 7 TOPIC 7
Matrices Similarity Matrices
TOPIC 5 TOPIC 5
Exponents and Exponential Functions Exponents and Exponential Functions
B.E.S.T. GEOMETRY

B.E.S.T. ALGEBRA 2

TOPIC 8
TOPIC 8 TOPIC 8
Probability
TOPIC 6 Right Triangles and Trigonometry TOPIC 6 Probability
Polynomials and Factoring Polynomials and Factoring

B.E.S.T. ALGEBRA 1
TOPIC 9
TOPIC 7 Coordinate Geometry TOPIC 7
Quadratic Functions Quadratic Functions
TOPIC 10
B.E.S.T. ALGEBRA 1

TOPIC 8

STUDENT COMPANION

STUDENT COMPANION
Circles TOPIC 8
Solving Quadratic Equations
Solving Quadratic Equations
TOPIC 11
TOPIC 9
Two- and Three-Dimensional Models TOPIC 9
Working With Functions
Working With Functions
TOPIC 10
Analyzing Data TOPIC 10

STUDENT COMPANION
Analyzing Data STUDENT COMPANION STUDENT COMPANION
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B.E.S.T. GEOMETRY B.E.S.T. ALGEBRA 2 FPO B.E.S.T. GEOMETRY
FPO STUDENT COMPANION B.E.S.T. ALGEBRA 2
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Student Edition Student Companion

Student Edition Student Companion Student Digital Experience on Savvas Realize

Includes all instructional content Encourages students’ active Robust digital assets offer interactive instruction
engagement in class that engages students and helps them develop
Balance of conceptual understanding, conceptual understanding.
procedural fluency, and applications Provides space for independent
note-taking, solution planning, and Embedded interactive digital math tools are easily
Focus on developing mathematical for recording work on Practice & accessible anytime during instruction.
thinking and reasoning skills Problem Solving exercises
Online practice powered by MathXL® for School
Supports the development offers ready-to-go auto-graded assignments with
of students’ expertise with built-in learning aids:
mathematical thinking and • Daily homework and practice
reasoning skills
• Differentiated learning assignments
• Additional practice

Adaptive Practice

Individualized study plans generated from the

Topic Readiness Assessments offer targeted
instruction and practice to remediate identified
areas of weakness.

Virtual Nerd tutorials, available online, or by

scanning QR code on the exercise or Student
Companion pages, offer engaging video tutorials.

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Family Engagement resources help families

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32
 Digital

TE TE
VOL 1 VOL 1
B.E.S.T. GEOMETRY

VOLUME 1 (See Volume 1 for Topics 1–6.)

Topic 1 Foundations of Geometry

Topic 2 Parallel and Perpendicular Lines
Topic 3 Transformations TE
VOL 1
B.E.S.T. ALGEBRA 1 Topic 4 Triangle Congruence
Topic 5 Relationships in Triangles
VOLUME 1 (See Volume 1 for Topics 1–5.)
Topic 6 Quadrilaterals
Topic 1 Solving Equations and Inequalities
Topic 2 Linear Equations
VOLUME 2 (See Volume 2 for Topics 7–11.)
Topic 3 Linear and Absolute Value Functions
Topic 4 Systems of Linear Equations and Inequalities Topic 7 Similarity

Topic 5 Exponents and Exponential Functions Topic 8 Right Triangles and Trigonometry
Topic 9 Coordinate Geometry
Topic 10 Circles
VOLUME 2 (See Volume 2 for Topics 6–10.) Topic 11 Two- and Three-Dimensional Models
Topic 6 Polynomials and Factoring
B.E.S.T. ALGEBRA 2

B.E.S.T. GEOMETRY

Topic 7 Quadratic Functions

Topic 8 Solving Quadratic Equations
Topic 9 Working With Functions
Topic 10 Analyzing Data
B.E.S.T. ALGEBRA 1

Teacher’s Edition Teacher’s Edition

VOLUME 1 VOLUME 1

Teacher’s Edition
Savvas.com Savvas.com

Program Overview
B.E.S.T. ALGEBRA 2 Teacher’s Edition
FPO FPO
B.E.S.T. GEOMETRY
VOLUME 1

Savvas.com

B.E.S.T. ALGEBRA 1
FPO
B.E.S.T. ALGEBRA 1

Teacher’s Edition
Teacher’s Edition
Program Overview

Teacher’s Edition Program

Teacher’s Edition Teacher Digital Experience on Savvas Realize
Overview

Available in two volumes Detailed program overview captures Robust digital assets offer instruction with embedded
key program and lesson features. interactives powered by Desmos that engage students and
Each topic begins with bring math content to life.
an overview of the focus,
connections, and MTR Customizable platform allows teachers to rearrange content
Standards. or add their own.

Each lesson includes a Embedded digital math tools powered by Desmos are easily
lesson overview that lists accessible anytime during instruction.
the lesson’s objectives,
makes connections within Teacher’s Edition eText mirrors print and includes links to all
the topic and program, digital assets.
and highlights skills,
Topics in Realize Reader are downloadable for offline use.
concepts, or applications.
Online practice powered by MathXL® for School includes
Robust teaching support
built-in learning aids and automatic feedback as students
offers probing questions
work through exercises and activities.
developed within the
framework of NCTM’s Assessment Resources include course-level, topic-level, and
Effective Mathematics lesson-level assessments that can be administered with paper
Teaching Practices. and pencil or online.
Data-driven differentiated Online assessments include assessment items, such as
intervention provides technology-enhanced items, and performance tasks.
engaging activities that
address students’ learning Differentiated lesson activities are auto-assigned based on
needs. students’ scores on the online lesson quiz. These activities
include:
• Mathematical Literacy and Vocabulary
• Reteach to Build Understanding
• Additional Practice
• Enrichment
• Adaptive Practice
• Answers and Solutions includes all of the answers and
solutions to the exercises in each course.
• ExamView Assessment Generator is available to build
tests and worksheets.

33
USING A LESSON
Get a lesson overview
LESSON OVERVIEW

LESSON 6-2
Exponential Models Glossary

A Lesson Overview C Vocabulary Builder

Objective REVIEW VOCABULARY English | Spanish
• compound interest | interés compuesto
Students will be able to:
• exponential function | función exponencial
✔ Rewrite exponential functions to identify rates.
✔ Interpret the parameters of an exponential function within NEW VOCABULARY
the context of compound interest problems. • compound interest formula | formula de interés compuesto
✔ Construct exponential models given two points or by using • continuously compounded interest formula | formula de
regression. interés compuestoen forma continua
Essential Understanding • natural base e | base natural e
Exponential models are useful for representing situations in VOCABULARY ACTIVITY
which the rate increases by the same percent for each period
Discuss why an exponential model can be used to represent
of time and for interpreting problems that involve compound
compound interest and work through a simple example. Without
interest. Exponential regression can be used to generate
using their books, have students write the definitions of the new
exponential models for real-world contexts.
vocabulary terms in their own words.

In previous courses, students:

• Interpreted key features of exponential functions.
D Student Companion
In this lesson, students:
• Analyze real-world situations using exponential models. Students can do their work for the lesson in their Student
Companion or in Savvas Realize.
Later in this course, students:
• Connect their understanding of exponential functions to
logarithmic functions as they evaluate and simplify common
and natural logarithms.

Skills Students practice applying the compound interest

formula and the continuously compounded interest formula.
Applications Students apply exponential models to solve
problems involving compound interest.

B Florida’s B.E.S.T. Standards and Benchmarks

In this lesson, students focus on these benchmarks: These standards are highlighted in this lesson:

MA.912.FL.3.2 Solve real-world problems involving simple, MA.K12.MTR.2.1 Demonstrate understanding by representing
compound, and continuously compounded interest. problems in multiple ways.

AR.5.4 Write an exponential function to represent a relationship Students build understanding through modeling and express
between two quantities from a graph, a written description or a connections between how money compounds over time with
table of values within a mathematical or real-world context. mathematical models.

AR.5.5 Given an expression or equation representing an MTR.3.1 Complete tasks with mathematical fluency.
exponential function, reveal the constant percent rate of change Students complete tasks involving exponential relationships
per unit interval using the properties of exponents. Interpret the accurately and with confidence.
constant percent rate of change in terms of a real-world context.

They also work with concepts related to these benchmarks:

AR.5.7, F.1.1, FL.3.1, FL.3.4, and DP.2.9

TOPIC 6 297A LESSON 2

HSM23_TEA2_FL_T06_L02.indd 1 12/04/21 3:10 AM

34
 STEP 2 STEP 3 STEP 4
LESSON STEP 1
UNDERSTAND PRACTICE & ASSESS &
OVERVIEW EXPLORE
& APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

A LESSON OVERVIEW

• Provides lesson objectives and essential understandings

• Includes overview of the focus, connections, and concepts, skills, or
applications in the lesson
• Allows for online lesson planning where you can customize the lesson
and use an online calendar

Plan a lesson online at SavvasRealize.com.

B FLORIDA’S B.E.S.T. STANDARDS AND BENCHMARKS

• Presents Benchmarks that are addressed in the lesson

• Showcases Select Mathematical Thinking and Reasoning Standards
of the lesson

C VOCABULARY BUILDER

• Lists both new and review vocabulary in English and Spanish for each lesson
• Includes a vocabulary activity to help all students build their mathematical
literacy.

D STUDENT COMPANION
6-6
Activity

MODEL & DISCUSS

Exponential
• Interactive worktext provides active engagement and helps students build
A store introduces two new models of fitness trackers to its product line.
A glance at the data is enough to see that sales of both types of fitness trackers and Logarithmic
are increasing. Unfortunately, the store has limited space for the merchandise. Equations
understanding during instruction. The manager decides that the store will sell both models until sales of
TrackSmart exceed those of FitTracker.
SavvasRealize.com

• Students take ownership of their learning by trying problems on their own, taking FitTracker TrackSmart

notes, and focusing on the skills in the Thinking and Reasoning questions.
Number Sold Number Sold
Week 4 228 130
Week 3 112 44
Week 2 54 17
Week 1 28 5
A. Apply Math Models Find an equation of an exponential that models the
sales for each fitness tracker. Describe your method.

B. Based on the equations that you wrote, determine when the store will stop
selling FitTracker.

THINKING AND REASONING

Communicate and Justify How do you know that the sales data is modeled by an
exponential function?

LESSON 6-6 Exponential and Logarithmic Equations 313

35
USING A LESSON In Step 1, encourage
mathematical discourse as
EXPLORE students share their work.

STEP 1 Explore
Activity

F MODEL & DISCUSS G

INSTRUCTIONAL FOCUS Students write exponential equations AVAILABLE ONLINE
to model sales data and determine when one model will exceed
the other. Students see the need for efficient ways to solve
exponential equations.
STUDENT COMPANION Students can complete the Model &
Discuss activity in their Student Companion.

Before  WHOLE CLASS

Implement Tasks that Promote Reasoning and

Problem Solving
Q: Why do you think an exponential model is better than a linear
or a quadratic model for this problem?
[Sales are not increasing at a constant rate, so a linear model
would not make sense. It also appears that sales will continue
to increase as the products become more popular. There is no
indication given in the data that sales will take a downward
turn, so a quadratic model would not make sense.]
 STUDENT EDITION

During
6-6
 SMALL GROUP
E MODEL & DISCUSS

Support Productive Struggle in Learning Exponential and

A store introduces two new models of fitness trackers to its product line.
A glance at the data is enough to see that sales of both types of fitness
Logarithmic trackers are increasing. Unfortunately, the store has limited space for the
Mathematics Equations merchandise. The manager decides that the store will sell both models until
sales of TrackSmart exceed those of FitTracker.
Q: How do you know that TrackSmart will outsell FitTracker at
some point? I CAN… solve exponential FitTracker TrackSmart
and logarithmic equations. Number Sold Number Sold
[The rate that TrackSmart is increasing is greater than the rate Week 4 228 130
VOCABULARY
that FitTracker is increasing. So eventually, sales of TrackSmart • exponential equation Week 3 112 44
• logarithmic equation Week 2 54 17
will be greater than sales of FitTracker.] Week 1 28 5
MA.912.AR.5.2–Solve one-
variable equations involving
Q: How can you use the equations to determine when the store logarithms or exponential
expressions. Interpret solutions as
A. Apply Math Models Find an equation of an exponential function that
models the sales for each fitness tracker. Describe your method.
will stop selling FitTracker? viable in terms of the context and
identify any extraneous solutions. B. Based on the equations that you wrote, determine when the store will
MA.K12.MTR.3.1, MTR.4.1, stop selling FitTracker.
[Evaluate the equation that models each tracker for different MTR.7.1

numbers of weeks until the week is found when sales of How do properties of exponents and
ESSENTIAL QUESTION
FitTracker are less than sales of TrackSmart.] logarithms help you solve equations?


CONCEPT Property of Equality for Exponential Equations
SAMPLE STUDENT WORK

For Early Finishers VOCABULARY Symbols Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y.
An exponential equation is an
Q: Graph the two equations you found. Does your graph verify A. equation
I used my calculator
that contains variables to findIfaretwo
Words
the exponential regression models
powers of the same base are equal, then their exponents
equal; if two exponents are equal, then the powers with
in the exponents.
your results? Explain. for each set of data. The exponential model for FitTracker is
the same base are equal.

[Yes; The intersection point in the graph indicates when the y = 13.6266(2.0180) x, and the exponential model for
EXAMPLE 1 Solve Exponential Equations Using a Common Base
sales of Tracksmart will begin to exceed those of FitTracker. TrackSmart is y = 1.8081 (2.9227) 1x.x+7 = 4 3x?
solution to (__
2)
What is the
The x-coordinate matches the result.] 1 x+7 = 4 3x
(2)
__ Write the original equation.
x+7 3x
(2 −1) = (2 2) Rewrite each side with a common base.
B. Graph both equations 2on
−x−7a
=2graphing
6x calculator.
Power of a Power Property

After  WHOLE CLASS The graphs intersect −x

near x = 6, so the
− 7 = 6x
store will
Property of Equality for Exponential Equations
−7 = 7x Add x to each side.
stop selling FitTracker after
−1 = x 6 weeks.Divide each side by 7.
Facilitate Meaningful Mathematical Discourse
Try It! 1. Solve each equation using a common base.
Ask students to share their answers to Part A.
a. 25 3x = 125 x+2 b. 0.001 = 10 6x

Q: Why did different methods result in the same answer for Use with MODEL & DISCUSS
Part B? THINKING AND REASONING
LESSON 6-6 Exponential and Logarithmic Equations 327
[Although the methods for finding the equations may be Communicate and Justify How do you know that the sales data is
different, the equations themselves should be fairly alike.] modeled by an exponential function?
[When calculating linear, quadratic, and exponential regression,
HSM23_SEA2_FL_T06_L06.indd 327 10/01/21 5:14 AM

the exponential model had the highest r value, so that is the

most accurate model for the data.]

TOPIC 6 327B LESSON 6

HSM23_TEA2_FL_T06_L06.indd 2 12/04/21 3:23 AM

36
 STEP 2 STEP 3 STEP 4
STEP 1
UNDERSTAND & PRACTICE & ASSESS &
EXPLORE
APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

PROBLEM-BASED LEARNING
Tips for Facilitating Problem-Based
Model & Discuss in the Student Edition Learning
E
• Engages students with a high-interest problem in which new math ideas • Set expectations. Make sure
are embedded students know you expect them to do
the thinking.
• Helps students connect prior knowledge to the new math ideas
• Give students time to struggle.
• Allows for multiple entries to the problem making the problem accessible Research shows that as they think,
to all learners conceptual understandings emerge.
• Students can complete the activity using existing math knowledge base. • Foster communication. Have
students share their thinking with a
partner, small group, or the whole
F Model & Discuss in the Teacher’s Edition class.

• Teaching support is provided for all phases of the activity: before, during, and after. • Be encouraging. Show that you
value students’ thinking especially
• Before: Introduce the activity. This is whole-class discussion. when they struggle.
• During: Students work in pairs or small groups. Questions in the Teacher’s Edition • Embed the Mathematical
can help students who are struggling. Thinking and Reasoning
• After: This is another whole-class discussion during which students are readied for Standards during classroom
the new concepts in the lesson. discussions.

• Effective Mathematics Teaching Practices labeled with

an ETP icon provide teachers with support to create a productive
learning environment that optimizes student learning.
See the Effective Mathematics Teaching Practices pages for more
6-6
Exponential and
MODEL & DISCUSS

A store introduces two new models of fitness trackers to its product line.

information. Logarithmic
A glance at the data is enough to see that sales of both types of fitness
trackers are increasing. Unfortunately, the store has limited space for the
Equations merchandise. The manager decides that the store will sell both models until
sales of TrackSmart exceed those of FitTracker.

I CAN… solve exponential

G Model & Discuss Online and logarithmic equations.
FitTracker
Number Sold
TrackSmart
Number Sold

VOCABULARY Week 4 228 130

Week 3 112 44
• Assignable online at SavvasRealize.com. Students do not have
• exponential equation
• logarithmic equation Week 2 54 17
access to the sample solutions. MA.912.AR.5.2–Solve one-
Week 1 28 5
variable equations involving
A. Apply Math Models Find an equation of an exponential function that
• Students can do the activity completely online on their laptop
logarithms or exponential
expressions. Interpret solutions as models the sales for each fitness tracker. Describe your method.
viable in terms of the context and
B. Based on the equations that you wrote, determine when the store will
or tablet.
identify any extraneous solutions.
MA.K12.MTR.3.1, MTR.4.1, stop selling FitTracker.
MTR.7.1

• Students can write their solutions on an interactive ESSENTIAL QUESTION

How do properties of exponents and

white board. logarithms help you solve equations?

• Students will also encounter Model & Discuss, which helps

CONCEPT Property of Equality for Exponential Equations

them become better modelers, and Explore & Reason, in which Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y.
VOCABULARY Symbols
An exponential equation is an
Words If two powers of the same base are equal, then their exponents
students explore and reason about concepts. equation that contains variables
in the exponents.
are equal; if two exponents are equal, then the powers with
the same base are equal.

• See the Welcome to the enVision Florida B.E.S.T. A|G|A User’s

Guide pages for more information about the opening activities.
EXAMPLE 1 Solve Exponential Equations Using a Common Base
x+7
What is the solution to (__
1
) = 4 3x?
2
1 x+7 = 4 3x
(2)
__ Write the original equation.
−1 x+7
(2 ) 2 3x
= (2 ) Rewrite each side with a common base.
2 −x−7 = 2 6x Power of a Power Property
−x − 7 = 6x Property of Equality for Exponential Equations
−7 = 7x Add x to each side.
−1 = x Divide each side by 7.

Try It! 1. Solve each equation using a common base.

a. 25 3x = 125 x+2 b. 0.001 = 10 6x

LESSON 6-6 Exponential and Logarithmic Equations 327

Model & Discuss and Cirtique & Explain, See the Welcome to the
HSM23_SEA2_FL_T06_L06.indd 327 10/01/21 5:14 AM

enVision Florida B.E.S.T. A|G|A User’s Guide pages.

37
USING A LESSON In Step 2, begin by making
UNDERSTAND & APPLY the math explicit!

6-6
Exponential and
MODEL & DISCUSS

A store introduces two new models of fitness trackers to its product line.
A glance at the data is enough to see that sales of both types of fitness
Logarithmic trackers are increasing. Unfortunately, the store has limited space for the
Equations merchandise. The manager decides that the store will sell both models until
sales of TrackSmart exceed those of FitTracker.

2
I CAN… solve exponential FitTracker TrackSmart
and logarithmic equations. Number Sold Number Sold

STEP Understand & Apply VOCABULARY Week 4 228 130

J
• exponential equation Week 3 112 44
• logarithmic equation Week 2 54 17
Week 1 28 5
MA.912.AR.5.2–Solve one- Activity Assess
variable equations involving
logarithms or exponential
A. Apply Math Models Find an equation of an exponential function that
expressions. Interpret solutions as models the sales for each fitness tracker. Describe your method.
viable in terms of the context and

INTRODUCE THE ESSENTIAL QUESTION identify any extraneous solutions.

MA.K12.MTR.3.1, MTR.4.1,
B. Based on the equations that you wrote, determine when the store will
stop selling FitTracker.
MTR.7.1
AVAILABLE ONLINE
Establish Mathematics Goals to Focus Learning ESSENTIAL QUESTION
How do properties of exponents and
logarithms help you solve equations?

Remind students that exponential functions and logarithmic CONCEPT Property of Equality for Exponential Equations
functions have an inverse relationship. They will discover why
H
VOCABULARY Symbols Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y.
this is useful when solving equations as they work through the An exponential equation is an
equation that contains variables
Words If two powers of the same base are equal, then their exponents
are equal; if two exponents are equal, then the powers with
examples in the lesson. in the exponents.
the same base are equal.

I
Solve Exponential Equations Using EXAMPLE 1 Solve Exponential Equations Using a Common Base
EXAMPLE 1
a Common Base
x+7
What is the solution to (__
1
) = 4 3x?
2
1 x+7 = 4 3x
(2)
__ Write the original equation.
x+7 3x
(2 −1) = (2 2) Rewrite each side with a common base.
Facilitate Meaningful Mathematical Discourse 2 −x−7 = 2 6x Power of a Power Property

Q: Why was each side of the equation written with an equivalent −x − 7 = 6x Property of Equality for Exponential Equations
−7 = 7x Add x to each side.
expression using 2 as the base? −1 = x Divide each side by 7.
[In order to solve an exponential equation, the bases must be
the same.] Try It! 1. Solve each equation using a common base.
a. 25 3x = 125 x+2 b. 0.001 = 10 6x
Q: Could the expressions have been rewritten by using a common
base of __12 , 4, or 10 rather than 2?
[The expressions could be rewritten with a common base of __12 LESSON 6-6 Exponential and Logarithmic Equations 327

or 4. They could also be rewritten with a common base of 10,

however, this would make the equation more complex rather HSM23_SEA2_FL_T06_L06.indd 327 10/01/21 5:14 AM

than drawing closer to a solution.]

K Try It! Answers

Elicit and Use Evidence of Student Thinking
Q: Which base did you use for Part a? For Part b?
1. a. 2 [5 for Part a because it is a factor of both 25 and 125; 10 for
b. −__12 Part b because it is a factor of both 0.001 and 10]

AVAILABLE ONLINE

L ADDITIONAL EXAMPLES

Example 5 Students practice solving logarithmic equations with Example 6 Students solve exponential equations by graphing
multiple logarithmic terms in this additional example. with this additional example.
Q: What is the first step to solve this problem? Q: How can you graph each side of the equation as its own
[Use the Product Property of Logarithms.] function?
[Set each side of the equation equal to y, then graph the two
equations.]

TOPIC 6 327 LESSON 6

HSM23_TEA2_FL_T06_L06.indd 327 12/04/21 3:23 AM

STEP 2 Understand & Apply

Activity Assess

Rewrite Exponential Equations

EXAMPLE 2
Using Logarithms AVAILABLE ONLINE

Implement Tasks that Promote Reasoning and CONCEPTUAL

Problem Solving UNDERSTANDING EXAMPLE 2 Rewrite Exponential Equations Using Logarithms

How can you rewrite the equation 17 = 4 x using logarithms?

Q: Why is 10 (log17) equivalent to 17? There is no common base for 17 and 4. Write each number as a power of 10.

[because exponentials and logarithms are inverse operations] 17 = 4 x Write the original equation.
x
10 log 17 = 10 log4 Write the equation using the powers of 10.

Try It! Answers log 17 = log 4 x Property of Equality for Exponential Equations

Rewriting expressions using logarithms can help you solve many types
2. log 5x = log 12 of problems.

M Use with EXAMPLE 2 Try It! 2. Rewrite the equation 5 x = 12 using logarithms.
THINKING AND REASONING
Communicate and Justify In order to set the exponents of two CONCEPT Property of Equality for Logarithmic Equations
exponential expressions equal to each other, what must be true Symbols If x > 0, then log b x = log b y if and only if x = y.
about the exponential expressions? Words If two logarithms (exponents) of the same base are equal, then
the quantities are equal; if two quantities are equal, and the
bases are the same, then the logarithms (exponents) are equal.
[The bases of the two expressions must be equivalent.]
EXAMPLE 3 Solve Exponential Equations Using Logarithms

Solve Exponential Equations What is the solution to 3 x+1 = 5 x?

EXAMPLE 3 3 x+1 = 5 x Write the original equation.
Using Logarithms COMMON ERROR log(3 x+1) = log(5 x) Property of Equality for Logarithmic Equations
The entire quantity of x + 1 is the
exponent, so it must be written as (x + 1) log 3 = x log 5 Power Property of Logarithms

Pose Purposeful Questions a quantity to be multiplied by the

logarithmic expression.
x log 3 + log 3 = x log 5 Use the Distributive Property.
x(log3 − log5) = −log 3 Move terms and factor out x.
Q: Why are log 3 and x log 5 subtracted from both sides of the x = __________
−log3
log3 − log5
Divide. y

equation? x ≈ 2.15 Evaluate. 30

20
[x log 3 and x log 5 have a common factor of x and log 3 is a Check
Substitute 2.15 into the equation: 10
constant. By subtracting those terms from both sides, x can be 3 x+1 = 3 2.15+1 ≈ 31.8 x
factored out and the equation can be solved.] 5 x = 5 2.15 ≈ 31.8
O 1 2 3

The point of intersection of the graphs is about (2.15, 31.8).

Try It! Answers
Try It! 3. What is the solution to 2 3x = 7 x+1?
log 7
3. _____________ , or approximately 14.57
(3 log 2 − log 7)

Elicit and Use Evidence of Student Thinking 328 TOPIC 6 Exponential and Logarithmic Functions

Q: Explain how the Distributive Property is used to solve the

equation. [The Distributive Property is used to multiply log 7 HSM23_SEA2_FL_T06_L06.indd 328 10/01/21 5:14 AM

through (x + 1). It’s used again to factor x out of (3xlog2 − xlog7).]

N Struggling Students Advanced Students

USE WITH EXAMPLE 2 Some students have difficulty rewriting USE WITH EXAMPLE 3 Have students explore solving multi-step
equations using logarithms. exponential equations.
• Have students rewrite the equations using logarithms. • Have students solve these equations and round their answers to
the nearest thousandth.
1. 5 x = 16
1. 15 = 6 2x − 9 [0.887]
[x log 5 = log 16]
2. 5 x − 5 + 1 = 4 [5.683]
2. 2 2x − 3 = 19
[(2x − 3) log 2 = log 19] 3. 2 2x + 1 − 6 = −3 [0.293]
3. 4 3x= 6x − 2 Q: What would you do first to solve these problems?
[3x log 4 = (x − 2) log 6] [Rewrite the equation with the exponential term on one side
and the constant term on the other side.]
Q: What property did you use to rewrite each equation?
[Power Property of Logarithms]

TOPIC 6 328 LESSON 6

HSM23_TEA2_FL_T06_L06.indd 328 12/04/21 3:23 AM

38
 STEP 2 STEP 3 STEP 4
STEP 1
UNDERSTAND & PRACTICE & ASSESS &
EXPLORE
APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

UNDERSTAND & APPLY

H Essential Question
• Makes the math explicit

I Examples
• Provides guided instruction of the math concepts in the lesson
• Includes at least one example that develops conceptual understanding
• Includes at least one example that applies the concepts of the lesson
• Teacher’s Edition offers robust probing questions designed to scaffold student learning.

J Digital Examples
• Provide audio support for struggling readers or English Language Learners
• Pauses throughout the interactives encourage student thinking and responses
• Include easy-to-find links to digital math tools and the bilingual glossary
A link to a specific math tool is provided when helpful.
• Offer highly interactive examples

K Try It!
• Provides a formative assessment opportunity after each example

L Additional Examples Available online

• Offers additional examples for students in need of more instruction

M Thinking and Reasoning

• Questions are designed to help students build on their mathematical thinking and reasoning skills.

N Differentiated Support
• Lesson adaptations or extensions found at point-of-use to maximize student learning

39
USING A LESSON
UNDERSTAND & APPLY (continued)

STEP 2 Understand & Apply

Concept
Summary Assess

O CONCEPT SUMMARY Logarithmic Functions

Q: How are the domain and the range of y = log x related to the AVAILABLE ONLINE
domain and the range of y = 10 x?
[The domain of y = log x is the same as the range of y = 10 x CONCEPT SUMMARY Logarithmic Functions

and the range of y = log x is the same as the domain of GRAPH 4

y

y = 10 x.] 2 The functions are

y = 10 x x inverses so their graphs
Q: How are the x-intercept and the asymptote of y = log x −4 −2 O 2 4 are reflections of each
−2 other across the line
affected if the graph is translated 3 units to the right? y = log x with equation y = x.
−4
[The x-intercept and asymptote are also translated 3 units
to the right. The x-intercept is 4 and the asymptote is x = 3.] EQUATIONS y = log x y = 10 x

KEY FEATURES Domain: {x | x > 0} Domain: all real numbers

Range: all real numbers Range: {y | y > 0}

Do You UNDERSTAND? | Do You KNOW HOW? x-intercept: 1 y-intercept: 1

Asymptote: y-axis Asymptote: x-axis

END BEHAVIOR As x → 0, y → −∞ As x → −∞, y → 0

Common Error As x → ∞, y → ∞ As x → ∞, y → ∞

Exercise 5 Students may ignore the −1 before adding 1__12 ,

writing the equation as g(x) = ln x + 1__12 , P Do You UNDERSTAND? Do You KNOW HOW? Q
1. ESSENTIAL QUESTION How is the 5. Graph the function y = log 4 x and identify
instead of g(x) = ln x + __12 . Explain that the graph of relationship between logarithmic and the domain and range. List any intercepts or
exponential functions revealed in the key asymptotes. Describe the end behavior.
f(x) = ln x − 1 is the graph of h(x) = ln x shifted down features of their graphs?
6. Write the equation for the function g(x),
1 unit, so shifting this function up by 1__12 units is the same 2. Error Analysis Raynard claims the domain which can be described as a vertical shift
1__12 units up from the function f(x) = ln x – 1.
as shifting h(x) up __12 unit.
of the function y = log 3 x is all real numbers.
Explain the error Raynard made.
7. The function y = 5 ln(x + 1) gives y, the
3. Communicate and Justify How are the number of downloads, in hundreds, x minutes
graphs of f(x) = log 5 x and g(x) = –log 5 x after the release of a song. Find the equation
related? of the inverse and interpret its meaning.
Answers 4. Use Patterns and Structure Explain the Downloading...
necessary steps to find the inverse y downloads
1. The domain of the logarithmic function is the same as of h(x) = log 6(x + 4). Find the inverse.
x minutes

the range of the exponential function, and the range of 8. Sketch the functions represented by the

the logarithmic function is the same as the domain of tables. Identify which graph is the logarithmic
function. Are the two functions inverses?
the exponential function. The logarithmic function has a.
x 10 −10 0.01 1 2 10
an x-intercept of 1, while the exponential function has a g(x) −8 0 2 2.301 3

y-intercept of 1. b. x −1 0 1 2 3 4
h(x) 0.001 0.01 0.1 1 10 100

2. This is the range of the function. The domain is {x | x > 0}. 318 TOPIC 6 Exponential and Logarithmic Functions

3. The graph of g(x) is the reflection of the graph of f(x) across

the x-axis. HSM23_SEA2_FL_T06_L04.indd Page 318 17/02/21 5:18 PM f-0869 /154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...

4. Rewrite the problem using x and y, then exchange the places

8. y
of the x and y. Next solve for the y. In this step use the 4 h
relationship between exponential and logartimic functions.
Finally, rewrite in inverse function notation. h −1(x) = 6x − 5. 2 g
5. y x
4 O 2 4
2
x g is the log function; yes
−2 O 2 4 6
−2

−4

domain: {x | x > 0}; range: all real numbers; intercept:

x-intercept 1; asymptote: y-axis; end behavior: As x → 0,
y → −∞. As x → ∞, y → ∞.
6. g(x) = ln x + __12
y
__
7. x = e 5 − 1; This function gives x, the number of minutes after
the release of a song, in terms of the number of downloads in
hundreds, y.

TOPIC 6 318 LESSON 4

HSM23_TEA2_FL_T06_L04.indd 318 12/04/21 3:16 AM

40
 STEP 2 STEP 3 STEP 4
STEP 1
UNDERSTAND & PRACTICE & ASSESS &
EXPLORE
APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

UNDERSTAND & APPLY

O Concept Summary
• Presents a summary of the main math concepts found in the lesson
CONCEPT SUMMARY Key Features of Exponential Functions

Exponential Growth Exponential Decay

• Teaching support includes guiding questions to monitor students’ GRAPHS

CONCEPTy SUMMARY
y
4
Writing
y Exponential
4
y Models
4 4
understanding. 2
b>1
2
b>1 0 < b < 10 < b < 1
2 2

O −22 O 4
x General
x
O −22 O 4
x x
−4 −2 −4 2 4 −4 −2 −4 2 4
−2 −2
Exponential−2Model
−2
Compound Interest
−4 −4 −4 −4

ALGEBRA
Growth factor: 1 + r
y=a∙ bx A = P(1 + __
n)
r nt
Decay factor: 1 − r

EQUATIONS y = a ∙ b x, for b >1, a ≠ 0 y = a ∙ b x, for 0 < b < 1

A necklace costs $250 A principal of $3,000 is
KEY FEATURES NUMBERS
Domain: All real numbers
and increases inDomain:
valueAllby
real numbers
invested at 5% annual
Range: {y | y ≥ 0} Range: {y | y ≥ 0}
Intercepts: (0, a) 2% per year. interest, compounded
Intercepts: (0, a)
Asymptote: x-axis Asymptote: x-axis
a = initial amount $250 monthly, for 4 years.
END BEHAVIOR As x → −∞, y → 0 As x → −∞, y → ∞

b = growth factor 1.02As x → ∞, y →P0 = 3,000

As x → ∞, y → ∞

Growth: A(t) = a(1 + r) t Decay: A(t) = a(1 − r) t

MODELS
x = number of years r = 5%

y= 250(1.02)x n=
per year
Do You UNDERSTAND? Do You KNOW HOW?
1. ESSENTIAL QUESTION How do graphs t = 4 years
5. Graph the function f(x) = 4(3) x. Identify the
(12)(4)
A = 3000(1 + ____
12 )
and equations reveal key features of domain, range, intercept, and asymptote,
and describe the end behavior.
0.05
exponential growth and decay functions?

P Do You UNDERSTAND? 2. Vocabulary How do exponential functions

differ from polynomial and rational
6. The exponential function f(x) = 2500(0.4) x
models the amount of money in Zachary’s
functions? savings account over the last 10 years. Is

• Always includes having students answer the lesson Essential Question

Zachary’s account balance increasing or
3. Error Analysis Charles claimed the function decreasing? Write the base in terms of the
x
f(x) = (__3 ) represents exponential decay. rate of growth or decay.
Do You UNDERSTAND?
2
Explain the error Charles made.
• Focus is on determining students’ understanding CONCEPT
of lessonSUMMARY
concepts.Writing Exponential Models 7. Describe how the graph of g(x) = 4(0.5) x−3

4. Analyze and Persevere How are compares to the graph of f(x) = 4(0.5) x.
1. ESSENTIAL
exponential growth functions similar to QUESTIONWhy do you
• Mathematical Thinking and Reasoning Standards are indicated with
exponential decay functions? How are they 8. Two trucks were purchased by a landscaping
different? develop exponential models
company in to represent
2016. Their values are modeled
General Continuously
by the functions f(x) = 35(0.85) x and g(x) =
and interpret situations? quarterly rates.
red run-in heads. Exponential Model Compound Interest
46(0.75) x where x is the number of years
Compounded
since 2016.Interest
Which function models the truck
that is worth the most after 5 years? Explain. 5. f(t) =
2. Error Analysis The exponential model
ALGEBRA y=a∙ bx A = P(1 + __
n ) y = 5,000(1.05) represents
r nt t
A = Pe rtthe amount 6. f(t) =
294
Yori earns in an account after t years when
TOPIC 6 Exponential and Logarithmic Functions
A necklace costs $250 A principal of $3,000 is$5,000 is invested. Yori said
A principal of the monthly
NUMBERS and increases in value by invested at 5% annual interest rate$3,000
of theisexponential
invested model is
2% per year. interest, compounded 5%. Explain at
HSM23_SEA2_FL_T06_L01.indd 294
Yori’s
5% error.
continuously 27/01/21 5:50 PM

7.
a = initial amount $250 monthly, for 4 years. compounded interest r = 3%, t = 5
3. Vocabulary Explain the similarities and
P = 3,000 for 4 years.
b = growth factor 1.02 differences between compound interest and 8.
r = 5% = 3,000
continuouslyP compounded interest.
x = number of years r = 1.5%, t =
n = 12 compounding4.periods r = 5%
y = 250(1.02)x Represent and Connect Write a math story
per year t = 4the
that would use years
general exponential
t = 4 years 3000e (0.05)(4)
A = Explain
model to solve. the meanings of the 9.
0.05 (12)(4)
A = 3000(1 + 12 )
____ variables and parameters in your model.
10.

11. Paul invests

Q Do You Know HOW?
Doskill
• Focus is on determining students’ proficiency with Youapplication.
UNDERSTAND? Do You KNOW HOW?
1. ESSENTIAL QUESTION Why do you The exponential function models the annual
• Teaching support includes Common Error box to develop
support all learners.
exponential models to represent rate of increase. Find the monthly and
and interpret situations? quarterly rates.
5. f(t) = 2,000(1.03) t
2. Error Analysis The exponential model
y = 5,000(1.05) t represents the amount 6. f(t) = 500(1.055) t
Yori earns in an account after t years when
$5,000 is invested. Yori said the monthly Find the total amount of money in an account
302 TOPIC 6 Exponential and Logarithmic Functions
interest rate of the exponential model is at the end of the given time period.
5%. Explain Yori’s error. 7. compounded monthly, P = $2,000,
3. Vocabulary Explain the similarities and r = 3%, t = 5 years
differences between compound interest and 8. continuously compounded, P = $1,500,
HSM23_SEA2_FL_T06_L02.indd 302

continuously compounded interest.

r = 1.5%, t = 6 years
4. Represent and Connect Write a math story
Write an exponential model given two points.
that would use the general exponential
model to solve. Explain the meanings of the 9. (3, 55) and (4, 70)
variables and parameters in your model.
10. (7, 12) and (8, 25)

11. Paul invests $6,450 in an account that earns

continuously compounded interest at an
annual rate of 2.8%. What is the value of the
account after 8 years?

41
USING A LESSON In Step 3, assign
PRACTICE & PROBLEM SOLVING powerful practice!

STEP 3 Practice & Problem Solving

Practice Tutorials Math Tools

R PRACTICE & PROBLEM SOLVING

S T
AVAILABLE ONLINE
AVAILABLE ONLINE

Lesson Practice You may opt to have students

complete the automatically scored Practice and PRACTICE & PROBLEM SOLVING
Problem Solving items online powered by MathXL UNDERSTAND PRACTICE
for School.
9. Use Patterns and Structure Are the logarithmic Graph each function and identify the domain and
and exponential functions shown inverses of range. List any intercepts or asymptotes. Describe
Choose from:  Lesson Practice
each other? Explain. the end behavior. SEE EXAMPLE 1
Adaptive Practice y 14. y = log 5 x 15. y = log 8 x
4
16. y = log __
3 x 17. y = log 0.1 x
You may also take advantage of the bank of 2 10

exercises for assigning additional practice. x Describe the graph in terms of transformations
−4 −2 O 2 4 of the parent function f(x) = log 6 x. Compare the
asymptote and x-intercept of the given function to
−2
the parent function. SEE EXAMPLE 2
−4 1 log x
18. g(x) = __ 19. g(x) = log 6 (–x)
2 6
Assignment Guide 20. Describe how the graph of g(x) = – In(x + 0.5) is
10. Communicate and Justify How is the graph
related to the graph of f(x) = In x. SEE EXAMPLE 2
of the logarithmic function g(x) = log 2 (x – 7)
On-Level Honors related to the graph of the function
Find the equation of the inverse of each function.
f(x) = log 2 x? Explain your reasoning.
7–10, 12–33 7–32 SEE EXAMPLE 3

11. Error Analysis Describe and correct the error 21. f(x) = 5 x–3 22. f(x) = 6 x+7
a student made in finding the inverse of the f −1(x) = log 5 x + 3 f −1(x) = log 6 x − 7
Item Analysis exponential function f(x) = 5 x–6 + 2. 23. f(x) = In (x + 3) – 1
f −1(x) = e x + 1 − 3
24. f(x) = 4 log 2 (x − 3) + 2
(x − 2)
_____
f −1(x) = 2 4 + 3
25. The altitude y, in feet, of a plane t minutes
y = 5x–6 + 2 Write in y = f(x) form.
Example Items DOK x = 5y–6 + 2 Interchange x and y.
after takeoff is approximated by the function
y = 5,000 ln(.05t) + 8,000. Solve for t in terms
x – 2 = 5y–6 Subtract 2 from each side. of y. SEE EXAMPLE 4
1 12–15 1 log5 x – 2 = y – 6 Rewrite in logarithmic form.
log5 x – 2 + 6 = y Add 6 to each side. For items 26–29, let f(x) = e x, g(x) = In x,
2 8, 11, 16–18, 31 2 log5 x + 4 = y Simplify.
2
h(x) = e −x , and j(x) = In x 2. Find a simplified

19–22, 32 1
f–1(x) = log5 x + 4 Rewrite as an
inverse function. ✗ expression for each composition of functions.
SEE EXAMPLE 5

3 26. f ∘ g 27. g ∘ f
x x
7, 9 2 12. Analyze and Persevere The number of 28. g ∘ h 29. f ∘ j
members m who joined a new workout center −x 2 x2
4 10, 23 2 30. Apply Math Models A scientist is conducting
w weeks after opening is modeled by the
an experiment with a pesticide. Use a calculator
equation m = 1.6 w+2, where 0 ≤ w ≤ 10. Find
to find an exponential model for the data in
24–27 1 the inverse of the function and explain what
the table. Use the model to determine how
the inverse tells you.
much pesticide remains after 180 days. Then
5 28 2 13. Use Patterns and Structure The graph shows a transform the function so it graphs a straight
transformation of the parent graph f(x) = log 3 x. line. SEE EXAMPLE 6
30 3 Write an equation for the graph. Day 0 20.00g

y Day 1 14.73g

Answers 4 Day 2 11.29g

Day 3 8.38g
2
9. no; The x-intercept of the blue function x Day 4 6.82g
is not equal to the y-intercept of the red O 4 6 Day 5 4.75g
function. −2 Day 6 3.15g

10. g(x) is shifted 7 units to the right; for

LESSON 6-4 Logarithmic Functions 319
each x, g(x) = f(x – 7), so the y-values of
the functions are equal for x-values that
are 7 units apart.
HSM23_SEA2_FL_T06_L04.indd Page 319 13/02/21 3:45 PM eteam /154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...

11. When rewriting in logarithmic form, 14. y 15. y

the student should have placed the 2 2
expression x – 2 in parenthesis. The
correct inverse function is f −1(x) = log 5 1 1
(x −2) + 6. x x
12. w = log 1.6 m − 2; the number of weeks O 10 20 O 10 20
after opening when the number of
members reaches m
domain: {x | x > 0}; range: all real domain: {x | x > 0}; range: all real
13. g(x) = log 3 (x − 2) + 1 numbers; intercept: x-intercept 1; numbers; x-intercept 1; asymptote: y-axis;
asymptote: y-axis; end behavior: As As x → 0, y → −∞. As x → ∞, y → ∞.
x → 0, y → −∞. As x → ∞, y → ∞. See answers for Exercises 16–20, 25, and 30 on the next page.

TOPIC 6 319 LESSON 4

HSM23_TEA2_FL_T06_L04.indd 319 12/04/21 3:16 AM

42
 STEP 2 STEP 3 STEP 4
STEP 1
UNDERSTAND & PRACTICE & ASSESS &
EXPLORE
APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

PRACTICE & PROBLEM SOLVING

R Practice and Problem Solving Exercises

Consists of four exercise sets:
• Understand exercises focus on students’ conceptual understanding of lesson
content.
• Practice exercises help students build procedural fluency with lesson content.
• Apply exercises have students apply the lesson content to real-world and
mathematical situations.
• Assessment Practice exercises help students prepare for different types of
standardized assessments. Every exercise set includes Florida’s B.E.S.T. and
SAT/ACT practice problems. Every exercise set includes one performance task
that requires students to synthesize their learning and justify their solution in
writing.

S Online Practice powered by MathXL® for School

• Lesson Practice corresponds to the Practice & Problem Solving
assignment in the print book and is auto-graded.

T Digital Math Tools

• A suite of digital math tools including a robust graphing calculator
and geometry tools to explore transformations, evaluate equations,
plot tables of data, and much more.
• Always available to students and teachers at SavvasRealize.com

43
USING A LESSON In Step 4, choose from a variety
of differentiated intervention
ASSESS AND DIFFERENTIATE activities!

STEP 4 Assess & Differentiate

Assess Tutorials Worksheets

U LESSON QUIZ
Use the Lesson Quiz to assess students’ understanding of the
mathematics in the lesson.
 ASSESSMENT RESOURCES AVAILABLE ONLINE
Students can take the Lesson Quiz online or you can download a
Name
printable copy from SavvasRealize.com. The Lesson Quiz is also B.E.S.T. ALGEBRA 2

SavvasRealize.com

available in the Assessment Sourcebook. 6-4 Lesson Quiz

Logarithmic Functions
Item Analysis
1. The graph y 4. An internet service provider uses
8
shows the the function R = 3 log(a + 2) + 15
Item DOK Skills Review & Benchmarks function 6
to relate its sales revenue R, in
Item DOK Practice Benchmarks f(x) = 3 x. 4 hundreds of dollars, to a, the cost
1 2 F-BF.2.4.c, F-IF.3.7.e Match each key 2 of advertising.
MA.912.AR.5.8
00.000.00.0.0 x
1 2 A-24
0-00 feature for the −4 −2 O 2 4 Part A
2 2 MA.912.F.3.7
00.000.0.0.0
F-BF.2.3, F-IF.3.7.e graph of f –1(x)

TK
If the provider earns 21 hundred
with its description.
dollars, how much did they spend
2 3 2 2 F-15
0-00 MA.912.F.2.2
00.000.0.0.0
F-BF.2.4.a All real on advertising? Round to the
(0, ∞) Never
numbers nearest dollar, if necessary.
3 4 2 2 F-16
0-00 00.000.0.0.0
MA.912.F.3.7
F-BF.2.4.a Domain ❑ ❑ ❑ $ 98
Part B
4 5 2 2 A-24
0-00 00.000.00.0.0
MA.912.AR.5.9
F-IF.2.6, F-IF.3.9
Range ❑ ❑ ❑
What constraints are needed to
Increasing ❑ ❑ ❑ make the logarithmic function
5 2 F-15
0-00 MA.912.F.1.7
00.000.0.0.0 a better model as the provider
Decreasing ❑ ❑ ❑
begins to earn revenue?
The domain should be restricted to
Use the student scores on the Lesson Quiz to prescribe 2. The logarithmic function
f(x) = log x is transformed to ◻ a ≥ 0. ◻ a ≥ 15.9.
differentiated assignments. g(x) = Iog(x + 1) + 3. Which ◻ a > −2. ◻ a > 0.
statement is true?
If students take the Lesson Quiz online, it will be automatically 𝖠 The graph of f(x) is translated 5. Consider y
scored and appropriate differentiated practice will be assigned 1 unit upward. f(x) = 3x – 1 x
O 2 4 6
based on student performance. 𝖡 The graph of f(x) is translated and the graph
−2
3 units downward. of g(x).
−4
𝖢 The vertical asymptote shifts Select the
• Reteach to Build Understanding from x = 0 to x = −1. function that −6
makes the
• Mathematical Literacy and 𝖣 The vertical asymptote shifts statement true.
Intervention 0–3 points from x = −1 to x = 3.
Vocabulary f(x) g(x) Both
• Additional Practice 3. Complete the equation of the Has the greater x-intercept ❑ ❑ ❑
inverse of the function
• Mathematical Literacy and f(x) = log 2 (9x). Has an asymptote of y = 1 ❑ ❑ ❑

Vocabulary __1 Is always increasing ❑ ❑ ❑

On-Level 4 points f –1(x) = 9 ( 2 )x
• Additional Practice Is positive over the interval
❑ ❑ ❑
(2, ∞)
• Enrichment
enVision® Florida B.E.S.T. Algebra 2 • Assessment Resources

Advanced 5 points • Enrichment

HSM23_ANCA2_FL_T06_L04_LQ.indd Page 2 01/05/21 9:25 AM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/Algebra ...

TOPIC 6 320A LESSON 4

HSM23_TEA2_FL_T06_L04.indd 1 12/04/21 3:17 AM

STEP 4 Assess & Differentiate

Assess Tutorials Worksheets

V DIFFERENTIATED RESOURCES
= Intervention = On-Level = Advanced

 = This activity is available as a digital assignment powered by MathXL® for School.

AVAILABLE ONLINE

Reteach to Build
Understanding  Additional Practice  Enrichment 
Provides scaffolded reteaching for Provides extra practice for Presents engaging problems and activities
the key lesson concepts. each lesson. that extend the lesson concepts.

Name B.E.S.T. ALGEBRA 2 Name B.E.S.T. ALGEBRA 2

Name B.E.S.T. ALGEBRA 2

SavvasRealize.com SavvasRealize.com SavvasRealize.com

6-4 Reteach to Build Understanding 6-4 Additional Practice 6-4 Enrichment

Logarithmic Functions Logarithmic Functions Logarithmic Functions

1. All exponential functions of the form f(x) = b x and logarithmic functions of the Graph the function below and identify the domain, range, x-intercept, y-intercept, According to the Richter scale, a magnitude + 1.0 109

form f(x) = log bx are invertible , because these functions have exactly one asymptote, and end behavior. Compare the graph to the parent function. 8.9
earthquake causes 10 times the ground motion of a
y-value for each x-value , so when the x and y are exchanged , the y
standard earthquake. The Richter scale equation is
inverses will also have exactly one x-value for each y-value . 1. f(x) = log 4(x – 2) + 2
R = log(___
A )
A GREAT

2 . A is the amplitude of a seismograph reading,

2. Find and correct the error a student made when finding the inverse of the domain: {x | x > 2}; x
0
in centimeters, taken 100 km from the epicenter of
108 8

logarithmic funtion f(x) = log 6(4x + 2) − 5. range: all real numbers; O 2 4 6 8 10 the earthquake. A 0 is the amplitude of a “standard
MAJOR

x-intercept: 2__1; 107 7

y = log 6(4x + 2) − 5 Write in y = f(x) from. 16 earthquake”, which is 1 micron, or 10 −4 cm. R is the 106
STRONG
6 MODERATE
y-intercept: none; asymptote: x = 2; magnitude of the earthquake on the Richter scale. 105
104
102 3
4
5 SMALL
1 2
x = log 6(4y + 2) − 5 Interchange x and y. end behavior: As x → 2, y → −∞. As x → ∞, y → ∞.
MINOR
10−1 −1 0 NOT FELT
Answer the following questions.
x + 5 = log 6(4y + 2) Add 5 on each side. The graph is shifted right 2 units and up 2 units. −1 0 1 2 3 4 5 6 7 8 9

6 x + 5 = 4y + 2; the Find the inverse of each function. 1. A scientist predicts there will be an earthquake with an epicenter in Palmdale,
6x + 5 = 4y + 2 Rewrite in exponent form. exponent should be x + 5. California and the magnitude of this earthquake will be an 8.2 on the Richter
2. f(x) = 4x + 1 3. f(x) = 2 log 0.5(−5x) + 4 4. f(x) = ln 3 x − 2 scale. For a seismograph 100 km from the epicenter, how many times as large
6 x + 3 = 4y Subtract 2 from each side. 6 x + 5 − 2 = 4y
f −1(x) = −__51 (0.5 ) f −1(x) = ____
___
x–4 e x+2 will the amplitude of the reading be as A 0?
________
6x + 5 − 2 f −1(x) = log4x−1
x
8.2 = log( AA ) ___ ___
(A0)
6 +3
______ 2
4
=y Divide by 4 on each side.
4
=y 3 A
= 10 8.2; 10 8.2 times as large
x 0
6 +3
The equation of the inverse of f(x) = log 6(4y + 2) − 5 is f −1(x) = _____ . 5. A hurricane center uses the function s = 95 logd + 75 to relate the wind speed
4 2. Earthquakes with magnitude greater than 5.0 may cause damage. A scientist
x+5
f −1(x) = 6 4 − 2 ________ in miles per hour s and distance in miles d a hurricane travels. How many miles
will the hurricane travel with a wind speed of approximately 320 mph?
calculates that if a seismograph is 100 km from an earthquake centered in
San Diego, the amplitude of the reading will be 10 3.1cm. Assuming she is right,
320 = 95 logd + 75 d ≈ 10 2.58 ≈ 380 mi what will the magnitude of the earthquake in San Diego be?

R = log(___ = log(____
−4 )
3. The f(x) = 3 (x – 2) – 1 and
A )
A 10 3.1
g(x) = log 3(x + 1) + 2 are inverse
f(x) = 3(x−2) − 1 g(x) = log3(x + 1) + 2
6. Which company’s profit shows a greater average rate of change between 10 R = 10 7.1 R = 7.1
010
functions shown on the graph at 2010 and 2015?
x y x y
Company A: $1.5 million profit in 2010; after 5 years, grew exponentially 3. Complete the following table to find out which city has the largest magnitude
the right. Complete the table
to $2.5 million. of the surface wave. The surface wave magnitude M of the earthquake is given
without using a calculator. −8 −8
0 9 9 0
Company B: profit, in million of dollars, modeled by P(B) = 1.3(1.15) x, where x is by this formula: M = log(__
T)
A
+ 1.66(log D) + 3.3.
y
4 1 −2 −2 1 the number of years after the end of 2010.
f(x) 3 3 The surface wave magnitude of the
2 2 0 0 2
Company A: 0.2 million per year; Company B: about City (A, T, D)
earthquake
g(x)
x 0.26 million per year; Company B has greater average
3 2 2 3 San Francisco (10 4, 5, 150°) 10.21
−4 −2 O 2 4 4 8 8 4 rate of change.
y = x −2 (10 3.1, 5, 100°)
San Diego 9.02
5 26 26 5 Let f(x) = 43x + 1, g(x) = log4x, and h(x) = 4x. Find a simplified expression for each
−4
Domain: All real numbers Domain: {x  x > −1} composition of functions. Palmdale (10 4.2, 6, 180°) 10.47
Range: {y  y > −1} Range: All real numbers
7. f ∘ g (f ∘ g)(x) = 3x + 1 8. h ∘ g (h ∘ g)(x) = x a. What is the magnitude of the earthquake in San Francisco?
R = log( AA ) = log( 10−4 )
x-intercept: 2; y-intercept: − 8 x-intercept: − 8 ; y-intercept: 2
9 9 ___ 4
____
10 R = 10 8 R=8
Asymptote: x = –1 Asymptote: y = –1 0 10
End Behavior: End Behavior: b. Compare the magnitude of an earthquake that hits San Diego and
As x −∞, y –1 As x −1, y –∞ an earthquake that hits San Francisco. Which city is hit with the most
As x ∞, y ∞ As x ∞, y ∞
damage? San Francisco

enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources

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Mathematical Literacy
and Vocabulary
Digital Resources and Video Tutorials
The Reteach to Build Understanding, Additional Practice, and Enrichment activities
W
Helps students develop and reinforce
are available as digital assignments powered by MathXL for School. These activities
understanding of key terms and concepts.
are automatically assigned when students complete the lesson quiz online and are
automatically scored.
Name B.E.S.T. ALGEBRA 2

6-4 Mathematical Literacy and Vocabulary

SavvasRealize.com
Students can access Virtual Nerd instructional tutorials online.
Logarithmic Functions

A jet takes off from an airport at sea level. Its altitude h, in feet, at time t, in
seconds, is modeled by the function h = 1,650 ln(t + 1) + 100. How long will it take
the jet to reach 10,000 ft?
Use the two sets of note cards to find the solution of this problem. The set on the
left explains the thinking. The set on the right shows the steps. Write the thinking
and the steps in the correct order.
Think Cards Step Cards
Divide each side by 1,650. h = 1,650 ln(t + 1) + 100

Write the original equation.

6 = ln(t + 1)
Subtract 1 from each side. t ≈ 402 sec
Write in exponential form.
10,000 = 1,650 ln(t + 1) + 100

Subtract 100 from each side. e6 = t + 1

Write the equation when the

jet reaches 10000 feet. 9,900 = 1,650 ln(t + 1)

Think Write
First, write the original equation. Step 1
h = 1,650 ln(t + 1) + 100
Second, substitute 10,000 for h. Step 2
10,000 = 1,650 ln(t + 1) + 100
Third, subtract 100 from Step 3
each side. 9,900 = 1,650 ln(t + 1)
Then, divide each side by 1,650. Step 4
6 = ln(t + 1)
Next, write in exponential form. Step 5
e6 = t +1
Finally, subtract 1 from each Step 6
side. t ≈ 402 sec
QR CODES Students also have direct access to select videos by scanning
enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources the QR code on the exercise pages or the Student Companion pages.
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TOPIC 6 320B LESSON 4

HSM23_TEA2_FL_T06_L04.indd 2 22/04/21 2:43 AM

44
 STEP 2 STEP 3 STEP 4
STEP 1
UNDERSTAND & PRACTICE & ASSESS &
EXPLORE
APPLY PROBLEM SOLVING DIFFERENTIATE

Digital

ASSESS AND DIFFERENTIATE

U Lesson Quiz
• Available for print or digital administration
• Robust differentiated intervention resources can be assigned to students based
on quiz score.
• Digital version is auto-scored and intervention or enrichment auto-assigned to students.

V Differentiated Intervention
• Robust resources address the needs of all students.

Reteach to Build Understanding**

• Guided reteaching helps students better understand lesson concepts.

Mathematical Literacy & Vocabulary

• Scaffolded support helps students build vocabulary.

Additional Practice**

• Offers more practice for each lesson

Enrichment**

• Offers activities for advanced learners

Print-based and Digital Activities:

All activites are available at SavvasRealize.com as PDFs and as editable files.

**Digital assignments powered by MathXL® for School are available and auto scored.

W Video Tutorials
• Virtual Nerd video tutorials are available for every lesson.
• Videos can be accessed online, or by scanning the QR code on the
exercise pages or the Student Companion pages.

45
 PREPARING FOR A TOPIC Insightful overview of the content
right at point of use!

TOPIC 6 TOPIC 6
Exponential and Logarithmic Functions Exponential and Logarithmic Functions
MATH BACKGROUND MATH BACKGROUND

Topic 6 focuses on extending previous understanding of exponential functions. Students Students learn best when concepts are connected through the curriculum. This coherence is
identify the key features of exponential functions. Students understand logarithms and achieved within topics, across topics, across strands, and across grade levels.
their properties. Students learn how to solve exponential and logarithmic equations.

IN THIS TOPIC
Understanding Logarithms Solving Exponential and Logarithmic MAKING MATHEMATICAL CONNECTIONS How is content connected within Topic 6? MAKING MATHEMATICAL CONNECTIONS
Convert Between Exponential and Logarithmic Form Students Equations Looking Back Looking Ahead
• Exponential and Logarithmic Functions
learn that the inverse of an exponential function is a logarithmic Solving Exponential Equations Using a Common Base Students How does Topic 6 connect to what Students use what they learn about the How does Topic 6 connect to what
function, and vice versa. Students use this to rewrite and solve rewrite each side of an equation as an exponent with a common students learned earlier? key features of exponential functions students will learn later?
equations. base. Then students solve the equation using the Property of in Lesson 6-1 to identify rates of
Equality for Exponential Equations. ALGEBRA 1 growth of exponential functions and Pre-Calculus
Evaluate Logarithms Students learn how to evaluate common • Properties of Exponents In Topic 5 of solve problems involving compound • Exponential and Logarithmic Functions
logarithms and natural logarithms using a calculator, allowing Algebra 1, students used the Product, interest in Lesson 6-2. They also apply In Topic 6, students study the behavior
Solve Exponential Equations Using Logarithms Students solve
students to estimate solutions of logarithmic equations. Quotient, and Power Properties of their knowledge from Lesson 6-1, along of exponential and logarithmic
exponential equations using logarithms. Students use their
Exponents to simplify exponential with the new concept of exponential functions. In pre-calculus, they apply
understanding of the Property of Equality for Logarithmic
expressions. In Topic 6 of Algebra 2, and logarithmic functions as inverses, their understanding of these effects to
log(900) Equations and the Product, Quotient, and Power Properties of
2.954242509 students will recognize that the to identify key features of the graphs the graphs of trigonometric functions.
Logarithms to solve equations. Students check their solutions by
ln(e) Product, Quotient, and Power of logarithmic functions in Lesson 6-4. They also learn how the value of b
1 graphing the equation.
ln(–1.87) Properties of Logarithms are very affects the period of the graph of a
Error y similar to the Properties of Exponents, trigonometric function.
• Exponential and Logarithmic Equations
and will use the new properties to
30 In Lesson 6-3, students learn how to
solve logarithmic equations. CALCULUS
convert between exponential and
20 logarthmic forms of an expression and • Differentiating Logarithmic Functions
• Transformations of Functions In
how to evaluate logarithms. In In Lesson 6-4, students learn to
Algebra 1, Lessons 7-1 and 7-2, students
Graph Logarithmic Functions Students learn to graph logarithmic 10 Lesson 6-5, they learn the Properties calculate the rate of change between
learned how a, h, and k affect the
functions and identify the key features of the graphs. This x of Logarithms. These two lessons two points of a logarithmic function.
graph of a function in the form
understanding allows students to easily graph transformations of O In Calculus, students will extend their
1 2 3 f(x) = a(x − h) 2 + k. In Topic 6, students provide the background knowledge
logarithmic functions. students need to solve exponential understanding of rate of change.
will apply that knowledge to graph
and logarithmic equations in Students will use differentiation to
y functions such as g(x) = a ∙ b (x − h) + k
4 Lesson 6-6. Students solve exponential calculate the rate of change of a
and j(x) = a ∙ log(x − h) + k.
Solve Logarithmic Equations Students solve logarithmic equations equations by rewriting expressions with logarithmic function at a single point.
2 using the Property of Equality for Logarithmic Equations. Students a common base and converting them
y = 2x
x understand they must check that each solution satisfies the to logarithmic form. They also solve
−4 −2 O 2 4 original equation to eliminate any extraneous solutions. TOPIC 2 equations containing exponential and
−2 y = log2 x • Linear-Quadratic Systems Students logarithmic expressions by graphing.
learned to solve systems of linear and
Solve Logarithmic and Exponential Equations by Graphing
−4 quadratic equations. In Topic 6,
Students solve equations with logarithmic or exponential • Exponential and Logarithmic Models
students will solve systems of linear
expressions on one side and linear expressions on the other side. Throughout the topic, students model
equations and exponential or
Students graph each side of the equation and find any points of real-life situations with exponential
logarithmic equations.
Properties of Logarithms Students expand and simplify intersection of the graphs. Students know that these points of and logarithmic functions.
logarithmic expressions using the Product, Quotient, and Power intersection are solutions to the equation.
Properties of Logarithms. Students use the Change of Base
formula to rewrite logarithms with different bases. Painting Value
80 (5, 64.4)

Value (thousands of dollars)

60 (0, 40)

40

20
0
0 1 2 3 4 5
Years

Sculpture Value

f(x) = 50(1.075)x
(thousands of dollars in x years)

TOPIC 6 286A TOPIC OVERVIEW TOPIC 6 286B TOPIC OVERVIEW

HSM23_TEA2_FL_T06_OV.indd 1 12/04/21 2:30 AM HSM23_TEA2_FL_T06_OV.indd 2 12/04/21 2:30 AM

Math Background: Focused Curriulum Math Background: Connected Curriculum

Each topic focuses on one or more content areas, such as Algebra This page summarizes the content connections throughout the
or Functions. As you prepare for a topic, familiarize yourself with course that help students learn new ideas by connecting those
the Topic Overview in the Teacher’s Edition. ideas to prior knowledge.
• Content Focus A discussion of the math content in the topic is • Looking Back In “Looking Back,” connections to previously
provided. Sample work and strategies illustrate the underlying taught concepts and skills are described. These connections
concepts so that you can anticipate the work students will do. may be within the grade across content areas or across grades.
See the Focused Curriculum page for more information.
• Looking Ahead In “Looking Ahead,” connections to content
• Professional Development Videos View Topic students will encounter later are described. Again, these
Overview Videos for important information about the connections may be within a grade or across grades.
content.
Additional Reading and Resources See the Connected
Curriculum page for more information.

46
 Digital

TOPIC 6 TOPIC 6
Exponential and Logarithmic Functions Exponential and Logarithmic Functions
MATH BACKGROUND MATHEMATICAL THINKING AND REASONING

Concepts Skills Mathematical Thinking and Reasoning Within Topic 6 Lessons

• Exponential Growth and Decay Students understand that for • Compound Interest Formula Students use the compound
The Mathematical Thinking and Reasoning Standards promote deeper learning
exponential functions of the form f(x) = a ∙ b x, the domain is interest formula and the continuously compounded interest
and understanding of mathematics and provide clear expectations for students
the set of is all real numbers. The value of a is the y-intercept; formula to find compound interest given the principal, the
when a is positive, the range is y > 0 and when a is negtive, the rate, and the time.
as they engage with mathematics. Opportunities to develop expertise with
range is y < 0. Students also differentiate between exponential these important skills exist throughout the topic and program. Here we focus
growth and exponential decay models based on the value of • Change of Base Formula Students learn and practice using on representing problems and using structure.
b. When b > 1 in f(x) = a ∙ b x, the function models exponential the Change of Base Formula as a method for evaluating
growth. When 0 < b < 1, it models exponential decay. logarithmic expressions. As students model with exponential and logarithmic functions, look for
the following behaviors to assess and identify students who engage with
• Logarithms Students understand that logarithms are • Solving Exponential and Logarithmic Equations Students solve mathematics through thinking and reasoning.
the inverse of exponentiation and that by applying this a variety of exponential and logarithmic equations. Some
relationship, they can graph logarithmic functions and solve exponential equations have different bases and contain a
exponential and logarithmic equations. Students learn that variable within the exponent. Other exponential equations Highlighted Mathematical Thinking and Reasoning Within Topic 6 Lessons
most calculators can only evaluate common logs and natural need to be rewritten in logarithmic form in order to be solved.
logs, so when they encounter logarithms with bases other than Students apply the Property of Equality for Logarithmic Equations
MTR.5.1 Use patterns and structure to help
10 and e, they need to apply the Properties of Logarithms to to rewrite logarithmic equations as polynomial equations in MTR.2.1 Demonstrate understanding by
understand and connect mathematical
rewrite them in a form the calculator can handle. order to solve. Finally, they solve equations that contain a representing problems in multiple ways.
logarithm and an exponent in the same expression by graphing. concepts.
y Mathematicians who demonstrate understanding by Mathematicians who use patterns and structure to help
4
representing problems in multiple ways: understand and connect mathematical concepts:
g
2 Applications • Graph transformations of exponential functions to explore • Apply general mathematical rules of transforming functions
x • Modeling With Exponential and Logarithmic Equations Students differences between the asymptotes and intercepts of parent to transform graphs of logarithmic functions and use structure
O 2 encounter many different types of situations and relationships and given functions. by identifying that because logarithmic functions are inverses
−2 f that lead to creating and solving exponential and logarithmic of exponential functions, their graphs are reflections of
equations. Students calculate interest earned using the compound exponential functions across the line y = x.
interest formula and the continuously compounded interest
formula. Students solve problems involving growth rates of • Contextualize exponential relationships by identifying • Look for patterns to determine whether they can use the
• Modeling Exponential and Logarithmic Functions Students
populations and the altitudes of airplanes. the monthly growth rate from an annual rate. They natural log or the common log to solve an exponential
understand that they can model several real-world situations
decontextualize exponential relationships by representing equation. They use structure to determine the annual rate
with exponential and logarithmic functions. They also realize
real-world situations as exponential functions. that was used to project the amount of money in an account.
they can compare exponential and logarithmic functions using
equations, tables, and graphs. • Utilize data points on a graph of an exponential model to • Look for overall structure and patterns in exponents and
write the model. logarithms, as well as apply general math rules to evaluate
• Geometric Sequences Students understand that geometric logarithms.
sequences are exponential functions with a domain limited to
the set of natural numbers. Students learn that sequences can A = 9,200 In t + 10,000 • Represent exponential functions in logarithmic form to solve for • Look for repeated multiplication in their calculations and
be defined recursively or explicitly. variables within exponents, such as determining the magnitude recognize that these problems can be represented by
of an earthquake from the energy produced by the earthquake. exponential functions.
t is time A is
in minutes altitude Create opportunities for students to represent exponential and logarithmic functions in multiple ways. You can also
after takeoff. in feet. support students as they develop an understanding of the patterns in exponents and logarithms.

You can use these questioning strategies throughout the topic.

Q: How can you use Properties of Logarithms to explain Q: What do you notice about the calculations that will help you
how two different logarithmic equations have the same recognize that a problem can be represented by an exponential
graph? function?
Q: How are exponential functions related to logarithmic Q: How does the Change of Base Formula for logarithms help you
functions? solve exponential equations?
Q: What is the relationship between the growth factor and the Q: What do you notice about the graphs of exponential and
growth rate of an exponential model? logarithmic functions that share an inverse relationship?
Q: How can you decide if a real-world problem can be modeled Q: In what ways does rewriting an exponential equation in
using a logarithmic function? logarithmic form connect to other mathematical concepts?

TOPIC 6 286C TOPIC OVERVIEW TOPIC 6 286D TOPIC OVERVIEW

HSM23_TEA2_FL_T06_OV.indd 3 12/04/21 2:30 AM HSM23_TEA2_FL_T06_MP.indd 4 12/04/21 2:34 AM

Math Background: Balanced Curriculum Mathematical Thinking and Reasoning

This page summarizes the three aspects of rigor that balance An overview of how the Mathematical Thinking and Reasoning
each topic. Standards are connected to content objectives in the topic(s) is
provided.
• Concepts An overview of the key concepts students will
encounter in the topic. • Examples are given of how Mathematical Thinking and
Reasoning Standards are reinforced in every topic.
• Skills The procedural skills taught in the topic(s) are discussed
and fluency expectations, if any, are addressed. Additional Reading and Resources See the Mathematical
Thinking and Reasoning Standards page for more information.
• Applications A description of the types of applications in
the topic(s) is provided.

Additional Reading and Resources See the Balanced

Curriculum page for more information.

47
PREPARING FOR A TOPIC (continued) Everything you need to plan for
teaching a topic!

TOPIC 6 TOPIC 6
Exponential and Logarithmic Functions Exponential and Logarithmic Functions Assess

TOPIC PLANNER TOPIC READINESS ASSESSMENT

AVAILABLE ONLINE

6-1 Key Features of 6-2 Exponential Models Mathematical Modeling in 3 6-3 Logarithms
Lesson

9. Simplify the expression (x 3 · x 3 ) .

1
_ 1 6
_
Name
Exponential Functions
B.E.S.T. ALGEBRA 2

Acts: The Crazy Conditioning SavvasRealize.com

15. What are the domain and range of
the function given by {(−2, 0),
6 Readiness Assessment 𝖠 ​x 4 𝖢𝖢𝖢𝖢 x6
(−1, 2), (0, 4), (1, 6), (2, 8)}?
𝖡 ​x 3 𝖣𝖣𝖣𝖣 x2
−1, 0, 1, 2}
3 DAYS (O) 3 DAYS (H) 3 DAYS (O) 3 DAYS (H) Domain: {−2,
3 DAYS (O) 3 DAYS (H) 1–2 DAYS (O) 1–2 DAYS (H)
10. Simplify the expression (9x 4) 2 .
3
_ Range: {0, 2, 4, 6, 8}
1. Evaluate the expression 10 x−3 for 6. Ina records the wait time y, in
x = 2. days, for popular library books 𝖠 ​27x 4 𝖢𝖢𝖢𝖢 81x 6
Benchmarks Benchmarks Benchmarks Benchmarks 𝖠𝖠𝖠𝖠𝖠𝖠 𝖠𝖠−10 𝖢𝖢𝖢𝖢 based on how many people are 16. The table shows a function f. What
0.1
on the wait list, x.
𝖡 27x 6 𝖣𝖣𝖣𝖣 81x 12 is the domain of f −1, the inverse of
MA.912.AR.5.4, AR.5.5, AR.5.7, AR.5.4, AR.5.5, AR.5.7, F.1.1, AR.5.4, AR.5.5 NSO.1.6, AR.5.2, AR.5.7, F.3.7
Benchmarks

𝖡 𝖠𝖠−1 𝖣𝖣𝖣𝖣 10 Which line best fits these data? function f?

F.1.1, F.1.7 FL.3.1, FL.3.2, FL.3.4, DP.2.9 MTR Standards MTR Standards x (0, 3), (1, 5), (2, 9), (3, 13), (4, 15)
11. Find the inverse of the function x 1 2 3 4 5
2. Evaluate f(x) = (_32 ) for x = −3 and f(x) = _23 x − 6. f(x) 1 4 9 16 25
MTR Standards MTR Standards MTR.7.1 MTR.2.1, MTR.4.1, MTR.5.1 x = 0. Select the function values. 𝖠𝖠𝖠𝖠𝖠𝖠 y = 2.52x + 4.29 3
__ 2
__ 2
−__ 6 4 9
◻𝖠𝖠 A. f(−3) = −2 ◻ 2 3 3 𝖠 {1, 2, 3, 4, 5}
MA.K12.MTR.1.1, MTR.2.1, MTR.2.1, MTR.3.1, MTR.6.1 D. f(0) = 0 𝖡 y = 3.11x + 2.71
−1 __2
◻𝖠𝖠 B. f(−3) = ___
8 ◻ E. f(0) = 1 𝖢𝖢𝖢𝖢 y = 3.63x + 2.26 f (x) = x+ 4 𝖡 {1, 4, 9, 16, 25}
MTR.7.1 27 3
𝖢𝖢𝖢𝖢 {1, 2, 3, 4, 5, 9, 16, 25}
◻𝖠𝖠 C. f(−3) = ___
27
8
◻ 2
F. f(0) = __
3
𝖣𝖣𝖣𝖣 y = 4.07x + 3.34
12. What is the inverse of the function 𝖣𝖣𝖣𝖣 {−5, −4, −3, −2, −1, 1, 2, 3, 4, 5}
7. How does the graph of f(x) = (x − 1) 3 + 8?
3. Evaluate −3 x+2 for x = −2. _____
y = (x + 2) 2 − 5 relate to the graph 𝖠 ​f −1(x) = √x − 7
3
17. The function f(x) = 2x 2 − 5x + 3
𝖠𝖠𝖠𝖠𝖠𝖠 𝖠𝖠−9 𝖢𝖢𝖢𝖢 0 of its parent function, y = x 2? __ represents the height in meters, y,
The rate of exponential Exponential models are useful Many real-world problem A logarithmic function is the 𝖡 𝖠𝖠−1 𝖣𝖣𝖣𝖣 1 𝖡 ​f −1(x) = −1 + √x 3
of a marble on a marble run after
𝖠𝖠𝖠𝖠𝖠𝖠 Translate 2 units left and reflect _____
𝖢𝖢𝖢𝖢 ​f −1(x) = 1 + √x − 2
3
growth or decay is the ratio for representing situations in situations can be represented inverse of an exponential 4. For the function g(x) = (−2) 0.5x,
over the y-axis
_____
x seconds. What is the average rate
of change of the function from
Essential Understanding

𝖣𝖣𝖣𝖣 ​f −1(x) = 1 + √x − 8
3

between two consecutive which the rate increases by the with a mathematical model, function. Logarithms are select all the equations that 𝖡 Translate 5 units down, stretch x = 0 to x = 3?
vertically by a factor of 2
are true.
output values in an same percent for each period but that model might not found by determining the 𝖢𝖢𝖢𝖢 Translate 2 units left and 13. The table shows the function f.
𝖠 ​−2 𝖢𝖢𝖢𝖢 1
◻ 1
A. g(−4) = −__ ◻ D. g(4) = 4 𝖡 ​−1 𝖣𝖣𝖣𝖣
exponential function. of time and for interpreting represent the real-world exponent that must be applied ◻ 1
B. g(−2) = __
4
◻ E. g(2) = −2
5 units down
Complete the table for the inverse
4

problems that involve situation exactly. to a base to yield a given 2 𝖣𝖣𝖣𝖣 Translate 2 units right and of function f.
◻ C. g(0) = −1 5 units up 18. The graph represents the
compound interest. Exponential result. x 0 1 2 3 4 population of a bacterium, y, after
f(x) 4 2 0 3 1 x seconds. Find the average rate of
5. The number of muffins, y, The Cozy 8. Select the graph of y = −_12 (x − 3) 3
regression can be used to Cafe bakes each day, x, increases as and its parent function.
f –1(x) 2 4 1 3 0 change of the function, from
generate exponential models their advertising campaign brings
4
y
4
y
x = −1 to x = 2.
in more customers. They recorded
for real-world contexts. how many muffins they baked
2 2 14. For the function f(x) = −3x + 1, 4
y
(2, 4)
𝖠𝖠𝖠𝖠𝖠𝖠 𝖢𝖢𝖢𝖢 what is the value of f −1(x) for x = 4?
x x
every second day. What is the line −4 −2 O 2 4 −4 −2 O 2 4
2
of best fit for these data?
−2 −2
−1 (−1, 12)
−4 −4
f (4) = −1 x
−4 −2 O 2 4
(0, 5), (2, 12), (4, 21), (6, 28), (8, 38),
y −2
(10, 45) 4
y 4

2 −4
𝖠𝖠𝖠𝖠𝖠𝖠 y = 0.12x − 0.54 2
x x
𝖡 −4 −2 O 2 4 𝖣𝖣𝖣𝖣 −4 −2 O 2 4
𝖡 y = 0.24x − 1.09 𝖠 __29 3
__
• Interpret key features • Rewrite exponential • Use mathematical modeling • Understand the inverse −2 −2 𝖢𝖢𝖢𝖢 2
𝖢𝖢𝖢𝖢 y = 4.07x + 4.48 −4 −4
𝖡 __67 19
of exponential functions functions to identify rates. to represent a problem and relationship between AVAILABLE ONLINE 𝖣𝖣𝖣𝖣 ___
5
𝖣𝖣𝖣𝖣 y = 8.14x + 4.47
represented algebraically, • Interpret the parameters to propose a solution. exponents and logarithms. Topic Readiness Assessment Assess students’ understanding of prerequisite
enVision® Florida B.E.S.T. Algebra 2 • Assessment Resources
concepts and
enVision Florida B.E.S.T. Algebra 2 • Assessment Resources ®

graphically, in tables, or in
Content Objective

of an exponential function • Test and verify the • Use logarithms to solve skills using the Topic Readiness Assessment found at SavvasRealize.com. These auto-scored
written descriptions. within the context of appropriateness of their exponential models. HSM23_ANCA2_FL_T06_RA.indd Page 1 01/05/21 7:08 PM f-0317a
online assessments provide students with a breadth of technology-enhanced item types.
/155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/Algebra ... HSM23_ANCA2_FL_T06_RA.indd Page 2 01/05/21 7:08 PM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/Algebra ...

• Graph transformations compound interest math models. • Evaluate logarithms using

of exponential functions problems. Individualized Study Plan Based on their performance, students will be assigned a study
• Explain why the results from technology.
showing intercepts and • Construct exponential their mathematical models plan tailored to their specific learning needs.
end behavior. models given two points or might not align exactly with
• Model quantities that have by using regression. the problem situation.
a constant percent rate of
Item Analysis for Diagnosis and Intervention
change using exponential Intervention Intervention
Item
Item DOK Skills ReviewDOK
& Practice Benchmarks
Benchmarks Item
Item DOK Skills ReviewDOK
& Practice Benchmarks
Benchmarks
function. Lesson Lesson

decay factor, exponential compound interest formula, none common logarithm, logarithm, 11 1 1 NS-011
00-00 MA.912.NSO.1.1
00.000.0.0.0
N-RN.1.2 1010 1 1 NS-02
00-001 MA.912.NSO.1.1
00.000.00.0.0
N-RN.1.2
Vocabulary

decay function, exponential continuously compounded logarithmic function, natural 22 1 2 NS-021

00-00 N-RN.1.2
MA.912.NSO.1.1
00.000.0.0.0 1111 2 1 0-002
F-15 F-BF.2.4
MA.912.F.3.7
00.0.0.0.0
function, exponential growth interest formula, natural base e logarithm
function, growth factor 33 1 1 NS-011
00-00 N-RN.1.2
MA.912.NSO.1.1
00.000.00.0.0 1212 2 1 0-002
F-16 F-BF.2.4
MA.912.F.3.7
00.000.00.0
4 1 1 N-RN.1.2
MA.912.NSO.1.1
00.000.0.0.0 13 1 2 F-BF.2.4
4
5
1
<new1 table TK>
2
NS-02
00-00
MA.912.F.1.2
00.000.0.0.
S-ID.2.6.a
13
14
<new
2
table
1
TK>
F-15
0-00
2
MA.912.F.3.7
00.000.00.0
F-BF.2.4
5 2 F-15
0-00 MA.912.DP.2.4
00.000.0.0.0 14 2 F-16
0-00 MA.912.F.3.7
00.0.00.0.0
6 1 2 S-ID.2.6.a 15 10 1 F-IF.1.1
6 2 F-16
0-00 MA.912.DP.2.4
00.000.0.0.0 15 1 R-19
0-00 MA.912.AR.2.4
00.0.00.0.0
7 1 2 F-BF.2.3 16 1 1 F-BF.2.4, F-IF.1.1
7 2 F-15
0-00 MA.912.F.2.1
00.0.0.0.0 16 1 A-24
0-00 MA.912.F.3.7
00.000.00.0
8 1 2 F-BF.2.3 17 1 2 F-IF.2.6
8 2 F-16
0-00 MA.912.F.2.5
00.0.0.0.0 17 2 F-15
0-00 MA.912.F.1.3
00.000.00.0
9 1 1 N-RN.1.2 18 1 2 F-IF.2.6
9 1 NS-01
00-00 MA.912.NSO.1.1
00.0.0.0.0.0 18 2 F-16
0-00 MA.912.F.1.3
00.000.00.0

TOPIC 6 286E TOPIC OVERVIEW TOPIC 6 286H TOPIC READINESS ASSESSMENT

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Topic Planner Topic Readiness Assessment

The Topic Planner provides an overview of the topic. This page presents an overview of the Topic Readiness
Assessment. Teachers can use this assessment to determine their
• Objective and Essential Understanding, These are about
students’ readiness for the content of the topic and prescribe
the content of the lesson.
intervention as needed.
• Vocabulary Any new vocabulary terms for the lesson are listed.
• Online Topic Readiness Assessment Students can take
• Pacing recommendations for each lesson. the Topic Readiness Assessment online where it is auto-scored
and intervention is auto-assigned.
• Benchmarks The benchmarks addressed in each lesson, and
highlighted Mathematical Thinking and Reasoning Standards,
are listed.

TOPIC 6 TOPIC 6
Exponential and Logarithmic Functions Exponential and Logarithmic Functions
TOPIC RESOURCES TOPIC OPENER
AVAILABLE ONLINE
PROGRAM

PROGRAM
DIGITAL

DIGITAL
LESSON

LESSON
TOPIC

TOPIC

TOPIC Exponential and

6
Resources Resources MATHEMATICAL MODELING IN 3 ACTS
Logarithmic Functions
Student and Teacher Edition Florida’s B.E.S.T. Assessment Practice Workbook
TOPIC ESSENTIAL QUESTION

• enVision® STEM Project • • Benchmark Practice • How do you use exponential and logarithmic functions to
model situations and solve problems?

• Mathematical Modeling in 3 Acts • • Practice Test •

Topic Overview Topic Vocabulary
• Practice & Problem Solving • OTHER RESOURCES enVision® STEM Project:
Analyze Elections
• Change of Base Formula
• common logarithm
6-1 Key Features of Exponential Functions • compound interest

• Concept Summary • • Concept Review • • • AR.5.4, AR.5.5, AR.5.7, F.1.1, F.1.7, MTR.1.1, MTR.2.1, MTR.7.1

6-2 Exponential Models

• continuously compounded
interest
AR.5.4, AR.5.5, AR.5.7, F.1.1, FL.3.1, FL.3.2, FL.3.4, DP.2.9,
MTR.2.1, MTR.3.1, MTR.6.1 • decay factor

• Topic Review • Desmos® Activities • Mathematical Modeling in 3 Acts: The Crazy Conditioning
AR.5.4, AR.5.5, MTR.7.1
• exponential equation
• exponential function
The Crazy Conditioning
Like all sports, soccer requires its players to be well trained. That is why
6-3 Logarithms • exponential decay function players often have to run sprints in practice.

• Visual Glossary (English and Spanish) • Desmos® Anytime Tools • • • NSO.1.6, AR.5.2, AR.5.7, F.3.7, MTR.2.1, MTR.4.1, MTR.5.1
• exponential growth
To make sprint drills more interesting, many coaches set up competitions.

TOPIC 6
6-4 Logarithmic Functions function
AR.5.7, AR.5.8, AR.5.9, F.1.7, F.2.2, F.2.3, F.2.5, F.3.7, MTR.1.1,
Coaches might split the players into teams and have them run relay races
• growth factor against each other. Or they might have the players sprint around cones
• • •
MTR.5.1, MTR.7.1
Resource Masters Digital Math Tools 6-5 Properties of Logarithms • logarithm and over barriers. What other ways would make doing sprints more fun?
NSO.1.6, NSO.1.7, AR.5.2, MTR.4.1, MTR.5.1, MTR.6.1 • logarithmic equation Think about this during the Mathematical Modeling in 3 Acts lesson.

• Reteach to Build Understanding • Additional Examples •

6-6 Exponential and Logarithmic Equations
AR.5.2, MTR.3.1, MTR.4.1, MTR.7.1
• logarithmic function
• natural base e
6-7 Geometric Sequences • natural logarithm

• •
AR.10.2, MTR.4.1, MTR.6.1, MTR.7.1

Additional Practice Virtual Nerd Videos •

Online Only

• Mathematical Literacy and Vocabulary • Answers & Solutions • Digital Experience

• Enrichment • Adaptive Practice •

INTERACTIVE STUDENT EDITION
Access online or offline.
VIDEOS Watch clips to support
Mathematical Modeling in 3 Acts Lessons
and enVision® STEM Projects.
ASSESSMENT Show what
you’ve learned.
FAMILY ENGAGEMENT TUTORIALS Get help from

• •
Involve family in your learning. ADAPTIVE PRACTICE Practice that is Virtual Nerd, right when you need it.
enVision® STEM Project ExamView® Test Generator • • • ACTIVITIES Complete Explore & Reason,
Model & Discuss, and Critique & Explain
just right and just for you.
GLOSSARY Read and listen to
MATH TOOLS Explore math
with digital tools and manipulatives.
activities. Interact with Examples and Try Its. English and Spanish definitions.
Assessment Sourcebook DESMOS Use Anytime and as embedded
Family Engagement • • • ANIMATION View and interact with
real-world applications.
CONCEPT SUMMARY Review
key lesson content through
Interactives in Lesson content.
QR CODES Scan with your mobile device
multiple representations.
• Course Readiness Test • Classroom Videos •
Go online | SavvasRealize.com
PRACTICE Practice what
you’ve learned.
for Virtual Nerd Video Tutorials and Math
Modeling Lessons.

• Progress Monitoring Assessments A, B, C • Topic Overview Professional Development Video •

286 TOPIC 6 Exponential and Logarithmic Functions TOPIC 6 Exponential and Logarithmic Functions 287

• Topic Readiness Assessment • Multilingual Handbook •

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• Lesson Quiz • Topic Essential Question Mathematical Modeling in 3 Acts

How do you use exponential and logarithmic functions to Generate excitement about the upcoming Mathematical
• Topic Assessment •
model situations and solve problems? Modeling in 3 Acts lesson by having students read about
• Topic Performance Tasks • the math modeling problem for this topic.
Revisit the Topic Essential Question throughout the topic.
• Cumulative Assessment • See the Teacher's Edition Topic Review for notes about See the Teacher's Edition lesson support for notes about
answering the Topic Essential Question. how to use the lesson video in your classroom.

TOPIC 6 286G TOPIC OVERVIEW TOPIC 6 286–287 TOPIC OVERVIEW

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Lesson Resources and Topic Resources Topic Opener

Digital

Print
A list of lesson and topic resources is shown after
the Topic Planner. You can customize the course and
This page presents a succinct overview of the topic, showing the
Topic Essential Question, a listing of the lessons, and the new
upload your own resources or district resources vocabulary for the topic.
at SavvasRealize.com.

48
FAMILY ENGAGEMENT
Digital

Family Engagement Resources

The Family Engagement Resources consist of program level, Topic Level resources include:
topic level, and lesson level support for families. • a topic summary in family-friendly language that explains the
math content taught in the topic and offers ideas for how to
Program Level resources include:
connect that math to everyday life experiences.
• a list of Florida’s B.E.S.T. Standards: Mathematics within
the course • a list of the lessons and Florida’s B.E.S.T Standards: Mathematics
taught in the topic
• a list of student digital resources—including glossaries, math
tools, and video tutorials—that families can use any time. Lesson Level resources include:
• lesson objectives that outline the lesson’s learning goals.
• links to video tutorials that reinforce concepts in each lesson.
• links to key vocabulary.

49
MATHEMATICAL MODELING Focus on the mathematical
modeling process

What is Mathematical Modeling?

TOPIC 6
The writing team of the GAIMME Report (2016) describe Mathematical Modeling in 3 Acts: Video

mathematical modeling as “[using] mathematics to answer big, The Crazy Conditioning

messy, reality-based questions” (p. 9) and as “a process that uses
Lesson Overview
mathematics to represent, analyze, make predictions or otherwise MATHEMATICAL
MODELING
IN 3 ACTS
Objective
provide insight into real-world phenomena” (p. 10). Students will be able to:
✓ Use mathematical modeling to represent a problem situation

The writing team recommends five guiding principles to help

and to propose a solution.
MA.912.AR.5.4–Write an
exponential function to represent
a relationship between two
quantities from a graph, a written

✓ Test and verify the appropriateness of their math models. description or a table of values
within a mathematical or
real-world context. Also AR.5.5

teachers begin to implement real mathematical modeling tasks in

✓ Explain why the results from their mathematical models MA.K12.7.1

might not align exactly with the problem situation.

The Crazy Conditioning

the classroom. These principles can be summarized as follows: Essential Understanding Like all sports, soccer requires its players to be well-trained. That is why
players often have to run sprints in practice.
To make sprint drills more interesting, many coaches set up competitions.
Many real-world problem situations can be represented with a Coaches might split the players into teams and have them run relay races
against each other. Or they might have the players sprint around cones and
mathematical model, but that model might not represent the over barriers. What other ways would make doing sprints more fun? Think

1. Modeling tasks are open-ended, with very little information

about this during this Mathematical Modeling in 3 Acts lesson.
real-world situation exactly.
ACT 1 Identify the Problem

given. Students need to be making assumptions, finding Previously in this topic, students:
• Solved problems using exponential functions.
1. Write down the Main Question you will answer.
2. Make an initial conjecture that answers this Main Question.
3. Explain how you arrived at your conjecture.

information, thinking about potential models that can be used.

4. Write a number that you know is too small.
In this lesson, students: 5. Write a number that you know is too high.
6. What information do you need to know to answer the main question?
• Develop a mathematical model to represent and propose How can you get it? How will you use that information?

a solution to a problem situation involving exponential

2. Modeling tasks require students to make genuine choices.

ACT 2 Develop a Model
functions. 1. Use the math that you have learned in this Topic to refine your conjecture.
2. Is your refined conjecture between the high and low estimates you came

Later in this topic, students: up with earlier?

3. Assessment of modeling tasks should focus on the process, not

ACT 3 Interpret the Results
• Refine their mathematical modeling skills using
1. Did your refined conjecture match the actual answer exactly? If not,
logarithmic functions. what might explain the difference?

the product. The nature of modeling tasks means that students Skills Students draw on their understanding of concepts related
to exponential functions to develop a representative model.
TOPIC 6 Mathematical Modeling in 3 Acts 305

may come up with different assumptions, different mathematical

HSM23_SEA2_FL_T06_MML.indd 305 10/01/21 5:15 AM

Applications Students apply their mathematical model to test

and validate its applicability to similar problem situations.
Student Companion
models. There may not be a single answer. Students can do their work for the lesson in their Student
Companion or in Savvas Realize.

4. Modeling tasks are best tackled in groups.

5. Modeling tasks come in all sizes. The important consideration is Florida’s B.E.S.T. Standards and Benchmarks

giving students lots of experiences with the process. In this lesson, students apply concepts and skills related to
Mathematics Florida Benchmarks MA.912.AR.5.4 and AR.5.5.
MA.K12.MTR.7.1 Apply mathematics to real-world contexts.
To solve the problem presented, students identify variables and
MA.912.AR.5.4 Write an exponential function to represent a the relationship among them, develop a model that represents

In enVision Florida B.E.S.T. A|G|A students learn to be good

relationship between two quantities from a graph, a written the situation, and use the model to propose a solution. Students
description, or a table of values within a mathematical or real- interpret their solutions and propose explanations for why their
world context. answer may not match the real-world answer.

modelers through a range of tasks. AR.5.5 Given an expression or equation representing an Students also engage in analyzing problems (MTR.1.1) and
representing problems in multiple ways (MTR.2.1). In testing their
exponential function, reveal the constant percent rate of change
models, students look for patterns in the structure of their models

• Mathematical Modeling in 3 Acts: These tasks present

per unit interval using the properties of exponents. Interpret the
constant percent rate of change in terms of a real-world context. (MTR.5.1).

to students interesting, real-world situation and ask them to

come up with a question to answer. They work in groups to 305
TOPIC 6 MATHEMATICAL MODELING IN 3 ACTS

determine what assumptions they can make, what information

they need, and what mathematics can model the situation. HSM23_TEA2_FL_T06_MML.indd 305 12/04/21 3:12 AM

• Model & Discuss: In these activities, students focus on

elements of the modeling process to help them become more
proficient modelers. Focus on Mathematical Modeling
• One Mathematical Modeling in 3 Acts lesson in each
topic

• Presents a high-interest situation that engages students in

the modeling process.

• Unlike traditional real-world problems, the situations

in Mathematical Modeling in 3 Acts lessons do not set up
the problem that students are to solve.

50
 Digital

TOPIC 6 Mathematical Modeling in 3 Acts TOPIC 6 Mathematical Modeling in 3 Acts

Video
Video

The Crazy Conditioning ACT 3 The Solution

In this mathematical modeling task, students will explore and apply concepts related to Play the video.The final video reveals the total time and distance
exponential equations and functions. Students will watch an athlete performing a running of the twentieth round of the drill.
drill. They will be tasked with extrapolating both the time and distance for a certain round of
the drill. To do so, they apply concepts they study in Topic 6. MAIN QUESTION ANSWER
The twentieth round of the drill is 29,789 miles and takes about
190 days.
Do the “post-game” analysis. Ask students why the actual
ACT 1 The Hook distance and time do not match the mathematical answer. The
first three rounds involve very few changes of direction, while
Play the video. The video shows an athlete running conditioning
later rounds involve frequent changes of direction. The benefit of
drills on a field. We see the athlete run three rounds of drills.
using regression is that it allows us to extrapolate.
After the question brainstorming, present to students the Main
Although completing this task is physically impossible, it's still
Question they will be tasked with answering. In this lesson,
good to have a conversation about how his speed would change
there are two main questions. If your class is advanced, ask both
over time.
questions. You could ask the first question and leave students to
predict the answer to the second question without making a
math model. Remind students to write down their questions
ONE POSSIBLE SOLUTION
and conjectures. Make a table to show the number of yards the runner runs in
each round for the first five rounds. See the table shown.
MAIN QUESTION Round Distance (yd)
Use a graphing calculator and regression to find an exponential
How far will the athlete run in the twentieth round?
function that models the data. The function in the form y = a • 2 x 1 200
How long will it take?
that best fits the data is y = 54.545 • 2 x.
2 300
Substitute 20 for x.
3 500
y = 54.545 • 2 20
y ≈ 57,194,578 4 900

The twentieth round will be a total distance of about 5 1,700

ACT 2 Modeling With Math 57,194,578 yards, or 32,497 miles, which is greater than the
Necessary Information circumference of Earth!
Think about the task. Ask students to speculate how they
Field length 100 yards
can determine how far the athlete runs in the twentieth METHOD 1: Use the Model for Distance
round and how long it will take. Then have them think Field divisions First round: no divisions
He runs a total of 1,000 yards in 4:08 for the first three rounds,
about what information they need. Second round: field divided in half
for a rate of 0.248 seconds per yard, so assuming the runner never
Third round: field divided in quarters
Reveal the information. As students ask, use the video to provide slows down or takes a break, it would take 57,199,821 • 0.248 ≈
the length of the field and the running times for the first three Completion times First round: 41 seconds 14,185,556 seconds, or about 164 days!
rounds. Students may ask for the athlete’s running speed, which Second round: 1 minute, 12 seconds
they will need to approximate. Third round: 2 minutes, 15 seconds METHOD 2: Develop a Model for Time
You can use the graphing calculator and regression to model Make a table like the one shown to represent the time for Round Time (s)
two functions: one relating distance to round number, and one each round.
1 41
relating time to round number. Students should use the form Use a graphing calculator and regression to find an exponential
y 1 ~ a • 2 x 1 in the graphing calculator for each model. function that models the data. The function in the form y = a • 2 x 2 72
What’s the connection? Give students time to think about how is y = 17.26 • 2 x.
their ideas might relate to exponential functions. If students have 3 135
Substitute 20 for x.
made a table, is there a pattern to the rows of the table?
y = 17.26 • 2 20
The number of divisions on the field doubles each time. Would a
linear, quadratic, or exponential model best fit the data? y ≈ 18,098,422 SEQUEL
The twentieth round will take about 18,098,422 seconds, or about Note: This sequel is best used after students have learned about
INTERESTING MOMENTS WITH STUDENTS
209 days. logarithms in Lesson 3.
It is easy to build a table but challenging to find a pattern
and then model it with a function. It may take students a long As the round number increases, both distance and time increase Tell students a runner wants to run the distance of a marathon
INTERESTING MOMENTS WITH STUDENTS
time to notice important patterns and make crucial conjectures exponentially. You could have students perform linear, quadratic, on the same field. Ask them to make a conjecture about which
about what they see. Suggest that students use a graphical and exponential regressions on their data for distance, and repeat Writing an equation for round n will challenge most students. number round she should use. [For the 100-yard field, the
representation to help them uncover interesting patterns. for time. In each case, the exponential model fits best, where the It relies on seeing the pattern that each round is runner should use the tenth round, which divides the field
function is of the form y = a • 2 x. d(n) = 100(2 n−1 + 1), where d is the distance in yards. into 1024 equal parts, to complete a marathon. She would run
approximately 29.1 miles.]

TOPIC 6 305A MATHEMATICAL MODELING IN 3 ACTS

TOPIC 6 305B MATHEMATICAL MODELING IN 3 ACTS

HSM23_TEA2_FL_T06_MML.indd 1 12/04/21 3:12 AM

HSM23_TEA2_FL_T06_MML.indd 2 12/04/21 3:12 AM

The Hook The Model

• Multimedia presentation of the problem situation provides • Students identify information needed to develop a
just enough information to engage students. mathematical model.

• Students brainstorm possible questions to answer. The • Students develop a mathematical model that represents
teacher then focuses students on the main question. the problem situation.

• Students propose conjectures that answer the main question. • Students use their models to revise and refine their
original conjectures.
• Brainstorming questions that the problem situation raises
foster participation from all students and help to lower barriers
to learning. The Resolution and Sequel
• Students are shown the solution through multimedia
presentation.

• Students look to explain disparities or differences between

their conjectures and the actual solution.

• Students work on the sequel, time permitting.

51
MATHEMATICAL THINKING AND REASONING Robust embedded support to
help students develop proficiency
STANDARDS IN THIS PROGRAM with these important thinking
habits

Building Expertise with MTR Standards TOPIC 6

Exponential and Logarithmic Functions
Students cannot be college- and career-ready without developing
the thinking and reasoning skills that are characteristic of MATHEMATICAL THINKING AND REASONING
mathematicians. These skills cannot be taught in isolation; rather Mathematical Thinking and Reasoning Within Topic 6 Lessons
they are integral to the math content that students learn. The Mathematical Thinking and Reasoning Standards promote deeper learning
and understanding of mathematics and provide clear expectations for students
as they engage with mathematics. Opportunities to develop expertise with

B.E.S.T. A|G|A
these important skills exist throughout the topic and program. Here we focus
on representing problems and using structure.
As students model with exponential and logarithmic functions, look for

A major emphasis of enVision Florida B.E.S.T. A|G|A is helping the following behaviors to assess and identify students who engage with
mathematics through thinking and reasoning.

students develop expertise with the MTR Standards. Throughout the Highlighted Mathematical Thinking and Reasoning Within Topic 6 Lessons
program are multiple opportunities for students to engage in the MTR.5.1 Use patterns and structure to help
mathematics through thinking and reasoning. These opportunities
MTR.2.1 Demonstrate understanding by
understand and connect mathematical
representing problems in multiple ways.
concepts.
are described on these pages. Mathematicians who demonstrate understanding by
representing problems in multiple ways:
Mathematicians who use patterns and structure to help
understand and connect mathematical concepts:

• Graph transformations of exponential functions to explore • Apply general mathematical rules of transforming functions
differences between the asymptotes and intercepts of parent to transform graphs of logarithmic functions and use structure
and given functions. by identifying that because logarithmic functions are inverses
of exponential functions, their graphs are reflections of
exponential functions across the line y = x.

• Contextualize exponential relationships by identifying • Look for patterns to determine whether they can use the

Mathematical Thinking and Reasoning the monthly growth rate from an annual rate. They
decontextualize exponential relationships by representing
natural log or the common log to solve an exponential
equation. They use structure to determine the annual rate
real-world situations as exponential functions. that was used to project the amount of money in an account.

• Mathematical Thinking and Reasoning Teacher Pages • Utilize data points on a graph of an exponential model to
write the model.
• Look for overall structure and patterns in exponents and
logarithms, as well as apply general math rules to evaluate

include:
logarithms.

• Represent exponential functions in logarithmic form to solve for • Look for repeated multiplication in their calculations and
variables within exponents, such as determining the magnitude recognize that these problems can be represented by
of an earthquake from the energy produced by the earthquake. exponential functions.

-- Descriptions of behaviors students may exhibit with regard to Create opportunities for students to represent exponential and logarithmic functions in multiple ways. You can also

two highlighted MTR Standards;

support students as they develop an understanding of the patterns in exponents and logarithms.

You can use these questioning strategies throughout the topic.

-- Guiding questions to help students develop expertise with the Q: How can you use Properties of Logarithms to explain
how two different logarithmic equations have the same
Q: What do you notice about the calculations that will help you
recognize that a problem can be represented by an exponential
two highlighted standards. graph?
Q: How are exponential functions related to logarithmic
function?
Q: How does the Change of Base Formula for logarithms help you
functions? solve exponential equations?
Q: What is the relationship between the growth factor and the Q: What do you notice about the graphs of exponential and
growth rate of an exponential model? logarithmic functions that share an inverse relationship?
Q: How can you decide if a real-world problem can be modeled Q: In what ways does rewriting an exponential equation in
using a logarithmic function? logarithmic form connect to other mathematical concepts?

TOPIC 6 286D TOPIC OVERVIEW

HSM23_TEA2_FL_T06_MP.indd 4 12/04/21 2:34 AM

52
 Digital

6-6
6-6
Activity

MODEL & DISCUSS MODEL & DISCUSS

A store introduces two new models of fitness trackers to its product line.
Exponential
A store introduces two new models of fitness trackers to its product line.
Exponential and A glance at the data is enough to see that sales of both types of fitness
A glance at the data is enough to see that sales of both types of fitness trackers and Logarithmic
are increasing. Unfortunately, the store has limited space for the merchandise. Equations
Logarithmic trackers are increasing. Unfortunately, the store has limited space for the The manager decides that the store will sell both models until sales of
Equations merchandise. The manager decides that the store will sell both models until TrackSmart exceed those of FitTracker.
sales of TrackSmart exceed those of FitTracker. SavvasRealize.com

I CAN… solve exponential FitTracker TrackSmart FitTracker TrackSmart

and logarithmic equations. Number Sold Number Sold Number Sold Number Sold

Week 4 228 130 Week 4 228 130

VOCABULARY
• exponential equation Week 3 112 44 Week 3 112 44
• logarithmic equation Week 2 54 17 Week 2 54 17
Week 1 28 5
Week 1 28 5
MA.912.AR.5.2–Solve one- A. Apply Math Models Find an equation of an exponential that models the
variable equations involving
logarithms or exponential
A. Apply Math Models Find an equation of an exponential function that sales for each fitness tracker. Describe your method.
expressions. Interpret solutions as models the sales for each fitness tracker. Describe your method.
viable in terms of the context and
identify any extraneous solutions. B. Based on the equations that you wrote, determine when the store will
MA.K12.MTR.3.1, MTR.4.1, stop selling FitTracker.
MTR.7.1

How do properties of exponents and

ESSENTIAL QUESTION
logarithms help you solve equations?

CONCEPT Property of Equality for Exponential Equations

VOCABULARY Symbols Suppose b > 0 and b ≠ 1, then b x = b y if and only if x = y.

B. Based on the equations that you wrote, determine when the store will stop
An exponential equation is an
Words If two powers of the same base are equal, then their exponents selling FitTracker.
equation that contains variables
are equal; if two exponents are equal, then the powers with
in the exponents.
the same base are equal.

EXAMPLE 1 Solve Exponential Equations Using a Common Base

x+7
What is the solution to (__
1
) = 4 3x?
2
1 x+7 = 4 3x
(2)
__ Write the original equation.
x+7 3x
(2 −1) = (2 2 ) Rewrite each side with a common base.
2 −x−7 = 2 6x Power of a Power Property
−x − 7 = 6x Property of Equality for Exponential Equations
−7 = 7x Add x to each side. THINKING AND REASONING
Communicate and Justify How do you know that the sales data is modeled by an
−1 = x Divide each side by 7. exponential function?

Try It! 1. Solve each equation using a common base.

a. 25 3x = 125 x+2 b. 0.001 = 10 6x

LESSON 6-6 Exponential and Logarithmic Equations 327 LESSON 6-6 Exponential and Logarithmic Equations 313

HSM23_SEA2_FL_T06_L06.indd 327 10/01/21 5:14 AM

Mathematical Thinking and Reasoning Features

These features in the Student Edition and the Student Companion • MTR Standards boxes during instruction offer students tips
focus on helping students develop their thinking and reasoning and questions that help students develop their thinking and
skills. reasoning skills as they learn to solve problems.
• Model & Discuss, Explore & Reason, Critique & • Thinking and Reasoning Questions in the Student
Explain The lesson opening activities offer rich opportunities Companion focus students’ thinking on particular aspects of the
for students to apply the MTR Standards. Mathematical Thinking and Reasoning Standards.
-- The Model & Discuss helps students become better modelers • MTR Standards Run-in Heads in the exercises remind
as they work through the different elements in the modeling students to apply the standards as they solve problems.
process.
-- The Explore & Reason focuses students on generalizations
and abstract and quantitative reasoning.
-- The Critique & Explain emphasizes mathematical
argumentation. Students construct arguments to defend a
claim or critique an argument.

53
EFFECTIVE MATHEMATICS Professional development
TEACHING PRACTICES at your service!

Focus On Teaching and Learning

In 2000, the National Council of Teacher of Mathematics (NCTM) Guiding Principles for School
published Principles and Standards for School Mathematics that
outlined essential features of high quality mathematics programs. It Mathematics
included a set of guiding principles and a set of standards – both
Teaching and Learning
content and processes -- for high quality programs.
An excellent mathematics program requires effective teaching
Later documents from NCTM refined learning expectations that engages students in meaningful learning through
(Curriculum Focal Points for Prekindergarten through Grade 8 individual and collaborative experiences that promote their
Mathematics: A Quest for Coherence and Focus in High School ability to make sense of mathematical ideas and reason
Mathematics: Reasoning and Sense Making) identifying the most mathematically.
significant mathematical concepts and skills at each level from
Access and Equity
prekindergarten through grade 8 and urging for a redefinition of
An excellent mathematics program requires that all students
high school mathematics to focus on reasoning and sense-making.
have access to a high-quality mathematics curriculum,
In Principles to Actions, published in 2014, the NCTM leadership effective teaching and learning, high expectations, and the
called on all stakeholders involved in educating students – support and resources needed to maximize their learning
teachers, principals, curriculum developers, and superintendents – potential.
to take action to ensure that all students benefit from a high quality
Curriculum
mathematics education.
An excellent mathematics program includes a curriculum that
The main message is that the key to high quality mathematics develops important mathematics along coherent learning
education and student success in mathematics resides primarily progressions and develops connections among areas of
in effective teaching. To that end, it provides an updated set mathematical study and between mathematics and the
of principles, grounded in research and best practices, which real world.
places major emphasis on teaching and learning. To highlight this
Tools and Technology
importance, it describes eight Mathematics Teaching Practices that
An excellent mathematics program integrates the use of
research has shown need to be part of every mathematics lesson.
mathematical tools and technology as essential resources to
help students learn and make sense of mathematical ideas,
B.E.S.T. A|G|A reason mathematically, and communicate their mathematical
The development of enVision Florida B.E.S.T. A|G|A thinking.
was guided by these principles with a particular focus on Assessment
offering teachers instructional support that aligns to the Effective An excellent mathematics program ensures that assessment
Mathematics Teaching Practices. is an integral part of instruction, provides evidence of
 Throughout each lesson are questions that focus on each of proficiency with important mathematics content and
the eight teaching practices. Look for the Effective Teaching practices, includes a variety of strategies and data sources,
Practices icon on the lesson pages. and informs feedback to students, instructional decisions,
and program improvement.
National Council of Teachers of Mathematics. (2014).
Effective Mathematics Teaching Practices Principles to Action.
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.

54
 Problem Solving
the equation of a quadratic function affects its vertex and axis of
symmetry.
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g, h, and
EXPLORE
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TOPIC 7 271B VOCABULARY
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APPLICATION
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determine in 0. that will
the domain 272 TOPIC 7 Quadratic
can solve for all of the coefficients and write the function in
showOnlythethe sign of ccentered
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y-intercept.]
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TOPIC 7 271
Q:55
LESSON 3
decrease.]
 Rich, high-interest projects
that incorporate math,
science, and engineering

TOPIC 6
Exponential and Logarithmic Functions Activity

Project

Overview of the Project

MAKING MATHEMATICAL CONNECTIONS
In this project, students will learn when and how to use Benford’s Law
to analyze data. … you determined the number of items a
In Topic 4 …
business must produce and sell to make a profit.
Next Generation Sunshine State Standards for Science
… you use Benford's law to analyze the results
SC.912.CS-CP.1.2, SC.912.CS-CP.1.3, SC.912.CS-CS.1.1
In this Topic … of several elections and determine which, if any,
Florida's B.E.S.T. Benchmarks for Mathematics may be fraudulent.
MA.912.AR.5.2, AR.5.7
Mathematical Thinking and Reasoning Standards
AVAILABLE ONLINE
MA.K12.MTR.1.1, MTR.6.1, MTR.7.1

TOPIC 6 PROJECT
Introducing the Project
Present the situation by discussing the difference between error Did You Know?
and fraud and how one might analyze data to find errors and/or While you might expect the digits 1 through 9
to lead off the numbers in a data set with equal
Data Sets That Follow Benford's Law

evidence of fraud.
2n
frequency, Benford’s Law shows that they do not.
Benford’s Law states that, in real-world data, the × 3

leading digit is 1 more than 30% of the time, while

The questions below can be used to guide the discussion. the leading digit is 9 less than 5% of the time.
Bacterial growth Expansion of 2n Price × Quantity

Q: What’s the difference between making a mistake and

committing fraud? Data Sets That Do Not Follow Benford’s Law

[Making a mistake is not done on purpose.] 3 000-55-5555

Q: An audit is a formal process of checking someone else’s data, Price Quantity Zip Codes Social
often financial data. Why do you think people may not want to Security
numbers
be audited?
[Look for and probe realistic reasons, such as being afraid of
having made a mistake, being afraid of being penalized for
that mistake, etc.]
Bacteria (plural of bacterium)

Q: Is it realistic to expect real-world data to follow a theoretical exist in soil, water, plants,
glaciers, hot springs, and the
model exactly? Why or why not? oceans. Bacteria grow by
duplicating themselves, so a
[No; Explanations will vary.] population grows by doubling.
In a laboratory, a population can
double at regular intervals. These
Have students read the task they will be asked to complete. intervals vary from about 12
minutes to as much as 24 hours.

Implementing the Project

Show the Topic 6 STEM video to generate interest in the project. Your Task: Analyze Elections
You and your classmates will use Benford's law to analyze
election results and determine which, if any, may be fraudulent.
One of the interesting discoveries about Benford’s Law is that it is
scale invariant—converting a data set from one unit of measure
to another does not change its distribution. Encourage students to
use at least one set of physical data, so that they can explore this 288 TOPIC 6 enVision STEM Project

through unit conversions (centimeters to inches, kilometers to miles,

etc.). If students use a source of monetary data, have them convert HSM23_SEA2_FL_T06_STEM.indd 288 10/01/21 5:15 AM

to other forms of currency (dollars to euros, dollars to yen, etc.). For students who are interested in analyzing multiple real-world
The connection to exponential and logarithmic functions occurs data sets, provide extra copies of Analyze a Data Set.
when students generate mathematical examples in the Analyze
Powers of b and Analyze Powers of 10 worksheets, although they
may be more interested in real-world examples.
Finishing the Project
You may wish to plan a day when students share their completed
You can download blackline masters for use with the project from
analyses. Encourage students to explain their process as well as
the Teacher Resource Center.
their results.

TOPIC 6 288 STEM

HSM23_TEA2_FL_T06_STEM.indd 288 12/04/21 2:57 AM

56
 Digital

enVision STEM Projects

• Activities foster coherence and connections across topics and courses.
• Projects are centered around key themes in science, technology,
engineering, and mathematics and present situations that address
real-world issues.
• Flexible implementation options allow for projects to be implemented
in a variety of ways.

Science and Engineering Practices

The Next Generation Sunchsine State Standards for Science
describe science and engineering practices that overlap with the problem
solving in mathematics.

• Asking Questions and Defining Problems Students specify

relationships between variables, and clarify problems.
• Developing and Using Models Students develop, use, and revise as
needed models to describe, test, and predict more abstract phenomena and
design systems.
• Analyzing and Interpreting Data Students analyze data, distinguishing
between correlation and causation, and use basic statistical techniques of
data and error analysis.
• Engaging in Argument from Evidence Students construct a
convincing argument that supports or refutes claims for either explanations
or solutions.

57
 BUILDING LITERACY IN MATHEMATICS Vocabulary, reading,
and writing support

LESSON 6-3 STEP 1 Explore

Logarithms Glossary

CRITIQUE & EXPLAIN

Activity

INSTRUCTIONAL FOCUS Students use their knowledge of AVAILABLE ONLINE

Lesson Overview Vocabulary Builder powers of ten to determine a method for solving an exponential
equation.

Objective REVIEW VOCABULARY English | Spanish STUDENT COMPANION Students can complete the Critique &
• exponential function | función exponencial Explain activity in their Student Companion.
Students will be able to:
• inverse function | función inversa
✔ Understand the inverse relationship between exponents and
logarithms. NEW VOCABULARY Before  WHOLE CLASS

✔ Use logarithms to solve exponential models. • common logarithm | logaritmo común Implement Tasks That Promote Reasoning and
✔ Evaluate logarithms using technology. • logarithm | logaritmo Problem Solving
Essential Understanding • logarithmic function | función logarítmica Q: What do you notice about the relationship between the
A logarithmic function is the inverse of an exponential • natural logarithmic function | function logarítmica natural magnitude and the amplitude of an earthquake?
function. Logarithms are found by determining the exponent [As x increases by 1, y increases by a power of 10.]
that must be applied to a base to yield a given result.
VOCABULARY ACTIVITY
Activate students' prior knowledge of inverse functions by
Previously in this topic, students: having them find the inverses of two or three linear or quadratic During  SMALL GROUP

functions. Then make the connection to the relationship between

• Interpreted the key features of exponential functions.
exponential and logarithmic functions. Support Productive Struggle in Learning Critique & Explain is powered by Desmos.
• Used exponential functions to model relationships. Mathematics  STUDENT EDITION

Q: How are Taylor and Chen’s work alike?

In this lesson, students:
6-3
Powered By

[Both students made estimates and both may have initially CRITIQUE & EXPLAIN

• Use the relationship between exponents and logarithms to

solve problems.
Student Companion thought that x would be halfway between 3 and 4. However,
Chen used the equation to check and refine his answer.]
Logarithms
Earthquakes make seismic waves
through the ground. The equation
y = 10 x relates the height, or
Magnitude, x
2
Amplitude, y
100
amplitude, in microns, of a seismic
Students can do their work for the lesson in their Student wave, y, and the power, or magnitude,
3 1,000

• Evaluate common and natural logarithms and solve equations Q: When would each student’s method of solving the problem be ? 5,500
Companion or in Savvas Realize. x, of the ground-shaking it can cause.
4 10,000
involving logarithms. more appropriate? I CAN… evaluate and
simplify logarithms.

[Use Taylor’s method when an estimate is acceptable. Use VOCABULARY

Taylor and Chen used different methods to find the magnitude of the

Later in this topic, students:

earthquake with amplitude 5,500.
Chen’s method when a more precise answer is needed.] • common logarithm
• logarithm
• Evaluate logarithmic expressions using properties of • logarithmic function Taylor Chen
• natural logarithm 5,500 is halfway between y = 10x
logarithms. For Early Finishers 1,000 and 10,000.
103 = 1,000
MA.912.AR.5.2–Solve one-
Q: Use the method you prefer to find the magnitude of an variable equations involving
logarithms or exponential
3.5 is halfway between 3 104 = 10,000
103.5 ≈ 3,162
Skills Students practice evaluating logarithms to understand earthquake with amplitude 400.
expressions. Interpret solutions as
viable in terms of the context and
and 4.
103.7 ≈ 5,012
the inverse relationship between exponents and logarithms. [10 2.4 ≈ 251, 10 2.5 ≈ 316, 10 2.6 ≈ 398; The magnitude of an
identify any extraneous solutions.
Also NSO.1.6, AR.5.7, F.3.7 The magnitude is about 3.5. 103.8 ≈ 6,310
MA.K12.MTR.2.1, MTR.4.1, 103.74 ≈ 5,500
MTR.5.1
Applications Students apply logarithms to solve problems earthquake with amplitude 400 is about 2.6.] The magnitude is about 3.74.

involving logarithmic scales, such as involving the magnitudes A. What is the magnitude of an earthquake with amplitude 100,000? How
do you know?
of earthquakes.
After  WHOLE CLASS B. Communicate and Justify Critique Taylor’s and Chen’s work. Is each
method valid? Could either method be improved?
C. Describe how to express the exact value of the desired magnitude.
Facilitate Meaningful Mathematical Discourse
What are logarithms and how are they
Facilitate discussion by asking students to share ideas of how to ESSENTIAL QUESTION
evaluated?
CONCEPTUAL
find the exact value of the desired magnitude Powered By
UNDERSTANDING EXAMPLE 1 Understand Logarithms
 SAMPLE STUDENT WORK
Solve the equations 2x = 8 and 2 x = 8.
Q: How would you solve the equation y = 10x? Division is the inverse of multiplication,

Florida’s B.E.S.T. Standards and Benchmarks

You can use inverse 8
2x = __
___ so you can divide both sides by 2 to
[Substitute 5,500 for y and solve for x.] operations to solve the 2 2
A. An earthquake with amplitude 100,000
x = 4 has a magnitude of 5
first equation.
solve the equation.

Q: Can you apply the same method to solve this equation? because 10 5 = 100,000.
The operation in 2 x = 8 is exponentiation. To solve this equation, you need
USE PATTERNS AND

In this lesson, students focus on these benchmarks: These standards are highlighted in this lesson: Explain. [No; y = 10 x cannot be solved by dividing both sides STRUCTURE
an inverse for exponentiation that answers the question, “To what exponent
would you raise the base 2 to get 8?”
Creating the notation log 2 x to
by 10 because x is an exponent, not a factor.] represent the exponent to which The inverse of applying an exponential
MA.912.AR.5.2 Solve one- variable equations involving MA.K12.MTR.2.1 Demonstrate understanding by representing B. Taylor’s method gives aisrough
applying aestimate,
you raise 2 to get x is similar to_
function logarithm but it This
is isnot valid if you
read “logarithm base 2 of 8”
creating the radical notation √x function. To solve the equation
or “log base 2 of 8.”
logarithms or exponential expressions. Interpret solutions as problems in multiple ways. Use with CRITIQUE & EXPLAIN need to find a more
2 x =precise
to represent one number you can
value.
8, you can write log 28 =Chen’s
x. method gives a much
THINKING AND REASONING square to get x. 3
Solving this gives log 28 = 3 because 2 = 8.
viable in terms of the context and identify any extraneous Students represent solutions to problems using logarithms and closer approximation, although guess and checkCONTINUED
is not ONthe most
THE NEXT PAGE

solutions. Communicate and Justify Taylor reasoned that since 5,500 was
express connections between logarithms and exponents. efficient way to solve problems.
halfway between 1,000 and 10,000 that the magnitude had to 306 TOPIC 6 Exponential and Logarithmic Functions

NSO.1.6 Given a numerical logarithmic expression, evaluate and MTR.5.1 Use patterns and structure to help understand and be halfway between 3 and 4. What is incorrect about Taylor’s
generate equivalent numerical expressions using the properties of connect mathematical concepts. reasoning? C. The exact value of the magnitude is the value of x when 10x = 5500.
HSM23_SEA2_FL_T06_L03.indd 306 27/01/21 3:40 PM

logarithms or exponents. Students look for overall structure and patterns in exponents [Exponential functions increase exponentially, so they do not
They also work with concepts related to these benchmarks: and logarithms, as well as apply general math rules to evaluate increase at a constant rate. There is more growth between
AR.5.7 and F.3.7 logarithms. 10 3.5 and 10 4 than between 10 3 and 10 3.5.]

TOPIC 6 306A LESSON 3 TOPIC 6 306B LESSON 3

HSM23_TEA2_FL_T06_L03.indd 1 12/04/21 3:14 AM HSM23_TEA2_FL_T06_L03.indd 2 12/04/21 3:14 AM

enVision Florida B.E.S.T. A|G|A was designed with a focus The activities that launch each lesson promote not just
on building students’ mathematical literacy. understanding of math concepts, but reinforce and build key
language skills of speaking and listening as students share and
At the beginning of every lesson in the Teacher’s Edition is a
defend their solution strategies.
Vocabulary Builder activity that focuses on both mathematical
vocabulary and academic vocabulary. • Model & Discuss explorations promote mathematical
discourse and require students to practice both speaking and
listening skills.

• Critique & Explain explorations promote argumentation and

reasoning. Students use reading and writing skills to construct
mathematical arguments.

58
 Digital

TOPIC
Topic Review
6
Name B.E.S.T. ALGEBRA 2

SavvasRealize.com

6-7 Mathematical Literacy and Vocabulary TOPIC ESSENTIAL QUESTION

Geometric Sequences
1. How do you use exponential and logarithmic functions to model
situations and solve problems?
For Items 1–9, match the recursive definition in Column A with the explicit
definition in Column B. An example has been given.
Vocabulary Review
Column A Column B
3, if n = 1 Choose the correct term to complete each sentence.
Ex. an = a. an = a1r n–1 • decay factor
an–1(6), if n > 1
2. A(n) has base e.
• exponential function
a , if n = 1 3. A(n) has the form f(x) = a ∙ b x.
1. an = a 1 (r), if n > 1 b. an = x(y)n–1 • logarithmic function
n–1 4. In an exponential function, when 0 < b < 1, b is a(n) .
• growth factor
25, if n = 1 5. The is helpful for evaluating logarithms with a base
2. an = 1 c. an = 2(2.6)n–1 • common logarithm
an–1 5 , if n > 1 other than 10 or e.
6. A(n) has base 10. • natural logarithm
2, if n = 1 1
d. an = 25 5 n–1 • Change of Base
3. an = 7. The inverse of an exponential function is a(n) .
an–1(4), if n > 1 Formula
4, if n = 1
4. an = e. an = 2(4)n–1
an–1(2), if n > 1
Concepts & Skills Review

TOPIC 6 REVIEW
2.6, if n = 1
5. an = f. an = 3(6)n–1
an–1(2), if n > 1
LESSON 6-1 Key Features of Exponential Functions
x, if n = 1
6. an = g. an = 1 (25)n–1
an–1(y), if n > 1 5
Quick Review Practice & Problem Solving
2, if n = 1
7. an = h. an = 4(2)n–1 An exponential function has the form Identify the domain, range, intercept, and
an–1(2.6), if n > 1
f(x) = a ∙ b x. When a > 0 and b > 1, the asymptote of each exponential function. Then
y, if n = 1 function is an exponential growth function. describe the end behavior.
8. an = i. an = 2.6(2)n–1
an–1(x), if n > 1
8. f(x) = 400 ∙ (__1 )
When a > 0 and 0 < b < 1, the function is an x
exponential decay function. 2
1 , if n = 1 9. f(x) = 2 ∙ (3) x
9. an = 5 j. an = y(x)n–1
an–1(25), if n > 1
Example 10. Represent and Connect Seth invests $1,400
Paul invests $4,000 in an account that pays 2.5% at 1.8% annual compound interest for
interest annually. How much money will be in 6 years. How much will Seth have at the
the account after 5 years? end of the sixth year?

Write and use the exponential growth function 11. Apply Math Models Bailey buys a car for
model. $25,000. The car depreciates in value 18%
per year. How much will the car be worth
A(t) = a(1 + r) n after 3 years?
A(5) = 4,000(1 + 0.025) 5
12. Identify the domain, y
A(5) = 4,000(1.025) 5 range, intercept, and
4
A(5) = 4,525.63 asymptote.
2
There will be about $4,525.63 in Paul’s account
after 5 years. x
−4 −2 O 2 4

enVision® Florida B.E.S.T. Algebra 2 • Teaching Resources

TOPIC 6 Topic Review 341

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HSM23_SEA2_FL_T06_TR.indd 341 15/01/21 1:04 AM

Literacy Support at the End of Lessons Literacy Support at the End of Topics
• Mathematical Literacy and Vocabulary activities offer • Vocabulary Review At the end of each topic in the Student
vocabulary support for all students, especially English Language Edition is a page of Vocabulary Review. It includes questions to
Learners and struggling readers. reinforce understanding of the vocabulary used in the topic and
asks students to use vocabulary in writing.
•  • Bilingual Glossary A bilingual glossary is always
available to students and teachers at SavvasRealize.com

59
 LESSON 6-4 Logarithmic Functions 315

SUPPORTING ENGLISH HSM23_SEA2_FL_T06_L04.indd Page 315 13/02/21 3:18 PM eteam

ELL instructional support

/154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...

LANGUAGE LEARNERS in 100% of the lessons!

X English Language Learners (Use with EXAMPLE 3)

SPEAKING BEGINNING In small groups, have LISTENING INTERMEDIATE Have students WRITING ADVANCED Have one student hold
students discuss the meanings of inter– listen to the definition of inverse and a card with X on it and another student
and change, and how they relate to the answer the question. hold a card with Y on it. Ask them to come
meaning of interchange. to the front of the room and exchange
The definition of inverse is the opposite or
the cards.
Q: Where do you hear the word reverse of something.
interchange used? Q: What does inverse mean in math? Q: What word from the example was just
[Sample: when merging from one [opposite operations such as addition demonstrated? [interchange]
highway to another] and subtraction; exchanging x and y in Q: Write definitions for interchange and
Q: What does the word interchange mean? functions] inverse in your own words. Explain how
[a mutual, or reciprocal, exchange] the two words are related.
[Check students’ work.]

TOPIC 6 315 LESSON 4

HSM23_TEA2_FL_T06_L04.indd 315 12/04/21 3:16 AM

STEP 2 Understand & Apply

Concept
Summary
Assess

Y CONCEPT SUMMARY Logarithmic Functions

Q: How are the domain and the range of y = log x related to the AVAILABLE ONLINE
domain and the range of y = 10 x?
[The domain of y = log x is the same as the range of y = 10 x CONCEPT SUMMARY Logarithmic Functions

and the range of y = log x is the same as the domain of GRAPH 4

y

y = 10 x.] 2 The functions are

y = 10 x x inverses so their graphs
Q: How are the x-intercept and the asymptote of y = log x −4 −2 O 2 4 are reflections of each
−2 other across the line
affected if the graph is translated 3 units to the right? y = log x with equation y = x.
−4
[The x-intercept and asymptote are also translated 3 units
to the right. The x-intercept is 4 and the asymptote is x = 3.] EQUATIONS y = log x y = 10 x

KEY FEATURES Domain: {x | x > 0} Domain: all real numbers

Range: all real numbers Range: {y | y > 0}

Do You UNDERSTAND? | Do You KNOW HOW? x-intercept: 1 y-intercept: 1

Asymptote: y-axis Asymptote: x-axis

END BEHAVIOR As x → 0, y → −∞ As x → −∞, y → 0

Common Error As x → ∞, y → ∞ As x → ∞, y → ∞

Exercise 5 Students may ignore the −1 before adding 1__12 ,

writing the equation as g(x) = ln x + 1__12 , Do You UNDERSTAND? Do You KNOW HOW?
1. ESSENTIAL QUESTION How is the 5. Graph the function y = log 4 x and identify
instead of g(x) = ln x + __12 . Explain that the graph of relationship between logarithmic and the domain and range. List any intercepts or
exponential functions revealed in the key asymptotes. Describe the end behavior.
f(x) = ln x − 1 is the graph of h(x) = ln x shifted down features of their graphs?
6. Write the equation for the function g(x),
1 unit, so shifting this function up by 1__12 units is the same 2. Error Analysis Raynard claims the domain which can be described as a vertical shift
1__21 units up from the function f(x) = ln x – 1.
as shifting h(x) up __12 unit.
of the function y = log 3 x is all real numbers.
Explain the error Raynard made.
7. The function y = 5 ln(x + 1) gives y, the
3. Communicate and Justify How are the number of downloads, in hundreds, x minutes
graphs of f(x) = log 5 x and g(x) = –log 5 x after the release of a song. Find the equation
related? of the inverse and interpret its meaning.
Answers 4. Use Patterns and Structure Explain the Downloading...
necessary steps to find the inverse y downloads
1. The domain of the logarithmic function is the same as of h(x) = log 6(x + 4). Find the inverse.
x minutes

the range of the exponential function, and the range of 8. Sketch the functions represented by the

the logarithmic function is the same as the domain of tables. Identify which graph is the logarithmic
function. Are the two functions inverses?
the exponential function. The logarithmic function has a.
x 10 −10 0.01 1 2 10
an x-intercept of 1, while the exponential function has a g(x) −8 0 2 2.301 3

y-intercept of 1. b. x −1 0 1 2 3 4
h(x) 0.001 0.01 0.1 1 10 100

2. This is the range of the function. The domain is {x | x > 0}. 318 TOPIC 6 Exponential and Logarithmic Functions

3. The graph of g(x) is the reflection of the graph of f(x) across

the x-axis. HSM23_SEA2_FL_T06_L04.indd Page 318 17/02/21 5:18 PM f-0869 /154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...

4. Rewrite the problem using x and y, then exchange the places

8. y
of the x and y. Next solve for the y. In this step use the 4 h
relationship between exponential and logartimic functions.
Finally, rewrite in inverse function notation. h −1(x) = 6x − 5. 2 g
5. y x
4 O 2 4
2
x g is the log function; yes
−2 O 2 4 6
−2

−4

domain: {x | x > 0}; range: all real numbers; intercept:

x-intercept 1; asymptote: y-axis; end behavior: As x → 0,
y → −∞. As x → ∞, y → ∞.

6. g(x) = ln x + __12
60 y
__
7. x = e 5 − 1; This function gives x, the number of minutes after
the release of a song, in terms of the number of downloads in
hundreds, y.
 Digital

X ENGLISH LANGUAGE LEARNERS (ELL) INSTRUCTION

• ELL support is provided for every lesson in the Teacher’s

Edition.
• For use during the instructional part of the lesson.
• Targeted support offers strategies that focus on key areas
of language development: listening, speaking, reading, and
writing for students at different proficiency levels.
• ELL Principles based on research and best practices frame the
instructional support.

VISUAL LEARNING

Y Visual Learning in Math Instruction

CONCEPT SUMMARY Key Features of Exponential Functions

• Concept Summary provides a concise summary of the Exponential Growth Exponential Decay

mathematical content in multiple representations. GRAPHS 4

y
4
y
4
y
4
y

b>1 b>1 0 < b < 10 < b < 1

Bilingual Glossary is always available to students and

2 2 2 2
x x x x
O −22 O 4
−4 −2 −4 2 4 O −22 O 4
−4 −2 −4 2 4

teachers while using digital resources. The glossary is in −2 −2 −2 −2

English and Spanish to help students connect Spanish math

−4 −4 −4 −4

Growth factor: 1 + r Decay factor: 1 − r

terms they may know to English equivalents. EQUATIONS y = a ∙ b x, for b >1, a ≠ 0 y = a ∙ b x, for 0 < b < 1

Domain: All real numbers Domain: All real numbers

• Multilingual handbook provides glossary in 10 different

KEY FEATURES
Range: {y | y ≥ 0} Range: {y | y ≥ 0}
Intercepts: (0, a) Intercepts: (0, a)

languages.
Asymptote: x-axis Asymptote: x-axis

END BEHAVIOR As x → −∞, y → 0 As x → −∞, y → ∞

As x → ∞, y → ∞ As x → ∞, y → 0

MODELS Growth: A(t) = a(1 + r) t Decay: A(t) = a(1 − r) t

Do You UNDERSTAND? Do You KNOW HOW?

1. ESSENTIAL QUESTION How do graphs 5. Graph the function f(x) = 4(3) x. Identify the
and equations reveal key features of domain, range, intercept, and asymptote,
exponential growth and decay functions? and describe the end behavior.

2. Vocabulary How do exponential functions 6. The exponential function f(x) = 2500(0.4) x

differ from polynomial and rational models the amount of money in Zachary’s
functions? savings account over the last 10 years. Is
Zachary’s account balance increasing or
3. Error Analysis Charles claimed the function
x decreasing? Write the base in terms of the
f(x) = (__3 ) represents exponential decay. rate of growth or decay.
2
Explain the error Charles made.
7. Describe how the graph of g(x) = 4(0.5) x−3
4. Analyze and Persevere How are compares to the graph of f(x) = 4(0.5) x.
exponential growth functions similar to
exponential decay functions? How are they 8. Two trucks were purchased by a landscaping
different? company in 2016. Their values are modeled
by the functions f(x) = 35(0.85) x and g(x) =
46(0.75) x where x is the number of years
since 2016. Which function models the truck
that is worth the most after 5 years? Explain.

294 TOPIC 6 Exponential and Logarithmic Functions

HSM23_SEA2_FL_T06_L01.indd 294 27/01/21 5:50 PM

61
PREPARING STUDENTS FOR SUCCESS

MATHEMATICAL
MODELING TOPIC 2 PROJECT
IN 3 ACTS

Did You Know?

MA.912.AR.3.4–Write a
quadratic function to represent
Cameras and RADAR precisely track everything that
the relationship between two
quantities from a graph, a written
happens in a professional baseball game, including
description or a table of values
within a mathematical or real- the speed and launch angle of every batted ball.
world context. Also AR.3.8
MA.K12.MTR.7.1

Swift Kick
Whether you call it soccer, football, or fùtbol, it’s the most popular sport in
the world by far. Even if you don’t play soccer, you probably know several
people who do.
There are many ways to kick a soccer ball: you can use any part of either
foot. If you want the ball to end up in the goal, you also need to try
different amounts of spin and power. You’ll see one person’s effort in the
Mathematical Modeling in 3-Acts lesson.

ACT 1 Identify the Problem

If a baseball player hits a
1. What is the first question that comes to mind after watching the video?
90-mph pitch with more
2. Write down the main question you will answer about what you saw in
the video.
than 8,000 pounds of force,
3. Make an initial conjecture that answers this main question.
the ball leaves the bat at a
4. Explain how you arrived at your conjecture. speed of 110 mph.
5. What information will be useful to know to answer the main question?
How can you get it? How will you use that information?

ACT 2 Develop a Model Fenway Park

Tropicana Field
6. Use the math that you have learned in this Topic to refine your conjecture. Wrigley Field

ACT 3 Interpret the Results

7. Did your refined conjecture match the actual answer exactly? If not,
what might explain the difference?

88 TOPIC 2 Mathematical Modeling in 3 Acts

You can model the flight of a hit baseball with

HSM23_SEA2_FL_T02_MML.indd 88 20/01/21 11:38 PM
a parabola. The initial vertical and horizontal
Each baseball park has unique features that help speed of the ball can be found using right triangle
determine whether a hit will be a home run. trigonometry and the launch angle of the hit.

Your Task: Hit a Home Run

You and your classmates will design a ballpark and determine what it
would take to hit a home run at that park.

58 TOPIC 2 enVision STEM Project

HSM23_SEA2_FL_T02_STEM.indd 58 30/12/20 11:38 PM

Personal Relevancy STEM Projects Relevant to Students’ Lives

• Mathematical Modeling in 3 Acts In each task’s • enVision STEM Projects Students carry out math-
accompanying videos, students see diverse people actively application projects designed for a range of interests.
engaged in a real-world problem situation. Students explore
interesting and engaging problems that they can imagine • Relevant Contexts enVision STEM Projects are centered
occurring in their everyday lives. In each lesson, students around key themes in science, technology, engineering, and
choose the strategy to solve the problem, leveraging mathematics and present situations that address real-world
student voice. issues.

• Images and Names used throughout the Student Edition • Collaboration During Learning enVision STEM
reflect the diversity of the world. Students are able to see Projects provide opportunities for collaborative work and
themselves and others in the images and names used discussion so that every student’s input is valued and used to
throughout the program. help the class build a collective understanding of new ideas.

62
 EXAMPLE 5 Solve Polynomial Equations
Build Procedural Fluency from Conceptual
Understanding
EXAMPLE 5
Q: How can the pattern in the coefficients help you to factor the What are the s

equation? Rewrite the eq

[The coeffiecients are 1, 3, 3, 1 which is the row in Pascal’s 2x 3 + 5x

Triangle corresponding to (a + b) 3.] VOCABULARY x 3 + 3x 2 +

DigitalAequation
polynomial equation is an
that can be written in
The roots are th
(x + 1) 3 = 0
Q: How could you find the roots of P(x) if you didn’t recognize the the form P(x) = 0, where P(x) is
a polynomial. x+1=0
identity x 3 + 3x 2y + 3xy 2 + y 3 = (x + y) 3? x=–
[Use a calculator to graph the function P(x) and see where the To check, write
as a separate p
graph appears to cross the x-axis. Then use synthetic division

1-4
Use the INTERS
to test whether that value is a root and find another factor in that the graphs

CRITIQUE & EXPLAIN order to simplify P(x).] Try It! 5. W

Yumiko and Hugo are looking at the a.
Try It! Answers
Arithmetic
Sequences
table of data about the number of
families attending a party and how many
Attendees

Items
0 1 2 3 4
Learn Together and Have a Growth Mindset
5. a. x = −3, x = 2 EXAMPLE 6
1 5 9 13 17 __ __
• Classroomb.Conversations
items are needed to fill piñatas.
x = −1, x = 0, x = √2 i, x = −√2To
i foster a culture of
What are the so
in Pinata
The solutions a
polynomial x 3 –
Yumiko writes f(1) = 1 + 4 = 5,
collaborative
Elicit learning, the Student
of StudentEdition
Thinking asks students
HAVE A GROWTH MINDSET
and Use Evidence How can you take on challenges
with positivity?
Factor to find t

I CAN… interpret arithmetic f(2) = f(1) + 4 = 5 + 4 = 9, x 3 – 16

sequences.
f(3) = f(2) + 4 = 9 + 4 = 13, to reflect on [Both
Learning Together and Having a Growth
Q: In part a, what do you notice about both sides of the equation?
sides of the equation have a x 3-term, which you can
x(x 2 – 1
x(x – 4)(x +
VOCABULARY
• arithmetic sequence
f(4) = f(3) + 4 = 13 + 4 = 17. Mindset. Thesubtract
Teacher’s
from bothEdition
sides beforeprovides
solving.] discussion points, By the Zero-Pro
Sketch the func

•
•
common difference
explicit definition
Hugo writes g(x) = 1 + 4x.
as well asEXAMPLE
questions 6
to encourage
Solve a Polynomial students to contribute
Inequality by 20

• recursive definition
A. Describe the pattern Yumiko found for finding items needed for the
piñatas. their knowledge and understandings
Graphing to the conversation. 10

• sequence −2 O

MA.912.AR.10.1–Given a
B. Describe the pattern Hugo found for finding items needed for the Student voice
Use and is elevated within the
Connect Mathematical classroom as they are
Representations −10

−20
piñatas.
mathematical or real-world context,
write and solve problems involving encouragedwhere to bring their knowledge
the graph of P(x) is below the x-axis?
and experiences to
Q: Why is the solution of the inequality the same as the intervals
arithmetic sequences. ASSESSMENT PRACTICE
C. Choose Efficient Methods Compare the two methods. Which method
MA.K12.MTR.3.1, MTR.5.1,
MTR.7.1 APPLICATION
would be more useful in finding the items needed when the 100th
EXAMPLE 4 Write a Polynomial Function
the discussion.[The inequality is x 3 − 16x < 0. You are trying to find the values
3
Try It! 6. Wh
a.
family attends? Why? of x − 16x that are less thanEXAMPLE
zero. The graph3 of P(x) shows
Identify all
Positive or Negative Intervals
The height, in 36. The graph of g(x) = 3(x − 2) 2 is a transformation values of x 3 − 16x. The intervals where the graph is below the 160 TOPIC 3 Polynomial Functions
Hachi makes Seminole Indian
= x 2. Describe in words the For what 3
of intervals
x − 16x <is0f(x) 2
=x − 9 positive? For what intervals is the
dolls of the
to sell graph
at the of f(x)
local street What is an arithmetic?sequence, and how
x-axis are the same as the solutions because the
ESSENTIAL QUESTION
sequence of transformations
market. do you represent
? find
?and
$that ×its terms?
takes the
y-value of any point below thefunction negative?
x-axis is less than 0.] HSM23_SEA2_FL_T03_L05.indd 160

Try It! Answers Use technology to graph the function:

CONCEPTUAL graph
As Hachi of fa to
produces the graph of g.Price F.2.2 Units sold
greater
UNDERSTANDING number
EXAMPLE of dolls,
1 she can lowerArithmetic Sequences
Understand __ __
6. a. x < −3 − √3 or −3 + √3 < x < 0 y
the price per unit. The function f(x) > 0 (above the
37.
A.
v(x) =SAT/ACT
Is the sequence
49 − 2x relatesWhich
arithmetic?
the priceofIfv the following
so, what
= functions
is the common $ difference? What
– b. x < −2 or −1 < x < 1 or x > 3 8 x-axis) when x < –3
is the next term in thex.sequence?
to therepresents
number a parabola
produced The with a vertex at (−3, 4) COMMON ERROR
with Positivity and when x > 3.
cost3,c of making x dolls
8, 13, 18, 23, … can be Be carefulHave
not to aconfuse
Growth Mindset: Take on Challenges
a 4
and
represented that passes
with the functionthroughProfit the pointRevenue(−1, −4)? Cost When
positive you struggle
function value andwith
a a challenge, you build connections xin
c(x) This
= 12x is a
+ sequence,
64. a function whose domain is the Natural numbers. your rate
positive brain. Ask: What
of change. can you tell yourself so you keep a positive
A positive −4 −2 O 2 4
Ⓐ
HowCreate
f(x)table
many aSeminole
2
= xthat− 5 dolls
shows
Indian
Ⓒ
the term number, f(x) = 2(x and
or domain,
should Hachi sell
+ to
each week
1) 2−4
the term,
maximize
rateattitude
of changeand don’t
means thegive up when you struggle? What can you do
y-values
−4
f(x) < 0 (below the
or range. when
of the something
function seemsbut
are increasing hard? x-axis) between
Ⓑ f(x) = −2(x + 3)
her profit P? 2
+4 Ⓓ f(x) = 2(x − 3) 2 − 32 are not necessarily greater than 0. −8 the x-intercepts of
–3 and 3.
Formulate WriteTerm
a function revenue R by multiplying the price v(x) = 49 − 2x of each
Numberfor Term f
item by the number sold x.
38. Performance
1 3 Task The Bluebird Bakery sells
R(x) = (49
+5 Struggling Students
8 − 2x)x
COMMON ERROR more2 alfajores when
+5 itThelowers
difference
its prices, but this The function is positive on (−∞, −3) and (3, ∞).
• Learn Together presents ideas that teachers can share
Then write the function for profit P. between
3 13 USE WITH EXAMPLE 6 Students may struggle with solving cubic 3. What values of x
The common difference is
also 4changes profits.
+5 consecutive inequalities. The function is negative on (−3, 3). [−__13 < x < __34 ]
always calculated by subtracting P(x) = 18
R(x) − c(x) Profit = Revenue − Cost
and discuss with students to help them learn and apply
+5 numbers in Have students practice solving quadratic inequalities.
a term from the next term; 5 23 the range The function is neither positive nor negative at the
4. What x-intercepts
values of x of −3
d = a n − a n − 1. = (49 − 2x)x
+5 − (12x + 64) Substitute for R(x) and c(x). 1. What values of x are solutions to the inequality
skills to successfully learn together. Learning together is an
6 ? is 5. [x < −6 or x > −3]
Compute Simplify the function. x 2 − 12x + 35 < 0?
g(x)= –192(x – 0.445)2 + 38 An arithmetic
P(x) sequence
= (49 − 2x)x − (12x +is64)
a sequence with a constant difference
Write the profit function. important lifelong skill that students develop over time.
[5 < x < 7] Try It! 3. a. For what interval(s) is h(x) = −| x | + 5 positive?
between consecutive terms. This difference is known as the common 2. What values of x are solutions to the inequality x 2 − 5x − 3 > 0? is the function negative?
= (49x −
difference,
2
d. ) − (12x + 64)
or2x Use the Distributive Property. b. For 2what interval(s)
[x < −__12 or x > 3]
Find three additional This =
sequence
2 − 12x − 64
49x − 2xis Distributedifference,
an arithmetic sequence with the common the factor of −1.
–2) d = 5=. −2x
The2next term
+ 37x in the sequence is 23 + 5, or 28.
− 64 Combine like terms.
EXAMPLE 4 Identify Where a Function Increases or Decrease
5). Hachi's profit function is y (9.25,
CONTINUED ON107.125)
THE NEXT PAGE TOPIC 3 160 LESSON 5
P(x) = −2x 2 + 37x − 64. 100 For what values of x is g(x) = 2 − | x | increasing? For what values is
LESSON 1-4 Arithmetic Sequences 31 it decreasing?
The curvature of theInterpret Hachi's profit is modeled by a quadratic 80
function. The domain of the function is the LEARN TOGETHER
Construct a table and sketch a graph to represent the absolute value
set of whole numbers. Her maximum profit 60 Do you seek help when needed?
$0.75 each HSM23_TEA2_FL_T03_L05.indd Page 160 19/06/21 2:33 PM f-0315
Profit ($)

cannot correspond to the vertex of the Do you offer help and support y
HSM23_SEA2_FL_T01_L04.indd 31
40 20/12/20 1:41 AM
others? x g(x) The greatest value a
graph because she cannot sell a part g(x) is increasing 4
of a doll. Consider P(9) =107 and P(10) =106 −3 −1 as x goes from
20 maximum
and choose the maximum profit of the two. −∞ to 0.
x −2 0
Hachi's best business plan is to produce O 4 8 12 16 20 −1 1 x attains is the
and sell 9 Seminole Indian dolls per week, –20 −6 −4 O 2 4 6
for a The 0 2
x – 52.5)² + 45 weeklyprofit
profit offunction
$107. for the
alfajores is −2
1 1 g
f(x) = −500(x − 0.45) 2 + 400. ThisNumber function
of Wind Chimes Sold g(x) is decreasing
2 0
Diverse Contexts
−4
represents the profit earned when the price 3 −1
as x goes from
0 to ∞.
Try of
It! an alfajor
4. The cost of is x dollars.
Hachi's materialsThe bakery
changes so that wants to
her new cost
• Real-Word Situations
maximize presented
function is c(x) = 4x +
their profits. 42. in the Critique & The values of g(x) are increasing on the interval (−∞, 0).
Explain, Explore & Reason, Model profit.
& Discuss, Examples,
Find the new profit function. Then find the quantity that • Have a Growth Mindset helps students understand the
The values of g(x) are decreasing on the interval (0, ∞).
Part Amaximizes
Whatprofit
is the
anddomain of the function?
calculate the
and Exercises are ones students may have experienced importance and qualities of mindfulness. Students develop
or learnedPart
about,
B Find providing
LESSON 3-2 opportunities
the daily profits
Adding, Subtracting,
for sellingfor students
alfajores
and Multiplying Polynomials 135 to an understanding that havingTry
a growth
It! 4. Graphmindset helps
the functions. people
For what values of x is each function
for $0.40 each and for $0.75 each.
share their acquired knowledge. Some contexts relate to reach their goals. increasing? For what values of x is it decreasing?
a. f (x) = x 2 − 4x b. f (x) = −2x − 3

HSM23_SEA2_FL_T03_L02.indd 135
geography, environment,
Part C What price should sights,the and enterprises.
bakery charge toIllustrations
27/01/21 7:56 PM

provide information
maximize their about
profitscontexts to support
from selling alfajores? English TOPIC 1 Functions, Inequalities, and Systems
8 CONCEPTUAL
EXAMPLE 3 Identify an Extraneous Solution
UNDERSTANDING
LanguagePart
Learners.
D What is the maximum profit? What is the solution of the equation _____
1
x−5
+ _____
x
= __________
2
?
2 x−3 x − 8x + 15
HAVE A GROWTH MINDSET
• Reflecting Students’ Experiences in the diverse In what ways do you give your
HSM23_SEA2_FL_T01_L01.indd 8
Step 1 Multiply each side of the equation by the common denominato
(x − 5) (x − 3).
best effort and persist?
contexts embedded throughout examples and exercises (x − 5)(x − 3)(_____
1 + _____
x
)=
2(x − 5)(x − 3)
____________
x−5 x−3 2 x − 8x + 15
provides opportunities for students to see mathematics as Step 2 Continue to simplify.
relevant to their lives. (x − 5)(x − 3) ____________
___________
x−5
+
x(x − 5)(x − 3) ____________
x−3
=
2(x − 5)(x − 3)
x 2 − 8x + 15
Step 3 Divide out common factors in the numerator and the denomina
(x − 5)(x − 3) ____________
___________ x(x − 5)(x − 3) ____________
2(x − 5)(x − 3)
+ =
x−5 x−3 x 2 − 8x + 15
Step 4 Solve the equation.
LESSON 2-1 Vertex Form of a Quadratic Function 65 (x − 3) + x(x − 5) = 2 assumption that _____ = 1 and
(x − 5) (x − 3)
(x − 5) (x − 3)
x − 3 + x 2 − 5x = 2 This is only true if x ≠ 5 or 3.
x 2 − 4x − 3 = 2
63
x 2 − 4x − 5 = 0
25/12/20 3:11 AM
(x − 5)(x + 1) = 0
DIFFERENTIATED INSTRUCTION
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enVision Florida B.E.S.T. A|G|A offers comprehensive differentiation and

intervention support that addresses the needs of all learners. This support provides
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of resources to support the teacher in personalizing instruction.

Study
Plan

Readiness Standard Practice Summative

B.E.S.T. A|G|A
or Lesson
Assessment Lesson Quiz Assessment
Adaptive Practice

Mathematical Additional
Reteaching Literacy and Enrichment
Practice
Vocabulary

= System-Driven = Data-Driven / Teacher-Driven

System-Driven Differentiated Instruction Teacher-Driven Differentiated Instruction and

and Intervention Intervention
Teachers can take advantage of a variety of opportunities to have Through reports and data analytics, teachers can have
the Savvas Realize system auto-adapt assignments based on an the information they need to adapt their instruction and
individual student’s needs. assignments for students.
• Flexibility for teachers to choose from a library of
Topic Level differentiation and intervention resources:
• Reteach to Build Understanding in print or online powered by
• A Individualized Study Plan is generated based on
MathXL for School
a student’s score on the Topic Readiness Assessment when
taken online. • Mathematical Literacy and Vocabulary
• Digital intervention lessons
Lesson Level • Enrichment Activities in print or online powered by MathXL
for School
• Students take the Lesson Quiz online at the end of the
lesson instruction. • Additional Practice in print and online powered by MathXL
for School
• Differentiation can be auto-assigned based on results
from the lesson quiz. Remediation or enrichment can include:
• Additional Practice
• Reteach to Build Understanding
• Enrichment activity
• Online Practice powered by MathXL® for School
• Mathematical Literacy and Vocabulary
• Adaptive Practice provides personalized practice based on
each student’s areas of strength and weakness.

64
 Digital

Individualized Study Plan

Topic Readiness Assessment screens every student on his or
her understanding of pre-requisite content for the topic.
Individualized Study Plans are generated from the results
of the Topic Readiness Assessment. Each student receives a study
plan with additional instruction and practice tailored to his or her
specific learning needs.
Intervention resources are designed to be completed by
students independently once assigned by their teacher. They can
also be completed with the guidance of a teacher or tutor.
• Skills Review & Practice provides students scaffolded Name A26
Materials: Graph paper

instruction and practice exercises to review critical prerequisite Solving Systems by Graphing

concepts and skills. Example Solve by graphing. y = 3x - 9

x + y = -1

STEP 1 Graph each equation on the same coordinate plane.

You can use what you know about slope-intercept form.
y = 3x – 9 m= b=

You can also make a table of values.

x + y = –1
x 0 –1
y –1 0

STEP 2 Determine the point of intersection of the two lines.

The point of intersection lies on both lines, so it is
a solution of both equations.
The lines intersect at the point ( 2 , –3 ).

STEP 3 Check the solution for both equations.

Substitute (2, –3) for (x, y).
y = 3x + 9 x + y = –1
2 =3 +9 + = –1

–3 = = –1

Exercise Solve each system of equations by graphing.

1. y = 5x - 2 2. 2x - y = 4 3. y = x + 2
y=x+6 y=x+2 x+y =

4. y = 3x + 2 5. y = 2x + 1 6. y = x - 3
y = 3x - 4 2x = 4x + 2 x+y =3

7. 5x - y = 1 8. y = x - 5 9. y = 3x - 1
y=x-3 y - 4x = 4x y = 3x - 4

Skills Review & Practice

• Examples provide explicit instruction, an opportunity to try

a problem with scaffolding and a solution, and an exercise to
assess understanding.

• Practice exercises follow the examples to help students

internalize the instruction.

65
DIFFERENTIATED INSTRUCTION TOPIC 6
Exponential and Logarithmic Functions
Built-in resources for
Assess

AND INTERVENTON
TOPIC READINESS ASSESSMENT
supporting all learners! AVAILABLE ONLINE

9. Simplify the expression (x 3 · x 3 ) .

1
_ 1 6
_
Name B.E.S.T. ALGEBRA 2

SavvasRealize.com
15. What are the domain and range of
the function given by {(−2, 0),
6 Readiness Assessment 𝖠 ​x 4 𝖢𝖢𝖢𝖢 x6
(−1, 2), (0, 4), (1, 6), (2, 8)}?
𝖡 ​x 3 𝖣𝖣𝖣𝖣 x2
−1, 0, 1, 2}
Domain: {−2,

10. Simplify the expression (9x 4) 2 .

3
_ Range: {0, 2, 4, 6, 8}
1. Evaluate the expression 10 x−3 for 6. Ina records the wait time y, in
x = 2. days, for popular library books 𝖠 ​27x 4 𝖢𝖢𝖢𝖢 81x 6
based on how many people are 16. The table shows a function f. What
𝖠𝖠𝖠𝖠𝖠𝖠 𝖠𝖠−10 𝖢𝖢𝖢𝖢 0.1
on the wait list, x.
𝖡 27x 6 𝖣𝖣𝖣𝖣 81x 12 is the domain of f −1, the inverse of
𝖡 𝖠𝖠−1 𝖣𝖣𝖣𝖣 10 Which line best fits these data? function f?
11. Find the inverse of the function x 1 2 3 4 5
x (0, 3), (1, 5), (2, 9), (3, 13), (4, 15)
2. Evaluate f(x) = (_23 ) for x = −3 and f(x) = _32 x − 6. f(x) 1 4 9 16 25
x = 0. Select the function values. 𝖠𝖠𝖠𝖠𝖠𝖠 y = 2.52x + 4.29 3
__ 2
__ 2
−__ 6 4 9
◻𝖠𝖠 A. f(−3) = −2 ◻ D. f(0) = 0 𝖡 y = 3.11x + 2.71 2 3 3 𝖠 {1, 2, 3, 4, 5}
◻𝖠𝖠 B. f(−3) = ___
8 ◻ E. f(0) = 1 𝖢𝖢𝖢𝖢 y = 3.63x + 2.26 f
−1
(x) = __2 x+ 4 𝖡 {1, 4, 9, 16, 25}
27 3
𝖢𝖢𝖢𝖢 {1, 2, 3, 4, 5, 9, 16, 25}
◻𝖠𝖠 C. f(−3) = ___
27
8
◻ 2
F. f(0) = __
3
𝖣𝖣𝖣𝖣 y = 4.07x + 3.34
12. What is the inverse of the function 𝖣𝖣𝖣𝖣 {−5, −4, −3, −2, −1, 1, 2, 3, 4, 5}
7. How does the graph of f(x) = (x − 1) 3 + 8?
3. Evaluate −3 x+2 for x = −2. _____
y = (x + 2) 2 − 5 relate to the graph 𝖠 ​f −1(x) = √x − 7
3
17. The function f(x) = 2x 2 − 5x + 3

Powerful System- and Teacher-Driven

𝖠𝖠𝖠𝖠𝖠𝖠 𝖠𝖠−9 𝖢𝖢𝖢𝖢 0 of its parent function, y = x 2? __ represents the height in meters, y,
𝖡 𝖠𝖠−1 𝖣𝖣𝖣𝖣 1 𝖡 ​f −1(x) = −1 + √x 3
of a marble on a marble run after
𝖠𝖠𝖠𝖠𝖠𝖠 Translate 2 units left and reflect _____
𝖢𝖢𝖢𝖢 ​f −1(x) = 1 + √x − 2
3
over the y-axis x seconds. What is the average rate
4. For the function g(x) = (−2) 0.5x, _____ of change of the function from
𝖣𝖣𝖣𝖣 ​f −1(x) = 1 + √x − 8
3

select all the equations that

𝖡 Translate 5 units down, stretch x = 0 to x = 3?
vertically by a factor of 2
are true. 𝖠 ​−2 𝖢𝖢𝖢𝖢 1
◻ 1 ◻ 𝖢𝖢𝖢𝖢 Translate 2 units left and 13. The table shows the function f.

Resources
A. g(−4) = −__ D. g(4) = 4 𝖡 ​−1 𝖣𝖣𝖣𝖣 4
4 5 units down
◻ 1
B. g(−2) = __ ◻ E. g(2) = −2 Complete the table for the inverse
2 𝖣𝖣𝖣𝖣 Translate 2 units right and of function f.
◻ C. g(0) = −1 5 units up 18. The graph represents the
x 0 1 2 3 4 population of a bacterium, y, after
f(x) 4 2 0 3 1 x seconds. Find the average rate of
5. The number of muffins, y, The Cozy 8. Select the graph of y = −_21 (x − 3) 3
f –1(x) 2 4 1 3 0 change of the function, from
Cafe bakes each day, x, increases as and its parent function.
their advertising campaign brings x = −1 to x = 2.
y y
4 4
in more customers. They recorded y
how many muffins they baked
2 2 14. For the function f(x) = −3x + 1, 4 (2, 4)
𝖠𝖠𝖠𝖠𝖠𝖠 𝖢𝖢𝖢𝖢 what is the value of f −1(x) for x = 4?
x x
every second day. What is the line −4 −2 O 2 4 −4 −2 O 2 4
2
of best fit for these data?
−2 −2
−1 (−1, 12)
f (4) = −1 x

• Use enVision Florida B.E.S.T. A|G|A resources to

−4 −4
−4 −2 O 2 4
(0, 5), (2, 12), (4, 21), (6, 28), (8, 38),
y −2
(10, 45) 4
y 4

2 −4
𝖠𝖠𝖠𝖠𝖠𝖠 y = 0.12x − 0.54 2
x x
𝖡 −4 −2 O 2 4 𝖣𝖣𝖣𝖣 −4 −2 O 2 4
𝖡 y = 0.24x − 1.09 −2 −2 𝖠 __92 𝖢𝖢𝖢𝖢 3
__
2
𝖢𝖢𝖢𝖢 y = 4.07x + 4.48 −4 −4
𝖡 __67 19
AVAILABLE ONLINE 𝖣𝖣𝖣𝖣 ___
5
𝖣𝖣𝖣𝖣 y = 8.14x + 4.47

provide a comprehensive personalized solution to meet the

Topic Readiness Assessment Assess students’ understanding of prerequisite concepts and
enVision Florida B.E.S.T. Algebra 2 • Assessment Resources ®
enVision® Florida B.E.S.T. Algebra 2 • Assessment Resources

skills using the Topic Readiness Assessment found at SavvasRealize.com. These auto-scored
HSM23_ANCA2_FL_T06_RA.indd Page 1 01/05/21 7:08 PM f-0317a
online assessments provide students with a breadth of technology-enhanced item types.
/155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/Algebra ... HSM23_ANCA2_FL_T06_RA.indd Page 2 01/05/21 7:08 PM f-0317a /155/SA00202_AGA_ANC/AGA/FL/Anc/2023/AGA/1418351970/Layout/Interior_Files/Algebra ...

Individualized Study Plan Based on their performance, students will be assigned a study

needs of all levels of learners and provide intervention activities.

plan tailored to their specific learning needs.

Item Analysis for Diagnosis and Intervention

Item Intervention Intervention
Item DOK Skills ReviewDOK
& Practice Benchmarks
Benchmarks Item
Item DOK Skills ReviewDOK
& Practice Benchmarks
Benchmarks

• Customize lessons by uploading your own resources.

Lesson Lesson
11 1 1 NS-011
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22 1 2 NS-021
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MA.912.F.3.7
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33 1 1 NS-011
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F-16 F-BF.2.4
MA.912.F.3.7
00.000.00.0
4 1 1 N-RN.1.2
MA.912.NSO.1.1
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4
5
1
<new1 table TK>
NS-02
00-00
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MA.912.F.1.2
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S-ID.2.6.a
13
14
<new
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table
1
TK>
F-15
0-00
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MA.912.F.3.7
00.000.00.0
F-BF.2.4
5 2 F-15
0-00 MA.912.DP.2.4
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6 1 2 S-ID.2.6.a 15 10 1 F-IF.1.1
6 2 F-16
0-00 MA.912.DP.2.4
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0-00 MA.912.AR.2.4
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7 1 2 F-BF.2.3 16 1 1 F-BF.2.4, F-IF.1.1
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TOPIC 6 286H TOPIC READINESS ASSESSMENT

HSM23_TEA2_FL_T06_TRA.indd 8 12/04/21 2:48 AM

Includes DOK (Depth of Knowledge)

DIFFERENTIATION RESOURCES WHERE TO FIND

ONGOING During a • Adaptations for below- and above-level
INTERVENTION LESSON students
Teacher’s Edition, print and digital
Teacher Driven • Common Errors

1
STRATEGIC At the end of a Differentiation Library
INTERVENTION LESSON • Reteach to Build Understanding
Teacher Resource Masters, available as
System and • Mathematical Literacy and Vocabulary

2
downloadable PDFs and downloadable
Teacher Driven • Additional Practice Word docs, and digital
• Enrichment Activities
• Online Practice with built-in Learning Aids
powered by MathXL® for School
• Adaptive Practice SavvasRealize.com

• Virtual Nerd Tutorial Videos

INTENSIVE As needed • Personalized Study Plans

INTERVENTION ANYTIME • Skills Review & Practice
during a Topic SavvasRealize.com
• Virtual Nerd Tutorial Videos
3 System and
Teacher Driven
• Online Practice with built-in Learning Aids
powered by MathXL for School

66
 Digital

Powerful, online, adaptive resources

• Differentiation before a topic is based on results of the
Topic Readiness Assessment. If students take the assessment
online, an Adaptive Study Plan with intervention lessons will be
automatically generated.
• Differentiation after a lesson is based on results of the
Lesson Quiz. If students take the quiz online, activities from the
Differentiation Library can be automatically assigned.
• Differentiation after a group of topics is based on results
of online Topic Assessments and online Cumulative Assessments.
Students are assigned remediation or enrichment that can
include the following:
-- Reteach to Build Understanding powered by MathXL®
for School
-- Online and Additional Practice with built-in Learning Aids
powered by MathXL for School
• Assignment Reports show the status of assigned resources.
• Usage Data lets you know how much time students are
spending in the online resources.

67
 Assessment resources to monitor
ASSESSMENT RESOURCES progress and prepare students
for high-stakes tests!

Assessments

• Types of Assessments The table on the next page describes

the variety of assessments offered in enVision Florida
B.E.S.T. A|G|A. Note the following:
-- Online assessments can be customized as needed.
• Types of Assessment Items The formats of the assessment
items include the following:
-- Technology-enhanced items, e.g., multiple choice, editing
task choice, selectable hot text, multiselect, graphic response
item display, equation editor, and matching
-- Performance tasks

B.E.S.T. A|G|A

Assessment Data

• Assessment Reports A variety of auto-generated assessment

reports are available for online assessments.
-- Individual and class views of progress are provided in an
easy-to-view format.
-- Standards mastery reports show individual students’ mastery
or classwide mastery for each standard.

Assessment Practice

• Assessment Practice in Lessons Every lesson includes

Florida’s B.E.S.T. Assessment practice items in formats that
students will encounter on Florida’s B.E.S.T. Assessment.

68
 Digital

All of the assessment listed below are available as both print and digital resources.
Most of the digital assessments are auto-scored.

ASSESSMENTS
DIAGNOSTIC At the start of the Course Readiness Assessment
ASSESSMENT YEAR Diagnose students’ areas of strength and weakness; results can be used to prescribe
differentiated intervention.

At the start of a Topic Readiness Assessment

TOPIC Diagnose students’ proficiency with topic prerequisite concepts and skills; results can be
used to generate personalized study plan.

FORMATIVE During a Try It!

ASSESSMENT LESSON Assess students’ understanding of concepts and skills presented in each example; results
can be used to modify instruction as needed.
Do You UNDERSTAND? and Do You Know HOW?
Assess students’ conceptual understanding and procedural fluency with lesson content;
results can be used to review or revisit content.

At the end of a Lesson Quiz

LESSON Assess students’ conceptual understanding and procedural fluency with lesson content;
results can be used to prescribe differentiated instruction.

SUMMATIVE At the end of a Topic Assessment, Form A and Form B

ASSESSMENT TOPIC Assess students’ conceptual understanding and procedural fluency with topic content.
Additional Topic Assessment with ExamView®
Topic Performance Task, Form A and Form B
Assess students’ ability to apply concepts learned and their proficiency with the
Mathematical Thinking and Reasoning Standards.

After a group of Cumulative Assessments

TOPICS Assess students’ understanding of and proficiency with concepts and skills taught
throughout the school year; results can be used to prescribe intervention.
Progress Monitoring Assessment
Assess students’ progress at checkpoints throughout the year. Results can be used to
monitor students’ growth.

At the end of the End-of-Course Assessment

YEAR Assess students’ understanding of and fluency with concepts and skills taught over the
full school year.

69
PROFESSIONAL DEVELOPMENT Professional development at
your service!

TOPIC 6

Professional Development Provided in the Program

Exponential and Logarithmic Functions

• Math Background in the Teacher’s Edition discusses the TOPIC OPENER

elements of a focused, connected, and balanced curriculum AVAILABLE ONLINE

contained in each topic. TOPIC Exponential and

6
MATHEMATICAL MODELING IN 3 ACTS
Logarithmic Functions
TOPIC ESSENTIAL QUESTION

How do you use exponential and logarithmic functions to

model situations and solve problems?

Topic Overview Topic Vocabulary

enVision® STEM Project: • Change of Base Formula
Analyze Elections • common logarithm
6-1 Key Features of Exponential Functions • compound interest
AR.5.4, AR.5.5, AR.5.7, F.1.1, F.1.7, MTR.1.1, MTR.2.1, MTR.7.1
• continuously compounded

LESSON 6-3
6-2 Exponential Models interest
AR.5.4, AR.5.5, AR.5.7, F.1.1, FL.3.1, FL.3.2, FL.3.4, DP.2.9,
MTR.2.1, MTR.3.1, MTR.6.1 • decay factor
Mathematical Modeling in 3 Acts: The Crazy Conditioning • exponential equation The Crazy Conditioning

Logarithms
AR.5.4, AR.5.5, MTR.7.1 • exponential function Like all sports, soccer requires its players to be well trained. That is why

• Lesson Overviews describe the focus of each lesson,

6-3 Logarithms • exponential decay function players often have to run sprints in practice.
NSO.1.6, AR.5.2, AR.5.7, F.3.7, MTR.2.1, MTR.4.1, MTR.5.1 Glossary
• exponential growth
To make sprint drills more interesting, many coaches set up competitions.

TOPIC 6
6-4 Logarithmic Functions function
AR.5.7, AR.5.8, AR.5.9, F.1.7, F.2.2, F.2.3, F.2.5, F.3.7, MTR.1.1,
Coaches might split the players into teams and have them run relay races
MTR.5.1, MTR.7.1 • growth factor against each other. Or they might have the players sprint around cones

connections of math concepts within the course, and aspects of

6-5 Properties of Logarithms • logarithm and over barriers. What other ways would make doing sprints more fun?
NSO.1.6, NSO.1.7, AR.5.2, MTR.4.1, MTR.5.1, MTR.6.1 • logarithmic equation Think about this during the Mathematical Modeling in 3 Acts lesson.
6-6 Exponential and Logarithmic Equations • logarithmic function
AR.5.2, MTR.3.1, MTR.4.1, MTR.7.1
• natural base e

Lesson Overview 6-7 Geometric Sequences

Vocabulary Builder
rigor that balance the lesson to provide ongoing professional
• natural logarithm
AR.10.2, MTR.4.1, MTR.6.1, MTR.7.1

Objective REVIEW VOCABULARY English | Spanish

development all year long. Students will be able to:

✔ Understand the inverse relationship between exponents and
Digital Experience
INTERACTIVE STUDENT EDITION
Access online or offline.
• exponential function | función exponencial
• inverse
VIDEOS Watch clips to support
function
Mathematical
®
| función
Modeling in 3 Acts Lessons
and enVision STEM Projects.
inversa
you’ve learned.
ASSESSMENT Show what

FAMILY ENGAGEMENT TUTORIALS Get help from

logarithms. Involve family in your learning.

ACTIVITIES Complete Explore & Reason, NEW
ADAPTIVE PRACTICE Practice that is
VOCABULARY
just right and just for you.
Virtual Nerd, right when you need it.
MATH TOOLS Explore math
Model & Discuss, and Critique & Explain GLOSSARY Read and listen to with digital tools and manipulatives.
✔ Use logarithms to solve exponential models.
activities. Interact with Examples and Try Its.
• common logarithm | logaritmo
English and Spanish definitions.
común
DESMOS Use Anytime and as embedded
ANIMATION View and interact with CONCEPT SUMMARY Review Interactives in Lesson content.
✔ Evaluate logarithms using technology. real-world applications.
• logarithm
key lesson content through
| logaritmo
multiple representations.
QR CODES Scan with your mobile device
PRACTICE Practice what for Virtual Nerd Video Tutorials and Math
Go online | SavvasRealize.com you’ve learned. Modeling Lessons.

Essential Understanding • logarithmic function | función logarítmica

286 TOPIC 6 Exponential and Logarithmic Functions TOPIC 6 Exponential and Logarithmic Functions 287
A logarithmic function is the inverse of an exponential • natural logarithmic function | function logarítmica natural
function. Logarithms are found by determining the exponent
HSM23_SEA2_FL_T06_TO.indd Page 286 13/02/21 2:00 PM eteam

that must be applied to a base to yield a given result.

/154/SA00202_AGA/enVision_Mathematics/FL/SE/2023/AGA/1418346527/Layout/Interior_F ...
VOCABULARY ACTIVITY
HSM23_SEA2_FL_T06_TO.indd 287 10/01/21 5:14 AM

Topic Essential Question Activate students'

Mathematical prior knowledge
Modeling of inverse functions by
in 3 Acts
Previously in this topic, students: having them find the inverses of two or three linear or quadratic
How do you use exponential and logarithmic functions to Generate excitement
functions. aboutthe
Then make theconnection
upcoming to
Mathematical
the relationship between
• Interpreted the keyand
model situations features
solve of exponential functions.
problems? Modeling in 3 Acts lesson by having students read about
exponential and logarithmic functions.
• Used exponential functions to model relationships. the math modeling problem for this topic.
Revisit the Topic Essential Question throughout the topic.
See the
In this Teacher's
lesson, Edition Topic Review for notes about
students: See the Teacher's Edition lesson support for notes about
answering the Topic Essential Question. how to use the lesson video in your classroom.
• Use the relationship between exponents and logarithms to
Student Companion
• Topic-level Professional Development videos are solve problems.

• Evaluate common and natural logarithms and solve equations

Students can do their work for the lesson in their Student

at SavvasRealize.com. In each Topic Overview Video,

Companion or in Savvas Realize.
involving logarithms.

Later in this topic, students:

an author highlights and gives helpful perspectives • Evaluate logarithmic expressions using properties of
logarithms.

on important mathematics concepts and skills in the Skills Students practice evaluating logarithms to understand

topic. The videos are quick, focused “Watch me first”

the inverse relationship between exponents and logarithms.

Applications Students apply logarithms to solve problems

experiences to help you plan for the topic.

involving logarithmic scales, such as involving the magnitudes
of earthquakes.

TOPIC 6 286–287 TOPIC OVERVIEW

Florida’s B.E.S.T. Standards and Benchmarks

HSM23_TEA2_FL_T06_TO.indd 287 12/04/21 2:52 AM

In this lesson, students focus on these benchmarks: These standards are highlighted in this lesson:

MA.912.AR.5.2 Solve one- variable equations involving MA.K12.MTR.2.1 Demonstrate understanding by representing
logarithms or exponential expressions. Interpret solutions as problems in multiple ways.
viable in terms of the context and identify any extraneous Students represent solutions to problems using logarithms and

• mySavvasTraining.com features many online tutorials and

solutions. express connections between logarithms and exponents.
NSO.1.6 Given a numerical logarithmic expression, evaluate and MTR.5.1 Use patterns and structure to help understand and

quick-start guides to help you jumpstart your enVision Florida

generate equivalent numerical expressions using the properties of connect mathematical concepts.
logarithms or exponents. Students look for overall structure and patterns in exponents

B.E.S.T. A|G|A training. Available 24/7!

They also work with concepts related to these benchmarks: and logarithms, as well as apply general math rules to evaluate
AR.5.7 and F.3.7 logarithms.

TOPIC 6 306A LESSON 3

HSM23_TEA2_FL_T06_L03.indd 1 12/04/21 3:14 AM

Go Deeper with Savvas Professional Development

Services
• A wide array of in-district and online services is
available to help you with your professional learning. Savvas’
research-based professional development is proven to help
schools and districts grow student achievement by:

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success
Learn more at Savvas.com!

70
 B.E.S.T. A|G|A CORRELATIONS
Correlation to Florida's B.E.S.T. Algebra 2 Standards . . . . . . . . . . . . . . . . . . . . 72
  Correlation to Florida's B.E.S.T. Standards for Mathematics . . . . . . . . . . . . . 72
  Correlation to Florida's B.E.S.T. Standards for English Language Arts . . . . . . . 86
  Correlation to Florida's B.E.S.T. English Language Development Standards . . . 88
   Correlation to Florida's B.E.S.T. Access Points . . . . . . . . . . . . . . . . . . . . . 88A
Florida's B.E.S.T. Standards for Mathematics to
enVision Florida B.E.S.T. A|G|A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in B.E.S.T. ALGEBRA 2

envision Florida
NUMBER SENSE AND OPERATIONS B.E.S.T.
Algebra 2

MA.912.NSO.1 Generate equivalent expressions and perform operations with expressions involving exponents,
radicals or logarithms.

NSO.1.3 Generate equivalent algebraic expressions involving radicals or rational exponents using the 5-1, 5-2
properties of exponents.
Clarifications:
Clarification 1: Within the Algebra 2 course, radicands are limited to monomial algebraic
expressions.

NSO.1.5 Add, subtract, multiply and divide algebraic expressions involving radicals. 5-2
Clarifications:
Clarification 1: Within the Algebra 2 course, radicands are limited to monomial algebraic
expressions.

NSO.1.6 Given a numerical logarithmic expression, evaluate and generate equivalent numerical 6-3, 6-5
expressions using the properties of logarithms or exponents.
Clarifications:
Clarification 1: Within the Mathematics for Data and Financial Literacy Honors course,
problem types focus on money and business.

NSO.1.7 Given an algebraic logarithmic expression, generate an equivalent algebraic expression using 6-5
the properties of logarithms or exponents.
Clarifications:
Clarification 1: Within the Mathematics for Data and Financial Literacy Honors course,
problem types focus on money and business.

MA.912.NSO.2: Represent and perform operations with expressions within the complex number system.

NSO.2.1 Extend previous understanding of the real number system to include the complex number 2–4
system. Add, subtract, multiply and divide complex numbers.

MA.912.NSO.4 Represent and perform operations with matrices.

NSO.4.1♦ Given a mathematical or real-world context, represent and manipulate data using matrices. 7–2

NSO.4.2♦ Given a mathematical or real-world context, represent and solve a system of two- or three- 7–4
variable linear equations using matrices.

NSO.4.3♦ Solve mathematical and real-world problems involving addition, subtraction and multiplication 7–2
of matrices.
Clarifications:
Clarification 1: Instruction includes identifying and using the additive and multiplicative
identities for matrices.

NSO.4.4♦ Solve mathematical and real-world problems using the inverse and determinant of matrices. 7–3, 7–4
♦ Benchmark required for Algebra 2 Honors course only.

72
 envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.1 Interpret and rewrite algebraic expressions and equations in equivalent forms.

AR.1.1 Identify and interpret parts of an equation or expression that represent a quantity in terms of 2-3, 3-1, 3-3, 8-3
a mathematical or real-world context, including viewing one or more of its parts as a single
entity.
1
Example: The expression 1.15t can be rewritten as (1.1512)12t which is approximately
equivalent to 1.01212t. This latter expression reveals the approximate equivalent monthly
interest rate of 1.2% if the annual rate is 15%.
Clarifications:
Clarification 1: Parts of an expression include factors, terms, constants, coefficients and
variables.
Clarification 2: Within the Mathematics for Data and Financial Literacy course, problem types
focus on money and business.

AR.1.3 Add, subtract and multiply polynomial expressions with rational number coefficients. 3-2, 3-3
Clarifications:
Clarification 1: Instruction includes an understanding that when any of these operations are
performed with polynomials the result is also a polynomial.
Clarification 2: Within the Algebra 1 course, polynomial expressions are limited to 3 or
fewer terms.

AR.1.5 Divide polynomial expressions using long division, synthetic division or algebraic 3-4
manipulation.

AR.1.6 Solve mathematical and real-world problems involving addition, subtraction, multiplication or 3-2, 3-4
division of polynomials.

AR.1.8 Rewrite a polynomial expression as a product of polynomials over the real or complex number 3-3, 3-4, 3-5, 3-6
system.
Clarifications:
Clarification 1: Instruction includes factoring a sum or difference of squares and a sum or
difference of cubes.

AR.1.9 Apply previous understanding of rational number operations to add, subtract, multiply and 4-3, 4-4
divide rational algebraic expressions.
Clarifications:
Clarification 1: Instruction includes the connection to fractions and common denominators.

AR.1.11♦ Apply the Binomial Theorem to create equivalent polynomial expressions. 3-3, 8-3
Clarifications:
Clarification 1: Instruction includes the connection to Pascal’s Triangle and to combinations.
♦ Benchmark required for Algebra 2 Honors course only.

73
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.3 Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

AR.3.2 Given a mathematical or real-world context, write and solve one-variable quadratic equations 2–3, 2–4, 2–5, 2–6
over the real and complex number systems.
Clarifications:
Clarification 1: Within this benchmark, the expectation is to solve by factoring techniques,
taking square roots, the quadratic formula and completing the square.

AR.3.3 Given a mathematical or real-world context, write and solve one-variable quadratic 1–5
inequalities over the real number system. Represent solutions algebraically or graphically.

AR.3.4 Write a quadratic function to represent the relationship between two quantities from a graph, 2–1, 2–2, 2–3, 2–5
a written description or a table of values within a mathematical or real-world context.
Algebra I Example: Given the table of values below from a quadratic function, write an
equation of that function.

x −2 −1 0 1 2
f (x) 2 −1 −2 −1 2

Clarifications:
Clarification 1: Within the Algebra 1 course, a graph, written description or table of values
must include the vertex and two points that are equidistant from the vertex.
Clarification 2: Instruction includes the use of standard form, factored form and vertex form.
Clarification 3: Within the Algebra 2 course, one of the given points must be the vertex or
an x-intercept.

AR.3.8 Solve and graph mathematical and real-world problems that are modeled with quadratic 1–1, 1–5, 2–1, 2–2,
functions. Interpret key features and determine constraints in terms of the context. 2–3, 2–5, 2–6
Algebra 1 Example: The value of a classic car produced in 1972 can be modeled by the
function V(t) = 19.25t 2 − 440t + 3500, where t is the number of years since 1972. In what
year does the car’s value start to increase?
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; vertex; and symmetry.
Clarification 2: Instruction includes the use of standard form, factored form and vertex form.
Clarification 3: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
Clarification 4: Within the Algebra 1 course, notations for domain, range and constraints are
limited to inequality and set-builder.

AR.3.9 Given a mathematical or real-world context, write two-variable quadratic inequalities to 2-7
represent relationships between quantities from a graph or a written description.
Clarifications:
Clarification 1: Instruction includes the use of standard form, factored form and vertex form
where any inequality symbol can be represented.

AR.3.10 Given a mathematical or real-world context, graph the solution set to a two-variable quadratic 2-7
inequality.
Clarifications:
Clarification 1: Instruction includes the use of standard form, factored form and vertex form
where any inequality symbol can be represented.
♦ Benchmark required for Algebra 2 Honors course only.
74
 envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.4 Write, solve and graph absolute value equations, functions and inequalities in one and two
variables.

AR.4.2 Given a mathematical or real-world context, write and solve one-variable absolute value 1-5
inequalities. Represent solutions algebraically or graphically.
AR.4.4 Solve and graph mathematical and real-world problems that are modeled with absolute value 1–1, 1–2
functions. Interpret key features and determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; vertex; end behavior and symmetry.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two
variables.
AR.5.2 Solve one-variable equations involving logarithms or exponential expressions. Interpret 6–3, 6–5, 6–6
solutions as viable in terms of the context and identify any extraneous solutions.
AR.5.4 Write an exponential function to represent a relationship between two quantities from a graph, 6–1, 6–2
a written description or a table of values within a mathematical or real-world context.
Clarifications:
Clarification 1: Within the Algebra 1 course, exponential functions are limited to the forms
f (x) = abx, where b is a whole number greater than 1 or a unit fraction, or f (x) = a(1 ± r)x,
where 0 < r < 1.
Clarification 2: Within the Algebra 1 course, tables are limited to having successive
nonnegative integer inputs so that the function may be determined by finding ratios between
successive outputs.
AR.5.5 Given an expression or equation representing an exponential function, reveal the constant 6–1, 6–2
percent rate of change per unit interval using the properties of exponents. Interpret the constant
percent rate of change in terms of a real-world context.
AR.5.7 Solve and graph mathematical and real-world problems that are modeled with exponential 6–1, 6–2, 6–3, 6–4
functions. Interpret key features and determine constraints in terms of the context.
Example: The graph of the function f (t) = e5t + 2 can be transformed into the straight line
y = 5t + 2 by taking the natural logarithm of the function’s outputs.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; constant percent rate of change; end
behavior and asymptotes.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
Clarification 3: Instruction includes understanding that when the logarithm of the dependent
variable is taken and graphed, the exponential function will be transformed into a linear
function.
Clarification 4: Within the Mathematics for Data and Financial Literacy course, problem types
focus on money and business.
♦ Benchmark required for Algebra 2 Honors course only.

75
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two
variables. (continued)
AR.5.8 Given a table, equation or written description of a logarithmic function, graph that function 6–4
and determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain and range inequality notation,
interval notation or set-builder notation.
AR.5.9 Solve and graph mathematical and real-world problems that are modeled with logarithmic 6–4
functions. Interpret key features and determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
MA.912.AR.6 Solve and graph polynomial equations and functions in one and two variables.
AR.6.1 Given a mathematical or real-world context, when suitable factorization is possible, solve 3–5, 3–6
one-variable polynomial equations of degree 3 or higher over the real and complex number
systems.
AR.6.2♦ Explain and apply the Remainder Theorem to solve mathematical and real-world problems. 3–4

AR.6.5 Sketch a rough graph of a polynomial function of degree 3 or higher using zeros, multiplicity 3–1, 3–5
and knowledge of end behavior.
MA.912.AR.7 Solve and graph radical equations and functions in one and two variables.
AR.7.1 Solve one-variable radical equations. Interpret solutions as viable in terms of context and 5–4
identify any extraneous solutions.
AR.7.2 Given a table, equation or written description of a square root or cube root function, graph 5–3
that function and determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and relative maximums
and minimums.
Clarification 2: Instruction includes representing the domain and range inequality notation,
interval notation or set-builder notation.
AR.7.3 Solve and graph mathematical and real-world problems that are modeled with square root or 5–3, 5–4
cube root functions. Interpret key features and determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and relative maximums
and minimums.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
♦ Benchmark required for Algebra 2 Honors course only.

76
 envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.8 Solve and graph radical equations and functions in one and two variables.

AR.8.1 Write and solve one-variable rational equations. Interpret solutions as viable in terms of the 4–5
context and identify any extraneous solutions.
Clarifications:
Clarification 1: Within the Algebra 2 course, numerators and denominators are limited to
linear and quadratic expressions.
AR.8.2 Given a table, equation or written description of a rational function, graph that function and 4–1, 4–2
determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain and range with inequality
notation, interval notation or set-builder notation.
Clarification 3: Within the Algebra 2 course, numerators and denominators are limited to
linear and quadratic expressions.
AR.8.3 Solve and graph mathematical and real-world problems that are modeled with rational 4–1, 4–2, 4–5
functions. Interpret key features and determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the
function is increasing, decreasing, positive or negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
Clarification 3: Instruction includes using rational functions to represent inverse proportional
relationships.
Clarification 4: Within the Algebra 2 course, numerators and denominators are limited to
linear and quadratic expressions.
MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe
quantities or relationships.

AR.9.2 Given a mathematical or real-world context, solve a system consisting of a two-variable linear 1–6, 2–8
equation and a non-linear equation algebraically or graphically.
AR.9.3 Given a mathematical or real-world context, solve a system consisting of two-variable linear or 1–6, 2–8
non-linear equations algebraically or graphically.
Clarifications:
Clarification 1: Within the Algebra 2 course, non-linear equations are limited to quadratic
equations.
AR.9.5 Graph the solution set of a system of two-variable inequalities. 1–6, 2–8
Clarifications:
Clarification 1: Within the Algebra 2 course, two-variable inequalities are limited to linear and
quadratic.
♦ Benchmark required for Algebra 2 Honors course only.

77
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

envision Florida
ALGEBRAIC REASONING B.E.S.T.
Algebra 2

MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe
quantities or relationships. (continued)

AR.9.7 Given a real-world context, represent constraints as systems of linear and non-linear equations 1–6, 2–8
or inequalities. Interpret solutions to problems as viable or non-viable options.
Clarifications:
Clarification 1: Instruction focuses on analyzing a given function that models a real-world
situation and writing constraints that are represented as non-linear equations or non-linear
inequalities.
Clarification 2: Within the Algebra 2 course, non-linear equations and inequalities are limited
to quadratic.
AR.9.10♦ Solve and graph mathematical and real-world problems that are modeled with piecewise 1–3
functions. Interpret key features and determine constraints in terms of the context.
Example: A mechanic wants to place an ad in his local newspaper. The cost, in dollars, of
an ad x inches long is given by the following piecewise function. Find the cost of an ad that
would be 16 inches long.
Cx = {12x, x < 5 60 + 8x − 5, x ≥ 5
Clarifications:
Clarification 1: Key features are limited to domain, range, intercepts, asymptotes and end
behavior.
Clarification 2: Instruction includes representing the domain, range and constraints with
inequality notation, interval notation or set-builder notation.
MA.912.AR.10 Solve problems involving sequences and series.

AR.10.1♦ Given a mathematical or real-world context, write and solve problems involving arithmetic 1–4
sequences.
Example: Tara is saving money to move out of her parent’s house. She opens the account with
$250 and puts $100 into a savings account every month after that. Write the total amount of
money she has in her account after each month as a sequence. In how many months will she
have at least $3,000?
AR.10.2♦ Given a mathematical or real-world context, write and solve problems involving geometric 6–7
sequences.
Example: A bacteria in a Petri dish initially covers 2 square centimeters. The bacteria grows at
a rate of 2.6% every day. Determine the geometric sequence that describes the area covered
by the bacteria after 0, 1, 2, 3… days. Determine using technology, how many days it would
take the bacteria to cover 10 square centimeters.
♦ Benchmark required for Algebra 2 Honors course only.

78
 envision Florida
FUNCTIONS B.E.S.T.
Algebra 2

MA.912.F.1 Understand, compare and analyze properties of functions.

F.1.1 Given an equation or graph that defines a function, determine the function type. Given an 1–1, 1–2, 2–1,
input-output table, determine a function type that could represent it. 2–2, 3–5, 4–1,
6–1, 6–2
Clarifications:
Clarification 1: Within the Algebra 1 course, functions represented as tables are limited to
linear, quadratic and exponential.
Clarification 2: Within the Algebra 1 course, functions represented as equations or graphs
are limited to vertical or horizontal translations or reflections over the x-axis of the following
parent functions: f (x) = x, f (x) = x2, f (x) = x3, f (x) = √x, f (x) = ∛x , f (x) = |x|, f (x) = 2x
and f (x) = ( 1 )x.
2

F.1.7 Compare key features of two functions each represented algebraically, graphically, in tables 1–2, 2–1, 3–2,
or written descriptions. 5–3, 6–1, 6–4
Clarifications:
Clarification 1: Key features include domain; range; intercepts; intervals where the function is
increasing, decreasing, positive or negative; end behavior and asymptotes.

F.1.9 Determine whether a function is even, odd or neither when represented algebraically, 5–7
graphically or in a table.

MA.912.F.2 Identify and describe the effects of transformations on functions. Create new functions given
transformations.

F.2.2 Identify the effect on the graph of a given function of two or more transformations defined by 1–2, 2–1, 3–7,
adding a real number to the x- or y-values or multiplying the x- or y-values by a real number. 4–1, 5–3, 6–4

F.2.3 Given the graph or table of f (x) and the graph or table of f (x) + k, kf (x), f (kx) and f (x + k), 1–2, 2–1, 3–7,
state the type of transformation and find the value of the real number k. 5–3, 6–4

F.2.5 Given a table, equation or graph that represents a function, create a corresponding table, 1–2, 2–1, 3–7,
equation or graph of the transformed function defined by adding a real number to the x- or 5–3, 6–4
y-values or multiplying the x- or y-values by a real number.

MA.912.F.3 Create new functions from existing functions.

F.3.2 Given a mathematical or real-world context, combine two or more functions, limited to linear, 3–2, 5–5
quadratic, exponential and polynomial, using arithmetic operations. When appropriate,
include domain restrictions for the new function.
Clarifications:
Clarification 1: Instruction includes representing domain restrictions with inequality notation,
interval notation or set-builder notation.
Clarification 2: Within the Mathematics for Data and Financial Literacy course, problem types
focus on money and business.

F.3.4 Represent the composition of two functions algebraically or in a table. Determine the domain 5–5, 5–6
and range of the composite function.

F.3.6 Determine whether an inverse function exists by analyzing tables, graphs and equations. 5–6
♦ Benchmark required for Algebra 2 Honors course only.

79
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

envision Florida
FUNCTIONS B.E.S.T.
Algebra 2

MA.912.F.3 Create new functions from existing functions. (continued)

F.3.7 Represent the inverse of a function algebraically, graphically or in a table. Use composition 5–6, 6–3, 6–4
of functions to verify that one function is the inverse of the other.
Clarifications:
Clarification 1: Instruction includes the understanding that a logarithmic function is the inverse
of an exponential function.

envision Florida
FINANCIAL LITERACY B.E.S.T.
Algebra 2

MA.912.FL.1 Build mathematical foundations for financial literacy

FL.3.1 Compare simple, compound and continuously compounded interest over time. 6–2
Clarifications:
Clarification 1: Instruction includes taking into consideration the annual percentage rate (APR)
when comparing simple and compound interest.

FL.3.2 Solve real-world problems involving simple, compound and continuously compounded interest. 6–2
Example: Find the amount of money on deposit at the end of 5 years if you started with $500
and it was compounded quarterly at 6% interest per year.
Example: Joe won $25,000 on a lottery scratch-off ticket. How many years will it take at 6%
interest compounded yearly for his money to double?
Clarifications:
Clarification 1: Within the Algebra 1 course, interest is limited to simple and compound.

FL.3.4 Explain the relationship between simple interest and linear growth. Explain the relationship 6–2
between compound interest and exponential growth and the relationship between continuously
compounded interest and exponential growth.
Clarifications:
Clarification 1: Within the Algebra 1 course, exponential growth is limited to compound
interest.
♦ Benchmark required for Algebra 2 Honors course only.

80
 envision Florida
DATA ANALYSIS AND PROBABILITY B.E.S.T.
Algebra 2

MA.912.DP.2 Solve problems involving univariate and bivariate numerical data.

DP.2.8 Fit a quadratic function to bivariate numerical data that suggests a quadratic association and 2–2
interpret any intercepts or the vertex of the model. Use the model to solve real-world problems
in terms of the context of the data.
Clarifications:
Problems include making a prediction or extrapolation, inside and outside the range of the
data, based on the equation of the line of fit.

DP.2.9 Fit an exponential function to bivariate numerical data that suggests an exponential 6–4
association. Use the model to solve real-world problems in terms of the context of the data.
Clarifications:
Clarification 1: Instruction focuses on determining whether an exponential model is
appropriate by taking the logarithm of the dependent variable using spreadsheets and other
technology.
Clarification 2: Instruction includes determining whether the transformed scatterplot has an
appropriate line of best fit, and interpreting the y-intercept and slope of the line of best fit.
Clarification 3: Problems include making a prediction or extrapolation, inside and outside the
range of the data, based on the equation of the line of fit.

MA.912.DP.4 Use and interpret independence and probability.

DP.4.1♦ Describe events as subsets of a sample space using characteristics, or categories, of the 8–1, 8–3
outcomes, or as unions, intersections or complements of other events.

DP.4.2♦ Determine if events A and B are independent by calculating the product of their probabilities. 8–1

DP.4.3♦ Calculate the conditional probability of two events and interpret the result in terms of its 8–2
context.

DP.4.4♦ Interpret the independence of two events using conditional probability. 8–2

DP.4.9♦ Apply the addition and multiplication rules for counting to solve mathematical and real-world 8–3
problems, including problems involving probability.

DP.4.10♦ Given a mathematical or real-world situation, calculate the appropriate permutation or 8–3
combination.
♦ Benchmark required for Algebra 2 Honors course only.

81
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

enVision Florida B.E.S.T.

MATHEMATICAL THINKING AND REASONING STANDARDS
Algebra 2

MA.K12.MTR.1.1 Actively participate in effortful learning both individually and collectively.

MTR.1.1 Mathematicians who participate in effortful learning both enVision Florida B.E.S.T. A|G|A provides numerous
individually and with others: instructional opportunities to help students engage with
• Analyze the problem in a way that makes sense given mathematics through the Mathematical Thinking and
the task. Reasoning (MTR) Standards.
• Ask questions that will help with solving the task.
Each lesson begins with a lesson-opening activity in
• Build perseverance by modifying methods as needed
which students participate in effortful learning both
while solving a challenging task.
individually and with others. Students interact with their
• Stay engaged and maintain a positive mindset when
peers and teachers to analyze a problem in a way that
working to solve tasks.
makes sense given the task. Another feature of each
• Help and support each other when attempting a new
lesson is the set of problem-solving exercises, in which
method or approach.
students build perseverance by modifying methods and
Clarifications: strategies as needed while solving challenging problems.
Teachers who encourage students to participate actively Each topic provides students opportunities to demonstrate
in effortful learning both individually and with others: positive and healthy social interactions.
• Cultivate a community of growth mindset learners.
1-STEM (p. 4), 1-3 (p. 28), 2-2 (p. 71), 2-3 (p. 75), 3-1
• Foster perseverance in students by choosing tasks that
(p. 131), 3-Mathematical Modeling in 3 Acts (p. 164),
are challenging.
4-2 (p. 201), 4-5 (p. 222), 5-3 (p. 254), 5-4 (p. 263),
• Develop students’ ability to analyze and problem
6-1 (p. 294), 6-3 (p. 306), 7-2 (p. 357), 7-2 (p. 371),
solve.
8-2 (p. 399), 8-3 (p. 411)
• Recognize students’ effort when solving challenging
problems.

MA.K12.MTR.2.1 Demonstrate understanding by representing problems in multiple ways.

MTR.2.1 Mathematicians who demonstrate understanding by enVision Florida B.E.S.T. A|G|A provides
representing problems in multiple ways: instruction throughout to help students demonstrate their
• Build understanding through modeling and using understanding of mathematics by representing problems
manipulatives. in multiple ways, using objects, drawings, tables, graphs
• Represent solutions to problems in multiple ways using and equations. Regularly, in Examples, students form
objects, drawings, tables, graphs and equations. connections between concepts and representations, and
• Progress from modeling problems with objects and as students progress through a lesson, they develop
drawings to using algorithms and equations. an understanding of how to choose a representation
• Express connections between concepts and based on a given context. Represent and Connect
representations. exercises remind students that various representations
• Choose a representation based on the given context or have different purposes and may be useful in different
purpose. situations.
Clarifications: 1-5 (p. 37), 1-6 (p. 50), 2-5 (p. 94), 2-8 (p. 109), 3-4
Teachers who encourage students to demonstrate (p. 154), 3-5 (p. 159), 4-2 (p. 198), 4-Review (p. 227),
understanding by representing problems in multiple 5-STEM (p. 230), 5-3 (p. 252), 6-2 (p. 298), 6-Review
ways: (p. 341), 7-1 (p. 355), 7-Mathematical Modeling in 3
• Help students make connections between concepts and Acts (p. 380), 8-3 (p. 410), 8-Review (p. 413),
representations.
• Provide opportunities for students to use manipulatives
when investigating concepts.
• Guide students from concrete to pictorial to abstract
representations as understanding progresses.
• Show students that various representations can have
different purposes and can be useful in different
situations.

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 enVision Florida B.E.S.T.
MATHEMATICAL THINKING AND REASONING STANDARDS
Algebra 2

MA.K12.MTR.3.1 Complete tasks with mathematical fluency.

MTR.3.1 Mathematicians who complete tasks with mathematical enVision Florida B.E.S.T. A|G|A provides
fluency: opportunities for students to complete tasks with
• Select efficient and appropriate methods for solving mathematical fluency. Students practice efficient and
problems within the given context. generalizable methods when performing calculations
• Maintain flexibility and accuracy while performing and solving problems. In numerous Examples throughout
procedures and mental calculations. the lessons, more than one method for solving a problem
• Complete tasks accurately and with confidence. is shown. This allows students to become more confident
• Adapt procedures to apply them to a new context. in their selection of efficient and appropriate methods for
• Use feedback to improve efficiency when performing solving problems. In Choose Efficient Methods exercises,
calculations. opportunities exist for students to select a procedure
that allows them to solve a problem efficiently and
Clarifications:
accurately.
Teachers who encourage students to complete tasks with
mathematical fluency: 1-4 (p. 31), 1-Review (p. 55), 2-Mathematical Modeling
• Provide students with the flexibility to solve problems in 3 Acts (p. 88), 2-8 (p. 104), 3-6 (p. 169), 3-7
by selecting a procedure that allows them to solve (p. 173), 4-1 (p. 188), 4-5 (p. 223), 5-1 (p. 233), 5-2
efficiently and accurately. (p. 241), 6-2 (p. 303), 6-6 (p. 330), 7-2 (p. 361), 7-3
• Offer multiple opportunities for students to practice (p. 367), 8-STEM (p. 386), 8-3 (p. 406)
efficient and generalizable methods.
• Provide opportunities for students to reflect on the
method they used and determine if a more efficient
method could have been used.

MA.K12.MTR.4.1 Engage in discussions that reflect on the mathematical thinking of self and others.

MTR.4.1 Mathematicians who engage in discussions that reflect In enVision Florida B.E.S.T. A|G|A, the lesson-
on the mathematical thinking of self and others: opening activities afford students opportunities to
• Communicate mathematical ideas, vocabulary and engage in discussions with classmates when analyzing
methods effectively. the mathematical thinking of self and others. Students
• Analyze the mathematical thinking of others. compare the efficiency of their solution methods and
• Compare the efficiency of a method to those expressed processes to those expressed by others. Throughout
by others. the lessons, students have many opportunities to
• Recognize errors and suggest how to correctly solve communicate their mathematical ideas effectively,
the task. recognize errors, and suggest how to solve a task
• Justify results by explaining methods and processes. correctly. Communicate and Justify exercises explicitly
• Construct possible arguments based on evidence. call for students to justify or explain their reasoning in
solutions. Students discuss questions and conjectures
Clarifications:
during the first stage of Mathematical Modeling in 3
Teachers who encourage students to engage in
Acts lessons. The ability to communicate mathematical
discussions that reflect on the mathematical thinking of
ideas, vocabulary, and methods effectively for a
self and others:
particular process is a stepping-stone to critical analysis
• Establish a culture in which students ask questions of
and reasoning of both the student’s own process and the
the teacher and their peers, and error is an opportunity
processes of others.
for learning.
• Create opportunities for students to discuss their 1-4 (p. 34), 1-6 (p. 49), 2-1 (p. 63), 2-2 (p. 66), 3-STEM
thinking with peers. (p. 124), 3-7 (p. 177), 4-1 (p. 190), 4-3 (p. 207), 5-1
• Select, sequence and present student work to advance (p. 231), 5-2 (p. 240), 6-Mathematical Modeling in 3
and deepen understanding of correct and increasingly Acts (p. 305), 6-5 (p. 318), 7-1 (p. 355), 7-3 (p. 370),
efficient methods. 8-2 (p. 396), 8-3 (p. 409)
• Develop students’ ability to justify methods and
compare their responses to the responses of their
peers.

83
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)
B.E.S.T. ALGEBRA 2

enVision Florida B.E.S.T.

MATHEMATICAL THINKING AND REASONING STANDARDS
Algebra 2

MA.K12.MTR.5.1 Use patterns and structure to help understand and connect mathematical concepts.

MTR.5.1 Mathematicians who use patterns and structure to help In enVision Florida B.E.S.T. A|G|A, students
understand and connect mathematical concepts: are encouraged to look for patterns and structure as
• Focus on relevant details within a problem. they connect mathematical concepts. There are many
• Create plans and procedures to logically order events, opportunities for students to relate learned concepts
steps or ideas to solve problems. to new concepts when developing solution plans. For
• Decompose a complex problem into manageable example, as students mature in their mathematical
parts. thinking, they focus on the given details of a problem
• Relate previously learned concepts to new concepts. and decompose a complex problem down into
• Look for similarities among problems. manageable parts to make it easier to solve. This
• Connect solutions of problems to more complicated focus helps them look for similarities among problems.
large-scale situations. Additionally, Use Patterns and Structure exercises
reinforce this standard. Throughout the lessons, students
Clarifications:
develop plans and procedures to solve problems,
Teachers who encourage students to use patterns and
logically ordering steps or ideas to make problems easier
structure to help understand and connect mathematical
to solve.
concepts:
• Help students recognize the patterns in the world 1-2 (p. 20), 1-Mathematical Modeling in 3 Acts (p. 51),
around them and connect these patterns to 2-3 (p. 76), 2-4 (p. 85), 3-1 (p. 125), 3-Review (p. 181),
mathematical concepts. 4-2 (p. 193), 4-4 (p. 210), 5-1 (p. 236), 5-4 (p. 255),
• Support students to develop generalizations based on 6-1 (p. 291), 6-6 (p. 329), 7-STEM (p. 348), 7-2
the similarities found among problems. (p. 360), 8-1 (p. 389), 8-2 (p. 400)
• Provide opportunities for students to create plans and
procedures to solve problems.
• Develop students’ ability to construct relationships
between their current understanding and more
sophisticated ways of thinking.

MA.K12.MTR.6.1 Assess the reasonableness of solutions.

MTR.6.1 Mathematicians who assess the reasonableness of In enVision Florida B.E.S.T. A|G|A, students are
solutions: provided opportunities to assess the reasonableness of
• Estimate to discover possible solutions. their solutions. In Examples, students can verify possible
• Use benchmark quantities to determine if a solution solutions by explaining the methods they used and
makes sense. checking their calculations when solving problems. They
• Check calculations when solving problems. have opportunities throughout the lessons to check for the
• Verify possible solutions by explaining the methods reasonableness of a solution as they estimate or predict
used. solutions prior to solving. Check for Reasonableness
• Evaluate results based on the given context. exercises remind students to check and reflect on their
work during and after a task to determine whether or
Clarifications:
not a solution makes sense. These exercises strengthen
Teachers who encourage students to assess the
students’ ability to verify solutions through justifications.
reasonableness of solutions:
• Have students estimate or predict solutions prior to 1-2 (p. 18), 1-6 (p. 43), 2-4 (p. 81), 2-7 (p. 106), 3-2
solving. (p. 138), 3-4 (p. 149), 4-STEM (p. 184), 4-2 (p. 200),
• Prompt students to continually ask, “Does this solution 5-Mathematical Modeling in 3 Acts (p. 264), 5-5
make sense? How do you know?” (p. 272), 6-1 (p. 293), 6-5 (p. 325), 7-2 (p. 359), 7-3
• Reinforce that students check their work as they (p. 365), 8-1 (p. 393), 8-1 (p. 394)
progress within and after a task.
• Strengthen students’ ability to verify solutions through
justifications.

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 enVision Florida B.E.S.T.
MATHEMATICAL THINKING AND REASONING STANDARDS
Algebra 2

MA.K12.MTR.7.1 Apply mathematics to real-world contexts.

MTR.7.1 Mathematicians who apply mathematics to real-world Students apply mathematics to real-world contexts
contexts: throughout enVision Florida B.E.S.T. A|G|A.
• Connect mathematical concepts to everyday Application Examples present real-world contexts,
experiences. providing opportunities for students to use given models.
• Use models and methods to understand, represent and Students are provided opportunities to use models and
solve problems. methods to understand, represent, and solve real-world
• Perform investigations to gather data or determine if a problems during Mathematical Modeling in 3 Acts
method is appropriate. lessons. Students are challenged to question the accuracy
• Redesign models and methods to improve accuracy or of their models and methods, validate their conclusions,
efficiency. and redesign their models to improve accuracy and
efficiency. In Apply Math Models exercises, students
Clarifications:
connect mathematical concepts to everyday experiences
Teachers who encourage students to apply mathematics
and determine when and if a particular method is
to real-world contexts:
appropriate. STEM Projects also provide opportunities
• Provide opportunities for students to create
for students to engage with mathematics in real-world
models, both concrete and abstract, and perform
situations.
investigations.
• Challenge students to question the accuracy of their 1-1 (p. 12), 1-4 (p. 36), 2-2 (p. 73), 2-4 (p. 87), 3-7
models and methods. (p. 175), 3-7 (p. 178), 4-2 (p. 197), 4-Mathematical
• Support students as they validate conclusions by Modeling in 3 Acts (p. 224), 5-1 (p. 235), 5-3 (p. 252),
comparing them to the given situation. 6-STEM (p. 288), 6-2 (p. 304), 7-3 (p. 372), 7-4
• Indicate how various concepts can be applied to other (p. 379), 8-1 (p. 394), 8-3 (p. 408)
disciplines.

85
FLORIDA’S B.E.S.T. STANDARDS FOR ENGLISH LANGUAGE ARTS
in (continued)
B.E.S.T. ALGEBRA 2

enVision Florida B.E.S.T.

ELA EXPECTATIONS
Algebra 2

ELA.K12.EE.1.1 Cite evidence to explain and justify reasoning.

EE.1.1 Clarifications: Students regularly justify their reasoning and cite evidence
K-1 Students include textual evidence in their oral to support arguments. This may occur when writing
communication with guidance and support from adults. geometric or algebraic proofs. This standard is also often
The evidence can consist of details from the text without emphasized during discussions in Explore & Reason,
naming the text. During 1st grade, students learn how to Critique & Explain, and Model & Discuss activities, or in
incorporate the evidence in their writing. written form in Communicate and Justify exercises. Some
examples are listed below.
2-3 Students include relevant textual evidence in their
written and oral communication. Students should name 1-4 (p. 31), 2-2 (p. 66), 2-3 (p. 74), 2-5 (p. 89), 3-6
the text when they refer to it. In 3rd grade, students should (p. 165), 4-4 (p. 209), 4-5 (p. 216), 5-2 (p. 239), 6-3
use a combination of direct and indirect citations. (p. 306), 7-4 (p. 373)
4-5 Students continue with previous skills and reference
comments made by speakers and peers. Students cite
texts that they’ve directly quoted, paraphrased, or used
for information. When writing, students will use the form
of citation dictated by the instructor or the style guide
referenced by the instructor.
6-8 Students continue with previous skills and use a style
guide to create a proper citation.
9-12 Students continue with previous skills and should be
aware of existing style guides and the ways in which they
differ.

ELA.K12.EE.2.1 Read and comprehend grade-level complex texts proficiently.

EE.2.1 Clarifications: This standard is consistently addressed in enVision STEM

See Text Complexity for grade-level complexity bands and Projects, Mathematical Modeling in 3 Acts activities, and
a text complexity rubric. numerous examples throughout the book. Some examples
are listed below.
1-1 (p. 5), 2-STEM (p. 58), 2-7 (p. 105), 3-5 (p. 159), 4-2
(p. 194), 5-Mathematical Modeling in 3 Acts (p. 264),
6-STEM (p. 288), 7-2 (p. 357), 8-Mathematical Modeling
in 3 Acts (p. 403)

ELA.K12.EE.3.1 Make inferences to support comprehension.

EE.3.1 Clarifications: This standard is consistently addressed in Critique &

Students will make inferences before the words infer Explain activities, Communicate and Justify exercises, and
or inference are introduced. Kindergarten students will numerous examples throughout the book. Some examples
answer questions like “Why is the girl smiling?” or make are listed below.
predictions about what will happen based on the title
1-4 (p. 35), 1-6 (p. 48), 2-5 (p. 89), 2-6 (p. 97), 3-2
page. Students will use the terms and apply them in 2nd
(p. 133), 3-5 (p. 163), 4-1 (p. 190), 4-3 (p. 207), 5-4
grade and beyond.
(p. 257), 5-5 (p. 270), 6-2 (p. 300), 6-5 (p. 325), 7-2
(p. 358), 7-4 (p. 378), 8-2 (p. 398), 8-3 (p. 409)

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 enVision Florida B.E.S.T.
ELA EXPECTATIONS
Algebra 2

ELA.K12.EE.4.1 Use appropriate collaborative techniques and active listening skills when engaging in discussions
in a variety of situations.

EE.4.1 Clarifications: This standard is consistently addressed in Explore &

In kindergarten, students learn to listen to one another Reason activities, Critique & Explain activities, and Model
respectfully. & Discuss activities. Some examples are listed below.
In grades 1-2, students build upon these skills by justifying 1-3 (p. 23), 1-5 (p. 38), 3-5 (p. 156), 4-1 (p. 185), 4-3
what they are thinking. For example: “I think ________ (p. 202), 5-5 (p. 265), 5-6 (p. 275), 6-6 (p. 327), 7-1
because _______.” The collaborative conversations are (p. 349)
becoming academic conversations.
In grades 3-12, students engage in academic
conversations discussing claims and justifying their
reasoning, refining and applying skills. Students build on
ideas, propel the conversation, and support claims and
counterclaims with evidence.

ELA.K12.EE.5.1 Use the accepted rules governing a specific format to create quality work.

EE.5.1 Clarifications: Students consistently use accepted rules to produce quality

Students will incorporate skills learned into work products work when completing STEM Projects. This standard is
to produce quality work. For students to incorporate these also addressed in Do You Understand?, Do You Know
skills appropriately, they must receive instruction. A 3rd How?, and Practice & Problem Solving exercises. Some
grade student creating a poster board display must have examples are listed below.
instruction in how to effectively present information to do
1-STEM (p. 4), 2-6 (p. 100), 3-3 (p. 147), 4-STEM
quality work.
(p. 184), 4-2 (p. 190), 5-3 (p. 252), 6-7 (p. 349),
7-1 (p. 354), 8-STEM (p. 386), 8-3 (p. 409)

ELA.K12.EE.6.1 Use appropriate voice and tone when speaking or writing.

EE.6.1 Clarifications: This standard is consistently addressed in Critique

In kindergarten and 1st grade, students learn the & Explain activities, Error Analysis exercises, and
difference between formal and informal language. For Communicate and Justify exercises. Some examples are
example, the way we talk to our friends differs from the listed below.
way we speak to adults. In 2nd grade and beyond,
1-1 (p.10), 1-4 (p.31), 2-1 (p.64), 2-2 (p.66), 3-6
students practice appropriate social and academic
(p.165), 3-7 (p.173), 4-1 (191), 4-4 (p.209), 5-1
language to discuss texts.
(p.236), 6-3 (p. 306)

87
FLORIDA’S B.E.S.T. ENGLISH LANGUAGE DEVELOPMENT
STANDARDS in B.E.S.T. ALGEBRA 2

ENGLISH LANGUAGE DEVELOPMENT FOR ENGLISH LANGUAGE enVision Florida B.E.S.T.

LEARNERS Algebra 2

ELD.K12.ELL.MA.1 English language learners communicate information, ideas and concepts necessary for
academic success in the content area of Mathematics.

ELL.MA.1 This standard is consistently addressed in Explore &

Reason activities, Communicate and Justify exercises, and
Do You Understand? exercises. Some examples are listed
below.
1-2 (p. 13), 2-1 (p. 63), 2-4 (p. 81), 3-7 (p. 171), 4-1
(p. 190), 5-3 (p. 252), 6-2 (p. 302), 6-5 (p. 324), 7-3
(p. 364), 8-1 (p. 387)

88
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in B.E.S.T. ALGEBRA 2

STRAND Number Sense and Operations (NSO)

Standard MA.912.NSO.1 Generate equivalent expressions and perform operations with expressions involving exponents, radicals
or logarithms.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.NSO.1.AP.3 Using properties of exponents, identify Lesson 5-2 Properties of Exponents and Radicals
equivalent algebraic expressions involving radicals and rational 5-2 Examples 1, 2, and 3
exponents. Radicands are limited to monomial algebraic 5-2 Virtual Nerd: What Are the Properties of Rational Exponents?
expression. 5-2 Reteach to Build Understanding
Skills Review and Practice A79

MA.912.NSO.1.AP.5 Add and subtract algebraic expressions Lesson 5-2 Properties of Exponents and Radicals
involving radicals. Radicands are limited to monomial algebraic 5-2 Example 4
expressions. 5-2 Virtual Nerd: What Are the Properties of Rational Exponents?
5-2 Reteach to Build Understanding
Skills Review and Practice A81

MA.912.NSO.1.AP.6 Given a numerical logarithmic expression, Lesson 6-5 Properties of Logarithms

identify an equivalent numerical expression using the properties of 6-5 Examples 2 and 3
logarithms or exponents. 6-5 Virtual Nerd: How Do You Expand a Logarithmic Expression Using
Properties of Logarithms?
6-5 Reteach to Build Understanding
Skills Review and Practice F71

MA.912.NSO.1.AP.7 Given an algebraic logarithmic Lesson 6-5 Properties of Logarithms

expression, identify an equivalent algebraic expression using the 6-5 Examples 2 and 3
properties of logarithms or exponents. 6-5 Virtual Nerd: How Do You Expand a Logarithmic Expression Using
Properties of Logarithms?
6-5 Reteach to Build Understanding
Skills Review and Practice F71

BENCHMARKS WITHIN STANDARD MA.912.NSO.1

MA.912.NSO.1.3 Generate equivalent algebraic expressions MA.912.NSO.1.7 Given an algebraic logarithmic expression,
involving radicals or rational exponents using the properties of generate an equivalent algebraic expression using the properties
exponents. of logarithms or exponents.

MA.912.NSO.1.5 Add, subtract, multiply and divide algebraic

expressions involving radicals.

MA.912.NSO.1.6 Given a numerical logarithmic expression,

evaluate and generate equivalent numerical expressions using the
properties of logarithms or exponents.

88A
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Number Sense and Operations (NSO) (continued)

Standard MA.912.NSO.2 Represent and perform operations with expressions within the complex number system.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.NSO.2.AP.1 Extend previous understanding of the real Lesson 2-4 Complex Numbers and Operations
number system to include the complex number system. Add and 2-4 Examples 1 and 2
subtract complex numbers. 2-4 Virtual Nerd: What Is the Difference Between Imaginary and
Complex Numbers?
2-4 Reteach to Build Understanding
Skills Review and Practice NS11

BENCHMARKS WITHIN STANDARD MA.912.NSO.2

MA.912.NSO.2.1 Extend previous understanding of the real

number system to include the complex number system. Add,
subtract, multiply and divide complex numbers.

88B
STRAND Algebraic Reasoning (AR)

Standard MA.912.AR.1 Interpret and rewrite algebraic expressions and equations in equivalent forms.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.1.AP.1 Identify a part(s) of an equation or Lesson 2-3 Factored Form of a Quadratic Function
expression and explain the meaning within the context of a Lesson 3-3 Polynomial Identities
problem. 2-3 Example 6
3-3 Example 3
2-3 Virtual Nerd: How Do You Solve a Quadratic Equation by Factoring?
3-3 Virtual Nerd: What Is the Formula for Factoring the Sum of Cubes?
2-3, 3-3 Reteach to Build Understanding
Skills Review and Practice A69, NS11

MA.912.AR.1.AP.3 Add, subtract and multiply polynomial Lesson 3-2 Adding, Subtracting, and Multiplying Polynomials
expressions with integer coefficients. 3-2 Examples 1A and 2
3-2 Virtual Nerd: How Do You Solve a Word Problem by Subtracting
Polynomials?
3-2 Reteach to Build Understanding
Skills Review and Practice A18, A20

MA.912.AR.1.AP.5 Divide polynomial expressions using long Lesson 3-4 Dividing Polynomials
division, synthetic division and algebraic manipulation where the 3-4 Examples 1 and 2
denominator is a linear expression. 3-4 Virtual Nerd: How Do You Divide a Polynomial by a Binomial Using
Synthetic Division When You Have a Remainder?
3-4 Reteach to Build Understanding
Skills Review and Practice A74, A75

MA.912.AR.1.AP.6 Solve mathematical and/or real-world Lesson 3-2 Adding, Subtracting, and Multiplying Polynomials
problems involving addition, subtraction, multiplication or division 3-2 Examples 4 and 5
of polynomials with integer coefficients. 3-2 Virtual Nerd: Do You Solve a Word Problem by Subtracting and
Multiplying Polynomials?
3-2 Reteach to Build Understanding
Skills Review and Practice A18, A20

MA.912.AR.1.AP.8 Select a polynomial expression as a product Lesson 3-5 Zeros of a Polynomial Function
of polynomials with integer coefficients over the real or complex 3-5 Example 1
number system. 3-5 Virtual Nerd: How Do You Find All the Zeros of a Polynomial Function
with Imaginary Zeros?
3-5 Reteach to Build Understanding
Skills Review and Practice F62

MA.912.AR.1.AP.9 Apply previous understanding of rational Lesson 4-4 Adding and Subtracting Rational Expressions
number operations with common denominators to add and 4-4 Examples 1, 2, 3, and 4
subtract rational expressions. 4-4 Virtual Nerd: How Do You Add Two Rational Expressions with
Different Denominators?
4-4 Reteach to Build Understanding
Skills Review and Practice A86

88C
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

BENCHMARKS WITHIN STANDARD MA.912.AR.1

MA.912.AR.1.1 Identify and interpret parts of an equation or MA.912.AR.1.6 Solve mathematical and real-world problems
expression that represent a quantity in terms of a mathematical or involving addition, subtraction, multiplication or division of
real-world context, including viewing one or more of its parts as a polynomials.
single entity.
MA.912.AR.1.8 Rewrite a polynomial expression as a product
MA.912.AR.1.3 Add, subtract and multiply polynomial of polynomials over the real or complex number system.
expressions with rational number coefficients.
MA.912.AR.1.9 Apply previous understanding of rational
MA.912.AR.1.5 Divide polynomial expressions using long number operations to add, subtract, multiply and divide rational
division, synthetic division or algebraic manipulation. algebraic expressions.

88D
STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.3 Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.3.AP.2 Solve mathematical one-variable quadratic Lesson 2-3 Factored Form of a Quadratic Function
equations with integer coefficients over the real and complex Lesson 2-5 Completing the Square
number systems. Lesson 2-6 The Quadratic Formula
2-3 Examples 3 and 4
2-5 Examples 1 and 3
2-6 Examples 1 and 2
2-3 Virtual Nerd: How Do You Solve a Quadratic Equation by Factoring?
2-5 Virtual Nerd: How Do You Solve a Quadratic Equation with Complex
Solutions by Completing the Square?
2-6 Virtual Nerd: How Do You Solve a Quadratic Equation with Complex
Solutions by Using the Quadratic Formula?
2-3, 2-5, 2-6 Reteach to Build Understanding
Skills Review and Practice A69, A71, A72

MA.912.AR.3.AP.3 Given a mathematical or real-world context, Lesson 1-5 Quadratic and Absolute Value Inequalities
select a one-variable quadratic inequality over the real number 1-5 Examples 1, 2, and 3
system that represents the solution algebraically or graphically. 1-5 Virtual Nerd: How Do You Solve a Quadratic Inequality by
Graphing?
1-5 Reteach to Build Understanding
Skills Review and Practice A52

MA.912.AR.3.AP.4 Select a quadratic function to represent the Lesson 2-1 Vertex Form of a Quadratic Function
relationship between two quantities from a graph. 2-1 Examples 3 and 4
2-1 Virtual Nerd: What Is Vertex Form of a Quadratic Equation?
2-1 Reteach to Build Understanding
Skills Review and Practice F16

MA.912.AR.3.AP.8 Solve mathematical problems that are Lesson 2-3 Factored Form of a Quadratic Function
modeled with quadratic functions, using key features and select 2-3 Example 4
the graph that represents this function. 2-3 Virtual Nerd: How Do You Solve a Quadratic Equation by Factoring?
2-3 Reteach to Build Understanding
Skills Review and Practice A69

MA.912.AR.3.AP.9 Select two-variable quadratic inequalities Lesson 2-7 Quadratic Inequalities

to represent relationships between quantities from a graph or a 2-7 Examples 1, 2, and 3
written description. 2-7 Virtual Nerd: What Is a Quadratic Inequality?
2-7 Reteach to Build Understanding
Skills Review and Practice A41

MA.912.AR.3.AP.10 Select the graph of the solution set to a Lesson 2-7 Quadratic Inequalities
two-variable quadratic inequality. 2-7 Examples 1, 2, and 3
2-7 Virtual Nerd: How Do You Graph a Quadratic Inequality?
2-7 Reteach to Build Understanding
Skills Review and Practice A41

88E
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

BENCHMARKS WITHIN STANDARD MA.912.AR.3

MA.912.AR.3.2 Given a mathematical or real-world context, MA.912.AR.3.8 Solve and graph mathematical and real-world
write and solve one-variable quadratic equations over the real problems that are modeled with quadratic functions. Interpret key
and complex number systems. features and determine constraints in terms of the context.

MA.912.AR.3.3 Given a mathematical or real-world context, MA.912.AR.3.9 Given a mathematical or real-world context,
write and solve one-variable quadratic inequalities over the real write two-variable quadratic inequalities to represent relationships
number system. Represent solutions algebraically or graphically. between quantities from a graph or a written description.

MA.912.AR.3.4 Write a quadratic function to represent the MA.912.AR.3.10 Given a mathematical or real-world context,
relationship between two quantities from a graph, a written graph the solution set to a twovariable quadratic inequality.
description or a table of values within a mathematical or real-
world context.

88F
STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.4 Write, solve and graph absolute value equations, functions and inequalities in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.4.AP.2 Solve a one-variable absolute value Lesson 1-5 Quadratic and Absolute Value Inequalities
inequality. Represent solutions algebraically or graphically. 1-5 Example 4
1-5 Virtual Nerd: How Do You Solve a Word Problem Using an AND
Absolute Value Inequality?
1-5 Reteach to Build Understanding
Skills Review and Practice A53

MA.912.AR.4.AP.4 Solve mathematical problems that are Lesson 1-1 Key Features of Functions
modeled with absolute value functions, using key features and Lesson 1-2 Transformations of Functions
select the graph that represents this function. 1-1 Examples 4 and 5
1-2 Examples 2 and 3
1-1, 1-2 Reteach to Build Understanding
Skills Review and Practice F54

BENCHMARKS WITHIN STANDARD MA.912.AR.4

MA.912.AR.4.2 Given a mathematical or real-world context, MA.912.AR.4.4 Solve and graph mathematical and real-world
write and solve one-variable absolute value inequalities. Represent problems that are modeled with absolute value functions. Interpret
solutions algebraically or graphically. key features and determine constraints in terms of the context.

88G
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.5.AP.2 Solve one-variable equations involving Lesson 6-3 Logarithms

logarithms or exponential expressions. Identify any extraneous Lesson 6-6 Exponential and Logarithmic Equations
solutions. 6-3 Example 5
6-6 Examples 1, 2, and 3
6-3 Virtual Nerd: How Do You Convert from Exponential Form to
Logarithmic Form?
6-6 Virtual Nerd: How Do You Solve an Exponential Equation Using
Logarithms?
6-3, 6-6 Reteach to Build Understanding
Skills Review and Practice F72

MA.912.AR.5.AP.4 Select an exponential function to represent Lesson 6-1 Key Features of Exponential Functions
two quantities from a graph or a table of values. 6-1 Example 5
6-1 Virtual Nerd: What’s an Exponential Function?
6-1 Reteach to Build Understanding
Skills Review and Practice F67

MA.912.AR.5.AP.5 Given an expression or equation Lesson 6-1 Key Features of Exponential Functions
representing an exponential function, reveal the constant percent 6-1 Example 4
rate of change per unit interval using the properties of exponents. 6-1 Virtual Nerd: What Is Exponential Growth?
6-1 Reteach to Build Understanding
Skills Review and Practice F67

MA.912.AR.5.AP.7 Solve and select the graph of mathematical Lesson 6-1 Key Features of Exponential Functions
exponential functions. 6-1 Examples 1 and 2
6-1 Virtual Nerd: What Is Exponential Growth?
6-1 Reteach to Build Understanding
Skills Review and Practice F67

MA.912.AR.5.AP.8 Given an equation of a logarithmic Lesson 6-4 Logarithmic Functions

function, select the graph of that function. 6-4 Examples 1 and 2
6-4 Virtual Nerd: How Do You Graph a Logarithmic Function by Making
a Table?
6-4 Reteach to Build Understanding
Skills Review and Practice F70

MA.912.AR.5.AP.9 Solve and select the graph of mathematical Lesson 6-4 Logarithmic Functions
logarithmic functions. 6-4 Examples 1 and 2
6-4 Virtual Nerd: How Do You Graph a Logarithmic Function by Making
a Table?
6-4 Reteach to Build Understanding
Skills Review and Practice F70

88H
BENCHMARKS WITHIN STANDARD MA.912.AR.5

MA.912.AR.5.2 Solve one-variable equations involving MA.912.AR.5.7 Solve and graph mathematical and real-world
logarithms or exponential expressions. Interpret solutions as viable problems that are modeled with exponential functions. Interpret
in terms of the context and identify any extraneous solutions. key features and determine constraints in terms of the context.

MA.912.AR.5.4 Write an exponential function to represent MA.912.AR.5.8 Given a table, equation or written description
a relationship between two quantities from a graph, a written of a logarithmic function, graph that function and determine its
description or a table of values within a mathematical or real- key features.
world context.
MA.912.AR.5.9 Solve and graph mathematical and real-world
MA.912.AR.5.5 Given an expression or equation representing problems that are modeled with logarithmic functions. Interpret
an exponential function, reveal the constant percent rate of key features and determine constraints in terms of the context.
change per unit interval using the properties of exponents.
Interpret the constant percent rate of change in terms of a
realworld context.

88I
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.6 Solve and graph polynomial equations and functions in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.6.AP.1 Solve one-variable polynomial equations of Lesson 3-5 Zeros of Polynomial Functions
degree 3 or higher in factored form, over the real number system. 3-5 Example 5
3-5 Virtual Nerd: How Do You Find All the Zeros of a Polynomial Function
with Imaginary Zeros?
3-5 Reteach to Build Understanding
Skills Review and Practice F62

MA.912.AR.6.AP.5 Create a rough graph of a polynomial Lesson 3-1 Graphing Polynomial Functions
function of degree 3 or higher in factored form using zeros, Lesson 3-5 Zeros of Polynomial Functions
multiplicity and knowledge of end behavior. 3-1 Examples 1, 2, and 3
3-5 Examples 1 and 2
3-1 Virtual Nerd: How Do You Use a Table to Estimate Where the Turning
Points of a Polynomial Function Occur?
3-1 Virtual Nerd: How Do You Sketch the Graph of a Polynomial Function
Given Characteristics of the Graph?
3-1, 3-5 Reteach to Build Understanding
Skills Review and Practice A73

BENCHMARKS WITHIN STANDARD MA.912.AR.6

MA.912.AR.6.1 Given a mathematical or real-world context, MA.912.AR.6.5 Sketch a rough graph of a polynomial function
when suitable factorization is possible, solve one-variable of degree 3 or higher using zeros, multiplicity and knowledge of
polynomial equations of degree 3 or higher over the real and end behavior.
complex number systems.

88J
STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.7 Solve and graph radical equations and functions in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.7.AP.1 Solve one-variable radical equations and Lesson 5-4 Solving Radical Equations
identify any extraneous solutions. 5-4 Examples 1, 2, 4, and 5
5-4 Virtual Nerd: How Do You Solve a Radical Equation?
5-4 Virtual Nerd: How Do You Solve a Radical Equation with an
Extraneous Solution?
5-4 Reteach to Build Understanding
Skills Review and Practice A84

MA.912.AR.7.AP.2 Given a table, equation or written Lesson 5-3 Graphing Radical Functions
description of a square root or cube root function, select the graph 5-3 Examples 1 and 2
that represents the function. 5-3 Virtual Nerd: How Do You Graph a Cube Root Function Using
a Table?
5-3 Reteach to Build Understanding
Skills Review and Practice F66

MA.912.AR.7.AP.3 Given a mathematical or real-world Lesson 5-3 Graphing Radical Functions

problem that is modeled with square root or cube root functions, 5-3 Examples 4 and 5
using key features in terms of the context, select the graph that 5-3 Virtual Nerd: How Do You Graph a Square Root Function Using
represents this model. a Table?
5-3 Reteach to Build Understanding
Skills Review and Practice F66

BENCHMARKS WITHIN STANDARD MA.912.AR.7

MA.912.AR.7.1 Solve one-variable radical equations. Interpret MA.912.AR.7.3 Solve and graph mathematical and real-
solutions as viable in terms of context and identify any extraneous world problems that are modeled with square root or cube root
solutions. functions. Interpret key features and determine constraints in terms
of the context.
MA.912.AR.7.2 Given a table, equation or written description
of a square root or cube root function, graph that function and
determine its key features.

88K
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.8 Solve and graph rational equations and functions in one and two variables.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.8.AP.1 Solve one-variable rational equations and Lesson 4-5 Solving Rational Equations
identify any extraneous solutions. 4-5 Examples 1 and 3
4-5 Virtual Nerd: How Do You Solve a Rational Equation with an
Extraneous Solution?
4-5 Reteach to Build Understanding
Skills Review and Practice A87

MA.912.AR.8.AP.2 Given a table, equation or written Lesson 4-1 Inverse Variation and the Reciprocal Function
description of a rational function, select the graph that represents 4-1 Example 4
the function. 4-1 Virtual Nerd: How Do You Graph an Inverse Variation Equation Using
a Table?
4-1 Reteach to Build Understanding
Skills Review and Practice F75

MA.912.AR.8.AP.3 Given a mathematical and/or real-world Lesson 4-2 Graphing Rational Functions
problem that is modeled with rational functions, using key features 4-2 Examples 1, 2, and 3
in terms of the context, select the graph that represents this model. 4-2 Virtual Nerd: How Do You Find the Horizontal Asymptotes of a
Rational Function?
4-2 Virtual Nerd: How Do You Find the Vertical Asymptotes of a Rational
Function?
4-2 Reteach to Build Understanding
Skills Review and Practice F76

BENCHMARKS WITHIN STANDARD MA.912.AR.8

MA.912.AR.8.1 Write and solve one-variable rational MA.912.AR.8.3 Solve and graph mathematical and real-world
equations. Interpret solutions as viable in terms of the context and problems that are modeled with rational functions. Interpret key
identify any extraneous solutions. features and determine constraints in terms of the context.

MA.912.AR.8.2 Given a table, equation or written description

of a rational function, graph that function and determine its key
features

88L
STRAND Algebraic Reasoning (AR) (continued)

Standard MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe quantities
or relationships.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.AR.9.AP.2 Solve a system consisting of a two-variable Lesson 2-8 Systems Involving Quadratic Equations and Inequalities
linear equation and a quadratic equation algebraically or 2-8 Examples 1, 2, and 3
graphically. 2-8 Virtual Nerd: How Do You Solve a Linear-Quadratic System Using
Substitution?
2-8 Reteach to Build Understanding
Skills Review and Practice A56

MA.912.AR.9.AP.3 Solve a system consisting of two-variable Lesson 2-8 Systems Involving Quadratic Equations and Inequalities
linear or quadratic equations algebraically or graphically. 2-8 Examples 1, 2, and 3
2-8 Virtual Nerd: How Do You Solve a Linear-Quadratic System Using
Substitution?
2-8 Reteach to Build Understanding
Skills Review and Practice A56

MA.912.AR.9.AP.5 Select the graph of the solution set of a Lesson 2-8 Systems Involving Quadratic Equations and Inequalities
system of two-variable inequalities. 2-8 Examples 4 and 5
2-8 Virtual Nerd: How Do You Solve a System of Equations by Graphing
if One Equation Is a Quadratic?
2-8 Reteach to Build Understanding
Skills Review and Practice A57

MA.912.AR.9.AP.7 Given a real-world context, as systems of Lesson 1-6 Linear Systems

linear and non-linear equations or inequalities with identified 1-6 Example 2
constraints, select a solution as a viable or non-viable option. 1-6 Reteach to Build Understanding
Skills Review and Practice A57

BENCHMARKS WITHIN STANDARD MA.912.AR.9

MA.912.AR.9.2 Given a mathematical or real-world context, MA.912.AR.9.5 Graph the solution set of a system of two-
solve a system consisting of a two-variable linear equation and a variable inequalities.
non-linear equation algebraically or graphically.
MA.912.AR.9.7 Given a real-world context, represent
MA.912.AR.9.3 Given a mathematical or real-world context, constraints as systems of linear and non-linear equations or
solve a system consisting of two-variable linear or non-linear inequalities. Interpret solutions to problems as viable or non-viable
equations algebraically or graphically. options.

88M
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Functions (F)

Standard MA.912.F.1 Understand, compare and analyze properties of functions.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.F.1.AP.1b Given an input-output table with an Lesson 6-1 Key Features of Exponential Equations
accompanying graph, determine a function type, either linear or 6-1 Example 1
quadratic, that could represent it. 6-1 Virtual Nerd: What’s an Exponential Function?
6-1 Reteach to Build Understanding
Skills Review and Practice F68

MA.912.F.1.AP.7 Compare key features of two functions each Lesson 5-3 Graphing Radical Functions
represented algebraically or graphically. 5-3 Example 3
5-3 Virtual Nerd: How Do You Graph a Square Root Function Using
a Table?
5-3 Reteach to Build Understanding
Skills Review and Practice F66

MA.912.F.1.AP.9 Select whether a function is even, odd or Lesson 3-7 Transformations of Polynomial Functions
neither when represented algebraically. 3-7 Example 2
3-7 Reteach to Build Understanding
Skills Review and Practice F94

BENCHMARKS WITHIN STANDARD MA.912.F.1

MA.912.F.1.1 Given an equation or graph that defines a MA.912.F.1.9 Determine whether a function is even, odd or
function, classify the function type. Given an input-output table, neither when represented algebraically, graphically or in a table.
determine a function type that could represent it.

MA.912.F.1.7 Compare key features of two functions each

represented algebraically, graphically, in tables or written
descriptions.

88N
STRAND Functions (F) (continued)

Standard MA.912.F.2 Identify and describe the effects of transformations on functions. Create new functions given transformations.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.F.2.AP.2 Identify the effect on the graph of a given Lesson 2-1 Vertex Form of a Quadratic Function
function of two or more transformations defined by adding a real Lesson 4-1 Inverse Variation and Reciprocal Functions
number to the x- or y-values. 2-1 Example 1
4-1 Example 5
2-1 Virtual Nerd: How Do You Vertically Translate a Quadratic Function?
4-1 Virtual Nerd: How Do You Graph an Inverse Variation Equation Using
a Table?
2-1, 4-1 Reteach to Build Understanding
Skills Review and Practice F15, F75

MA.912.F.2.AP.3 Given the graph of a given function after Lesson 5-3 Graphing Radical Functions
replacing f(x) by f(x) + k and f(x + k), kf(x), for specific values of Lesson 6-4 Logarithmic functions
k select the type of transformation and find the value of the real 5-3 Example 4
number k. 6-4 Example 2
5-3 Virtual Nerd: How Do You Graph a Square Root Function Using
a Table?
6-4 Virtual Nerd: What Does the Constant ‘k’ Do to the Graph of
f(x) = log(x) + k?
5-3, 6-4 Reteach to Build Understanding
Skills Review and Practice F66, F70

MA.912.F.2.AP.5 Given a table, equation or graph that Lesson 3-7 Transformations of Polynomial Functions
represents a function, select a corresponding table, equation 3-7 Examples 3 and 4
or graph of the transformed function defined by adding a real 3-7 Virtual Nerd: How Do You Translate a Polynomial Function
number to the x- or y-values. Horizontally?
3-7 Reteach to Build Understanding
Skills Review and Practice F60

BENCHMARKS WITHIN STANDARD MA.912.F.2

MA.912.F.2.2 Identify the effect on the graph of a given MA.912.F.2.5 Given a table, equation or graph that represents
function of two or more transformations defined by adding a real a function, create a corresponding table, equation or graph of the
number to the x- or y-values or multiplying the x- or y-values by a transformed function defined by adding a real number to the x- or
real number. y-values or multiplying the x- or y-values by a real number.

MA.912.F.2.3 Given the graph or table of f(x) and the graph

or table of f(x) + k, kf(x), f(kx) and f(x + k), state the type of
transformation and find the value of the real number k.

88O
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Functions (F) (continued)

Standard MA.912.F.3 Create new functions from existing functions.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.F.3.AP.2 Given a mathematical and/or real-world Lesson 5-5 Function Operations

context, combine two or more functions, limited to linear, 5-5 Examples 1 and 2
quadratic, and polynomial, using arithmetic operations of 5-5 Virtual Nerd: Function Operations-What Are Function Operations?
addition, subtraction, or multiplication. 5-5 Reteach to Build Understanding
Skills Review and Practice F64

MA.912.F.3.AP.4 Given a composite function within a Lesson 5-5 Function Operations

mathematical or real-world context, identify the domain and range 5-5 Examples 4, 5, and 6
of the composite function. 5-5 Virtual Nerd: What Are Compositions of Functions?
5-5 Reteach to Build Understanding
Skills Review and Practice F64

MA.912.F.3.AP.6 Determine whether an inverse function exists Lesson 5-6 Inverse Relations and Functions
by analyzing graphs and equations. 5-6 Examples 1, 2, and 3
5-6 Virtual Nerd: What’s the Inverse of a Relation?
5-6 Reteach to Build Understanding
Skills Review and Practice F65

MA.912.F.3.AP.7 Represent the inverse of a function Lesson 5-6 Inverse Relations and Functions
algebraically. Use composition of functions to verify that one 5-6 Example 5
function is the inverse of the other. 5-6 Reteach to Build Understanding
Skills Review and Practice F65

BENCHMARKS WITHIN STANDARD MA.912.F.3

MA.912.F.3.2 Given a mathematical or real-world context, MA.912.F.3.6 Determine whether an inverse function exists by
combine two or more functions, limited to linear, quadratic, analyzing tables, graphs and equations.
exponential and polynomial, using arithmetic operations. When
appropriate, include domain restrictions for the new function. MA.912.F.3.7 Represent the inverse of a function algebraically,
graphically or in a table. Use composition of functions to verify
MA.912.F.3.4 Represent the composition of two functions that one function is the inverse of the other.
algebraically or in a table. Determine the domain and range of
the composite function.

88P
STRAND Financial Literacy (FL)

Standard MA.912.FL.3 Describe the advantages and disadvantages of short-term and long-term purchases.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.FL.3.AP.1 Compare simple and compound interest Lesson 6-2 Exponential Models
over time. 6-2 Example 4
6-2 Virtual Nerd: How Do You Use the Formula for Compound Interest?
6-2 Reteach to Build Understanding
Skills Review and Practice F67

MA.912.FL.3.AP.2 Solve real-world problems involving simple Lesson 6-2 Exponential Models
and compound interest. 6-2 Example 4
6-2 Virtual Nerd: How Do You Use the Formula for Compound Interest?
6-2 Reteach to Build Understanding
Skills Review and Practice F67

BENCHMARKS WITHIN STANDARD MA.912.FL.3

MA.912.FL.3.1 Compare simple, compound and continuously MA.912.FL.3.2 Solve real-world problems involving simple,
compounded interest over time. compound and continuously compounded interest.

88Q
FLORIDA ACCESS POINTS FOR MATHEMATICS, GRADES 9–12
in (continued)
B.E.S.T. ALGEBRA 2

STRAND Data Analysis and Probability (DP)

Standard MA.912.DP.2 Solve problems involving univariate and bivariate numerical data.

ACCESS POINTS WHERE ACCESS POINTS ARE ADDRESSED

MA.912.DP.2.AP.8 Given a scatter plot, select a quadratic Lesson 2-2 Standard Form of a Quadratic Function
function that fits the data the best. 2-2 Example 5
2-2 Reteach to Build Understanding
Skills Review and Practice F58

MA.912.DP.2.AP.9 Given a scatter plot, select an exponential Lesson 6-4 Logarithmic Functions
function that fits the data the best. 6-4 Example 6
6-4 Reteach to Build Understanding
Skills Review and Practice A42

BENCHMARKS WITHIN STANDARD MA.912.F.3

MA.912.DP.2.8 Fit a quadratic function to bivariate numerical MA.912.DP.2.9 Fit an exponential function to bivariate
data that suggests a quadratic association and interpret any numerical data that suggests an exponential association. Use the
intercepts or the vertex of the model. Use the model to solve real- model to solve real-world problems in terms of the context of the data.
world problems in terms of the context of the data.

88R
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in

NUMBER SENSE AND OPERATIONS Algebra 1 Geometry Algebra 2

MA.912.NSO.1 Generate equivalent expressions and perform operations with expressions involving exponents,
radicals or logarithms.

NSO.1.1 Extend previous understanding of the Laws of Exponents to 5-1

include rational exponents. Apply the Laws of Exponents to
evaluate numerical expressions and generate equivalent numerical
expressions involving rational exponents.
Clarifications:
Clarification 1: Instruction includes the use of technology when
appropriate.
Clarification 2: Refer to the K-12 Formulas (Appendix E) for the Laws
of Exponents.
Clarification 3: Instruction includes converting between expressions
involving rational exponents and expressions involving radicals.
Clarification 4: Within the Mathematics for Data and Financial
Literacy course, it is not the expectation to generate equivalent
numerical expressions.

NSO.1.2 Generate equivalent algebraic expressions using the properties of 5-1, 5-4
exponents.

NSO.1.3 Generate equivalent algebraic expressions involving radicals or rational 5-1, 5-2
exponents using the properties of exponents.

NSO.1.4 Apply previous understanding of operations with rational numbers to add, 5-2
subtract, multiply and divide numerical radicals.
Clarifications:
Clarification 1: Within the Algebra 1 course, expressions are limited to a
single arithmetic operation involving two square roots or two cube roots.

NSO.1.5 Add, subtract, multiply and divide algebraic expressions involving 5-2
radicals.

NSO.1.6 Given a numerical logarithmic expression, evaluate and generate 6-3, 6-5
equivalent numerical expressions using the properties of logarithms
or exponents.

NSO.1.7 Given an algebraic logarithmic expression, generate an equivalent 6-5

algebraic expression using the properties of logarithms or
exponents.

89
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.1 Interpret and rewrite algebraic expressions and equations in equivalent forms.

AR.1.1 Identify and interpret parts of an equation or expression that 2-3, 5-4, 2-3, 3-1,
represent a quantity in terms of a mathematical or real-world context, 6-7, 7-4, 3-3, 8-3
including viewing one or more of its parts as a single entity. 8-4
Clarifications:
Clarification 1: Parts of an expression include factors, terms,
constants, coefficients and variables.
Clarification 2: Within the Mathematics for Data and Financial
Literacy course, problem types focus on money and business.

AR.1.2 Rearrange equations or formulas to isolate a quantity of interest. 1-3, 2-2,

2-3, 7-3,
Clarifications:
8-2, 8-4
Clarification 1: Instruction includes using formulas for temperature,
perimeter, area and volume; using equations for linear (standard, slope-
intercept and point-slope forms) and quadratic (standard, factored and
vertex forms) functions.
Clarification 2: Within the Mathematics for Data and Financial Literacy
course, problem types focus on money and business.

AR.1.3 Add, subtract and multiply polynomial expressions with rational number 6-1, 6-2, 3-2, 3-3
coefficients. 6-3
Clarifications:
Clarification 1: Instruction includes an understanding that when any
of these operations are performed with polynomials the result is also a
polynomial.
Clarification 2: Within the Algebra 1 course, polynomial expressions are
limited to 3 or fewer terms.

AR.1.4 Divide a polynomial expression by a monomial expression with 6-4

rational number coefficients.
Clarifications:
Clarification 1: Within the Algebra 1 course, polynomial expressions
are limited to 3 or fewer terms.

AR.1.5 Divide polynomial expressions using long division, synthetic division 3-4
or algebraic manipulation.

AR.1.6 Solve mathematical and real-world problems involving addition, 3-2, 3-4
subtraction, multiplication or division of polynomials.

AR.1.7 Rewrite a polynomial expression as a product of polynomials over 6-4, 6-5,

the real number system. 6-6, 6-7
Clarifications:
Clarification 1: Within the Algebra 1 course, polynomial expressions
are limited to 4 or fewer terms with integer coefficients.

90
ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.1 Interpret and rewrite algebraic expressions and equations in equivalent forms. (continued)

AR.1.8 Rewrite a polynomial expression as a product of polynomials over 3-3, 3-4,

the real or complex number system. 3-5, 3-6
Clarifications:
Clarification 1: Instruction includes factoring a sum or difference of
squares and a sum or difference of cubes.

AR.1.9 Apply previous understanding of rational number operations to add, 4-3, 4-4
subtract, multiply and divide rational algebraic expressions.
Clarifications:
Clarification 1: Instruction includes the connection to fractions and
common denominators.

AR.1.11 Apply the Binomial Theorem to create equivalent polynomial 3-3, 8-3
expressions.
Clarifications:
Clarification 1: Instruction includes the connection to Pascal’s
Triangle and to combinations.

MA.912.AR.2 Write, solve and graph linear equations, functions and inequalities in one and two variables.

AR.2.1 Given a real-world context, write and solve one-variable multi-step 1–1, 1–2
linear equations.

AR.2.2 Write a linear two-variable equation to represent the relationship 2–1, 2–2,
between two quantities from a graph, a written description or a 2–3
table of values within a mathematical or real-world context.
Clarifications:
Clarification 1: Instruction includes the use of standard form, slope-
intercept form and point-slope form, and the conversion between
these forms.

AR.2.3 Write a linear two-variable equation for a line that is parallel or 2–4
perpendicular to a given line and goes through a given point.
Clarifications:
Clarification 1: Instruction focuses on recognizing that perpendicular
lines have slopes that when multiplied result in -1 and that parallel
lines have slopes that are the same.
Clarification 2: Instruction includes representing a line with a pair of
points on the coordinate plane or with an equation.
Clarification 3: Problems include cases where one variable has a
coefficient of zero.

91
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.2 Write, solve and graph linear equations, functions and inequalities in one and two
variables. (continued)

AR.2.4 Given a table, equation or written description of a linear function, 2–1, 2–2,
graph that function, and determine and interpret its key features. 2–3, 3–2
Clarifications:
Clarification 1: Key features are limited to domain, range, intercepts
and rate of change.
Clarification 2: Instruction includes the use of standard form, slope-
intercept form and point-slope form.
Clarification 3: Instruction includes cases where one variable has a
coefficient of zero.
Clarification 4: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 5: Within the Algebra 1 course, notations for domain
and range are limited to inequality and set-builder notations.

AR.2.5 Solve and graph mathematical and real-world problems that are 2–2, 2–3,
modeled with linear functions. Interpret key features and determine 3–1, 3–2
constraints in terms of the context.
Algebra 1 Example: Lizzy’s mother uses the function
C(p) = 450 + 7.75p, where C(p) represents the total cost of a rental
space and p is the number of people attending, to help budget
Lizzy’s 16th birthday party. Lizzy’s mom wants to spend no more
than $850 for the party. Graph the function in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain, range, intercepts
and rate of change.
Clarification 2: Instruction includes the use of standard form, slope-
intercept form and point-slope form.
Clarification 3: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.
Clarification 4: Within the Algebra 1 course, notations for domain,
range and constraints are limited to inequality and set-builder.
Clarification 5: Within the Mathematics for Data and Financial
Literacy course, problem types focus on money and business.

AR.2.6 Given a mathematical or real-world context, write and solve one- 1–4, 1–5
variable linear inequalities, including compound inequalities.
Represent solutions algebraically or graphically.
Algebra 1 Example: The compound inequality 2x ≤ 5x + 1 < 4 is
equivalent to −1 ≤ 3x and 5x < 3, which is equivalent to
−1 ≤ x ≤ 3.
3 5

92
ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.2 Write, solve and graph linear equations, functions and inequalities in one and two
variables. (continued)

AR.2.7 Write two-variable linear inequalities to represent relationships 4–4

between quantities from a graph or a written description within a
mathematical or real-world context.
Clarifications:
Clarification 1: Instruction includes the use of standard form, slope-
intercept form and point-slope form and any inequality symbol can
be represented.
Clarification 2: Instruction includes cases where one variable has a
coefficient of zero.

AR2.8 Given a mathematical or real-world context, graph the solution set to 4–4
a two-variable linear inequality.
Clarifications:
Clarification 1: Instruction includes the use of standard form, slope-
intercept form and point-slope form and any inequality symbol can
be represented.
Clarification 2: Instruction includes cases where one variable has a
coefficient of zero.

MA.912.AR.3 Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

AR.3.1 Given a mathematical or real-world context, write and solve one- 8-2, 8-3, 8-4,
variable quadratic equations over the real number system. 8-5
Clarifications:
Clarification 1: Within the Algebra 1 course, instruction includes the
concept of non-real answers, without determining non-real solutions.
Clarification 2: Within this benchmark, the expectation is to solve by
factoring techniques, taking square roots, the quadratic formula and
completing the square.

AR.3.2 Given a mathematical or real-world context, write and solve one- 2–3, 2–4,
variable quadratic equations over the real and complex number 2–5, 2–6
systems.
Clarifications:
Clarification 1: Within this benchmark, the expectation is to solve by
factoring techniques, taking square roots, the quadratic formula and
completing the square.

AR.3.3 Given a mathematical or real-world context, write and solve 1–5

one-variable quadratic inequalities over the real number system.
Represent solutions algebraically or graphically.

93
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.3 Write, solve and graph quadratic equations, functions and inequalities in one and two
variables. (continued)

AR.3.4 Write a quadratic function to represent the relationship between two 7-1, 7-2, 2–1, 2–2,
quantities from a graph, a written description or a table of values 7-3, 7-4 2–3, 2–5
within a mathematical or real-world context.
Algebra I Example: Given the table of values below from a
quadratic function, write an equation of that function.

x −2 −1 0 1 2
f (x) 2 −1 −2 −1 2

Clarifications:
Clarification 1: Within the Algebra 1 course, a graph, written
description or table of values must include the vertex and two points
that are equidistant from the vertex.
Clarification 2: Instruction includes the use of standard form,
factored form and vertex form.
Clarification 3: Within the Algebra 2 course, one of the given points
must be the vertex or an x-intercept.

AR.3.5 Given the x-intercepts and another point on the graph of a quadratic 8-2
function, write the equation for the function.

AR.3.6 Given an expression or equation representing a quadratic function, 8-1, 8-2,

determine the vertex and zeros and interpret them in terms of a real- 8-4, 8-5
world context.

AR.3.7 Given a table, equation or written description of a quadratic 7-1, 7-2,

function, graph that function, and determine and interpret its key 7-3, 8-1,
features. 8-2, 9-3
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; vertex; and symmetry.
Clarification 2: Instruction includes the use of standard form,
factored form and vertex form, and sketching a graph using the
zeros and vertex.
Clarification 3: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 4: Within the Algebra 1 course, notations for domain
and range are limited to inequality and set-builder.

94
ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.3 Write, solve and graph quadratic equations, functions and inequalities in one and two
variables. (continued)

AR.3.8 Solve and graph mathematical and real-world problems that 7-2, 7-4, 1–1, 1–5,
are modeled with quadratic functions. Interpret key features and 8-1, 8-2 2–1, 2–2,
determine constraints in terms of the context. 2–3, 2–5,
2–6
Algebra 1 Example: The value of a classic car produced in 1972
can be modeled by the function V(t) = 19.25t 2 − 440t + 3500,
where t is the number of years since 1972. In what year does the
car’s value start to increase?
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; vertex; and symmetry.
Clarification 2: Instruction includes the use of standard form,
factored form and vertex form.
Clarification 3: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.
Clarification 4: Within the Algebra 1 course, notations for domain,
range and constraints are limited to inequality and set-builder.

AR.3.9 Given a mathematical or real-world context, write two-variable 2-7

quadratic inequalities to represent relationships between quantities
from a graph or a written description.
Clarifications:
Clarification 1: Instruction includes the use of standard form,
factored form and vertex form where any inequality symbol can be
represented.

AR.3.10 Given a mathematical or real-world context, graph the solution set to 2-7
a two-variable quadratic inequality.
Clarifications:
Clarification 1: Instruction includes the use of standard form,
factored form and vertex form where any inequality symbol can be
represented.

95
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.4 Write, solve and graph absolute value equations, functions and inequalities in one and two
variables.

AR.4.1 Given a mathematical or real-world context, write and solve one- 1-6
variable absolute value equations.

AR.4.2 Given a mathematical or real-world context, write and solve 1-6 1-5
one-variable absolute value inequalities. Represent solutions
algebraically or graphically.

AR.4.3 Given a table, equation or written description of an absolute value 3-4, 3-5,
function, graph that function and determine its key features. 9-3
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; vertex; end behavior and symmetry.
Clarification 2: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 3: Within the Algebra 1 course, notations for domain
and range are limited to inequality and set-builder.

AR.4.4 Solve and graph mathematical and real-world problems that are 1–1, 1–2
modeled with absolute value functions. Interpret key features and
determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; vertex; end behavior and symmetry.
Clarification 2: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.

MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two
variables.

AR.5.2 Solve one-variable equations involving logarithms or exponential 6–3, 6–5,

expressions. Interpret solutions as viable in terms of the context and 6–6
identify any extraneous solutions.

AR.5.3 Given a mathematical or real-world context, classify an exponential 5-4

function as representing growth or decay.
Clarifications:
Clarification 1: Within the Algebra 1 course, exponential functions
are limited to the forms f (x) = abx, where b is a whole number
greater than 1 or a unit fraction, or f (x) = a(1 ± r)x, where 0 < r < 1.

96
ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two
variables. (continued)

AR.5.4 Write an exponential function to represent a relationship between 5-3, 5-4 6–1, 6–2
two quantities from a graph, a written description or a table of
values within a mathematical or real-world context.
Clarifications:
Clarification 1: Within the Algebra 1 course, exponential functions
are limited to the forms f (x) = abx, where b is a whole number
greater than 1 or a unit fraction, or f (x) = a(1 ± r)x, where 0 < r < 1.
Clarification 2: Within the Algebra 1 course, tables are limited to
having successive nonnegative integer inputs so that the function
may be determined by finding ratios between successive outputs.

AR.5.5 Given an expression or equation representing an exponential 6–1, 6–2

function, reveal the constant percent rate of change per unit interval
using the properties of exponents. Interpret the constant percent rate
of change in terms of a real-world context.

AR.5.6 Given a table, equation or written description of an exponential 5-3, 9-3

function, graph that function and determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive
or negative; constant percent rate of change; end behavior and
asymptotes.
Clarification 2: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 3: Within the Algebra 1 course, notations for domain
and range are limited to inequality and set-builder.
Clarification 4: Within the Algebra 1 course, exponential functions
are limited to the forms f (x) = abx, where b is a whole number
greater than 1 or a unit fraction, or f (x) = a(1 ± r)x, where 0 < r < 1.

AR.5.7 Given a mathematical or real-world context, write and solve one- 6–1, 6–2,
variable quadratic equations over the real and complex number 6–3, 6–4
systems.
Clarifications:
Clarification 1: Within this benchmark, the expectation is to solve by
factoring techniques, taking square roots, the quadratic formula and
completing the square.

AR.5.8 Given a table, equation or written description of a logarithmic 6–4

function, graph that function and determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain and
range inequality notation, interval notation or set-builder notation.

97
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.5 Write, solve and graph exponential and logarithmic equations and functions in one and two
variables. (continued)

AR.5.9 Solve and graph mathematical and real-world problems that are 6–4
modeled with logarithmic functions. Interpret key features and
determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.

MA.912.AR.6 Solve and graph polynomial equations and functions in one and two variables.

AR.6.1 Given a mathematical or real-world context, when suitable 3–5, 3–6

factorization is possible, solve one-variable polynomial equations of
degree 3 or higher over the real and complex number systems.

AR.6.2 Explain and apply the Remainder Theorem to solve mathematical 3–4
and real-world problems.

AR.6.5 Sketch a rough graph of a polynomial function of degree 3 or higher 3–1, 3–5
using zeros, multiplicity and knowledge of end behavior.

MA.912.AR.7 Solve and graph radical equations and functions in one and two variables.

AR.7.1 Solve one-variable radical equations. Interpret solutions as viable in 5–4

terms of context and identify any extraneous solutions.

AR.7.2 Given a table, equation or written description of a square root 5–3

or cube root function, graph that function and determine its key
features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and relative maximums and minimums.
Clarification 2: Instruction includes representing the domain and
range inequality notation, interval notation or set-builder notation.

AR.7.3 Solve and graph mathematical and real-world problems that are 5–3, 5–4
modeled with square root or cube root functions. Interpret key
features and determine constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and relative maximums and minimums.
Clarification 2: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.

98
ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.8 Solve and graph radical equations and functions in one and two variables.

AR.8.1 Write and solve one-variable rational equations. Interpret solutions 4–5
as viable in terms of the context and identify any extraneous
solutions.
Clarifications:
Clarification 1: Within the Algebra 2 course, numerators and
denominators are limited to linear and quadratic expressions.

AR.8.2 Given a table, equation or written description of a rational function, 4–1, 4–2
graph that function and determine its key features.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 3: Within the Algebra 2 course, numerators and
denominators are limited to linear and quadratic expressions.

AR.8.3 Solve and graph mathematical and real-world problems that are 4–1, 4–2,
modeled with rational functions. Interpret key features and determine 4–5
constraints in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.
Clarification 3: Instruction includes using rational functions to
represent inverse proportional relationships.
Clarification 4: Within the Algebra 2 course, numerators and
denominators are limited to linear and quadratic expressions.

MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe
quantities or relationships.

AR.9.1 Given a mathematical or real-world context, write and solve a 4–1, 4–2,
system of two-variable linear equations algebraically or graphically. 4–3
Clarifications:
Clarification 1: Within this benchmark, the expectation is to solve
systems using elimination, substitution and graphing.
Clarification 2: Within the Algebra 1 course, the system is limited to
two equations.

AR.9.2 Given a mathematical or real-world context, solve a system 1–6, 2–8

consisting of a two-variable linear equation and a non-linear
equation algebraically or graphically.

99
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe
quantities or relationships. (continued)

AR.9.3 Given a mathematical or real-world context, solve a system 1–6, 2–8

consisting of two-variable linear or non-linear equations
algebraically or graphically.
Clarifications:
Clarification 1: Within the Algebra 2 course, non-linear equations
are limited to quadratic equations.

AR.9.4 Graph the solution set of a system of two-variable linear inequalities. 4–5
Clarifications:
Clarification 1: Instruction includes cases where one variable has a
coefficient of zero.
Clarification 2: Within the Algebra 1 course, the system is limited to
two inequalities.

AR.9.5 Graph the solution set of a system of two-variable inequalities. 1–6, 2–8
Clarifications:
Clarification 1: Within the Algebra 2 course, two-variable
inequalities are limited to linear and quadratic.

AR.9.6 Given a real-world context, represent constraints as systems of linear 4–3, 4–5
equations or inequalities. Interpret solutions to problems as viable or
non-viable options.
Clarifications:
Clarification 1: Instruction focuses on analyzing a given function
that models a real-world situation and writing constraints that are
represented as linear equations or linear inequalities.

AR.9.7 Given a real-world context, represent constraints as systems of 1–6, 2–8

linear and non-linear equations or inequalities. Interpret solutions to
problems as viable or non-viable options.
Clarifications:
Clarification 1: Instruction focuses on analyzing a given function
that models a real-world situation and writing constraints that are
represented as non-linear equations or non-linear inequalities.
Clarification 2: Within the Algebra 2 course, non-linear equations
and inequalities are limited to quadratic.

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ALGEBRAIC REASONING Algebra 1 Geometry Algebra 2

MA.912.AR.9 Write and solve a system of two- and three-variable equations and inequalities that describe
quantities or relationships. (continued)

AR.9.10 Solve and graph mathematical and real-world problems that are 1–3
modeled with piecewise functions. Interpret key features and
determine constraints in terms of the context.
Example: A mechanic wants to place an ad in his local newspaper.
The cost, in dollars, of an ad x inches long is given by the following
piecewise function. Find the cost of an ad that would be 16 inches long.

Cx = { 12x, x < 5
60 + 8x − 5, x ≤ 5
Clarifications:
Clarification 1: Key features are limited to domain, range, intercepts,
asymptotes and end behavior.
Clarification 2: Instruction includes representing the domain, range
and constraints with inequality notation, interval notation or set-
builder notation.

MA.912.AR.10 Solve problems involving sequences and series.

AR.10.1 Given a mathematical or real-world context, write and solve 1–4

problems involving arithmetic sequences.
Example: Tara is saving money to move out of her parent’s house.
She opens the account with $250 and puts $100 into a savings
account every month after that. Write the total amount of money she
has in her account after each month as a sequence. In how many
months will she have at least $3,000?

AR.10.2 Given a mathematical or real-world context, write and solve 6–7

problems involving geometric sequences.
Example: A bacteria in a Petri dish initially covers 2 square
centimeters. The bacteria grows at a rate of 2.6% every day.
Determine the geometric sequence that describes the area covered
by the bacteria after 0, 1, 2, 3… days. Determine using technology,
how many days it would take the bacteria to cover 10 square
centimeters.

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FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

FUNCTIONS Algebra 1 Geometry Algebra 2

MA.912.F.1 Understand, compare and analyze properties of functions.

F.1.1 Given an equation or graph that defines a function, determine the 5-3, 7-5, 1–1, 1–2,
function type. Given an input-output table, determine a function type 9-1, 9-2, 2–1, 2–2,
that could represent it. 9-3 3–5, 4–1,
6–1, 6–2
Clarifications:
Clarification 1: Within the Algebra 1 course, functions represented
as tables are limited to linear, quadratic and exponential.
Clarification 2: Within the Algebra 1 course, functions represented
as equations or graphs are limited to vertical or horizontal
translations or reflections over the x-axis of the following parent
functions:
f (x) = x, f (x) = x2, f (x) = x3, f (x) = √x, f (x) = ∛x , f (x) = |x|,
f (x) = 2x and f (x) = ( 1 )x.
2

F.1.2 Given a function represented in function notation, evaluate the 3-2, 3-4,
function for an input in its domain. For a real-world context, interpret 5-3, 5-4,
the output. 7-1, 7-4,
9-1, 9-2
Algebra 1 Example: The function f (x) = x − 8 models Alicia’s
2
position in miles relative to a water stand x minutes into a marathon.
Evaluate and interpret for a quarter of an hour into the race.
Clarifications:
Clarification 1: Problems include simple functions in two-variables,
such as f (x, y) = 3x − 2y.
Clarification 2: Within the Algebra 1 course, functions are limited to
one-variable such as f (x) = 3x.

F.1.3 Calculate and interpret the average rate of change of a real-world 5-4, 7-1,
situation represented graphically, algebraically or in a table over a 9-1
specified interval.
Clarifications:
Clarification 1: Instruction includes making the connection to
determining the slope of a particular line segment.

F.1.5 Compare key features of linear functions each represented 3-2, 3-3,
algebraically, graphically, in tables or written descriptions. 9-3
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
slope and end behavior.

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FUNCTIONS Algebra 1 Geometry Algebra 2

MA.912.F.1 Understand, compare and analyze properties of functions. (continued)

F.1.6 Compare key features of linear and nonlinear functions each 3-4, 7-5,
represented algebraically, graphically, in tables or written 9-3
descriptions.
Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior and asymptotes.
Clarification 2: Within the Algebra 1 course, functions other than
linear, quadratic or exponential must be represented graphically.
Clarification 3: Within the Algebra 1 course, instruction includes
verifying that a quantity increasing exponentially eventually exceeds
a quantity increasing linearly or quadratically.

F.1.7 Compare key features of two functions each represented 1–2, 2–1,
algebraically, graphically, in tables or written descriptions. 3–2, 5–3,
6–1, 6–4
Clarifications:
Clarification 1: Key features include domain; range; intercepts;
intervals where the function is increasing, decreasing, positive or
negative; end behavior and asymptotes.

F.1.8 Determine whether a linear, quadratic or exponential function best 5-3, 7-5
models a given real-world situation.
Clarifications:
Clarification 1: Instruction includes recognizing that linear functions
model situations in which a quantity changes by a constant amount
per unit interval; that quadratic functions model situations in which
a quantity increases to a maximum, then begins to decrease or a
quantity decreases to a minimum, then begins to increase; and that
exponential functions model situations in which a quantity grows or
decays by a constant percent per unit interval.
Clarification 2: Within this benchmark, the expectation is to identify
the type of function from a written description or table.

F.1.9 Determine whether a function is even, odd or neither when 5–7

represented algebraically, graphically or in a table.

MA.912.F.2 Identify and describe the effects of transformations on functions. Create new functions given
transformations.

F.2.1 Identify the effect on the graph or table of a given function after 3-3, 3-4,
replacing f (x) by f (x) + k, kf (x), f (kx) and f (x + k) for specific values 3-5, 7-1,
of k. 7-2
Clarifications:
Clarification 1: Within the Algebra 1 course, functions are limited to
linear, quadratic and absolute value.
Clarification 2: Instruction focuses on including positive and
negative values for k.

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FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

FUNCTIONS Algebra 1 Geometry Algebra 2

MA.912.F.2 Identify and describe the effects of transformations on functions. Create new functions given
transformations. (continued)

F.2.2 Identify the effect on the graph of a given function of two or more 1–2, 2–1,
transformations defined by adding a real number to the x- or 3–7, 4–1,
y-values or multiplying the x- or y-values by a real number. 5–3, 6–4

F.2.3 Given the graph or table of f (x) and the graph or table of f (x) + k, 9-3 1–2, 2–1,
kf (x), f (kx) and f (x + k), state the type of transformation and find the 3–7, 5–3,
value of the real number k. 6–4

F.2.5 Given a table, equation or graph that represents a function, create a 1–2, 2–1,
corresponding table, equation or graph of the transformed function 3–7, 5–3,
defined by adding a real number to the x- or y-values or multiplying 6–4
the x- or y-values by a real number.

MA.912.F.3 Create new functions from existing functions.

F.3.1 Given a mathematical or real-world context, combine two functions, 9-4

limited to linear and quadratic, using arithmetic operations. When
appropriate, include domain restrictions for the new function.
Example: The quotient of the functions f (x) = 3x2 − 7x + 3 and
2
g(x) = 6x – 1 can be expressed as h(x) = (3x − 7x + 2, where the
(6x − 1)
domain of h(x) is −∞ ≤ x ≤ 1 and 1 ≤ x ≤ ∞.
6 6
Clarifications:
Clarification 1: Instruction includes representing domain restrictions
with inequality notation, interval notation or set-builder notation.
Clarification 2: Within the Algebra 1 Honors course, notations for
domain and range are limited to inequality and set-builder.

F.3.2 Given a mathematical or real-world context, combine two or more 3–2, 5–5
functions, limited to linear, quadratic, exponential and polynomial,
using arithmetic operations. When appropriate, include domain
restrictions for the new function.
Clarifications:
Clarification 1: Instruction includes representing domain restrictions
with inequality notation, interval notation or set-builder notation.
Clarification 2: Within the Mathematics for Data and Financial
Literacy course, problem types focus on money and business.

F.3.4 Represent the composition of two functions algebraically or in a 5–5, 5–6

table. Determine the domain and range of the composite function.

F.3.6 Determine whether an inverse function exists by analyzing tables, 5–6

graphs and equations.

F.3.7 Represent the inverse of a function algebraically, graphically or in a 5–6, 6–3,

table. Use composition of functions to verify that one function is the 6–4
inverse of the other.
Clarifications:
Clarification 1: Instruction includes the understanding that a
logarithmic function is the inverse of an exponential function.

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FINANCIAL LITERACY Algebra 1 Geometry Algebra 2

MA.912.FL.1 Build mathematical foundations for financial literacy

FL.3.1 Compare simple, compound and continuously compounded interest 6–2

over time.
Clarifications:
Clarification 1: Instruction includes taking into consideration
the annual percentage rate (APR) when comparing simple and
compound interest.

FL.3.2 Solve real-world problems involving simple, compound and 3-2, 5-4 6–2
continuously compounded interest.
Example: Find the amount of money on deposit at the end of 5 years
if you started with $500 and it was compounded quarterly at 6%
interest per year.
Example: Joe won $25,000 on a lottery scratch-off ticket. How
many years will it take at 6% interest compounded yearly for his
money to double?
Clarifications:
Clarification 1: Within the Algebra 1 course, interest is limited to
simple and compound.

FL.3.4 Explain the relationship between simple interest and linear growth. 3-2, 5-4 6–2
Explain the relationship between compound interest and exponential
growth and the relationship between continuously compounded
interest and exponential growth.
Clarifications:
Clarification 1: Within the Algebra 1 course, exponential growth is
limited to compound interest.

105
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.1 Prove and apply geometric theorems to solve problems.

GR.1.1 Prove relationships and theorems about lines and angles. Solve 1-1, 1-6,
mathematical and real-world problems involving postulates, 2-1, 2-2,
relationships and theorems of lines and angles. 5-1
Clarifications:
Clarification 1: Postulates, relationships and theorems include
vertical angles are congruent; when a transversal crosses parallel
lines, the consecutive angles are supplementary and alternate
(interior and exterior) angles and corresponding angles are
congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
Clarification 2: Instruction includes constructing two-column proofs,
pictorial proofs, paragraph and narrative proofs, flow chart proofs
or informal proofs.
Clarification 3: Instruction focuses on helping a student choose a
method they can use reliably.

GR.1.2 Prove triangle congruence or similarity using Side-Side-Side, Side- 4-3, 4-4,
Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and 4-5, 4-6,
Hypotenuse-Leg. 7-3, 7-4,
7-5
Clarifications:
Clarification 1: Instruction includes constructing two-column proofs,
pictorial proofs, paragraph and narrative proofs, flow chart proofs
or informal proofs.
Clarification 2: Instruction focuses on helping a student choose a
method they can use reliably.

GR.1.3 Prove relationships and theorems about triangles. Solve 2-3, 4-2,
mathematical and real-world problems involving postulates, 5-2, 5-3,
relationships and theorems of triangles. 5-4, 5-5,
7-4, 7-5,
Clarifications:
8-1, 9-2
Clarification 1: Postulates, relationships and theorems include
measures of interior angles of a triangle sum to 180°; measures of
a set of exterior angles of a triangle sum to 360°; triangle inequality
theorem; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet
at a point.
Clarification 2: Instruction includes constructing two-column proofs,
pictorial proofs, paragraph and narrative proofs, flow chart proofs
or informal proofs.
Clarification 3: Instruction focuses on helping a student choose a
method they can use reliably.

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GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.1 Prove and apply geometric theorems to solve problems. (continued)

GR.1.4 Prove relationships and theorems about parallelograms. Solve 6-2, 6-3,
mathematical and real-world problems involving postulates, 6-4, 6-5
relationships and theorems of parallelograms.
Clarifications:
Clarification 1: Postulates, relationships and theorems include
opposite sides are congruent, consecutive angles are supplementary,
opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and rectangles are parallelograms with congruent
diagonals.
Clarification 2: Instruction includes constructing two-column proofs,
pictorial proofs, paragraph and narrative proofs, flow chart proofs
or informal proofs.
Clarification 3: Instruction focuses on helping a student choose a
method they can use reliably.

GR.1.5 Prove relationships and theorems about trapezoids. Solve 6-1

mathematical and real-world problems involving postulates,
relationships and theorems of trapezoids.
Clarifications:
Clarification 1: Postulates, relationships and theorems include the
Trapezoid Midsegment Theorem and for isosceles trapezoids: base
angles are congruent, opposite angles are supplementary and
diagonals are congruent.
Clarification 2: Instruction includes constructing two-column proofs,
pictorial proofs, paragraph and narrative proofs, flow chart proofs
or informal proofs.
Clarification 3: Instruction focuses on helping a student choose a
method they can use reliably.

GR.1.6 Solve mathematical and real-world problems involving congruence 4-4, 4-5,
or similarity in two-dimensional figures. 4-6, 7-2,
7-3
Clarifications:
Clarification 1: Instruction includes demonstrating that two-
dimensional figures are congruent or similar based on given
information.

107
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.2 Apply properties of transformations to describe congruence or similarity.

GR.2.1 Given a preimage and image, describe the transformation and 3-1, 3-2,
represent the transformation algebraically using coordinates. 3-3, 7-1
Example: Given a triangle whose vertices have the coordinates
(−3, 4), (2, 1.7) and (−0.4, −3). If this triangle is reflected across
the y-axis the transformation can be described using coordinates
as (x, y) → (−x, y) resulting in the image whose vertices have the
coordinates (3, 4), (−2, 1.7) and (0.4, −3).
Clarifications:
Clarification 1: Instruction includes the connection of transformations
to functions that take points in the plane as inputs and give other
points in the plane as outputs.
Clarification 2: Transformations include translations, dilations,
rotations and reflections described using words or using coordinates.
Clarification 3: Within the Geometry course, rotations are limited to
90°, 180° and 270° counterclockwise or clockwise about the center
of rotation, and the centers of rotations and dilations are limited to
the origin or a point on the figure.

GR.2.2 Identify transformations that do or do not preserve distance. 3-1, 3-2,

3-3, 3-4,
Clarifications:
7-1
Clarification 1: Transformations include translations, dilations,
rotations and reflections described using words or using coordinates.
Clarification 2: Instruction includes recognizing that these
transformations preserve angle measure.

GR.2.3 Identify a sequence of transformations that will map a given figure 3-1, 3-2,
onto itself or onto another congruent or similar figure. 3-3, 3-4,
3-5, 4-1,
Clarifications:
7-2
Clarification 1: Transformations include translations, dilations,
rotations and reflections described using words or using coordinates.
Clarification 2: Within the Geometry course, figures are limited to
triangles and quadrilaterals and rotations are limited to 90°, 180°
and 270° counterclockwise or clockwise about the center of rotation.
Clarification 3: Instruction includes the understanding that when a
figure is mapped onto itself using a reflection, it occurs over a line of
symmetry.

GR.2.4 Determine symmetries of reflection, symmetries of rotation and 3-5

symmetries of translation of a geometric figure.
Clarifications:
Clarification 1: Instruction includes determining the order of each
symmetry.
Clarification 2: Instruction includes the connection between
tessellations of the plane and symmetries of translations.

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GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.2 Apply properties of transformations to describe congruence or similarity. (continued)

GR.2.5 Given a geometric figure and a sequence of transformations, draw 3-1, 3-2,
the transformed figure on a coordinate plane. 3-3, 7-1,
7-2
Clarifications:
Clarification 1: Transformations include translations, dilations,
rotations and reflections described using words or using coordinates.
Clarification 2: Instruction includes two or more transformations.

GR.2.6 Apply rigid transformations to map one figure onto another to justify 4-1
that the two figures are congruent.
Clarifications:
Clarification 1: Instruction includes showing that the corresponding
sides and the corresponding angles are congruent.

GR.2.7 Justify the criteria for triangle congruence using the definition of 4-3, 4-4
congruence in terms of rigid transformations.

GR.2.8 Apply an appropriate transformation to map one figure onto another 7-2, 7-3
to justify that the two figures are similar.
Clarifications:
Clarification 1: Instruction includes showing that the corresponding
sides are proportional, and the corresponding angles are congruent.

GR.2.9 Justify the criteria for triangle similarity using the definition of 7-3
similarity in terms of non-rigid transformations.

MA.912.GR.3 Use coordinate geometry to solve problems or prove relationships.

GR.3.1 Determine the weighted average of two or more points on a line. 1–3
Clarifications:
Clarification 1: Instruction includes using a number line and
determining how changing the weights moves the weighted average
of points on the number line.

GR.3.2 Given a mathematical context, use coordinate geometry to classify 9–1, 9–2,
or justify definitions, properties and theorems involving circles, 9–3
triangles or quadrilaterals.
Example: Given Triangle ABC has vertices located at (−2, 2), (3, 3)
and (1, −3), respectively, classify the type of triangle ABC is.
Example: If a square has a diagonal with vertices (−1, 1) and
(−4, −3), find the coordinate values of the vertices of the other
diagonal and show that the two diagonals are perpendicular.
Clarifications:
Clarification 1: Instruction includes using the distance or midpoint
formulas and knowledge of slope to classify or justify definitions,
properties and theorems.

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FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.3 Use coordinate geometry to solve problems or prove relationships. (continued)

GR.3.3 Use coordinate geometry to solve mathematical and real-world 1–3, 2–4,
geometric problems involving lines, circles, triangles and quadrilaterals. 5–3, 9–1,
9–2, 10–2
Example: The line x + 2y = 10 is tangent to a circle whose center
is located at (2, −1). Find the tangent point and a second tangent
point of a line with the same slope as the given line.
Example: Given M(−4, 7) and N(12, −1), find the coordinates of
point P on MN so that P partitions MN in the ratio 2: 3.
Clarifications:
Clarification 1: Problems involving lines include the coordinates of a
point on a line segment including the midpoint.
Clarification 2: Problems involving circles include determining points
on a given circle and finding tangent lines.
Clarification 3: Problems involving triangles include median and
centroid.
Clarification 4: Problems involving quadrilaterals include using
parallel and perpendicular slope criteria.

GR.3.4 Use coordinate geometry to solve mathematical and real-world 9–1

problems on the coordinate plane involving perimeter or area of
polygons.
Example: A new community garden has four corners. Starting at the
first corner and working counterclockwise, the second corner is 200
feet east, the third corner is 150 feet north of the second corner and
the fourth corner is 100 feet west of the third corner. Represent the
garden in the coordinate plane, and determine how much fence is
needed for the perimeter of the garden and determine the total area
of the garden.

MA.912.GR.4 Use geometric measurement and dimensions to solve problems.

GR.4.1 Identify the shapes of two-dimensional cross-sections of three- 11-1

dimensional figures.
Clarifications:
Clarification 1: Instruction includes the use of manipulatives and
models to visualize cross-sections.
Clarification 2: Instruction focuses on cross-sections of right
cylinders, right prisms, right pyramids and right cones that are
parallel or perpendicular to the base.

GR.4.2 Identify three-dimensional objects generated by rotations of two- 11-1

dimensional figures.
Clarifications:
Clarification 1: The axis of rotation must be within the same plane
but outside of the given two-dimensional figure.

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GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.4 Use geometric measurement and dimensions to solve problems. (continued)

GR.4.3 Extend previous understanding of scale drawings and scale factors 7-1, 11-2,
to determine how dilations affect the area of two-dimensional figures 11-3
and the surface area or volume of three-dimensional figures.
Example: Mike is having a graduation party and wants to make sure
he has enough pizza. Which option would provide more pizza for
his guests: one 12-inch pizza or three 6-inch pizzas?

GR.4.4 Solve mathematical and real-world problems involving the area of 7-1, 9-1,
two-dimensional figures. 11-3, 11-4
Example: A town has 23 city blocks, each of which has dimensions
of 1 quarter mile by 1 quarter mile, and there are 4500 people in
the town. What is the population density of the town?
Clarifications:
Clarification 1: Instruction includes concepts of population density
based on area.

GR.4.5 Solve mathematical and real-world problems involving the volume 11-3, 11-4,
of three-dimensional figures limited to cylinders, pyramids, prisms, 11-5
cones and spheres.
Example: A cylindrical swimming pool is filled with water and has
a diameter of 10 feet and height of 4 feet. If water weighs 62.4
pounds per cubic foot, what is the total weight of the water in a full
tank to the nearest pound?
Clarifications:
Clarification 1: Instruction includes concepts of density based on
volume.
Clarification 2: Instruction includes using Cavalieri’s Principle to give
informal arguments about the formulas for the volumes of right and
non-right cylinders, pyramids, prisms and cones.

GR.4.6 Solve mathematical and real-world problems involving the surface 11-2
area of three-dimensional figures limited to cylinders, pyramids,
prisms, cones and spheres.

MA.912.GR.5 Make formal geometric constructions with a variety of tools and methods.

GR.5.1 Construct a copy of a segment or an angle. 1-2

Clarifications:
Clarification 1: Instruction includes using compass and straightedge,
string, reflective devices, paper folding or dynamic geometric
software.

GR.5.2 Construct the bisector of a segment or an angle, including the 1-2, 5-1
perpendicular bisector of a line segment.
Clarifications:
Clarification 1: Instruction includes using compass and straightedge,
string, reflective devices, paper folding or dynamic geometric
software.

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FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.5 Make formal geometric constructions with a variety of tools and methods. (continued)

GR.5.3 Construct the inscribed and circumscribed circles of a triangle. 5-2

Clarifications:
Clarification 1: Instruction includes using compass and straightedge,
string, reflective devices, paper folding or dynamic geometric
software.

GR.5.4 Construct a regular polygon inscribed in a circle. Regular polygons 10-3

are limited to triangles, quadrilaterals and hexagons.
Clarifications:
Clarification 1: When given a circle, the center must be provided.
Clarification 2: Instruction includes using compass and straightedge,
string, reflective devices, paper folding or dynamic geometric
software.

GR.5.5 Given a point outside a circle, construct a line tangent to the circle 10-2
that passes through the given point.
Clarifications:
Clarification 1: When given a circle, the center must be provided.
Clarification 2: Instruction includes using compass and straightedge,
string, reflective devices, paper folding or dynamic geometric
software.

MA.912.GR.6 Use properties and theorems related to circles.

GR.6.1 Solve mathematical and real-world problems involving the length of 10-2, 10-3,
a secant, tangent, segment or chord in a given circle. 10-4, 10-5
Clarifications:
Clarification 1: Problems include relationships between two chords;
two secants; a secant and a tangent; and the length of the tangent
from a point to a circle.

GR.6.2 Solve mathematical and real-world problems involving the measures 10-1, 10-4,
of arcs and related angles. 10-5
Clarifications:
Clarification 1: Within the Geometry course, problems are limited
to relationships between inscribed angles; central angles; and
angles formed by the following intersections: a tangent and a secant
through the center, two tangents, and a chord and its perpendicular
bisector.

GR.6.3 Solve mathematical problems involving triangles and quadrilaterals 5-2, 10-4
inscribed in a circle.
Clarifications:
Clarification 1: Instruction includes cases in which a triangle
inscribed in a circle has a side that is the diameter.

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GEOMETRY REASONING Algebra 1 Geometry Algebra 2

MA.912.GR.6 Use properties and theorems related to circles. (continued)

GR.6.4 Solve mathematical and real-world problems involving the arc length 10-1
and area of a sector in a given circle.
Clarifications:
Clarification 1: Instruction focuses on the conceptual understanding
that for a given angle measure the length of the intercepted arc is
proportional to the radius, and for a given radius the length of the
intercepted arc is proportional is the angle measure.

GR.6.5 Apply transformations to prove that all circles are similar. 7-2

MA.912.GR.7 Apply geometric and algebraic representations of conic sections.

GR.7.2 Given a mathematical or real-world context, derive and create the 9-3
equation of a circle using key features.
Clarifications:
Clarification 1: Instruction includes using the Pythagorean Theorem
and completing the square.
Clarification 2: Within the Geometry course, key features are limited
to the radius, diameter and the center.

GR.7.3 Graph and solve mathematical and real-world problems that are 9-3
modeled with an equation of a circle. Determine and interpret key
features in terms of the context.
Clarifications:
Clarification 1: Key features are limited to domain, range,
eccentricity, center and radius.
Clarification 2: Instruction includes representing the domain and
range with inequality notation, interval notation or set-builder
notation.
Clarification 3: Within the Geometry course, notations for domain
and range are limited to inequality and set-builder.

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FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

TRIGONOMETRY Algebra 1 Geometry Algebra 2

MA.912.T.1 Define and use trigonometric ratios, identities or functions to solve problems.

T.1.1 Define trigonometric ratios for acute angles in right triangles. 8-2
Clarifications:
Clarification 1: Instruction includes using the Pythagorean Theorem
and using similar triangles to demonstrate that trigonometric ratios
stay the same for similar right triangles.
Clarification 2: Within the Geometry course, instruction includes
using the coordinate plane to make connections to the unit circle.
Clarification 3: Within the Geometry course, trigonometric ratios are
limited to sine, cosine and tangent.

T.1.2 Solve mathematical and real-world problems involving right triangles 8-1, 8-2,
using trigonometric ratios and the Pythagorean Theorem. 8-3
Clarifications:
Clarification 1: Instruction includes procedural fluency with the
relationships of side lengths in special right triangles having angle
measures of 30°-60°-90° and 45°-45°-90°.

T.1.3 Apply the Law of Sines and the Law of Cosines to solve 8-4, 8-5
mathematical and real-world problems involving triangles.

T.1.4 Solve mathematical problems involving finding the area of a triangle 8-3
given two sides and the included angle.
Clarifications:
Clarification 1: Problems include right triangles, heights inside of a
triangle and heights outside of a triangle.

114
DATA ANALYSIS AND PROBABILITY Algebra 1 Geometry Algebra 2

MA.912.DP.1 Summarize, represent and interpret categorical and numerical data with one and two variables.

DP.1.1 Given a set of data, select an appropriate method to represent the 10-1, 10-2,
data, depending on whether it is numerical or categorical data and 10-3
on whether it is univariate or bivariate.
Clarifications:
Clarification 1: Instruction includes discussions regarding the
strengths and weaknesses of each data display.
Clarification 2: Numerical univariate includes histograms, stem-
and-leaf plots, box plots and line plots; numerical bivariate includes
scatter plots and line graphs; categorical univariate includes bar
charts, circle graphs, line plots, frequency tables and relative
frequency tables; and categorical bivariate includes segmented bar
charts, joint frequency tables and joint relative frequency tables.
Clarification 3: Instruction includes the use of appropriate units and
labels and, where appropriate, using technology to create data
displays.

DP.1.2 Interpret data distributions represented in various ways. State 10-1, 10-2,
whether the data is numerical or categorical, whether it is univariate 10-3
or bivariate and interpret the different components and quantities in
the display.
Clarifications:
Clarification 1: Within the Probability and Statistics course,
instruction includes the use of spreadsheets and technology.

DP.1.3 Explain the difference between correlation and causation in the 10-4, 10-5
contexts of both numerical and categorical data.
Algebra 1 Example: There is a strong positive correlation between
the number of Nobel prizes won by country and the per capita
chocolate consumption by country. Does this mean that increased
chocolate consumption in America will increase the United States of
America’s chances of a Nobel prize winner?

DP.1.4 Estimate a population total, mean or percentage using data from 10-1
a sample survey; develop a margin of error through the use of
simulation.
Algebra 1 Example: Based on a survey of 100 households in Twin
Lakes, the newspaper reports that the average number of televisions
per household is 3.5 with a margin of error of ±0.6. The actual
population mean can be estimated to be between 2.9 and 4.1
television per household. Since there are 5,500 households in Twin
Lakes the estimated number of televisions is between 15,950 and
22,550.
Clarifications:
Clarification 1: Within the Algebra 1 course, the margin of error will
be given.

115
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

DATA ANALYSIS AND PROBABILITY Algebra 1 Geometry Algebra 2

MA.912.DP.2 Solve problems involving univariate and bivariate numerical data.

D.P.2.4 Fit a linear function to bivariate numerical data that suggests a linear 10-4
association and interpret the slope and y-intercept of the model. Use
the model to solve real-world problems in terms of the context of
the data.
Clarifications:
Clarification 1: Instruction includes fitting a linear function both
informally and formally with the use of technology.
Clarification 2: Problems include making a prediction or
extrapolation, inside and outside the range of the data, based on
the equation of the line of fit.

DP.2.5 Given a scatter plot that represents bivariate numerical data, assess 10-4
the fit of a given linear function by plotting and analyzing residuals.
Clarifications:
Clarification 1: Within the Algebra 1 course, instruction includes
determining the number of positive and negative residuals; the
largest and smallest residuals; and the connection between outliers
in the data set and the corresponding residuals.

DP.2.6 Given a scatter plot with a line of fit and residuals, determine the 10-4
strength and direction of the correlation. Interpret strength and
direction within a real-world context.
Clarifications:
Clarification 1: Instruction focuses on determining the direction by
analyzing the slope and informally determining the strength by
analyzing the residuals.

DP.2.8 Fit a quadratic function to bivariate numerical data that suggests a 2–2
quadratic association and interpret any intercepts or the vertex of
the model. Use the model to solve real-world problems in terms of
the context of the data.
Clarifications:
ˆProblems include making a prediction or extrapolation, inside and
outside the range of the data, based on the equation of the line of fit.

DP.2.9 Fit an exponential function to bivariate numerical data that suggests 6–4
an exponential association. Use the model to solve real-world
problems in terms of the context of the data.
Clarifications:
Clarification 1: Instruction focuses on determining whether an
exponential model is appropriate by taking the logarithm of the
dependent variable using spreadsheets and other technology.
Clarification 2: Instruction includes determining whether the
transformed scatterplot has an appropriate line of best fit, and
interpreting the y-intercept and slope of the line of best fit.
Clarification 3: Problems include making a prediction or
extrapolation, inside and outside the range of the data, based on
the equation of the line of fit.

116
DATA ANALYSIS AND PROBABILITY Algebra 1 Geometry Algebra 2

MA.912.DP.3 Solve problems involving categorical data.

DP.3.1 Construct a two-way frequency table summarizing bivariate 10-3, 10-5

categorical data. Interpret joint and marginal frequencies and
determine possible associations in terms of a real-world context.
Algebra 1 Example: Complete the frequency table below.

Has an A Doesn’t have Total

in math an A in math
Plays an
20 90
instrument
Doesn’t play
20
an instrument
Total 350

Using the information in the table, it is possible to determine that the

second column contains the numbers 70 and 240. This means that
there are 70 students who play an instrument but do not have an
A in math and the total number of students who play an instrument
is 90. The ratio of the joint frequencies in the first column is 1 to 1
and the ratio in the second column is 7 to 24, indicating a strong
positive association between playing an instrument and getting an A
in math.

DP.3.2 Given marginal and conditional relative frequencies, construct a 10-5

two-way relative frequency table summarizing categorical bivariate
data.
Algebra 1 Example: A study shows that 9% of the population have
diabetes and 91% do not. The study also shows that 95% of the
people who do not have diabetes, test negative on a diabetes
test while 80% who do have diabetes, test positive. Based on the
given information, the following relative frequency table can be
constructed.

Positive Negative Total

Has diabetes 7.2% 1.8% 9%
Doesn’t have
4.55% 86.45% 91%
diabetes

Clarifications:
Clarification 1: Construction includes cases where not all frequencies
are given but enough are provided to be able to construct a two-way
relative frequency table.
Clarification 2: Instruction includes the use of a tree diagram when
calculating relative frequencies to construct tables.

117
FLORIDA’S B.E.S.T. STANDARDS FOR MATHEMATICS
in (continued)

DATA ANALYSIS AND PROBABILITY Algebra 1 Geometry Algebra 2

MA.912.DP.3 Solve problems involving categorical data. (continued)

DP.3.3 Given a two-way relative frequency table or segmented bar graph 10-5
summarizing categorical bivariate data, interpret joint, marginal and
conditional relative frequencies in terms of a real-world context.
Algebra 1 Example: Given the relative frequency table below, the
ratio of true positives to false positives can be determined as 7.2
to 4.55, which is about 3 to 2, meaning that a randomly selected
person who tests positive for diabetes is about 50% more likely to
have diabetes than not have it.

Positive Negative Total

Has diabetes 7.2% 1.8% 9%
Doesn’t have
4.55% 86.45% 91%
diabetes

Clarifications:
Clarification 1: Instruction includes problems involving false positive
and false negatives.

MA.912.DP.4 Use and interpret independence and probability.

DP.4.1 Describe events as subsets of a sample space using characteristics, 8–1, 8–3
or categories, of the outcomes, or as unions, intersections or
complements of other events.

DP.4.2 Determine if events A and B are independent by calculating the 8–1

product of their probabilities.

DP.4.3 Calculate the conditional probability of two events and interpret the 8–2
result in terms of its context.

DP.4.4 Interpret the independence of two events using conditional 8–2

probability.

DP.4.9 Apply the addition and multiplication rules for counting to solve 8–3
mathematical and real-world problems, including problems involving
probability.

DP.4.10 Given a mathematical or real-world situation, calculate the 8–3

appropriate permutation or combination.

118
LOGIC AND DISCRETE THEORY Algebra 1 Geometry Algebra 2

MA.912.LT.4 Develop an understanding of the fundamentals of propositional logic, arguments and methods
of proof.

LT.4.3 Identify and accurately interpret “if…then,” “if and only if,” “all” and 1-4, 1-5,
“not” statements. Find the converse, inverse and contrapositive of a 1-6
statement.
Clarifications:
Clarification 1: Within the Geometry course, proofs are limited to
geometric statements within the course.

LT.4.8 Construct proofs, including proofs by contradiction. 1-7, 5-4,

5-5
Clarifications:
Clarification 1: Within the Geometry course, proofs are limited to
geometric statements within the course.

LT.4.10 Judge the validity of arguments and give counterexamples to 1-4

disprove statements.
Clarifications:
Clarification 1: Within the Geometry course, instruction focuses on
the connection to proofs within the course.

119
NOTES
PROFESSIONAL DEVELOPMENT
Teaching for Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Eric Milou
Facilitating Mathematical Discourse . . . . . . . . . . . . . . . . . . . . . . . . . 125
Christine Thomas
Developing Mathematical Modelers . . . . . . . . . . . . . . . . . . . . . . . . . 128
Rose Zbiek
Mathematical Thinking and Reasoning . . . . . . . . . . . . . . . . . . . . . . 136
Al Cuoco
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT

TEACHING FOR UNDERSTANDING

Eric Milou, Ed.D.

Building Understanding
It is very clear that effective mathematics instruction begins with effective teaching. But over the
course of history, effective mathematics teaching has been defined in many ways. In the early
half of the 20th century, proficiency was defined by facility with computation, while in the later
half of the century, the standards-based movement emphasized problem solving and reasoning.
Such debate has often been acrimonious and has led to many false beliefs about successful
mathematics teaching. At the turn of the 21st Century, however, the National Research Council
published Adding it Up (NAP, 2001) in which it defined mathematical proficiency as having
five interwoven components:

• Conceptual understanding. Conceptual understanding “reflects a student’s ability

“
to reason in settings involving the careful application of concept definitions, relations, or
It is this transfer representations of either”1. With conceptual understanding, students are able to transfer their
knowledge to new situations and contexts in order to solve the problem presented. It is this
of knowledge transfer of knowledge that is so vital for success not only in mathematics, but in all disciplines
and in the workplace. The authors of Principles and Standards for School Mathematics
that is so vital for (NCTM, 2000) summarize it best: “Students must learn mathematics with understanding,
actively building new knowledge from experience and prior knowledge.”2
success not only in
• Procedural fluency. In a position page on procedural fluency, the National Council
mathematics but in of Teachers of Mathematics (NCTM) defines procedural fluency as “the ability to apply
all disciplines and in procedures accurately, efficiently, and flexibly; to transfer procedures to different problems

”
and contexts; to build or modify procedures from other procedures; and to recognize when
the workplace. one strategy or procedure is more appropriate to apply than another”.3

It should be noted that procedural fluency is more than memorizing procedures and facts.
Procedural fluency builds on the foundation of conceptual understanding, so knowledge of
procedures is no guarantee of conceptual understanding. For example, many secondary
students learn to use the “FOIL” routine for the multiplication of binomials without realizing
that multiplying two binomials is a function of the distributive property.

• Strategic competence. Strategic competence is the ability to “formulate mathematical

problems, represent them, and solve them.”4 While some may see this strand as similar to
what has been called problem solving and problem formulation in mathematics education,
it is important to point out that strategic competence involves authentic problem solving—
problems for which students must formulate a mathematical model to represent the problem
context and then determine the operations necessary to come up with a viable solution.

1 NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Online at nces.ed.gov/nationsreportcard/
mathematics/abilities.asp.
2 http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Principles,-Standards,-and-Expectations/
3 http://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/
4 National Research Council (2001). Adding It Up: Helping Children Learn Mathematics, Washington, DC: National
Academy Press, p. 124

122
 Digital

Learning to solve these authentic problems is the essence of mathematics and developing
such ability should be the primary goal of mathematics teaching. Many would argue that a
primary goal of mathematics teaching and learning is to develop the ability to solve a wide
variety of complex mathematics problems. Thus, mathematics instruction should be designed
so that students experience mathematics as problem solving.

• Adaptive reasoning. Adaptive reasoning is “the capacity to think logically about

the relationships among concepts and situations.”5 Adaptive reasoning is the “glue that
holds everything together, the lodestar that guides learning.”6 The importance of adaptive
reasoning cannot be understated. Students with adaptive reasoning can think logically about
the math and they can explain and justify what they are doing.

• Productive disposition. Productive disposition is “the tendency to see sense in

mathematics, to perceive it as both useful and worthwhile, to believe that steady effort
in learning mathematics pays off, and to see oneself as an effective learner and doer of
mathematics…Developing a productive disposition requires frequent opportunities to make
sense of mathematics, to recognize the benefits of perseverance, and to experience the
rewards of sense making in mathematics.”7

This balance of all five components is crucial to successful and effective mathematics teaching
and ultimately teaching for student understanding. An effective mathematics program must
“ An effective
mathematics
program must focus
focus on building students’ mathematical proficiency by helping them develop these five critical
components. on building students’
Teaching for Understanding in enVision Florida B.E.S.T. A|G|A mathematical
In planning for this new program, the enVision Florida B.E.S.T. A|G|A author team proficiency by
focused on implementing an instructional model that would provide students with rich and
varied opportunities to become mathematically proficient students. The team looked to ensure
helping them
that each component in the proficiency strand was adequately represented. develop these five
Each lesson opens with a problem based Launch activity in which students work collaboratively
to draw on their existing math knowledge base to consider possible models for the problem
presented (Model & Discuss), to respond to a mathematical argument presented (Critique &
Explain), or draw generalizations through an exploration (Explore & Reason). In the whole class
discussion that follow the collaborative work, the probing questions provided in the Teacher’s
critical components.
”
Edition help the teacher set students up for the new learning of the lesson, presented in the next
part of the lesson.

These activities target three of the competencies: adaptive reasoning, strategic competence,
productive disposition. Because these activities are student-centered activities, students need to
think logically about relationships presented in the activities (adaptive reasoning), to formulate
and represent mathematical problems (strategic competence), and see sense and value in
mathematics (productive disposition).

In part 2, of the lesson, Understand & Apply, the focus is on conceptual understanding and
procedural fluency as new concepts are presented through a series of visually-rich examples.
Every lesson has a Conceptual Understanding example that focuses on making explicit the
mathematical concept presented. In addition, skill examples are designed to help students
develop procedural fluency based on conceptual understanding. Each lesson ends with a
formative assessment activity. The questions in the Do You Understand? section target
students’ conceptual understanding while the Do You Know How? exercises focus on
procedural fluency.

5 National Research Council (2001). Adding It Up: Helping Children Learn Mathematics, Washington, DC: National
Academy Press, p. 129
6 ibid
7 National Research Council (2001). Adding It Up: Helping Children Learn Mathematics, Washington, DC: National
Academy Press, p. 131

123
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

Other opportunities in the program to help students develop these important competencies
include the Thinking and Reasoning questions and the Mathematical Thinking and Reasoning
(MTR) Standards call out boxes. The Thinking and Reasoning questions encourage students
to pause and reflect on their learning, and to make connections among concepts, emphasizing
logical thinking and sense making (adaptive reasoning and productive disposition).

The MTR Standards call out boxes often include open-ended questions that help students
make sense of the content they are learning (adaptive reasoning), formulate appropriate
questions (strategic competence), or see patterns in the mathematics (productive disposition).

Strategic competence and productive disposition are two areas of focus for the
enVision Florida B.E.S.T. A|G|A authorship team, many of whom are former high school
math teachers who appreciate the challenge of working with teens who often lack interest
in learning math or fail to see the relevance of mathematics in their daily lives. The program
includes two particular opportunities for students to perceive math as useful and worthwhile: the
Mathematical Modeling in 3 Act tasks and the enVision STEM projects.

“
The Mathematical Modeling in 3 Acts tasks ask students to develop a mathematical
enVision Florida model that can explain a real-world phenomenon and to use that model to answer the main
question posed. These high-interest, engaging activities also help develop students’ conceptual
B.E.S.T. A|G|A sets understanding, procedural fluency, and adaptive reasoning as they test out different models
and conjectures to answer the question posed.
as a program goal
The enVision STEM projects are also high-interest, engaging activities that are more open-
to provide multiple ended, encouraging students to explore real-world applications of math concepts. Like the
Mathematical Modeling in 3 Acts, these activities help students develop all of the competencies
opportunities for that make up mathematical proficiency.
students to develop Mathematics teaching in the United States too often produces students more proficient in
procedural fluency and less proficient in the other four strands. To achieve the goal of student
these essential understanding of mathematics, then all five strands––conceptual understanding, procedural
competencies fluency, strategic competence, adaptive reasoning, and productive disposition––must receive
attention in our classrooms. enVision Florida B.E.S.T. A|G|A sets as a program goal to
and become provide multiple opportunities for students to develop these essential competencies and become
mathematically proficient students.
mathematically

”
References
proficient students.
National Research Council. (2001). Adding it up: Helping children learn mathematics.
J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee,
Center for Education, Division of Behavioral and Social Sciences and Education.
Washington, DC: National Academy Press.
NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Online at nces.
ed.gov/nationsreportcard/mathematics/abilities.asp.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics. Reston, VA: NCTM.

124
 Digital

FACILITATING MATHEMATICAL DISCOURSE

Christine Thomas, Ph.D.

Introduction
The math achievement of U.S. students, especially secondary students, has been the focus of
concern and even angst among math educators and school administrators for many years. Not
only were U.S. students’ scores on internationally benchmarked assessment lower than those
of students in other industrialized or even emerging countries, but U.S. businesses chronically
complained that high school and even college graduates were ill-prepared for career
expectations.

Research findings suggest that the under performance of U.S. students on international
assessment can be attributed to, among other factors, students’ superficial and primarily
procedural understanding of key math concepts. Students know how to carry out an operation
or a procedural, for example, how to factor a quadratic equation or calculate measure of center
and spread, but are not always able to articulate when completing the square is a more useful
process than using the quadratic formula.

The National Council for Teachers of Mathematics (NCTM) leadership has been at the
forefront in looking to bring about a change in focus in the high school mathematics classroom
to emphasize conceptual understanding and reasoning over skill practice. This shift away
from memorizing procedures and facts—what to do—reflects the belief that students need a
deep understanding of mathematics—how to do it and why—in order to make use of their
mathematical knowledge and skill to solve unfamiliar problems in new settings or context; that
is, to be effective problem solvers. In its 2009 publication, Focus in High School Mathematics:
Reason and Sense Making, the NCTM leadership argued that high school mathematics
curricula and instruction should be grounded in reasoning and sense-making. Arguing that
reasoning and sense-making are the “foundations for the process of mathematics––problem
solving, reasoning and proof, connections, communication, and representation,”1 the authors
of the publication maintain that these processes should be pervasive through the high school
mathematics curriculum. With such an emphasis, they insist, students are able to develop deep
conceptual understanding of math content, allowing them to succeed in college and career
endeavors.

Why Discourse in the Mathematics Classroom

What then is the connection between discourse in the mathematics classroom and student
achievement in mathematics? Mathematical discourse is foundational to student learning and
is related to student achievement. It is anchored in communication that promotes students’
understanding of key ideas. While engaging in mathematical discourse, students share ideas,
which then helps them clarify their understanding of concepts; as they construct convincing
arguments, they develop precision of language for expressing mathematical ideas. Through
listening to the ideas and arguments of their classmates, they learn to see things from other
perspectives. In short it is through these mathematical conversations that deep understanding
emerges. (NCTM 1991, 2000).

Just as important is the nature of the tasks that students encounter and about which they engage
in mathematical discourse. These tasks need to focus on cognitively challenging tasks that
encourage multiple entries to the problem and allow for multiple approaches to its solution.
When tasks present interesting challenges and potential missteps, students benefit from working
their ways out of the missteps , leading to real understanding. (Hatano & Inagaki, 1991;
Michaels, O’Connor, & Resnick, 2008).

1 National
Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense
Making. Reston, VA, p. 3.

125
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

Meaningful Discourse in the Mathematical Thinking and Reasoning Standards

Facilitating meaningful discourse is integral to the Mathematical Thinking and Reasoning
(MTR) Standards. These standards call for students to engage in communicating, questioning,
explaining, analyzing, justifying, making conjectures, representing, sharing ideas, and
reflecting on the thinking of others.

MTR.1.1 calls for active student participation in effortful learning and promotes discourse by
engaging students in asking questions that will help with solving tasks and in promoting students
to help and support each other with attempting a new method or approach.

MTR.2.1 calls for students to demonstrate understanding by representing problems in

multiple ways whereby students are expected to express connections between concepts and
representations.

MTR. 3.1 calls for student use of feedback and posits that student feedback improves efficiency
when performing calculations.

MTR.4.1 calls for an explicit focus in discourse by promoting student engagement in

discussions that reflect on the mathematical thinking of self and others and specifies that
students (a) communicate mathematical ideas, vocabulary, and methods effectively; (b) analyze
the mathematical thinking of others: (c) compare the efficiency of a method to those expressed
by others; (d) recognize errors and suggest how to correctly solve a task; (e) justify results by
explaining methods and processes, and (f) construct possible arguments based on evidence.

MTR Clarifications in Facilitating Meaningful Discourse

The MTR Standards are explicit in defining teacher moves for effective facilitation of meaningful
discourse in the mathematics classroom. Teacher moves, stated as clarifications in the MTR
Standards, situate the teacher’s role in discourse as establishing a culture in which students ask
questions, creating opportunities for students to discuss, selecting, sequencing, and presenting
student work and developing students’ ability to justify and compare their responses to
their peers.

Effective teachers of mathematics recognize discourse as a major influence of the mathematics

students learn and understand effective teaching of mathematics requires teachers to promote
and orchestrate discourse in ways that build shared understanding of mathematical ideas while
fostering student reasoning and sense making (NCTM 1991, 2009, 2014). Central to the MTR
Standards is engagement in meaningful discourse in the mathematics classroom.

Facilitating and Promoting Meaningful Mathematical Discourse

The challenge that many teachers face is not with defining mathematical discourse or describing
what it should look like in the classroom; rather it is in successfully establishing an environment
to foster that discourse and in providing for students the cognitively rich tasks that encourage
the kind of discourse that leads students to deep conceptual understanding. Once again,
NCTM has been a leader in offering guidance to teachers to help them achieve a mathematics
classroom that fosters mathematical discourse. In 2014, NCTM published Principles to
Actions: Ensuring Mathematical Success for All, a “framework for strengthening the teaching
and learning of mathematics.” The authors identified eight effective teaching practices for
mathematics. They are as follows:

1. Establish mathematics goals to focus learning.

2. Implement tasks that promote reasoning and problem solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.

126
 Digital

Of particular relevance to this discussion are two practices: Practice 4–Facilitate meaningful
mathematical discourse; and Practice 5–Pose purposeful questions. Meaningful discourse
takes place in well-orchestrated classroom discussions and purposeful questions help to
drive well-orchestrated discussions. Providing teachers with guidance and purposeful
questions to facilitate meaningful discussions is one of the key goals of the teacher support in
enVision Florida B.E.S.T. A|G|A.

The instructional design of enVision Florida B.E.S.T. A|G|A is structured in a way to help
teachers facilitate mathematical discourse throughout the program. Each lesson opens with a
learning activity that is designed to promote reasoning and problem solving, to give students
opportunities for engage in productive struggle and in mathematical discourse. In the Teacher’s
Edition are questions to facilitate discussion after students have worked on these activities to
draw out their thinking about the problem and to help them connect the new concepts of the
lesson to the activities.

During Step 2, students are introduced to new concepts in the examples. The questions found in
the Teacher’s Edition are purposefully geared towards helping students develop understanding
of the core concepts that underline the mathematical procedures. The MTR Standards side
column boxes focus students’ attention on particular aspects of the concepts.

At the topic level, two special features emphasize mathematical discourse: the Mathematical
Modeling in 3 Acts tasks and enVision STEM projects. Both of these tasks present high-
interest, low-entry tasks that require student interaction and discourse. They both allow students
at all levels of mathematical proficiency to engage in the tasks and through the rich classroom
discussion, to grow in their understanding of the key concepts used in these tasks.

In enVision Florida B.E.S.T. A|G|A, throughout each topic, lesson, and project teachers
are provided questions that focus on each of the eight teaching practices. These are highlighted
by an icon “ETP.” The development of enVision Florida B.E.S.T. A|G|A was guided
by these Effective Mathematics Teaching Practices and the teacher support will help bring
purposeful and meaningful discourse into classrooms.

References
Hatano, G., & Inagaki, K. (1991). Sharing cognition through collective comprehension
activity. In L. B. Resnick, J. M. Levine, & S. D. Teasley (Eds.), P`erspectives on socially
shared cognition (p. 331–348). American Psychological Association. https://doi.
org/10.1037/10096-014
Michaels, S., O’Connor, C., & Resnick, L.B. (2008). Deliberative Discourse Idealized and
Realized: Accountable Talk in the Classroom and in Civic Life. Studies in Philosophy
and Education 27(4), 283-297. DOI: 10.1007/s11217-007-9071-1
National Council of Teachers of Mathematics (NCTM). Professional Standards for Teaching
Mathematics, Reston, VA: NCTM, 1991.
National Council of Teachers of Mathematics (NCTM). Focus in High School Mathematics:
Reasoning and Sense Making, Reston, VA: NCTM, 2009.
National Council of Teachers of Mathematics (NCTM). Principles to Actions, Reston, VA: NCTM,
2014.

127
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

DEVELOPING MATHEMATICAL MODELERS

Rose Mary Zbiek, Ph.D

Introduction
Mathematical modeling moved into the foreground for many teachers, curriculum developers,
teacher educators, and others in the United States largely due to its inclusion as both a high
school conceptual category and a mathematical practice. Despite the attention, mathematical
modeling is not yet clearly understood or accurately present in many mathematics classrooms.
Ensuring that all involved in mathematics instruction have a clear understanding of what
mathematical modeling is and where it fits in high school mathematics is warranted.

The essence of mathematical modeling is the pursuit of authentic questions that originate outside
mathematics. To understand mathematical modeling means to know it as a process and to
distinguish between mathematical modeling problems and application problems as well as

“
between mathematical modeling and modeling mathematics.
The essence of
Mathematical Modeling Defined and Described
mathematical Model and modeling are common terms used in different ways in everyday language and in
modeling is the mathematics. In mathematics teaching and learning, definitions—or at least descriptions—of
mathematical modeling abound and appear in multiple venues. Like mathematicians whose
pursuit of authentic work is mathematical modeling, the authors of Guidelines for Assessment and Instruction in
Mathematical Modeling Education (GAIMME) (Garfunkel et al., 2016), describe mathematical
questions that modeling as “a process that uses mathematics to represent, analyze, make predictions or
otherwise provide insight into real-world phenomena” (p. 10).
originate outside

”
Perhaps because mathematical modeling is commonly referred to as simply “modeling,” it is
mathematics. often confused with other forms of “modeling” in the STEM fields, such as statistical modeling or
engineering design. Peck, Gould, and Miller (2013) describe the essence of how mathematical
models and statistical models differ:

Bivariate, or two-variable, populations require functions of the form y = f(x) to model the
structure, which is the overall trend or relationship between two quantitative variables.
(p. 15)

Whereas

Statistical models extend the mathematical model by including a variability component.

Statistical measures of variability, such as the standard deviation, give an indication of
how much, on average, data values deviate from the structural part of the model. (p. 17)

Both statistical and mathematical modeling involve variables. However, attention to variance
in data and ways to represent variance as part of a model is a key difference between
mathematical modeling and statistical work.

Some scholars argue that a mathematical model should be defined more broadly as a system
of elements that captures what matters about a familiar system for the purposes of describing,
explaining, or predicting the familiar system (Doerr & English, 2003). Mathematical modeling
so defined seems reminiscent of engineering design, which now is an essential feature of the
curriculum in many schools due to its inclusion in Next Generation Science Standards (National
Science Teachers Association, 2012). Engineering design differs from mathematical modeling
as defined above in that the former represents systems or parts of systems. In addition,
development and evaluation of a design can be based in non-mathematical thinking and
reasoning, including scientific theory. Interestingly, despite differences among work in statistics,
engineering work, and mathematical modeling work, statistical work can be part of what is
used in mathematical modeling and mathematical modeling can inform engineering design.

128
 b (−8)
h = − ___ h = −_____ = −2
2a 2(−2)
h for x and solve for y = −2(−2) 2 − 8(−2) + 1
y-coordinate of
= −8 + 16 + 1
=9
The vertex is (−2, 9). Digital

x = −2
Mathematical(0, 1)
modeling also must be distinguished from modeling mathematics. As Cirillo,
Pelesko, Felton-Koestler, and Rubel (2016) observe, “modeling mathematics refers to using
representations of mathematics to communicate mathematical concepts or ideas” (p. 4). Actions
y (−2, 9) 8 y
y = ax +such
2 bx + cas using algebra tiles to illustrate completing the square or folding paper to create a
parabola, or using4 straws and marshmallows to build physical representations of polyhedral
c)
x are examples(−4, 1)of modeling
(0, 1) mathematics,
x and not instances of mathematical modeling.
O Modeling −8 mathematics
−4 O 4 begins
8 and ends in mathematics; mathematical modeling begins and
ends in the real world.
−4 Problems without a real-world context (see Figure 1) and problems in
x=−
b
2a
which the context remains
−8 completely in the background during the entire solution process
(see Figure 2) are not mathematical modeling problems—they serve other purposes in high
school mathematics. Mere inclusion of a real-world context in a problem does not make it a
PRACTICE & PROBLEM SOLVING
mathematical modeling problem—the problem might be an application task. To be a modeling
Do You KNOW HOW?
task requires the context to be essentially inseparable from APPLY the mathematics. ASSESSMENT PRACTICE
What key Find the vertex and y-intercept of the
quadratic function. 35. Mathematical Connections A streamer is 37. Select all equations that you can solve easily by
5. y = 3x 2 − 12x + 40 6. y = −x 2 + 4x + 7 launched 3 s after a fuse is lit and lands 8 s factoring. AR.3.1
after it is lit. ◻ A. x 2 + 6x = −8
For 7 and 8, find the maximum or minimum of
y
the parabola. 100 ◻ B. 2x 2 + x = 5
7. y = −2x 2 − 16x + 20 8. y = x2 + 12x − 15 Height (ft) 80 ◻ C. x 2 + 2x = 8
9. Find the equation in standard form of the 60 ◻ D. 2x 2 + 5x = 10
parabola that passes through the points
(0, 6), (−3, 15), and (−6, 6). 40 ◻ E. 2x 2 − 11x = −12

Graph1:
theNo
parabola. 20 38. SAT/ACT A quadratic equation of the form
Figure real-world context
x x 2 + bx + c = 0 has a solution of −2. Its related
10. y = 3x 2 + 6x − 2 0
0 2 4 6 8 10 function has a vertex at (2.5, −20.25). What is the
11. y = −2x 2 + 4x + 1 Time (s) other solution to the equation?

a. What is a quadratic equation in factored Ⓐ −11

form that models the situation? Ⓑ −4.5
Ⓒ 0.5
Figure b.
2:What
Real-world context
is the vertex essentially
of the irrelevant
function related to
your equation? How does this compare with Ⓓ7
the vertex of the graph?
Arguably, the vast majority of problems that appear in most school mathematics materials Ⓔ9
LESSON 2-2 Standard Form of a Quadratic Function
71
are applications problems, not mathematical modeling tasks. c. WhatHenry
can youPollak,
multiplyrenowned
your factored form 39. Performance Task An engineer is designing a
by to get the function for the graph? Explain
mathematician and modeler, succinctly distinguished mathematicalyour answer.
modeling from applications water fountain that starts 1 ft off of
the edge of a 10 ft wide pool. The water
of mathematics, noting that mathematical modeling includes: “(1) explicit attention at the from the fountain needs to project into the
beginning of the process of getting from25/12/20
the3:32
problem
AM
outside
36. Apply of mathematics
Math Models A 15 ft tolong
its mathematical
cable is
center of the pool. The path of the water from
connected from a hook to the top of a pole
formulation, and (2) an explicit reconciliation between the mathematics and the real-world
that has an unknown height. The distance from
the fountain is in the shape of a parabola.
situation at the end” (2003, p. 649).

“
the hook to the base of the pole is 3 ft shorter
than the height of the pole.
Mathematical Modeling as a Process Mathematical
1 ft
Pollak’s allusion to the beginning and the end of modeling work positions mathematical modeling 10 ft
is not
modeling as a process, consistent with the aforementioned definitions of mathematical
modeling. The process is inherently iterative—but not in the sense of simple recursion. a procedure or
Mathematical modeling is not a procedure or algorithm. There is no fixed or firm set of steps
that take one from the beginning to the end of a mathematical modeling event. The stops, algorithm. There
15
x ft

ft

restarts, and do-overs of mathematical modeling often come from the need to “reflect on Part A Let the the point (1, 0) be the location

whether the results make sense, possibly improving the model if it has not served its purpose” is nopointfixed
of the starting orWrite
of the water. firm
quadratic equation to model the path of
a

(NGA Center & CCSSO, 2010, p. 7). Revisions can be many, and the process might x – need to
set of steps that
3 ft
the water.
be called to a close when a useful, though not perfect, model is achieved.
Part B What is the maximum height of the

Many diagrams have been offered as ways to capture mathematical

the pole?
a. What can you use to find the height of
modeling as an iterative take
water? Use one from
your equation fromPart A.the

process grounded in the real world. They may differ in the number
b. Write
include or in how these elements are labeled, but they are the
of nodes
and solve or connections
a quadratic
height of the pole.
similar
they
equation to find
in that they involve probing a
beginning
the water to ofthe
Part C What is the equation for the path of
if the maximum height end
the water
must be 4 ft?
real-world situation, carrying out some mathematical work, andfarvalidating
c. How a model.
is the hook from the base of
of a mathematical

”
the pole?
Because most of school mathematics has long focused almost exclusively on learning
312 TOPIC 8 Solving Quadratic Equations
and doing the almost purely mathematical work, a diagram that articulates aspects of
modeling event.
mathematical modeling can prompt images of the critical role that the real-world contexts
play in mathematical modeling. The centrality of the real-world context is key to what makes
mathematical modeling different from applied mathematics problems and typical mathematical
HSM23_SEA1_FL_T08_L02.indd 312 26/02/21 11:11 PM

problem solving. Bliss, Fowler, and Galluzzo (2014) provide such a diagram (Figure 3).

129
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

Like most modeling diagrams, it shows that the mathematical modeling process begins with a
real-world problem, ends with reporting results, and provides a definite space for Analysis
& Model Assessment. Most importantly, the diagram highlights the three Research
and Brainstorming activities of defining the problem, defining variables, and making
assumptions. The arrows emphasize the back and forth movement among these elements.
This aspect of the diagram directly hits the goal for students to develop skill in making and
adjusting assumptions or approximations that they might use to simply a complicated real-world
situation to a problem that they can approach, and perhaps revisit. Getting a solution—the
mathematical work that does not distinguish mathematical modeling from other mathematical
work—is a notably small part of the overall process.

Real-World Problem

Defining
the Getting a
Problem Solution

Repeat as needed
Researching & or as time allows
Brainstorming
Analysis
Making Defining & Model
Assumptions Variables Assessment

Reporting Results

Figure 3: Diagram illustrating mathematical modeling as a process, adapted from Getting Started,
Getting Solutions (Bliss, Fowler, & Galluzzo, 2014, p. 6)

Mathematical Modeling and School Mathematics

The process of mathematical modeling is time-consuming and messy. When encountering a
mathematical modeling problem and entering into the activity of research & brainstorming, a
modeler (or teacher) might not know how long the work will take or what mathematics will be
needed. Multiple valid solutions, dead ends, and revisions are not in sync with the popular
conception of mathematics as a well-defined rapid run from a problem to its single correct
solution that has dominated mathematics for decades (Schoenfeld 1988). Different sets of
assumptions can lead to drastically different—yet similarly valid—models and solutions. The
differences are not merely different strategies for solving problems, such as using substitution,
elimination, and graphing as different strategies for solving a system of linear equations.
Different solutions and different models might arise from making different assumptions,
identifying different variables, and choosing among related mathematical objects. In some
cases, competing models might be different linear functions. Differences, in other cases, can
be noticeably drastic. For example, identifying different assumptions might lead to equations or
function, non-linear equations or fucntions, inequalities, or even new or different variables.

The non-predetermined nature of mathematical modeling contributes to the messiness of

mathematical modeling that lies in contrast to the organization of mathematical ideas in the
school curriculum. Multiple valid solutions, in addition to multiple strategies, compete with
common expectations of what many people have long believed good mathematics should be:
quickly solved problems with single correct answers (Schoenfeld, 1988) arranged according
to topics presented in a predetermined order. Mathematical modeling and school mathematics
curricula also have competing goals in that the former is open to whatever mathematics is
needed while the latter typically has a clearly articulated set of topics and expectations.

130
 Digital

Despite these competing goals, student engagement in mathematical modeling has the potential
to help students develop an understanding of curricula mathematics—the mathematics to be
learned in the classroom at the students’ current point in the curriculum (Zbiek & Conner, 2006).

Teachers and other educators can benefit from more productive beliefs about the doing,
learning, and teaching of mathematical modeling (Zbiek, 2016a) as they enact curriculum
materials that take a distributed atomistic approach to mathematical modeling.

Transitioning to Mathematical Modeling Work

Ideally, high school students should engage in big messy mathematical modeling problems,
but implementing a curriculum that fully embraces mathematical modeling can be challenging.
Geiger, Ärlebäck, and Frejd (2016) contrast a holistic approach with an atomistic approach
to mathematical modeling in the classroom. With a holistic approach, students engage in
the complete mathematical modeling process; within an atomistic approach, students attend
Practice & Problem Solving
to particular aspects of the modeling process at different times. An atomistic approach seems
Graph the followingSUMMARY
CONCEPT functions. What are the
Zeros of Polynomials
productive
domain and when available
range? Is time
the function and
increasing student
or familiarity with mathematical modeling are issues.
decreasing?
Teachers may
FUNCTION
__ want to consider distributed
a2(x
f(x) = (x –__ a) – b) 3(x – c) atomistic approach that involves opportunities
32. f(x) = √x − 1 33. f(x) = √x + 2
3

to make various _____

approaches explicit as they arise (Zbiek, 2016b). In a distributed atomistic
__
approach, students
1 √x + 1
34. f(x) = __
2 engage
35. f(x) =frequently
2 √x − 1
3
with particular
y
components of the modeling process over
The zeros are a, b, and c.
time, with√GRAPHthe goal of helping√_____
_____ them become competent modelers able to engage in full-fledged
36. f(x) = x − 3
3
37. f(x) = x + 4 − 2 2 a has multiplicity 2.
mathematical modeling work. Incorporating such an approach into high school classrooms
requires careful and
38. Communicate consideration
Justify Explain ofhow
_______ thetotypes of problems1 students are asked to solve and
b has multiplicity 3. what
√8x − 24 + 1 to
students are
rewrite theexpected to=do.
function g(x)3
(a, 0) (b, 0) (c, 0) x c has multiplicity 1.
identify the transformations from the parent O
__ −3 −2 −1 1 2 3
graph f(x) = 3√x .
A transition from a typical high school mathematics −1 experience to one that fully embraces
Multiplicity 1,
g Multiplicity 2, has
mathematical
39. Represent andmodeling
Connect The workspeedshould
a turning point at
start
s, in miles with abandoning commonly held
crossesexpectations
the x-axis for
−2
mathematics
per hour, oflearning.
a car when For
the x-axis example,
skid canabefocus on procedures, a common mainstay of the high
it starts to_______
estimated using the formula s = √30 ∙ 0.5d ,
school mathematics classroom, is a problem not only in terms of the absence of mathematical
where d is the length of the skid marks, in
f practices—including mathematical
feet. Graph the function. modeling—but
If a car’s skid marks also 3,in the way in which procedures are
Multiplicity
crosses the x-axis
x oftenmeasure
taught40using
ft in aworked
zone where examples.
the speed While such examples can be powerful tools for learners
8 (Starlimit is 25 mph, was the car speeding?
et al., 2015), they could also be barriers to mathematical modeling work if teachers and
Explain.
students anticipate worked examples (Davis, 2009). If students only follow the work of others,
withoutDo making their own decisions and conducting their
You UNDERSTAND? Do You ownKNOW
analyses,
HOW?they are not doing
mathematical modeling.

“
1. ESSENTIAL QUESTION How are the 4. If the graph of the function f has a multiple
One activity in of
zeros building mathematical
a polynomial modelers
function related to is explicit
zero at x =
work in2,using
what isora analyzing other’s
possible exponent
factor x − 2? Justify your reasoning.
of
One step in building
Practice &the
models Problem
or attempts Solving
equation toand graph models.
provide Students may be the
of a function? asked to use a function model and
Solve each radical equation. Check for
evaluate 2.the
extraneous
output
Error for Ina order
Analysis
solutions.
giventoinput value
identify (see example in Figure 4a)
the zeros 4
y or to solve an equation or mathematical
inequality—orof the system
function, ofa equations
student
___ or
factoredinequalities—given
the cubic an input
f or output value (see example
modelers is explicit
__
40. √x − 2 = 7 √2x =212
f(x) =41.
3
. in Figure 4b).
function x3 −
Alternatively, 3xstudents
− 10x asmay be asked to calculate or
follows: 2 measure and observe or
______
verify relationships based
43. 13 on
− √xprescribed
__
geometry figures superimposed onx a photo or sketches
work analyzing
42. √25 +x +5=9 4
= 10
−2 O
of a real-world
______ object
3 (see
2 example
_______
f(x) = x − 3x √ − 10x in Figure 4c). These −4
tasks 2
are seemingly 4
sufficient in
44. √5x + 1 + 1 = x 45. 6x − 20 − x = − 6 −2
meeting the expectation that students will mathematically analyze these relationships and draw
relevant conclusions.
46. Communicate
= x(x2 − 3x
= x(x
and
− 10)
− 5)(xGive
Justify + 2)an example of −4
models of others or
x = x.
a radical equation
x = 0, xthat hasx no
= −5, = 2real solutions.
from others’ attempts

”
Explain your reasoning.
5. Energy Solutions manufactures LED light
47. Analyze
_____
√
15w
and Persevere The formula bulbs. The profit p, in thousands of dollars
earned, is a function of the number of bulbs
to provide models.
d = _____ gives the diameter d, in inches, of
3.14Describe and correct the error the sold, x, in ten thousands. Profit is modeled by
a rope needed
studentto lift a weight of w, in tons.
made.
How much weight can be lifted with a rope the function −x 3 + 9x 2 − 11x − 21.
that 3.
hasAnalyze
a diameter
andofPersevere
4 in? Explain how For what number of bulbs manufactured
you can determine that the function does the company make a profit?
f(x) = x 3 + 3x 2 + 4x + 2 has both real and
Figure 4a: complex
Examplezeros.
of tasks for which students Figure 4b: Example of tasks for which students
work with others’ models work with others’ models

1/11/21 5:14 PM

LESSON 3-5 Zeros of Polynomial Functions 161 131

 trinomials. What is the value of f(0) − g(0)?
1ft 1ft Ⓐ −256
FROM THE AUTHORS Ⓑ −64
Ⓒ0
PROFESSIONAL DEVELOPMENT (continued) Ⓓ 32
51. Performance Task An electronics
manufacturer designs a smartphone with an
aspect ratio (the ratio of the screen’s height
h to its width w) of 16 : 9.
47. Use Patterns and Structure The profile of a
satellite dish is shaped like a parabola. The
bottom of the dish can be modeled byPracticethe Tutorial
10.15AM

PRACTICE & PROBLEM SOLVING function shown, where x and f(x) are measured 0.6 in.
Mixed Review Available Online
in meters. Use the vertex form of the quadratic
function to determine PRACTICE
h & PROBLEM SOLVING
APPLY ASSESSMENT PRACTICE the vertex or the lowest
point of the dish. How wide is the dish at 10 m
34. Model With Mathematics A small company offDoes
37. of the ground?
each Explain.
quadratic equation have two 0.2 in.APPLY
shows the profits from their business with the solutions? Select Yes or No. w
32. Represent and Connect A piece of tile artwork is 36.
function P(x) = −0.01x2 + 60x + 500, where x is
in the shape of a triangle. The second row has
the number of units they sell and P is the profit Yes No
3 tiles, and the third row has 5 tiles. If the last
in dollars. Part A Write the width in terms of h. What is
0= 2x2 + 1 ❑ ❑ row of the artwork has 27 tiles, how many rows
the area of the phone, including the border,
a. How many units are sold by the company to are there?
1 x2 1 x + 10 in terms of h?
earn the maximum profit? 0 = 2x2 + 5x + 1 ❑ f(x) =
❑
200
–
10
33. Apply Math Models A race car driver travels
Part B The total area of the screen and border
0 = 2x2 + 5x 34 ft in the first second of a race. If the driver
b. How many units are sold before the ❑ ❑ is about 21.48 in. 2. What is the value of h?
travels 3.5 additional feet each subsequent given.
company shows a profit?
0 = 4x2 – 4x + 1 second, how long will
and itwidth
take of
thethe
drive to reach
{a n−1 +
❑ ❑ Part C What are the height 1994,
a speed
screen? What of total
is the 50 ft/s?
height and width of an =
35. Make Sense and Persevere A pattern of
0= 4x2 –
4x – 1 ❑ ❑for which students the phone including the border?
triangular numbers is shown. The first is 1, the Figure 4c:
48. Higher Example
Order of tasks
Thinking The kicker on a football 34. Communicate and Justify A school board
second is 3, the third is 6, and so on.
work with
team others’
uses models
the function, h = −16t 2 + v 0t + h 0, committee has decided to spend its annual sequence.
38. SAT/ACT What are the solutions of
to model the height of a football being kicked technology budget this year on 90 student
x2 + 2x − 15 = 0 using the graph shown? 37. SAT/ACT
In ainto
distributed
the air.
y
atomistic approach, students should alsolaptops
be asked to modify
and plans to buy 40or evaluate
new given
laptops each
year from now on.
models
a. Show given. 16 Davis (2009) captures one aspect of the emphasis on real-world problems
that for any values of v 0 and

1 3 6 10 15
in a way
h 0, the
2
that 8 supports
maximum height modeler development. He distinguishes
of the object between
a. The school Full Domain
board decided that each(FD)
student
(v 0) ina
the school should have accessDomain
to a laptop(LD)
in
graphs of
is ____
64 functions
+ h 0 . as those
x that convey all key features of function and Limited
The formula 0.5n2
+ 0.5n can be used to find the next ten years. If there are 500 students,
the nth triangular number. Is 50 a triangular
graphs—graphs
−8 −4 O of4 otherwise 8
b. The kicker performs an experiment. He thinks
common functions with domains will thethat relate to
technology a realistic
coordinator meet context.
this tenth day?
number? Explain. Prospective modelers
−8 can be asked
if he can double the initial upward velocity about the implications when
goal? LD
Explain.graphs are used (see Figure
5). The ability
of the football to kicked
work fromwiththe
LDground,
graphs theis necessary to achieve the high school modeling goal of Ⓐ 45 Ⓑ 55
36. Make Sense and Persevere The equation b. What are some pros and cons of buying
−16x2 + 10x + 15 = 0 represents the height, identifying
maximum important
height willquantities
also double. in a practical situation andstudent
Is the map laptops
their relationships
in this manner? using such
If you could 38.
kicker
Ⓐ −3, correct? If not, how does
Ⓑ −5, the maximum
in feet, of a flotation device above the water tools height
as diagrams,
3 two-way
change? Explain.
3tables, graphs, flowcharts andchange formulas” (p.would
the plan, 7). you? If so, how
after x seconds. The linear term represents the Ⓒ −8, 5 Ⓓ −16, 0 would you change it?
initial velocity. The constant term represents LESSON 8-4 Completing the Square 325
39. Performance Task A human catapult is used 35. Analyze and Persevere On October 1, Nadia
the initial height.
to launch a person into a lake. The height of starts a push-up challenge by doing 18 push-ups.
the person is modeled as shown, where x is the On October 2, she does 21 push-ups. On October
time in seconds from the launch. 3, she does 24 push-ups. She continues until same rate.
HSM23_SEA1_FL_T08_L04.indd 325 24/01/21 12:29 AM
October 16, when she does the final push-ups in
Part A
f(x) = –16x2 + 50x + 20 the challenge.

a. Write an explicit definition to model the sequence.

number of push-ups Nadia does each day.
Part B
b. Write a recursive definition to model the
number of push-ups Nadia does each day.
c. How many push-ups will Nadia do on
October 16?
16-Day Push-Up Challenge Part C
Part A What equation can you use to find Day 1 18 push-ups
when the person touches the lake? Find the Day 2 21 push-ups
a. If the initial velocity is 0, when should the Day 3 24 push-ups
solution.
flotation device land in the water? Day 4 27 push ups

b. If the initial height is 0, when does the Part B Are your solutions the same for the
flotation device land in the water? equation and problem? Why or why not?

Part C What is the greatest height reached?

Figure 6: Example of expressing a model based
362 TOPIC 9 Solving Quadratic Equations
Figure 5: Example of reflection on Limited
Go Online | PearsonRealize.com on an identified pattern
Domain graph for model

Building on the atomistic approach of bringing modeling into daily lessons, teachers
HSM18_SEA1_CC_T09_L01.indd Page 362 13/02/17 7:32 am f-0224 could /130/PE02680/HSM_NEXT_GEN_2018/NA/SE/HSM_NEXT_GEN_2018/Algebra_1/XXXXXXXXXX/Layou
provide opportunities for students to develop models ... that take several forms, such as
representing patterns—or terms of a pattern—with linear functions or arithmetic
HSM23_SEA2_FL_T01_L04.indd Page 37 08/12/20 2:12 AM f-0260 sequences
(see Figure 6), or using particular algebraic forms to determine parameters to match particular
settings (see Figure 7). All of these tasks prescribe the type of model, engage students in making
few decisions, and often end with “correct” models.

132
 Residual Plot Residual Plot
Example y y
1 1
The scatter plot shows the percentage of
American adults with a high school diploma or 0.5 0.5
higher from 1940 to 2010. Based on the residual x x
LESSON 3-2 Linear 0Functions 0
plot below the scatter plot, how appropriate is 2 4 2 4
the linear model for the data? −0.5 −0.5
Quick Review Practice and Problem Solving Digital
Educational Attainment
1940–2010 −1 whose graph is a
A linear function is a function −11.
1 Use the simple interest formula
straight line. It represents a linear relationship A(t) = P(1 + rt). Evaluate the function
With High School Diploma

100.0
y
Adult Population in U.S.

between two variables.22. A linear

Applyfunction for t =the
writtenThe table shows
Math Models 7 to determine how much money
or Higher (Percent)

80.0 in function notation is f(x) = mx + btimes

and f(x)
winning for isthe 100-meterOliva
run inwill
thehave if she invests $850, which
read “f of x.” Olympics since 1928. What is theearns 4% annual
equation of simple interest for 7 years.
60.0
f(x) = 0.9176x + 25.298 the line of best fit for the data?
12. What
Analyzedoand
the Persevere Melissa runs a graphic
40.0 Example slope and y-intercept represent?design
Estimate the She charges by the page, and
business.
winning time in 2010, and predict hasthe
a winning
setup fee. The table shows her earnings
20.0 A taxi company charges $3.50 plus $0.85 per mile.
x time in 2020. for the last few projects. What is her per-page
0.0 What linear function can be used to determine
0 20 40 60 80 rate, and what is her setup fee?
the cost of a taxi ride of x miles?
YearHow much
Time (s) Year Time (s)
Years Since 1940
would a 3.5-mile taxi ride cost?
1928 10.80 1980 10.25 Cost ($) 185 335 485 635
Residual Plot Let d = distance of the taxi ride.
1932 10.30 1984 9.99 Page totals 2 4 6 8
y
4.00 Cost of taxi ride = cost × distance + fee
1936 10.30 1988 9.92
13. Use Patterns and Structure Tia’s Computer
f(d) = 0.85d + 3.5
2.00 1948 10.30 1992 9.96 Repair Shop charges the labor rates shown for
Use the function to determine the cost of a
x 1952 10.40 1996 9.84 computer repairs. What linear function can
0.00 3.5-mile ride. she use to determine the cost of a repair that
20 40 60 80 1956 10.50 2000 9.87
f(3.5) = (0.85)(3.5) + 3.5 takes 5.5 hours and includes $180 in parts?
−2.00 1960 10.20 2004 9.85
= 6.475
1964 10.00 2008 9.69 Hours 1 1.5 2 2.5
The cost of a 3.5-mile taxi ride is $6.48.
Years Since 1940 Labor ($) 85 127.5 170 212.5
1968 9.95 2012 9.63
The residual plot shows the residuals distributed 1972 10.14 2016 9.81
above and below the x-axis and clustered 1976 10.06
somewhat close to the x-axis. The linear model
LESSON 3-3 is Transforming Linear Functions
likely a good fit for the data.
Figure 7: Example of expressing a model based
Quick Review on an identified pattern Practice & Problem Solving
A transformation of a function f maps each point Given the function f(x) = x, how does the
Interestingly,
of itsGould’s
graph to a(2016) study suggests
distinct location. that the addition
A translation modeling activitiesoftoa constant
or subtraction which students
to the are
shifts each point of the graph of a function the output affect the graph?
exposed are mainly analyze and interpret tasks—tasks
same distance horizontally, vertically, or both.
that fit mainly in the Getting a Solution
TOPIC 10 Analyzing Data = x –argues f(x) = x +need
part of Bliss, Fowler,
Stretches and Galluzzo’s
and compressions diagram
scale each point of(Figure
a
14.3).f(x)She 2 that15.students 5 more
experience determining
graph variables
either horizontally and assumptions; Given
or vertically. they need to– identify
f(x) = 4x 5, describetasks
how thethatgraph
target
of the
g compares with the graph of f.
Research & Brainstorm components. Part A of the task in Figure 8 is an open-ended problem
that requires students to attend to real-world questions.
Example 16. g(x) = 4(x – 3) –09/02/21
5 17.PMg(x) = 2(4x – 5)
5:41

Let f(x) = 2x – 1. If g(x) = (2x – 1) + 3, how does

1-5
18. Generalize Given f(x) = –3x + 9, how does
EXPLOREthe graph of
& REASON g compare to the graph of f? multiplying the output of f by 2 affect the
Hana has some blue paint. She y wants slope and y-intercept of the graph?
Compound 4 mixes in
to lighten the shade, so she
Inequalities 1 cup of white paint. The color is plus 4 c plus 1 c 19. Apply Math Models A hotel business center
white paint white paint
2 mixing
still too dark, so Hana keeps
f in charges $40 per hour to rent a computer plus
1 cup of white paint atga time. After
x a $65 security deposit. The total rental charge
adding 4 cups, she decides the color
I CAN… write and solve is too light.−4 −2 O 2 is represented by f(x) = 40x + 65. How would
compound inequalities.
A. Explain in words how much paint
the equation change if the business center
The graph of g is the translation of the graph of increased the security deposit by $15?
VOCABULARY Hana should have added initially to
• compound inequality get thef shade
threesheunits
wants.up.
• element
• subset
B. Apply Math Models Represent your Figure 9: Example of expressing a model based
answer to part A with one or more inequalities.
on an identified pattern
C. Hana decides that she likes the shades of blue that appear in between
MA.912.AR.2.6–Given a
mathematical or real-world context,
Figure adding
8: Example
120 1 cup and 3
TOPIC
of
4 cups expressing
of white
Linear andpaint. How
Absolute
a can
model based
you represent
Value the
Functions
write and solve one-variable linear
inequalities, including compound on an number
identified pattern
of cups of white paint that yield the shades Hana prefers?
inequalities. Represent solutions
algebraically or graphically.
MA.K12.MTR.5.1, MTR.6.1,
MTR.7.1 Students
ESSENTIALalso need validation
QUESTION tasks.
What are compound The and
inequalities validation task in Figure 9 does not ask whether the
how are their solutions represented?
model is the “right”
HSM23_SEA1_FL_T03_TR.indd 120 model; rather students consider the aptness of the model by looking at how 08/02/21 1:07 PM

wellEXAMPLE
it might 1
fit an altered situation with different parameters and variables.
Understand Compound Inequalities

CONCEPTUAL How can you use inequalities to describe the sets of numbers graphed below?
UNDERSTANDING
Mathematical
A.
−3 0 modeling
2 as a process includes revisiting initial solutions and reiterating
components of the cycle. The The
The graph shows the solutions of two inequalities. problems that
two inequalities form launch lessons offer opportunities for students to
a compound

revisit the problem with altered situations or assumptions. These alterations should lead students
inequality. A compound inequality is made up of two or more inequalities.

to realize the need for new mathematical tools. A simple change in assumption from an hourly
Write an inequality to represent the solutions shown in each part of
the graph.
CHECK FOR rate of $8 x ≤ −3
per hour −3
to0 1.5 times the hourly rate for overtime presents a need for a piecewise-
REASONABLENESS
There is no number that can defined function (see Figure 10).x >The 2
need to organize and then manipulate data related to
both size and color inequalitycreates atheneed
graph isfor aormatrix rather than a table (see Figure 11).
be less than −3 AND greater 0 2
than 2. So it makes sense to The compound that describes x ≤ −3 x > 2.
use OR to write the compound B.
inequality. −4 0 1
The solutions shown in the graph are greater than or equal to –4. They
are also less than 1. Write two inequalities to represent this.

x ≥ −4 −4 0 1
x<1
−4 0 1
The compound inequality that describes the graph is –4 ≤ x and x < 1.
You can also write this as – 4 ≤ x < 1.

Try It! 1. Write a compound inequality for the graph.

−2 0 6

LESSON 1-5 Compound Inequalities 31

133
 1-3 MODEL & DISCUSS

FROM THE AUTHORS

A music teacher needs to buy
Piecewise-Defined guitar strings for her class.
Gui
Gui
Gu tar
taritar
ita rngs
Strings
Stri ings
Strngs
Stri Guitar Strings
Functions Gu
At store A, the guitar strings cost
$6 each. At store B, the guitar

PROFESSIONAL DEVELOPMENT (continued)

$6
strings are $20 for a pack of 4.

A. Make graphs that show

I CAN… graph and interpret

$20
piecewise-defined functions. the income each store
receives if the teacher
VOCABULARY needs 1–20 guitar strings
• piecewise-defined function B. Describe the shape of the graph for store A. Describe the shape of the
• step function graph for store B. Why are the graphs different?

MA.912.AR.9.10–Solve and graph

mathematical and real-world
problems that are modeled with
C. Communicate and Justify Compare the graphs for stores A and B. For
what numbers of guitar strings is it cheaper to buy from store B? Explain
how you know.
7-1
Operations
MODEL & DISCUSS
This screen shows the number of
piecewise functions. Interpret key Small, Medium, Large, and Extra CATEGORIES SEARCH Cart (0)

features and determine constraints

in terms of the context.
With Matrices Large limited-edition silkscreen shirts Online Shopping Club

on sale at an online store. Exclusive Edition T-Shirts on Sale!

MA.K12.MTR.1.1, MTR.5.1, How do you model a situation in which a
MTR.7.1 ESSENTIAL QUESTION function behaves differently over different A. Construct a table to summarize
parts of its domain? the inventory that is on sale.
I CAN… interpret the parts
CONCEPTUAL
3:45

of a matrix and use matrices B. At the end of the day, the

UNDERSTANDING EXAMPLE 1 Model With a Piecewise-Defined Function
for addition, subtraction, and store has sold this many of
Size S M L XL Size S M L XL
In Stock 23 53 21 32 In Stock 11 45 25 28
scalar multiplication.
each T-shirt from the sale items:
Alani has a summer job as a lifeguard. She makes $8/h for up to 40 h each
VOCABULARY red: 4 S, 6 M, 3 L, 5 XL; blue: 2 S,
week. If she works more than 40 h, she makes 1.5 times her hourly pay, or
8 M, 4 L, 0 XL. Make two new tables, one showing the merchandise
$12/h, for each hour over 40 h. How could you make a graph• and
equalwrite a
matrices
sold and one showing the inventory that is left.
function that shows Alani’s weekly earnings based on the number of hours
• scalar
she worked? • scalar multiplication C. Use Patterns and Structure What relationships did you use in creating
• zero matrix the two tables in Part B?
Step 1 Make a table of values and a graph.
Figure 10: Example ythat introduces the need MA.912.NSO.4.1-Given a
STUDY TIP How can you interpret matrices and
for piecewise-defined functions as aWhen
new Figure 11: Lesson launch that introduce the need
Hours Worked Pay
600 x > 40, Alani’s mathematical or real-world context, ESSENTIAL QUESTION
Remember that Alani makes represent and manipulate data operate with matrices?
20 160 pay is P(x) = ($8)(40) + using matrices. Also NSO.4.3
$8/h for the first 40 h and
mathematical
25 200 tool ($12)(x − 40), or P(x) = MA.K12.MTR.2.1, MTR.3.1, for matrix as a new mathematical tool

Pay (dollars)
$12/h for any additional hours EXAMPLE 1 Represent Data With a Matrix
30 240 12x − 160.
400
MTR.5.1
after that.
35 280 A. What could the data values in the matrix represent?
The atomistic
40 320 approach focuses students on discrete components of the mathematical modeling
[2 1 4]
200 When 0 ≤ x ≤ 40, 5 3 0

process. As they develop modeling timesskills, emerging modelers need toarrayorchestrate the elements
45 380 Alani’s pay P(x) is $8/h
50 440 the number of A matrix is a rectangular of values. Matrices S M L
60modeling cycles. The Mathematical Modeling Shirts in 3-Acts
x
into one 55
or 500
more mathematical
0 hours worked, or 8x. help organize information. For example, if the

“
0 20 40 5 3 0
columns of this matrix represent sizes (Small, Sweaters 2 1 4
tasks found in each topic are one way to provide this bridge. These tasks are not full-fledged
Hours Worked
Medium, Large) and the rows represent clothing

A atomistic
Step 2 Notice that the plot contains two linear segments with a slope that items (Shirts, Sweaters), then “4” can represent
modeling activities—they
changes generally
slightly at x = 40. A function have
that has different rules foran intended solution path and a singular correct
4 large sweaters.

answer. Still, these tasks do fit the atomistic view in that they allow students to put together
different parts of its domain is called a piecewise-defined function. AND
COMMUNICATE B. How can you refer to an entire matrix or to the elements of the matrix?
USE PATTERNS AND
JUSTIFY

approach focuses
STRUCTURE
Step 3 Write an equation for each piece of the graph.
elements of the modeling process on their path to becoming proficient modelers. Judicious use
The dimensions of a matrix are 7 5
[ 4 8]
This notation is used for piecewise- given in the form of row by A = −2
{12x −160,
8x, 0 ≤ x ≤ 40 0
defined functions to indicate the P(x) = column, often written as r × c.
different functions at different
of these tasks is consistent x > 40
with the goalWhat arefor students
the dimensions of this not only to solve problems in real world
students on discrete parts of the domain. A capital letter is used to refer to the entire matrix. This matrix is
CONTINUED matrix?
contexts but also to interpret their mathematical results in those contexts and to determinej k
ON THE NEXT PAGE
referred to as Matrix A, or just A.
A specific element can be referred to using subscripts to
whether and how to revise their responses. g a a

components of
LESSON 1-3 Piecewise-Defined Functions 23 indicate the row and column where the element is located. gj gk
h a a
In general, a ij indicates the element in row i and column j. hj hk
i aij aik
The subscript lists the row number, then the column
Students move into more authentic mathematical modeling experiences as they engage in
the mathematical
number. In a general matrix, a 32 refers to the element in row 3,
HSM23_SEA2_FL_T01_L03.indd 23 05/12/20 6:15 AM column 2, which is the number 8 in matrix A.
components of the mathematical modeling process beyond mathematical manipulations

”
CONTINUED ON THE NEXT PAGE

and ideas—the Getting a Solution part in Figure 3. Important in the success of the atomistic
modeling process. approach is the extent to which students orchestrate components and make decisions.
Routinely infusing curricular mathematics problems with modeling components singularly
LESSON 7-1 Operations With Matricesand
349

in combination allows students to decide such things as which variables to use and what
assumptions to make, which mathematical techniques to use, how the model will be analyzed,
HSM23_SEA2_FL_T07_L01.indd 349 20/01/21 1:18 AM

and what will be reported. Also important are opportunities for students to engage in full
modeling activities.

Summary
A distributed atomistic approach incorporates mathematical modeling into school mathematics
in a way that not only develops students’ modeling capacity but also supports students’ learning
of curricular mathematics as they focus on components of the modeling process. Students can
experience modeling moments within daily lesson with problems that are infused with modeling
components. This approach makes mathematical modeling a familiar mathematical practice and
supports student learning of high school mathematics content.

References
Bliss, K. M., Fowler, K. R., & Galluzzo, B. J. (2014). Getting started, getting solutions.
Philadelphia, Pa.: Society for Industrial and Applied Mathematics. http://m3challenge.
siam.org/ resources/modeling-handbook
Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. (2016). Perspectives on modeling
in school mathematics. In C. Hirsch (Ed.), Annual perspectives in mathematics
education 2016: Mathematical modeling and modeling mathematics (pp. 3–16).
Reston, VA: National Council of Teachers of Mathematics.
Davis, J. D. (2009). Understanding the influence of two mathematics textbooks on prospective
secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12,
365–389. doi: 10.1007/s10857-009-9115-2
Garfunkel, S., et al. (2016). Guidelines for assessment and instruction in mathematical
modeling education. Boston/Philadelphia: Consortium for Mathematics and its
Applications/Society for Industrial and Applied Mathematics. http://www.siam.org/
reports/gaimme.php

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Geiger, V., Ärlebäck, J. B., & Frejd, P. (2016). Interpreting curricula to find opportunities
for modeling: Case studies from Australia and Sweden. In C. Hirsch (Ed.), Annual
perspectives in mathematics education 2016: Mathematical modeling and modeling
mathematics (pp. 207-215). Reston, VA: National Council of Teachers of Mathematics.
Gould, H. (2016). What a modeling task looks like. In C. Hirsch (Ed.), Annual perspectives in
mathematics education 2016: Mathematical modeling and modeling mathematics
(pp. 179–186). Reston, VA: National Council of Teachers of Mathematics.
National Science Teachers Association (NSTA). (2012). Next generation science standards.
Washington, D.C.: NSTA. http://www.nextgenscience.org/
Peck, R., Gould, R., & Miller, S. (2013). Developing essential understanding of statistics for
teaching mathematics in grades 9–12. Essential Understanding Series (P. Wilson,
Vol. Ed.; R. M. Zbiek, Series Ed.). Reston, Va.: National Council of Teachers of
Mathematics.
Pollak, H. O. (2003). A history of the teaching of modeling.” In G. M. A. Stanic & J. Kilpatrick
(Eds.), A history of school mathematics (pp. 647–669). Reston, Va.: National Council
of Teachers of Mathematics.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of
“well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.
10.127/s15326985ep2302_5
Star, J. R., Caronongan, P., Foegen, A., Furgeson, J., Keating, B., Larson, M. R., Lyskawa,
J., McCallum, W. G., Porath, J., & Zbiek, R. M. (2015). Teaching strategies for
improving algebra knowledge in middle and high school students (NCEE 2014-
4333). Washington, DC: National Center for Education Evaluation and Regional
Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education,
2014). Retrieved from the NCEE website: http://ies.ed.gov/ncee/wwc/practiceguide.
aspx?sid=20
Zbiek, R. M. (2016a). Supporting teachers’ development as modelers and teachers of
modelers. In C. Hirsch (Ed.), Annual perspectives in mathematics education 2016:
Mathematical modeling and modeling mathematics (pp. 263–272). Reston, VA:
National Council of Teachers of Mathematics.
Zbiek, R. M., (2016b, November). Ensuring that your tasks help your students be mathematical
modelers. National Council of Teachers of Mathematics (NCTM) Regional Meeting,
Philadelphia, PA.
Zbiek, R. M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a
context for deepening students’ understandings of curricular mathematics. Educational
Studies in Mathematics, 63(1), 89–112. doi: 10.1007/s10649-005-9002-4

135
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

MATHEMATICAL THINKING AND REASONING

Al Cuoco Ph.D.

Introduction
There is a practice of mathematics, just as there is a practice of medicine or a practice of
teaching. There’s been a great deal of recent discussion about “mathematical practice” as an
important aspect of K—12 mathematics. For example, the Florida standards for Mathematical
Thinking and Reasoning and the Process Standards from the National Council for Teachers
of Mathematics (NCTM) both provide concrete examples of how mathematical habits and
reasoning can bring coherence and parsimony to K—12 mathematics. These standards
exemplify the ways of work employed by proficient users of mathematics—they are pillars

“
supporting a whole practice rather than a taxonomy that covers the whole field.
Standards exemplify Lists of standards are useful, but they are meant to accentuate a much larger style of work and
the ways of work an interconnected web of mathematical habits used by mathematicians in their own work.
Hyman Bass, an eminent American mathematician, puts it this way:
employed by It will be helpful to name and (at least partially) specify some of the things—practices,
proficient users dispositions, sensibilities, habits of mind—entailed in doing mathematics... These are
things that mathematicians typically do when they do mathematics. At the same time,
of mathematics— most of these things, suitably interpreted or adapted, could apply usefully to elementary
mathematics no less than to research.1
they are pillars
The implication of the italicized sentence in Bass’ quote—that there’s a role in precollege
supporting a whole mathematics for learning to do mathematics in ways similar to the styles of work employed
by mathematics professionals—has been gaining traction for over a decade. It’s now widely
practice rather than a accepted that a modern high school program contains opportunities for students to both learn
mathematics as an established body of knowledge and develop the ways of thinking that are
taxonomy that covers

”
indigenous to the discipline.
the whole field. This essay gives some examples of how a focus on mathematical thinking and reasoning can
help high school teachers bring mathematical coherence to the topics they teach. The examples
are couched in the context of mathematical tasks or areas of the curriculum, but mathematical
practice lives in the approach to a task, not in the task. Rather than saying “here’s a task that
exhibits structure,” it’s more accurate to say “here’s a task where it’s useful to use abstraction as
a way to head towards of a solution.”

Let me illustrate with four examples. Each of these examples starts with a rather mundane task,
and each of these tasks can be approached with many different methods. I want to illustrate
specific methods that exemplify mathematical thinking. The last example shows how different
aspects of mathematical thinking comes together over one investigation.

Example 1: Perseverance
This is a recount of an actual investigation conducted by a group of high school teachers
in 2011. There are several morals to the story; one is discussed at the end.

The problem was to find an exact value for sin 15°. After some time, the first solution came from
Amato who took a 30-60-90 triangle of hypotenuse length 2 and bisected the 30° angle:
C

B A

1 Bass, Hyman (2011). ”Vignette of Doing Mathematics: A Meta-cognitive Tour of the

Production of Some Elementary Mathematics,” The Mathematics Enthusiast: Vol. 8: No. 1,
Article 2. Available at: http://scholarworks.umt.edu/tme/vol8/iss1/2

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 Digital

Every trigonometry teacher has seen this: the assumption is that the angle bisector bisects the
opposite leg. From this, you get a value for sin 15°. Several people pointed out right away that
the assumption wasn’t correct (and they verified this in dynamic geometry software). But rather
than simply calling this a mistake, Sarah got up and said that she saw something. She drew an
altitude from D to AC and noted that this formed a ⊿ADP that’s congruent to ⊿ADB with a little
30-60-90 ⊿DCP leftover.
C
P

B A

Several students jumped on this idea, arguing along these lines:

__ __
• Since AC = 2 and AP = AB = √
​​ 3 ​​, PC = 2 − √
​​ 3 ​​
__
• Since ⊿DCP is 30-60-90, CD = 2, PC = 4 − 2​​√3 ​​.
__ __
• But BC = 1, so BD = 1 − CD = 2​​√3 ​​ − 3, and DP = 2​​√3 ​​ − 3, too.
__________
__
The Pythagorean theorem makes AD = ​√24
   − 12​√3 ​ ​​.
Hence
__
2​√3 ​ − 3
sin 15° = ___________
​​    
__________ __ ​​
​√24
   − 12​√3 ​ ​
This worked out numerically the same as the calculator approximation, but the feeling was that,

“
to see some structure that might lead to a generalization, this could be simplified considerably.
So, groups starting working on this. After a few minutes, Alicia worked out a board full of
algebraic simplifications, stopping at each step to make sure that everyone understood. She
Alicia’s work was a
ended up with
_______
__ real tour de force,
​√2 − √
​ 3 ​ ​
sin 15° = ​​ _______​​ the kind of old-
2
This again agreed numerically, and everyone thought it was a great simplification.
fashioned algebraic
Then Kevin entered sin15° into his calculator and it produced
__ __ calculations that
√ √
​​  ​ 6 ​ − ​ 2 ​​​
_______
4 many teachers

”
Were these two things the same? Pat suggested squaring both, and sure enough, they were.
love.
It was time to go home, but the group felt sure that this idea could be used to get the general
half-angle formulas.
_______
__
And, for the next meeting, they decided to investigate how and when one
can rewrite ​​√a + √
​ b ​ ​​ as the sum of two square roots of rational expressions in a and b.

These results certainly are not new or profound. It is the nature of the work itself— abstracting
from numericals, using special cases to inspire generalizations, salvaging false starts—that
makes it so faithful to real mathematical practice.

Moral: Perseverance has many faces. One of them is that an incorrect conjecture often
contains the germ of a good idea. Rather than abandoning a false start, mathematicians
often dig into what went wrong, sometimes for long periods of time, seeing if the basic idea
can be repaired or used in some other way to come to understanding. It often does. And the
repair itself can often be used to launch new investigations—in this case, an investigation
into equivalent algebraic expressions. This aspect of perseverance is extremely motivating for
learners—the idea that not giving up sometimes leads one from incorrect assumptions to valid
results can bootstrap the very practice of “sticking with it.”

137
FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

Example 2: Abstraction
Here’s the kind of problem that often stumps students:

A music store offers piano lessons at a discount for customers

buying new pianos. The costs for lessons and a one-time fee (for
music books, CDs, software,) are shown in the advertisement.
How much is the one-time fee?

Every teacher knows that the obstacle here is coming up with the
equation that models the situation—solving it is another (often
easier) matter. Here’s a method that builds on students’ ability
to solve similar problems in middle school: Take a guess. This is

“
not to guess-and-check, getting closer to the answer each time.
The underlying We want an equation, not a number. Of course, students need to
understand that the cost per lesson doesn’t depend on the number of lessons taken.
method has roots • Take a guess, say $20.
in the genetic • Check it: (∗)
epistemology of 300 − 20 ​​ = 46​​ __
(∗) ​​ _________
6
2 ​​
3
Jean Piaget.2 It’s a ​​  480 − 20 ​​ = 38​​ __
(∗) _________
12
1 ​​
3
habit of mind that 2 ​​ ≠ 38​​ __
(∗) 46​​ __
3
1 ​​
3

mathematicians use • That wasn’t right, but that’s okay— just keep track of your steps.
• Take another guess, say $70, and check it:
throughout their work:
​​  300 − 70 ​​ = _________
_________ ​​  480 − 70 ​​
abstracting regularity 6 12
• Keep it up, until you get a “guess checker”
from repeated

”
300 − guess 480 − guess
​​ ___________
   ​​ =    ___________
​​  ​​
calculations. 6 12
• The equation is
​​  300 − x ​​ = _______
_______ ​​  480 − x ​​
6 12
This method for finding an equation that models a situation builds on students’ ability to do
numerical calculations that lie behind pre-algebra problems. The idea is to check enough
guesses so that you get the regularity of the calculations needed to check the guesses;
a generic “guess checker” is the desired equation. So, it involves:

1) working through several specific examples, concentrating on the “rhythm” of their

calculations, and then
2) expressing this regularity in precise mathematical language.
The method captures a very common habit that is useful throughout algebra: carry out several
concrete examples of a process that you don’t quite “have in your head” in order to find
regularity and to build a generic algorithm that describes every instance of the calculation.

Moral: “Try it with numbers” is a slogan used by many teachers as a way to help students get
started on a problem when they don’t quite see how to model it with an algebraic expression.
This advice can be used to help students build the actual expression by learning to orchestrate
their numerical examples in a way that highlights the operations rather than the inputs to those
operations. And the habit is useful all over mathematics, from building Cartesian equations that
characterize geometric objects to building functions that model financial phenomena.

2 Piaget, Jean (1977). The Development of Thought. Equilibration of Cognitive Structures. New
York: Viking

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 Digital

Example 3: Structure in Expressions

Factoring has gotten a bad name in recent years. But polynomial algebra sits at the historical
core of algebra, and it plays a central role in current research.

Some factoring methods that can be introduced in elementary algebra are extensible, preview
important ideas, and give students a chance to develop the habits of reasoning about
calculations in algebraic structures and transforming a calculation in an invertible way
to one that is easier to carry out.
Because
(x + r)(x + s) = ​​x​​  2​​ + (r + s)x + rs,
factoring a quadratic polynomial, ​​x​​  2​​ + bx + c (with leading coefficient 1) amounts to finding
numbers r and s such that
r + s = b and rs = c
This “sum-product” approach is fairly tractable to beginning students. To factor ​​x​​  2​​ + 17x + 72,
students look for two numbers that add to 17 and multiply to 72, so that
​​x​​  2​​ + 17 + 72 = (x + 9)(x + 8)
Many students who can factor monic (leading coefficient of 1) quadratics by the sum-product
approach often have much more difficulty when the leading coefficient is not 1. Here again,
there are general purpose methods that live on beyond their utility for developing this particular
skill. One such method starts with the observation that 4​​x​​  2​​ + 36x + 45 can more easily be
factored if one “chunks” the terms and writes it as
(2x​​)​​  2​​ + 18(2x) + 45
One can think of this as a “quadratic in 2x,” thinking of 2x as the variable. One can even
replace 2x by some symbol, say z, and write the quadratic as
​​z​​  2​​ + 18z + 45
This factors by the sum-product method:
(z + 15)(z + 3)
Replacing z by 2x gives the factorization of the original quadratic.

The coefficients in this example were especially suited for this technique. What if one is faced
with something like
6​​x​​  2​​ + 11x − 10?
One can reason like this:

• Multiply the polynomial by 6 to make the leading coefficient a perfect square, remembering
that we have to divide by 6 at some point to get back to where we started.
6(6​​x​​  2​​ + 11x − 10) = 36​​x​​  2​​ + 11·6x − 60
• This is a quadratic in 6x; let z = 6x, so the right-hand side becomes monic
​​z​​  2​​ + 11z – 60
• This factors by the sum-product method
(z + 15)(z − 4)
• But z = 6x, so we have
(6x + 15)(6x − 4)
• Factor out common factors—3 from the first binomial and 2 from the second––producing
6(2x + 5)(3x − 2)
• Dividing by 6 gives the factorization of the original polynomial.

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FROM THE AUTHORS
PROFESSIONAL DEVELOPMENT (continued)

“ ”Seeing structure
in expressions” is
often described as
Notice that the insight here did not come from abstracting regularity from numerical examples;
rather it came from seeing a certain structure in the quadratic polynomial. “Seeing structure in
expressions” is often described as looking for hidden meaning.

Moral: This “scaling method” is a general-purpose tool that has applications all over algebra
and calculus. It amounts to changing the variable in order to hide complexity. It can be used,
looking for hidden for example, to transform a polynomial of any degree to one whose leading coefficient is 1.

”
More generally, chunking—treating a piece of an expression as a single object—is another
meaning. useful form of abstraction. For example, treating x − 2 as a single chunk in the expression
5 − (x − 2​​)​​  2​​
tells you that the maximum value for this expression (for all real values of x) is 5 (and that this
occurs when x = 2).

Example 4: Using Mathematical Thinking

A triangle is determined by the lengths of its sides—this is the SSS theorem in geometry. Hence,
one should be able to figure out the area of a triangle from its three side-lengths. There is a
formula for this, called Heron’s formula, but it often seems mysterious to students.

Here’s one approach to the formula that makes use of several aspects of mathematical practice
in one task.

First of all, if I’m in the habit of abstracting from numericals, I’ll want to find the area of a
specific triangle and see if I can do it in a way that doesn’t depend on the actual numbers. So,
I start with the famous 13-14-15 triangle:

​​ 1 ​​times the base times the height, so I draw in a height and label some
I know that the area is __
2
new pieces:
C

13 14

A B
15

“ Chunking again:
Think of ​​x​​  2​​ + ​​h​​  2​​ as
Even here, if I’m attuned to looking for structure, I see a method lurking
that will work for any numerical example. Plowing on, I set up two
equations, using the Pythagorean theorem:
C

”
13 14
h
​​x​​  2​​ + ​​h​​  2​​ = 169
a single thing.
(15 – x) + ​​h​​  2​​ = 196
A B
Expand the second equation: x 15 − x
225 − 30x + ​​x​​  2​​ + ​​h​​  2​​ = 196
Ah, but ​​x​​  2​​ + ​​h​​  2​​ = 169 from the first equation, so
225 − 30x + 169 = 196
From here, you can find x, then h, and then finally the area.

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If I were doing this for real, I’d try it with some other triangles, trying to get the rhythm of the
method and then forcing it into precise language. Even in this one example, if you stare long
enough, you can see hidden meaning: the roles played by 13, 14, and 15 in the calculations,
especially if you “delay the evaluation” and write the last equation as
​​15​​  2​​ − 2 · 15x + ​​13​​  2​​ = ​​14​​  2​​
This delayed evaluation is one way to reveal hidden structure. And, if the triangle had side-
lengths a, b, and c, you’d end up with
​​c​​  2​​ − 2cx + ​​a​​  2​​ = ​​b​​  2​​
Here’s where the structure of the expressions becomes dominant, and by careful transformations,
one can use this to carry out a generic version of your numerical calculations and transform
them into a derivation of Heron’s formula, as this student did:

Moral: It’s impossible to put this approach into one of the buckets given by lists of standards.
It employs in inextricable ways abstracting regularity, using precision, perseverance, and
exploiting structure, all at once. Sometimes one habit becomes more dominant (as in the final
transformations), but they are all here, all the time. And this is typical of how the practice of
mathematics is exercised.

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