UNIT 1

Real Numbers, MODULE

MODULE 1
Exponents, Real Numbers
8.NS.1.1, 8.NS.1.2,
8.EE.1.2

and Scientific MODULE

MODULE2
Notation Exponents and
Scientific Notation
8.EE.1.1, 8.EE.1.3,
8.EE.1.4

CAREERS IN MATH
Astronomer An astronomer is a scientist
who studies and tries to interpret the universe Unit 1 Performance Task
beyond Earth. Astronomers use math to At the end of the unit, check
calculate distances to celestial objects and out how astronomers use
to create mathematical models to help them math.
understand the dynamics of systems from stars
and planets to black holes. If you are interested
in a career as an astronomer, you should study
the following mathematical subjects:
• Algebra
• Geometry
• Trigonometry
• Calculus
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Larry Landolfi/Getty Images

Research other careers that require creating

mathematical models to understand physical
phenomena.

Unit 1 1
 C 729 = x3
_ _
729 =√x3
3
√
3
Solve for x by taking the cube root of both sides.
_
√
3
729 = x Apply the definition of cube root.

9=x Think: What number cubed equals 729?

The solution is 9.
D x3 = ___
8
125
_ _
√ x =√
3 ___
8
3 3
125
Solve for x by taking the cube root of both sides.
_
x =√
3 ___
8
125
Apply the definition of cube root.

x = _25 8
Think: What number cubed equals ____?
125

The solution is _25.

YOUR TURN
Solve each equation for x.
Personal
9
Math Trainer 7. x2 = 196 8. x2 = ___
256
Online Assessment
and Intervention
64
my.hrw.com 9. x3 = 512 10. x3 = ___
343

EXPLORE ACTIVITY 8.NS.1.2, 8.EE.1.2

Estimating Irrational Numbers

Irrational numbers are numbers that are not rational. In other words, they
cannot be written in the form _ba , where a and b are integers and b is not 0.

© Houghton Mifflin Harcourt Publishing Company

Square roots of perfect squares are rational numbers. Square
_ roots of numbers
√
that are not perfect squares are irrational. The number 3 is irrational because
3 is not a perfect square of any rational number.
_
Estimate the value of √2.
_
A Since 2 is not a perfect square, √2 is irrational.
_
B To estimate √2, first find two consecutive perfect squares that 2 is
between. Complete the inequality by writing these perfect squares in < 2 <
the boxes. _ _

√ √
_
C Now take the square root of each number. < √2 <

D Simplify the square roots of perfect squares. _

_ < √2 <
√ 2 is between and .

10 Unit 1
 √2 ≈ 1.5
_
E Estimate that √2 ≈ 1.5.
0 1 2 3 4
B
F To find a better estimate, first choose some numbers between
1 and 2 and square them. For example, choose 1.3, 1.4, and 1.5.

1.32 = 1.42 = 1.52 =

_
Is √2 between 1.3 and 1.4? How do you know?

_
Is √2 between 1.4 and 1.5? How do you know?

_ _
√ 2 is between and , so √2 ≈ .
G Locate and label this value on the number line.

1.1 1.2 1.3 1.4 1.5

Reflect _
11. How could you find an even better estimate of √ 2?

_
12. Find a better estimate of √2. Draw a number line
and locate and label your estimate.
_ _
√ 2 is between and , so √2 ≈ .
© Houghton Mifflin Harcourt Publishing Company

_
13. Estimate the value of √7 to two decimal places. Draw
a number line and locate and label your estimate.
_ _
√ 7 is between and , so √7 ≈ .

Lesson 1.1 11
Guided Practice
Write each fraction or mixed number as a decimal. (Example 1)
1. _25 2. _89 3. 3 _34
7
4. __
10
5. 2 _38 6. _56

Write each decimal as a fraction or mixed number in simplest form. (Example 2)

7. 0.675 8. 5.6 9. 0.44

_ _ _
10. 0.4 11. 0.26 12. 0.325

10x = 100x = 1000x =

-x - -x - -x -
_______________ ___________________ _______________________

x= x= x=

x= x= x=

Solve each equation for x. (Example 3)

25
13. x2 = 144 14. x2 = ___
289
15. x3 = 216

√
__ __ __

x= ± √ =±
x = ± __________ = ± _____
x= √
3
=

Approximate each irrational number to two decimal places without a calculator.

© Houghton Mifflin Harcourt Publishing Company

(Explore Activity)
_ _ _
16. √5 ≈ 17. √3 ≈ 18. √ 10 ≈

?
? ESSENTIAL QUESTION CHECK-IN

19. What is the difference between rational and irrational numbers?

12 Unit 1
 Name Class Date

1.1 Independent Practice Personal

Math Trainer
Online
8.NS.1.1, 8.NS.1.2, 8.EE.1.2 Assessment and
my.hrw.com Intervention

20. A __
7
16
-inch-long bolt is used in a machine. 21. The weight of an object on the moon is _16
What is the length of the bolt written as a its weight on Earth. Write _61 as a decimal.
decimal?

22. The distance to the nearest gas station is 23. A baseball pitcher has pitched 98 _32 innings.
2 _45 kilometers. What is this distance written What is the number of innings written as a
as a decimal? decimal?

24. A heartbeat takes 0.8 second. How many 25. There are 26.2 miles in a marathon. Write
seconds is this written as a fraction? the number of miles using a fraction.

26. The average

_ score on a biology test 27. The metal in a penny is worth about
was 72.1. Write the average score using a 0.505 cent. How many cents is this written
fraction. as a fraction?

28. Multistep An artist wants to frame a square painting with an

area of 400 square inches. She wants to know the length of the
© Houghton Mifflin Harcourt Publishing Company • ©Photodisc/Getty Images

wood trim that is needed to go around the painting.

a. If x is the length of one side of the painting, what equation can

you set up to find the length of a side?

b. Solve the equation you wrote in part a. How many solutions
does the equation have?

c. Do all of the solutions that you found in part b make sense in the
context of the problem? Explain.

d. What is the length of the wood trim needed to go around the painting?

Lesson 1.1 13
 _
29. Analyze Relationships To find √15, Beau _found 3 = 9 and 4 = 16. He
2 2

9 and 16, √ 15 must be between 3 and 4. He

said that since 15 is between_
3+4
thinks a good estimate for √15 is ____
2
= 3.5. Is Beau’s estimate high, low,
or correct? Explain.

30. Justify Reasoning What is a good estimate for the solution to the
equation x3 = 95? How did you come up with your estimate?

31. The volume of a sphere is 36π ft3. What is the radius of the sphere? Use
the formula V = _43 πr3 to find your answer.

FOCUS ON HIGHER ORDER THINKING

Work Area

32. Draw Conclusions Can you find the cube root of a negative number? If
so, is it positive or negative? Explain your reasoning.

© Houghton Mifflin Harcourt Publishing Company • © Ilene MacDonald/Alamy Images

33. Make a Conjecture Evaluate and compare the following expressions.
_ _ _ _ _ _

√ 4
__
25
√4
and ____
_
√ 25 √ 16
__
81
√ 16
and ____
_
√ 81 √ 36
__
49
√ 36
and ____
_
√ 49

Use your results to make a conjecture about a division rule for square
roots. Since division is multiplication by the reciprocal, make a conjecture
about a multiplication rule for square roots.

34. Persevere in Problem Solving The difference between the solutions to

the equation x2 = a is 30. What is a? Show that your answer is correct.

14 Unit 1
 LESSON 8.NS.1.1

1.2 Sets of Real Numbers Know that numbers that

are not rational are called
irrational. …

? ESSENTIAL QUESTION
How can you describe relationships between sets of real numbers?

Classifying Real Numbers Animals

Biologists classify animals based on shared Vertebrates
characteristics. A cardinal is an animal, a vertebrate, Birds
a bird, and a passerine.
Math On the Spot
Passerines
You already know that the set of rational numbers my.hrw.com
consists of whole numbers, integers, and fractions.
The set of real numbers consists of the set of
rational numbers and the set of irrational numbers.

Real Numbers
Rational Numbers Irrational
27
4
0.3 -6 Numbers
7
Integers
-3 √17
Passerines, such
Whole -2
Numbers - √11 as the cardinal,
are also called
-1
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Wikimedia

0 √2 “perching birds.”
1 3
√4
4.5 π

EXAMPL 1
EXAMPLE 8.NS.1.1

Write all names that apply to each number.

_
A √5 5 is a whole number that is not a perfect square. Animated
irrational, real Math
my.hrw.com
B –17.84 –17.84 is a terminating decimal.
Commons

rational, real
Math Talk
_ _
√ 81
C ____
9
√ 81
_____ 9
= __ =1 Mathematical Practices
9 9
whole, integer, rational, real What types of numbers are
between 3.1 and 3.9 on a
number line?

Lesson 1.2 15
 YOUR TURN
Write all names that apply to each number.
Personal
Math Trainer 1. A baseball pitcher has pitched 12_23 innings.
Online Assessment
and Intervention
my.hrw.com
2. The length of the side of a square that has an

area of 10 square yards.

Understanding Sets and Subsets

of Real Numbers
By understanding which sets are subsets of types of numbers, you can verify
Math On the Spot whether statements about the relationships between sets are true or false.
my.hrw.com

EXAMPLE 2 8.NS.1.1

Tell whether the given statement is true or false. Explain your choice.
A All irrational numbers are real numbers.
True. Every irrational number is included in the set of real numbers.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Digital Image copyright
Irrational numbers are a subset of real numbers.
B No rational numbers are whole numbers.
Math Talk
Mathematical Practices False. A whole number can be written as a fraction with a denominator
Give an example of a of 1, so every whole number is included in the set of rational numbers.
rational number that is a Whole numbers are a subset of rational numbers.
whole number. Show that
the number is both whole
and rational.

YOUR TURN
Tell whether the given statement is true or false. Explain your choice.

3. All rational numbers are integers.

©2004 Eyewire

Personal 4. Some irrational numbers are integers.

Math Trainer
Online Assessment
and Intervention
my.hrw.com

16 Unit 1
 Identifying Sets for Real-World
Situations
Real numbers can be used to represent real-world quantities. Highways have
posted speed limit signs that are represented by natural numbers such as Math On the Spot
55 mph. Integers appear on thermometers. Rational numbers are used in many my.hrw.com
daily activities, including cooking. For example, ingredients in a recipe are often
given in fractional amounts such as _23 cup flour.

EXAMPL 3
EXAMPLE 8.NS.1.1

Identify the set of numbers that best describes each situation. Explain
your choice. My Notes
A the number of people wearing glasses in a room
The set of whole numbers best describes the situation. The number of
people wearing glasses may be 0 or a counting number.
B the circumference of a flying disk has a diameter of 8, 9, 10, 11, or
14 inches

The set of irrational numbers best describes the situation. Each

circumference would be a product of π and the diameter, and any
multiple of π is irrational.

YOUR TURN
Identify the set of numbers that best describes the situation. Explain
your choice.
© Houghton Mifflin Harcourt Publishing Company

5. the amount of water in a glass as it evaporates

6. the number of seconds remaining when a song is playing, displayed as

a negative number

Personal
Math Trainer
Online Assessment
and Intervention
my.hrw.com

Lesson 1.2 17
Guided Practice
Write all names that apply to each number. (Example 1)
_
1. _78 2. √ 36

_
3. √ 24 4. 0.75

_
5. 0 6. - √ 100

_
18
7. 5.45 8. - __
6

Tell whether the given statement is true or false. Explain your choice.
(Example 2)

9. All whole numbers are rational numbers.

10. No irrational numbers are whole numbers.

Identify the set of numbers that best describes each situation. Explain your
choice. (Example 3)
11. the change in the value of an account when given to the nearest dollar
1
inch

© Houghton Mifflin Harcourt Publishing Company

16

12. the markings on a standard ruler

IN. 1

?
? ESSENTIAL QUESTION CHECK-IN

13. What are some ways to describe the relationships between sets of
numbers?

18 Unit 1
 Name Class Date

1.2 Independent Practice Personal

Math Trainer
Online
8.NS.1.1 Assessment and
my.hrw.com Intervention

Write all names that apply to each number. Then place the numbers in the
correct location on the Venn diagram.
_
14. √9 15. 257
_
16. √ 50 17. 8 _12

_
18. 16.6 19. √ 16

Real Numbers

Rational Numbers Irrational Numbers

Integers

Whole Numbers

Identify the set of numbers that best describes each situation. Explain
your choice.
© Houghton Mifflin Harcourt Publishing Company

20. the height of an airplane as it descends to an airport runway

21. the score with respect to par of several golfers: 2, – 3, 5, 0, – 1

22. Critique Reasoning Ronald states that the number __ 1

11 is not rational
because, when converted into a decimal, it does not terminate. Nathaniel
says it is rational because it is a fraction. Which boy is correct? Explain.

Lesson 1.2 19
 UNIT 1

Vocabulary Preview
Use the puzzle to preview key vocabulary from this unit. Unscramble
the circled letters to answer the riddle at the bottom of the page.

1. TCREEFP
SEAQUR

2. NOLRATAI
RUNMEB

3. PERTIANEG
MALCEDI

4. LAER
SEBMNUR

5. NIISICFTCE
OITANTON

1. Has integers as its square roots. (Lesson 1.1)

2. Any number that can be written as a ratio of two integers. (Lesson 1.1)
3. A decimal in which one or more digits repeat infinitely. (Lesson 1.1)

© Houghton Mifflin Harcourt Publishing Company

4. The set of rational and irrational numbers. (Lesson 1.2)
5. A method of writing very large or very small numbers by
using powers of 10. (Lesson 2.2)

Q: What keeps a square from moving?

A: !

2 Vocabulary Preview
23. Critique Reasoning The circumference of a circular region is shown. π mi
What type of number best describes the diameter of the circle? Explain

your answer.

24. Critical Thinking A number is not an integer. What type of number

can it be?

25. A grocery store has a shelf with half-gallon containers of milk. What type
of number best represents the total number of gallons?

FOCUS ON HIGHER ORDER THINKING Work Area

26. Explain the Error Katie said, “Negative numbers are integers.” What was
her error?

27. Justify Reasoning Can you ever use a calculator to determine if a

number is rational or irrational? Explain.

© Houghton Mifflin Harcourt Publishing Company

_
28. Draw Conclusions _1
_ The decimal _0.3 represents 3 . What type of number
best describes 0.9, which is 3 · 0.3? Explain.

29. Communicate Mathematical Ideas Irrational numbers can never be

precisely represented in decimal form. Why is this?

20 Unit 1
 LESSON
Ordering Real 8.NS.1.2

1.3 Numbers
Use rational approximations
of irrational numbers to
compare the size of irrational
numbers, locate them
approximately on a number
line diagram, and estimate
the value of expressions
(e.g., π2).

? ESSENTIAL QUESTION
How do you order a set of real numbers?

Comparing Irrational Numbers

Between any two real numbers is another real number. To compare and order
real numbers, you can approximate irrational numbers as decimals.

Math On the Spot

EXAMPL 1
EXAMPLE 8.NS.1.2 my.hrw.com
_ _
Compare √3 + 5 3 + √5 . Write <, >, or =. My Notes
_ Use perfect squares to estimate
STEP 1 First approximate √3 . square roots.
_
√ 3 is between 1 and 2. 12 = 1 22 = 4 32 = 9
_
Next approximate √5 .
_
√ 5 is between 2 and 3.

STEP 2 Then use your approximations to simplify the expressions.

_
√3 + 5 is between 6 and 7.
_
3 + √5 is between 5 and 6.
_ _
So, √ 3 + 5 > 3 + √ 5 .

Reflect
© Houghton Mifflin Harcourt Publishing Company

_ _
1. If 7 + √5 is equal to √5 plus a number, what do you know about the
number? Why?

_
2. What are the closest two integers that √300 is between?

YOUR TURN
Personal
Math Trainer
Compare. Write <, >, or =.
Online Assessment
_ _ _ _ and Intervention
3. √2 + 4 2 + √4 4. √ 12 +6 12 + √6
my.hrw.com

Lesson 1.3 21
 Ordering Real Numbers
You can compare and order real numbers and list them from least to greatest.

Math On the Spot EXAMPLE 2 8.NS.1.2

my.hrw.com _
Order √22 , π + 1, and 4 _12 from least to greatest.
_
My Notes STEP 1 First approximate √22 .
_
√ 22is between 4 and 5. Since you don’t know where it falls
_
between 4 and 5, you need to find a better estimate for √ 22 so
you can compare it to 4 _12 .
Since 22 is closer to 25 than 16, use squares
_ of numbers between
4.5 and 5 to find a better estimate of √22 .
4.52 = 20.25 4.62 = 21.16 4.72 = 22.09 4.82 = 23.04
_
Since 4.72 = 22.09, an approximate value for √22 is 4.7.
An approximate value of π is 3.14. So an approximate value
of π +1 is 4.14.
_
STEP 2 Plot √ 22 , π + 1, and 4 _21 on a number line.
1
π+1 42 √22

4 4.2 4.4 4.6 4.8 5

Read the numbers from left to right to place them in order from
least to greatest.
_
From least to greatest, the numbers are π + 1, 4 _12 , and √22 .

YOUR TURN

© Houghton Mifflin Harcourt Publishing Company

Order the numbers from least to greatest. Then graph them on the
number line.
_ _
5. √ 5 , 2.5, √ 3

Math Talk
Mathematical Practices

0 0.5 If real numbers a, b, and c

1 1.5 2 2.5 3 3.5 4
are in order from least to
greatest, what is the order
_ of their opposites from
6. π2, 10, √75 least to greatest?
Explain.

Personal
Math Trainer 8 8.5 9 9.5 10 10.5 11 11.5 12
Online Assessment
and Intervention
my.hrw.com

22 Unit 1
 Ordering Real Numbers in
a Real-World Context
Calculations and estimations in the real world may differ. It can be important
to know not only which are the most accurate but which give the greatest or Math On the Spot
least values, depending upon the context. my.hrw.com

EXAMPL 3
EXAMPLE 8.NS.1.2

Four people have found the distance in kilometers across a canyon using
different methods. Their results are given in the table. Order the distances
from greatest to least.

Distance Across Quarry Canyon (km)

Juana Lee Ann Ryne Jackson
_ _
√ 28 23
__
4
5.5 5_12

STEP 1 Write each value as a decimal.

_
√28 is between 5.2 and 5.3. Since 5.32 = 28.09, an approximate
_
value for √28 is 5.3.
23
__
4
= 5.75
_ _
5.5 is 5.555…, so 5.5 to the nearest hundredth is 5.56.

5 _12 = 5.5
_
23 _
STEP 2 Plot √28 , __
4
, 5.5, and 5 _21 on a number line.

1 23
√28 5 2 5.5 4

5 5.2 5.4 5.6 5.8 6

© Houghton Mifflin Harcourt Publishing Company

From greatest to least, the distances are:

_ _
__
23 km, 5.5 km, _
5 21 km, √28 km.
4

YOUR TURN
7. Four people have found the distance in miles across a crater using
different methods. Their results are given below.
_ _
Jonathan: __
10
3_1
3 , Elaine: 3.45, José: 2 , Lashonda:
√ 10
Personal
Order the distances from greatest to least. Math Trainer
Online Assessment
and Intervention
my.hrw.com

Lesson 1.3 23
Guided Practice
Compare. Write <, >, or =. (Example 1)
_ _ _ _
1. √3 +2 √3 + 3 2. √ 11 + 15 √ 8 + 15

_ _ _ _
3. √6 +5 6+ √5 4. √9 + 3 9+ √3

_ _ _ _
5. √ 17 - 3 -2 + √5 6. 10 - √ 8 12 - √2
_ _ _ _
7. √7 + 2 √ 10 - 1 8. √ 17 + 3 3 + √ 11
_
9. Order √ 3 , 2π, and 1.5 from least to greatest. Then graph them on the
number line. (Example 2)
_ _
√ 3 is between and , so √ 3 ≈ .

π ≈ 3.14, so 2π ≈ .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

From least to greatest, the numbers are , ,

.
10. Four people have found the perimeter of a forest Forest Perimeter (km)
using different methods. Their results are given
in the table. Order their calculations from Leon Mika Jason Ashley

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Elena

_
greatest to least. (Example 3) √ 17 -2 π
1 + __ 12
___ 2.5
2 5

?
? ESSENTIAL QUESTION CHECK-IN

11. Explain how to order a set of real numbers.

Elisseeva/Alamy Images

24 Unit 1
 Name Class Date

1.3 Independent Practice Personal

Math Trainer
Online
8.NS.1.2 Assessment and
my.hrw.com Intervention

Order the numbers from least to greatest.

_
_ √8 _
12. √ 7 , 2, ___ 13. √ 10 , π, 3.5
2

_ _ _
√ 220 , -10, √ 100 , 11.5 9
√ 8 , -3.75, 3, _
14. 15. 4

16. Your sister is considering two different shapes for her garden. One is a
square with side lengths of 3.5 meters, and the other is a circle with a
diameter of 4 meters.
a. Find the area of the square.
b. Find the area of the circle.
c. Compare your answers from parts a and b. Which garden would give
your sister the most space to plant?

17. Winnie measured the length of her father’s ranch

Distance Across Father’s Ranch (km)
four times and got four different distances.
Her measurements are shown in the table. 1 2 3 4
_ _
a. To estimate the actual length, Winnie first √ 60 58
__
8
7.3 7 _35
approximated each distance to the nearest
© Houghton Mifflin Harcourt Publishing Company

hundredth. Then she averaged the four

numbers. Using a calculator, find Winnie’s estimate.

_
b. Winnie’s father estimated the distance across his ranch to be √ 56 km.
How does this distance compare to Winnie’s estimate?

Give an example of each type of number.

_ _
18. a real number between √13 and √14

19. an irrational number between 5 and 7

Lesson 1.3 25
20. A teacher asks his students to write the numbers shown
in order from least to greatest. Paul thinks the numbers
are already in order. Sandra thinks the order should be
reversed. Who is right? _
√ 115 , ___
115
11 , and 10.5624

21. Math History There is a famous irrational number called Euler’s number,
symbolized with an e. Like π, its decimal form never ends or repeats. The
first few digits of e are 2.7182818284.
a. Between which two square roots of integers could you find this
number?

b. Between which two square roots of integers can you find π?

FOCUS ON HIGHER ORDER THINKING Work Area

22. Analyze Relationships There are several approximations used for π,

including 3.14 and __
7 . π is approximately 3.14159265358979 . . .
22

a. Label π and the two approximations on the number line.

© Houghton Mifflin Harcourt Publishing Company Image Credits: ©3DStock/

3.140 3.141 3.142 3.143

b. Which of the two approximations is a better estimate for π? Explain.

c. Find a whole number x so that the ratio ___

x
113 is a better estimate for π

than the two given approximations.

23. Communicate Mathematical Ideas If a set of six numbers that include

both rational and irrational numbers is graphed on a number line, what is
iStockPhoto.com

the fewest number of distinct points that need to be graphed? Explain.

_
24. Critique Reasoning Jill says that 12.6 is less than 12.63. Explain her error.

26 Unit 1
 MODULE QUIZ

Ready Personal
Math Trainer
1.1 Rational and Irrational Numbers Online Assessment
and Intervention

Write each fraction as a decimal or each decimal as a fraction. my.hrw.com

___
7
1. __
20 2. 1.27 3. 1_78

Solve each equation for x.

1
4. x2 = 81 5. x3 = 343 6. x2 = ___
100

7. A square patio has an area of 200 square feet. How long is each side

of the patio to the nearest 0.05?

1.2 Sets of Real Numbers

Write all names that apply to each number.
121
8. ____
____
† 121
π
9. __
2

10. Tell whether the statement “All integers are rational numbers” is true
or false. Explain your choice.

1.3 Ordering Real Numbers

Compare. Write <, >, or =.
__ __ __ ___
© Houghton Mifflin Harcourt Publishing Company

11. † 8 + 3 8 + †3 12. † 5 + 11 5 + † 11

Order the numbers from least to greatest.

___ __ ___ __
13. † 99, π2, 9.8 14. † 1 _
__ , 1, 0.2
25 4

ESSENTIAL QUESTION

15. How are real numbers used to describe real-world situations?

Module 1 27
 MODULE 1 MIXED REVIEW Personal
Math Trainer
Assessment Online

Readiness
Assessment and
my.hrw.com Intervention

Selected Response 6. Which of the following is not true?

___
17
‡ 27 + 3 > __
1. The square root of a number is 9. What is A π2 < 2π + 4 C
___ 2
the other square root? B 3π > 9 D 5 – ‡ 24 < 1
___
A –9 C 3 7. Which number is between ‡ 21 and __
3π
2?
B –3 D 81 14
A __ C 5
3 __
2. A square acre of land is 4,840 square yards. B 2‡ 6 D π+1
Between which two integers is the length
of one side? 8. What number is shown on the graph?

A between 24 and 25 yards

6 6.2 6.4 6.6 6.8 7
B between 69 and 70 yards
___
C between 242 and 243 yards A π+3 C ‡ 20 + 2
___ ___
D between 695 and 696 yards B ‡ 4 + 2.5 D 6.14

3. Which of the following is an integer but 9. Which is in order from least to greatest?
not a whole number? 10 11 10 __
A 3.3, __, π, __
3 4
C π, __, 11 , 3.3
3 4
A – 9.6 C 0 10 11
B __, 3.3, __, π
11 10
D __, π, 3.3, __
3 4 4 3
B –4 D 3.7

4. Which statement is false? Mini-Task

A No integers are irrational numbers. 10. The volume of a cube is given by V = x3,
B All whole numbers are integers.
where x is the length of an edge of the
cube. The area of a square is given by
C No real numbers are irrational A = x2, where x is the length of a side of

© Houghton Mifflin Harcourt Publishing Company

numbers. the square. A given cube has a volume of
D All integers greater than 0 are whole 1728 cubic inches.
numbers.
a. Find the length of an edge.
5. Which set of numbers best describes the
displayed weights on a digital scale that
shows each weight to the nearest half b. Find the area of one side of the cube.
pound?
A whole numbers
B rational numbers
c. Find the surface area of the cube.

C real numbers
D integers d. What is the surface area in square feet?

28 Unit 1
 Real
Numbers
MODULE

1
?
LESSON 1.1
ESSENTIAL QUESTION
Rational and
How can you use Irrational Numbers
real numbers to solve 8.NS.1.1, 8.NS.1.2,
real-world problems? 8.EE.1.2

LESSON 1.2
Sets of Real Numbers
8.NS.1.1

LESSON 1.3
Ordering Real
Numbers
8.NS.1.2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Daniel

Real-World Video
Living creatures can be classified into groups. The
sea otter belongs to the kingdom Animalia and
Hershman/Getty Images

class Mammalia. Numbers can also be classified into

my.hrw.com groups such as rational numbers and integers.

my.hrw.com my.hrw.com Math On the Spot Animated Math Personal Math Trainer
Go digital with your Scan with your smart Interactively explore Get immediate
write-in student phone to jump directly key concepts to see feedback and help as
edition, accessible on to the online edition, how math works. you work through
any device. video tutor, and more. practice sets.

3
 Are YOU Ready?
Complete these exercises to review skills you will need Personal
for this module. Math Trainer
Online
Find the Square of a Number my.hrw.com
Assessment and
Intervention

EXAMPLE Find the square of _23.

2 _ 2×2
_
3
× 23 = ____
3×3
Multiply the number by itself.

= _49 Simplify.

Find the square of each number.

1. 7 2. 21 3. -3 4. _45

5. 2.7 6. -_14 7. -5.7 8. 1_25

Exponents
EXAMPLE 53 = 5 × 5 × 5 Use the base, 5, as a factor 3 times.
= 25 × 5 Multiply from left to right.
= 125

Simplify each exponential expression.

( _13 )
2
9. 92 10. 24 11. 12. (-7)2

13. 43 14. (-1)5 15. 4.52 16. 105

Write a Mixed Number as an Improper Fraction

© Houghton Mifflin Harcourt Publishing Company

EXAMPLE 2_25 = 2 + _25 Write the mixed number as a sum of a whole number and
a fraction.
10 _
= __
5
+ 25 Write the whole number as an equivalent fraction with the
same denominator as the fraction in the mixed number.
12
= __
5
Add the numerators.

Write each mixed number as an improper fraction.

17. 3_13 18. 1_58 19. 2_37 20. 5_56

4 Unit 1
 Reading Start-Up Vocabulary
Review Words
Visualize Vocabulary integers (enteros)
✔ negative numbers
Use the ✔ words to complete the graphic. You can put more (números negativos)
than one word in each section of the triangle. ✔ positive numbers
(números positivos)
✔ whole number (número
entero)
Integers
1, 45, 192 Preview Words
cube root (raiz cúbica)
irrational numbers (número
0, 83, 308 irracional)
perfect cube (cubo
perfecto)
perfect square (cuadrado
-21, -78, -93
perfecto)
principal square root (raíz
cuadrada principal)

Understand Vocabulary rational number (número

racional)
Complete the sentences using the preview words. real numbers (número real)
repeating decimal (decimal
1. One of the two equal factors of a number is a . periódico)
square root (raíz cuadrada)
2. A has integers as its square roots. terminating decimal
(decimal finito)
3. The is the nonnegative square root
of a number.
© Houghton Mifflin Harcourt Publishing Company

Active Reading
Layered Book Before beginning the lessons in this
module, create a layered book to help you learn the
concepts in this module. Label the flaps “Rational
Numbers,” “Irrational Numbers,” “Square Roots,” and
“Real Numbers.” As you study each lesson, write
important ideas such as vocabulary, models, and
sample problems under the appropriate flap.

Module 1 5
 MODULE 1

Unpacking the Standards

Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to
learn in this module.

8.NS.1.1
Know that numbers that are What It Means to You
not rational are called irrational. You will recognize a number as rational or
Understand informally that irrational by looking at its fraction or decimal form.
every number has a decimal
expansion; for rational numbers UNPACKING EXAMPLE 8.NS.1.1
show that the decimal expansion Classify each number as rational or irrational.
repeats eventually, and convert a _
decimal expansion which repeats 0.3 = _13 0.25 = _41
eventually into a rational number. These numbers are rational because they can be written as ratios
of integers or as repeating or terminating decimals.
Key Vocabulary
_
rational number (número π ≈ 3.141592654… √ 5 ≈ 2.236067977…
racional)
A number that can be These numbers are irrational because they cannot be written as
expressed as a ratio of two ratios of integers or as repeating or terminating decimals.
integers.
irrational number (número
irracional)
A number that cannot be
expressed as a ratio of two
integers or as a repeating or
terminating decimal.

© Houghton Mifflin Harcourt Publishing Company

8.NS.1.2
Use rational approximations of
What It Means to You
irrational numbers to compare You will learn to estimate the values of irrational numbers.
the size of irrational numbers,
UNPACKING EXAMPLE 8.NS.1.2
locate them approximately _
on a number line diagram, Estimate the value of √8.
and estimate the value of 8 is not a perfect square. Find the two perfect squares closest to 8.
expressions (e.g., π2). 8 is between the perfect squares 4 and 9.
_ _ _
So √_8 is between √4 and √9.
√ 8 is between 2 and 3.
Visit my.hrw.com
_
to see all Florida
Math Standards
8 is close to 9, so √8 is close to 3.
_ = 7.84
2.8 2.852 = 8.1225 2.92 = 8.41
2
unpacked.
√ 8 is between 2.8 and 2.9, but closer to 2.8.
_
my.hrw.com A good estimate for √8 is 2.8.
6 Unit 1
 LESSON
Rational and Irrational 8.NS.1.1

1.1 Numbers
Know that numbers that
are not rational are called
irrational. Understand
informally that every number
has a decimal expansion; ...
Also 8.NS.1.2, 8.EE.1.2

? ESSENTIAL QUESTION
How do you rewrite rational numbers and decimals, take square
roots and cube roots, and approximate irrational numbers?

Expressing Rational Numbers

as Decimals
A rational number is any number that can be written as a ratio in the form _ba ,
where a and b are integers and b is not 0. Examples of rational numbers are Math On the Spot
6 and 0.5. my.hrw.com

6 can be written as _61 . 0.5 can be written as _12 .

Every rational number can be written as a terminating decimal or a repeating

decimal. A terminating decimal, such as 0.5, has a finite number of digits.
A repeating decimal has a block of one or more digits that repeat indefinitely.

EXAMPL 1
EXAMPLE 8.NS.1.1

Write each fraction as a decimal.

My Notes
A _14
0.25 Remember that the fraction bar means “divided by.”
⎯
4⟌ 1.00 Divide the numerator by the denominator.
-8
Divide until the remainder is zero, adding zeros after
20
the decimal point in the dividend as needed.
© Houghton Mifflin Harcourt Publishing Company

-20
0
1
— = 0.3333333333333...
1
_
4
= 0.25 3

1
B _3
0.333
⎯
3⟌ 1.000
−9 Divide until the remainder is zero or until the digits in
10 the quotient begin to repeat.
−9
Add zeros after the decimal point in the dividend as
10 needed.
−9
1 When a decimal has one or more digits that repeat
_ indefinitely, write the decimal with a bar over the
1
_
3
= 0.3 repeating digit(s).
Lesson 1.1 7
 YOUR TURN
Write each fraction as a decimal.
Personal
Math Trainer 5
Online Assessment
1. __
11
2. _18 3. 2_13
and Intervention
my.hrw.com

Expressing Decimals as
Rational Numbers
You can express terminating and repeating decimals as rational numbers.
Math On the Spot
my.hrw.com
EXAMPLE 2 8.NS.1.1

Write each decimal as a fraction in simplest form.

My Notes A 0.825
The decimal 0.825 means “825 thousandths.” Write this as a fraction.
825
____
1000 To write “825 thousandths”, put 825 over 1000.

Then simplify the fraction.

825 ÷ 25
________ 33
1000 ÷ 25
= __
40 Divide both the numerator and the denominator by 25.
33
0.825 = __
40
_
B 0.37
_ _
Let x = 0.37. The number
_ 0.37 has 2 repeating digits, so multiply each side
of the equation x = 0.37 by 102, or 100.
_
x = 0.37

© Houghton Mifflin Harcourt Publishing Company

_
(100)x = 100(0.37)
_ _ _
100x = 37.37 100 times 0.37 is 37.37.
_ _
Because x = 0.37, you can subtract x from one side and 0.37 from the
other.
_
100x = 37.37
_
−x −0.37
_ _
99x = 37 37.37 minus 0.37 is 37.

Now solve the equation for x. Simplify if necessary.

99x __
___
99
= 37
99 Divide both sides of the equation by 99.
37
x = __
99

8 Unit 1
 YOUR TURN
Write each decimal as a fraction in simplest form.
Personal
_ Math Trainer
4. 0.12 5. 0.57 6. 1.4 Online Assessment
and Intervention
my.hrw.com

Finding Square Roots and Cube Roots

The square root of a positive number p is x if x2 = p. There are two square
roots for every positive number. For example, the square roots of 36 are
6 and −6 because 62 = 36 and (−6)2 = 36. The square roots _ of 25 are 5 and − 5.
1
__ _1 _1
1
You can write the square roots of __ as ±_5. The symbol √5 indicates the positive,
1 Math On the Spot
25
my.hrw.com
or principal square root.

A number that is a perfect square has square roots that are integers. The
number 81 is a perfect square because its square roots are 9 and −9.

The cube root of a positive number p is x if x3 = p. There is one cube root for
every positive number. For example, the cube root of 8 is 2 because 23 = 8.
()
3 _
1
The cube root of __ is _1 because _1 = __
1
. The symbol √3
1 indicates the
27 3 3 27
cube root.

A number that is a perfect cube has a cube root that is an integer. The number
125 is a perfect cube because its cube root is 5.

EXAMPL 3
EXAMPLE 8.EE.1.2

Solve each equation for x.

A x2 = 121

x2 = 121 Solve for x by taking the square root of both sides.

Math Talk
Mathematical Practices
© Houghton Mifflin Harcourt Publishing Company

_
Can you square an integer
x = ±√ 121 Apply the definition of square root. and get a negative number?
What does this indicate
x = ±11 Think: What numbers squared equal 121? about whether negative
numbers have square
The solutions are 11 and −11. roots?
16
B x2 = ___
169
16
x2 = ___
169
Solve for x by taking the square root of both sides.
_
x = ±√ ___
16
169
Apply the definition of square root.
4
x = ±__ 16
Think: What numbers squared equal ____ ?
13 169

The solutions are __

4
13
and −__
4
13
.

Lesson 1.1 9