Check out the

Dynamic Classroom.
Learning Target
Convert between different forms
Laurie’s Notes BigIdeasMath.com

COMMON
STATE STANDARDS
of rational numbers. CORE 8.NS.A.1

Success Criteria
● Explain the meaning of Preparing to Teach
rational numbers. ● Students have written fractions and mixed numbers as decimals. They have
● Write fractions and mixed also written terminating decimals as fractions or mixed numbers. Now they
numbers as decimals. will extend their understanding to writing repeating decimals as fractions or
● Write repeating decimals as mixed numbers.
fractions or mixed numbers. ● MP7 Look for and Make Use of Structure: Mathematically proficient students
discern a pattern or structure. Recognizing the equivalence of equations
written in different forms and using the equations to write and solve a third
equation requires that students be able to manipulate equations.
Warm Up
Cumulative, vocabulary, and Motivate
prerequisite skills practice 1 2 3 10
● Ask students to use a calculator to help write the fractions —, —, —,…,— as
opportunities are available in 11 11 11 11
the Resources by Chapter or decimals. To save time, have students work in groups on different fractions
at BigIdeasMath.com. instead of trying to write decimal equivalents for all of them.
● Record the results on the board.
“What patterns do you observe?” Sample answers: All of the decimals have
two repeating digits, and the sum of those two digits is equal to 9. Also, the
ELL Support
first of the two repeating digits is one less than the numerator of the fraction.
Students may know the word ● Explain that today students will learn how to do the reverse of this process,
rational as meaning “reasonable” writing a repeating decimal as a fraction.
or “sensible.” Explain that when
rational is used in math to describe
a number, it has a very specific
Exploration 1
meaning. A rational number is a ● Allow time for students to complete part (a).
a “What was the result of multiplying each number in the first column by 10?”
number that can be written as —,
b Answers will vary. Students should recognize that the decimal points moved
where a and b are integers and b
one place to the right, but the decimals are still repeating.
does not equal 0.
● For part (b), work through the first row as a class. Help students understand
that when you subtract the first equation from the second, 10x = 3.333…
the repeating decimal subtracts out and you are left with a −(x = 0.333…)
Exploration 1 simple equation to solve. 9x = 3
a. 10x = 6.666. . . ; 10x = 1.111. . . ; ● After completing part (b), have students check their 1
x=—
10x = 2.444. . . answers with a calculator. 3
—
● In part (c), work through x = 0.12 as a class. Begin by multiplying the equation
b. See Additional Answers.
by 10 to get 10x = 1.212121….
4 5 3 931 Ask, “When you subtract x = 1.212121… from 10x = 1.212121…, does the
c. —; —; —; —; The procedure
33 11 11 990 repeating decimal subtract out? Explain.” No, you need to multiply
involves writing and solving an
10x = 1.212121… by 10 again, or multiply x = 0.121212… by 100.
equation that does not have a
repeating decimal; The procedure
● Complete the solution and then allow time for students to finish part (c) with
involves multiplying each side their partners. Have students share their work with the class.
of the original equation by 100 ● Three-Minute Pause: Have students discuss and write their explanations for
instead of 10. part (d). See the Formative Assessment Tip on page T-396 for a description of
Three-Minute Pause.
d. Let x equal the repeating decimal
d. Subtract the equation x = d
from the equation 10nx = 10nd,
where n is the number of
repeating digits. Then solve for x.

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 9.4 Rational Numbers
Learning Target: Convert between different forms of rational numbers.
Success Criteria: • I can explain the meaning of rational numbers.
• I can write fractions and mixed numbers as decimals.
• I can write repeating decimals as fractions or mixed numbers.

EXPLORATION 1 Writing Repeating Decimals as Fractions

Work with a partner.
a. Complete the table.

x 10x
x 5 0.333 . . . 10x 5 3.333 . . .

x 5 0.666 . . .

x 5 0.111 . . .

x 5 0.2444 . . .

Calculator
b. For each row of the table, use the two
Math Practice equations and what you know about 1÷3
=0.33333
Look for Structure solving systems of equations to write 33333
Why was it helpful to a third equation that does not involve C ÷ ×
multiply each side of a repeating decimal. Then solve
7 8
the equation x 5 d the equation. What does your 9 о
by 10 in part (a)? solution represent? 4 5 6 +
1 2 3 ()
0 . +Шо
c. Write each repeating decimal below =
as a fraction. How is your procedure
similar to parts (a) and (b)? How is
it different?

x 5 0.—
12 x 5 0.—
45

x 5 0.—
27 x 5 0.9—
40

d. Explain how to write a repeating decimal

with n repeating digits as a fraction.

Section 9.4 Rational Numbers 395

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 9.4 Lesson
a
Recall that a rational number is a number that can be written as —, where
b
a and b are integers and b ≠ 0. Every rational number can be written as a
decimal that will either terminate or repeat.
of a
You can think Terminating Decimals Repeating Decimals
ecimal as
terminating d has
t 0.25, 4.736, 21.03 5.222 . . . , 24.—
38, 12.—
015
a decimal tha
s at
repeating zero a
A rational number that can be written as —, where a is an integer and b is a
the end. b
power of 10, has a decimal form that terminates.

EXAMPLE 1 Writing Fractions and Mixed Numbers as Decimals

4
a. Write 1— as a decimal.
25
4 29 29 a
Notice that 1— = —. Because — can be written as —, where a is an integer
25 25 25 b
and b is a power of 10, the decimal form of the number terminates.

29 29 3 4 116
— 5 — 5 — 5 1.16
25 25 3 4 100

4
So, 1— 5 1.16.
25

5
b. Write — as a decimal.
33
5 a
Because — cannot be written as —, where a is an integer and b is a
33 b
Calculator power of 10, the decimal form of the number does not terminate.
5 ÷ 33
51515
=0.15151 Use long division
Divide 5 by 33. 0.1515
to divide 5 by 33. 33 )‾
5.0000
÷ ×
C −33
8 9 о
7 1 70
5 6 + − 1 65
4
2 3 () 50
1
+Шо = − 33
0 .
170
The remainder repeats.
− 165
So, — 5 0.—
5 So, you can stop.
15. 5
33

Try It Write the fraction or mixed number as a decimal.

3 2 3 6
1. — 2. −— 3. 4 — 4. 2 —
15 9 8 11

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 Laurie’s Notes
Scaffold instruction to support all
Scaffolding Instruction students in their learning. Learning is
● Students explored writing repeating decimals as fractions. Now they will individualized and you may want to
continue converting between different forms of rational numbers. group students differently as they move
● Emerging: Students may need to review writing fractions and mixed numbers in and out of these levels with each skill
as decimals. They also need more work with writing repeating decimals as and concept. Student self-assessment
fractions or mixed numbers. They will benefit from guided instruction for the and feedback help guide your
examples. instructional decisions about how and
● Proficient: Students have demonstrated that they can write repeating when to layer support for all students to
decimals as fractions. They can also write fractions and mixed numbers as become proficient learners.
decimals. These students should read the information at the tops of
pages 396 and 397 before proceeding to the Try It Exercises 5–8 and the
Self-Assessment exercises. Formative Assessment Tip
Three-Minute Pause
Discuss This technique is used when a
● Review the definition of a rational number. Emphasize that every rational lesson involves a particularly
number can be written as a decimal that will either terminate or repeat. large amount of information or a
lengthy process. After a period of
EXAMPLE 1 instruction, you take a
three-minute break and students
● This example reviews skills from the previous course.
work with partners or small groups
● Work through each part as shown, asking students to provide the reasoning.
to process the information they
● Teaching Tip: This is another opportunity to remind students to analyze the
have just learned. They might
problem before solving it. Determining if the decimal will terminate is more
ask one another questions or
efficient than jumping into long division only to find that the fraction can be
clarify what their understanding
written as a terminating decimal.
is, relative to the exploration
Try It or instruction they have just
experienced.
● Neighbor Check: Have students work independently and then have their
neighbors check their work. Have students discuss any discrepancies. The break in instruction allows
students to resolve questions and
ask for feedback from peers. Once
ELL Support the three minutes are up, the
instruction continues until the next
After demonstrating Example 1, have students work in groups to discuss
Three-Minute Pause. After the final
and complete Try It Exercises 1–4. Provide guiding questions such as, “Is
Three-Minute Pause, any lingering
the number a fraction or mixed number? Can the denominator be written as
questions can be written down and
a power of 10? Will the decimal terminate? What is the decimal?” Expect
used to clarify concepts taught in
students to perform according to their language levels.
the lesson.
Beginner: Write out the steps of writing the decimal and answer “yes” or “no.”
Intermediate: Use simple sentences to answer the guiding questions and
contribute to discussion. Extra Example 1
3
Advanced: Use detailed sentences and help guide discussion. a. Write −3 — as a decimal. −3.15
20
4 —
b. Write — as a decimal. 0.36
11

Try It
1. 0.2 2. −0.—2
—
3. 4.375 4. 2.54

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 Laurie’s Notes
Discuss
● Have students read the information at the top of the page. Then have students
Turn and Talk with a neighbor to explain the process of writing a repeating
decimal as a fraction.
● Use Popsicle Sticks to select students to explain the process to the class.
Extra Example 2
— 4
Write 5.12 as a mixed number. 5 —
33
EXAMPLE 2
1
“What is 1.25 written as a mixed number in simplest form?” 1—
4
— — 1
Try It ●
“What does this mixed number tell you about 1.25?” 1.25 is approximately 1—.
—
Let x = 1.25. Because there are two repeating digits, multiply by 100.
4
8 — —
5. — “If x = 1.25, then what is 100x ?” 125.25
9 ● Subtract the equations and solve for x. Note that you are solving a system
—
6. 2—
1 of two equations in one variable. Equation 1 is x = 1.25 and Equation 2 is
15 —
100x = 125.25.
64 ● Use a calculator to check the answer.
7. —
99 “Could you multiply by a power of 10 other than 100? Explain.” Yes, you can
50 multiply by any power of 10 that aligns the repeating decimal portions (any
8. −4 —
99 even power of 10).
● Ask students why multiplying by 10n works in the second step. Tell students
to focus on the portion that repeats and how the decimal point moves.
Encourage them to try some examples with more repeating digits. They need
ELL Support to make sure that the repeating portion is “aligned” when subtracting in the
fourth step. So, the repeating decimal is eliminated.
Allow students to work in pairs
● Extension: If time permits, you can explore with students what happens when
for the Self-Assessment for
you use the given steps for a repeating decimal that is not written in its “most
Concepts & Skills exercises. Have —
abbreviated” form. For example, a repeating decimal written as 0.44 instead
two pairs form a group to share —
of 0.4.These steps still work, but you must simplify a more difficult fraction.
their answers to Exercise 9. Have
each pair display their answers for
Exercises 10–17 on a whiteboard for
Try It
your review. ● Give students sufficient time to work through the exercises.
● For Exercise 8, students can ignore the negative sign. The negative sign can
be “put back in” after converting the decimal.
Ask volunteers to explain their work at the board.
Self-Assessment
●

for Concepts & Skills

9. A rational number is any Self-Assessment for Concepts & Skills
number that can be written ● Students should work on these exercises independently.
as a terminating decimal, a
a After students solve the exercises, they should use Turn and Talk to
repeating decimal, or as —,
b explain their solutions to each other.
where a and b are integers
and b ≠ 0. The Success Criteria Self-Assessment chart can be found in the
10. 0.18 11. −0.3—8 Student Journal or online at BigIdeasMath.com.
12. 3.—4 13. −12.1—6
7 2
14. −1— 15. —
9 9
31 233
16. 8 — 17. −6 —
33 990

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 All terminating decimals and all repeating decimals are rational numbers,
so you can write them as fractions.
You have previously written terminating decimals as fractions. To write
a repeating decimal d as a fraction, subtract the equation x 5 d from the
equation 10nx 5 10nd, where n is the number of repeating digits. Then
solve for x.

EXAMPLE 2 Writing a Repeating Decimal as a Fraction

—
Write 1.25 as a mixed number.
—
Calculator
Let x = 1.25.
124 ÷ 99 —
x = 1.25 Write the equation.
=1.2525252525
×
⋅
100 x = 100 1.25
—
⋅ There are 2 repeating digits, so multiply
each side by 102 5 100.
C ÷
—
7 8 9 о 100x = 125.25 Simplify.
+ —
4 5 6 − (x = 1.25) Subtract the original equation.
2 3 ()
1 99x = 124 Simplify.
0 . +Шо =
124
x=— Solve for x.
99

— 124 25
So, 1.25 = — = 1 —.
99 99

Try It Write the decimal as a fraction or a mixed number.

— — —
5. 0.888 . . . 6. 2.06 7. 0.64 8. −4.50

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria
in your journal.

9. VOCABULARY How can you identify a rational number?

WRITING FRACTIONS OR MIXED NUMBERS AS DECIMALS Write the

fraction or mixed number as a decimal.
9 7 4 1
10. — 11. −— 12. 3— 13. −12—
50 18 9 6

WRITING A REPEATING DECIMAL AS A FRACTION Write the repeating

decimal as a fraction or a mixed number.
14. −1.—7 15. 0.—2 16. 8.—
93 —
17. −6.235

Section 9.4 Rational Numbers 397

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 EXAMPLE 3 Modeling Real Life
The weight of an object on the moon is about 0.1— 6 times its weight
on Earth. An astronaut weighs 192 pounds on Earth. How much
does the astronaut weigh on the moon?
Write 0.1—6 as a fraction. Then use the fraction to find the astronaut’s
weight on the moon.

Let x = 0.1—6.
x = 0.1—6 Write the equation.

⋅
10 x = 10 (0.1—
6) ⋅ There is 1 repeating digit, so multiply
each side by 101 5 10.
10x = 1.—6 Simplify.
− (x = 0.1—6) Subtract the original equation.
9x = 1.5 Simplify.
1.5
x=— Solve for x.
9
1.5 15 1
The weight of an object on the moon is about — 5 — 5 — times its
9 90 6
weight on Earth.

So, an astronaut who weighs 192 pounds on Earth weighs

1
⋅
about — 192 5 32 pounds on the moon.
6

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria
in your journal.

18. A fun house mirror distorts the image it reflects. Objects reflected in
the mirror appear 1.—3 times taller. When a five-foot-tall person looks
in the mirror, how tall does he appear?

19. An exchange rate represents the value of one currency relative to

another. Your friend visits a country that uses a local currency with an
exchange rate of 1.2—
65 units of the local currency to $1. If a bank charges
$2 to change currency, how many units of the local currency does your
friend receive when she gives the bank $200?

20. DIG DEEPER A low fuel warning appears when a particular car has
—
0.0146 of a tank of gas remaining. The car holds 18.5 gallons of gas and
can travel 36 miles for each gallon used. How many miles can the car
travel after the low fuel warning appears?

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 Laurie’s Notes
EXAMPLE 3 Extra Example 3
—
The weight of a kitten is 0.27 times
● Ask a volunteer to read the problem. Ask another volunteer to explain what the weight of the mother. The mother
the problem is asking. weighs 9 pounds. How much does the
● Have students complete the first two squares of a Four Square for the 1
problem-solving plan. kitten weigh? 2— Ib
2
“How can you find the astronaut’s weight on the moon?” Multiply the
—
astronaut’s weight on Earth by 0.16.
“Why would you want to convert the repeating decimal to a fraction first?”
Sample answer: To find the exact weight on the moon.
—
● Work through the process of writing 0.16 as a fraction. Students should
complete the third square of their Four Squares.
● You may want to write a few more digits of the repeating decimals.
x = 0.1666…
⋅ ⋅
10 x = 10 (0.1666…)
10x = 1.666…
−(x = 0.166…)
9x = 1.5
1.5
x=—
9
1
● After writing the repeating decimal as a fraction, multiply — by 192 pounds to
6
find the astronaut’s weight on the moon.
● Check: Students should complete the last square of their Four Squares. They
—
could divide 32 pounds by 192 pounds to verify that the quotient is 0.16.

Self-Assessment for Problem Solving Self-Assessment

for Problem Solving
● Encourage students to use a Four Square to complete these exercises.
2
By now, most students should be able to get past the first square. 18. 6 — ft
3
● Give students sufficient time to work through the exercises. Ask
volunteers to explain their work at the board. 19. 250.6 units
20. 9.768 mi
The Success Criteria Self-Assessment chart can be found in the
Student Journal or online at BigIdeasMath.com.

Closure
—
● Quick Write: Explain the steps of writing 2.35 as a mixed number.

Learning Target
Convert between different forms
of rational numbers.

Success Criteria
● Explain the meaning of
rational numbers.
● Write fractions and mixed
numbers as decimals.
● Write repeating decimals as
fractions or mixed numbers.

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 Check out the Dynamic
Assessment System.
Review & Refresh Assignment Guide BigIdeasMath.com
1. 5
and Concept Check
2. −1 Scaffold assignments to support all students in their learning progression.
The suggested assignments are a starting point. Continue to assign additional
3. 23
exercises and revisit with spaced practice to move every student toward
4. 15°, 75°, 90° proficiency.
5. 40°, 60°, 80°
Level Assignment 1 Assignment 2
6. 10°, 30°, 140°
3, 6, 7, 10, 11, 12, 13, 17, 14, 15, 16, 21, 22, 23, 25,
Concepts, Skills, Emerging
19, 20 26, 28, 31, 35, 38
& Problem Solving Proficient
3, 6, 8, 10, 12, 15, 18, 20, 21, 23, 25, 26, 27, 28, 29,
7 22, 24 31, 32, 34, 35, 38
7. —
9
3, 6, 9, 10, 12, 15, 20, 21, 25, 26, 27, 28, 29, 30, 32,
85 Advanced
8. — 22, 24 33, 36, 37, 38
99
23
• Assignment 1 is for use after students complete the Self-Assessment for
9. — Concepts & Skills.
99
• Assignment 2 is for use after students complete the Self-Assessment for
10. −0.15
Problem Solving.
11. 9.08—3 • The red exercises can be used as a concept check.
12. 0.13—8
Review & Refresh Prior Skills
13. 6.025
Exercises 1–3 Evaluating Expressions
14. 0.14—6 Exercises 4–6 Using Interior Angle Measures
15. −2.3—8
16. 0.3125 in. Common Errors
● Exercises 10–15 When using long division, students may divide the
5
17. − — denominator by the numerator. Remind them to always divide the numerator
9
by the denominator, regardless of the size of the numbers.
1 ● Exercises 17–22 Students may ignore the repeating bar and write the decimal
18. 4 —
9 as a fraction or mixed number with a denominator that is a power of 10. For
353 — 5 1
19. − — example, in Exercise 17, they may write −0.5 = −— = −— instead of
990 10 2
— 5
−0.5 = −—. Remind students that a repeating decimal cannot be written as a
89 9
20. 6 — fraction with a denominator that is a power of 10.
990
103
21. —
550
170
22. 11—
333
8
23. —
9

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 9.4 Practice Go to BigIdeasMath.com to get
HELP with solving the exercises.

Review & Refresh

Evaluate the expression.
3— 3— 3—
1. 2 + √ 27 2. 1 − √8 3. 7√125 − 12

Find the measures of the interior angles of the triangle.

4. x° 5. 6. 140°
(x − 5)°
15°
60°
2x°

(x + 40)° x°

Concepts, Skills, & Problem Solving

WRITING REPEATING DECIMALS AS FRACTIONS Write the repeating decimal as a
fraction. (See Exploration 1, p. 395.)
7. 0.777 . . . 8. 0.858585 . . . 9. 0.232323 . . .

WRITING FRACTIONS OR MIXED NUMBERS AS DECIMALS Write the fraction or

mixed number as a decimal.
3 1 5
10. −— 11. 9 — 12. —
20 12 36
1 11 7
13. 6 — 14. — 15. −2—
40 75 18

5
16. PRECISION Your hair is — inch long. Write this length as
16
a decimal.

WRITING A REPEATING DECIMAL AS A FRACTION Write the repeating decimal as a

fraction or a mixed number.
17. −0.—5 18. 4.—1 19. −0.3—56

20. 6.0—
89 21. 0.18—
72 22. 11.—
510

23. STRUCTURE A forecast cone defines the probable

path of a tropical cyclone. The probability that the center
of a particular tropical cyclone remains within the
forecast cone is 0.—8. Write this probability as a fraction.

Section 9.4 Rational Numbers 399

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 24. STRUCTURE Describe how to write a decimal with 12 repeating digits
as a fraction.
22
25. STRUCTURE An approximation for the value of π is —. Write this
7
number as a repeating decimal.

26. MODELING REAL LIFE The density of iodine is about 6.28—1 times
the density of acetone. The density of acetone is about 785 kilograms
per cubic meter. What is the density of iodine? Write your answer as
a repeating decimal.

27. MODELING REAL LIFE A disinfectant manufacturer suggests

that its product kills 99.9—8% of germs. Write this percent as a repeating
decimal and then as a fraction. How many germs would survive when
the disinfectant is applied to an object with 18,000 germs?

28. MODELING REAL LIFE You and your friend are making
7
pear tarts for a bake sale. Your recipe uses — times the weight
6
of the diced pears used in your friend’s recipe. Your friend’s
recipe calls for 0.3 pound of diced pears. How many pounds of
pears should you buy to have enough for both recipes?

29. PROBLEM SOLVING The table shows the principal and interest earned
per year for each of three savings accounts with simple annual interest. Which
account has the greatest interest rate? Justify your answer.

Principal Interest Earned

Account A $90.00 $4.00

Account B $120.00 $5.50
Account C $100.00 $4.80

30. DIG DEEPER The probability that an athlete makes a half-court

basketball shot is 22 times the probability that the athlete makes a
three-quarter-court shot. The probability that the athlete makes
a three-quarter-court shot is 0.00—9. What is the probability that the
athlete makes a half-court shot? Write your answer as a percent.

NUMBER SENSE Determine whether the numbers are equal. Justify your answer.

31. — and 0.4—

9 1 135
09 32. — and 0 33. — and 1.5
22 999 90

ADDING AND SUBTRACTING RATIONAL NUMBERS Add or subtract.

34. 0.4—
09 + 0.6— 35. −0.— 36. — − 0.— 37. 0.—
03 − 0.—
5 11
81 63 + — 27 04
99 6
9 10
38. STRUCTURE Write a repeating decimal that is between — and —.
7 7
Justify your answer.

400 Chapter 9 Real Numbers and the Pythagorean Theorem

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 Mini-Assessment Concepts, Skills,
Write the fraction or mixed number as a decimal. & Problem Solving
7
1. — 0.35 24. Let x equal the repeating decimal
20
11 — d. Subtract the equation x = d
2. −3 — −3.61 from the equation 1012x = 1012d.
18
Write the repeating decimal as a fraction or a mixed number. Then solve for x.
— 1 25. 3.—
142857
3. 0.1 —
26. 4930.67—2 kg
9
— 34
4. 0.75 —
27. 0.999—8; —; 2 germs
45 8999
— 9 9000
5. −3.81 −3 —
11
28. 0.65 lb
29. Account C;
Account A interest rate:
$4.00 — —
— = 0.04 = 4.4%
$90.00
Account B interest rate:
$5.50 — —
— = 0.04583 = 4.583%
$120.00
Account C interest rate:
$4.80
— = 0.048 = 4.8%
$100.00
30. 22%
31. equal; 9 ÷ 22 = 0.40909. . .
32. not equal; 1 ÷ 999 = 0.001001. . .
33. equal; 135 ÷ 90 = 1.5

34. 1— or 1.—
1
09
11

35. − — or −0.—
58
58
99

36. 1— or 1.5—
37
60
66

37. − — or −0.—
1
01
99
38. Sample answer: 1.—3;
9 — 10 —
Section Resources — = 1.285714 and — = 1.428571
7 7
Surface Level Deep Level
Resources by Chapter Resources by Chapter
• Extra Practice • Enrichment and Extension
• Reteach Graphic Organizers
• Puzzle Time Dynamic Assessment System
Student Journal • Section Practice
• Self-Assessment
• Practice
Differentiating the Lesson
Tutorial Videos
Skills Review Handbook
Skills Trainer

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