Download 8 TE - Unit 1 Mod 1

Real
Numbers
?
ESSENTIAL QUESTION
How can you use
real numbers to solve
real-world problems?
MODULE
Since every rational and
irrational number is a real
number, any real-world
problem that can be
modeled and solved with
rational or irrational numbers
can be modeled and solved
with real numbers.
1
LESSON 1.1
Rational and
Irrational Numbers
8.NS.1, 8.NS.2, 8.EE.2
LESSON 1.2
Sets of Real Numbers
8.NS.1
LESSON 1.3
Ordering Real Numbers
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Daniel Hershman/Getty Images
8.NS.2
Real-World Video
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3
Module 1
Living creatures can be classified into groups. The
sea otter belongs to the kingdom Animalia and
class Mammalia. Numbers can also be classified into
groups such as rational numbers and integers.
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Animated Math
Personal Math Trainer
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write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
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Interactively explore
key concepts to see
how math works.
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practice sets.
3
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Are You Ready?
Are YOU Ready?
Assess Readiness
Complete these exercises to review skills you will need
for this module.
Use the assessment on this page to determine if students need
intensive or strategic intervention for the module’s prerequisite skills.
Find the Square of a Number
2
1
Enrichment
Access Are You Ready? assessment online, and receive
instant scoring, feedback, and customized intervention
or enrichment.
Online Assessment
and Intervention
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Multiply the number by itself.
Simplify.
Find the square of each number.
Intervention
Personal
Math Trainer
Find the square of _23.
2 _
2 × 2
_
× 23 = ____
3
3 × 3
= _49
Response to
Intervention
1. 7
49
2. 21
5. 2.7
7.29
6. _14
EXAMPLE
Differentiated Instruction
• Skill 22 Write a Mixed
Number as an Improper
Fraction
1
__
16
9
3. -3
7. -5.7
53 = 5 × 5 × 5
= 25 × 5
= 125
16
__
25
24
__
or 1.96
2 1
_
8. 15 25
4. _45
32.49
Use the base, 5, as a factor 3 times.
Multiply from left to right.
Simplify each exponential expression.
• Skill 11 Find the Square of a • Challenge worksheets
PRE-AP
Number
• Skill 12 Exponents
441
Exponents
Online and Print Resources
Skills Intervention worksheets
Online Practice
and Help
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9. 92
13. 4
3
Extend the Math PRE-AP
Lesson Activities in TE
81
10. 24
16
64
14. (-1)
5
-1
11.
( _13 )
2
15. 4.5
2
1
_
9
20.25
12. (-7)2
16. 10
5
49
100,000
Write a Mixed Number as an Improper Fraction
EXAMPLE
2_25 = 2 + _25
10 _
+ 25
= __
5
12
= __
5
Real-World Video Viewing Guide
Write the mixed number as a sum of a whole number and
a fraction.
Write the whole number as an equivalent fraction with the
same denominator as the fraction in the mixed number.
Add the numerators.
Write each mixed number as an improper fraction.
After students have watched the video, discuss the following:
• What are some different ways that biologists classify animals?
• What are some classifications of numbers mentioned in the video?
natural numbers, integers, rational numbers
17. 3_13
4
10
__
3
18. 1_58
13
__
8
19. 2_37
17
__
7
20. 5_56
35
__
6
© Houghton Mifflin Harcourt Publishing Company
3
EXAMPLE
Personal
Math Trainer
Unit 1
8_MCAAESE206984_U1MO01.indd 4
23/05/13 4:48 PM
PROFESSIONAL DEVELOPMENT VIDEO
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Author Juli Dixon models successful
teaching practices as she explores the
concept of real numbers in an actual
eighth-grade classroom.
Online Teacher Edition
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resources online—plan,
present, and manage classes
and assignments.
Professional
Development
ePlanner
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access all your resources online.
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Interactive Answers and
Solutions
Customize answer keys to print
or display in the classroom.
Choose to include answers only
or full solutions to all lesson
exercises.
Interactive Whiteboards
Engage students with interactive
whiteboard-ready lessons and
activities.
Personal Math Trainer:
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Intervention
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and intervention activities.
Prepare your students with
updated practice tests aligned
with Common Core.
Real Numbers
4
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Reading Start-Up
EL
Reading Start-Up
Have students complete the activities on this page by working alone
or with others.
Visualize Vocabulary
Use the ✔ words to complete the graphic. You can put more
than one word in each section of the triangle.
Strategies for English Learners
Each lesson in the TE contains specific strategies to help English
Learners of all levels succeed.
Emerging: Students at this level typically progress very quickly,
learning to use English for immediate needs as well as beginning to
understand and use academic vocabulary and other features of
academic language.
Expanding: Students at this level are challenged to increase their
English skills in more contexts, and learn a greater variety of vocabulary
and linguistic structures, applying their growing language skills in more
sophisticated ways appropriate to their age and grade level.
Bridging: Students at this level continue to learn and apply a range of
high-level English language skills in a wide variety of contexts, including comprehension and production of highly technical texts.
Integers
1, 45, 192
0, 83, 308
whole numbers
-21, -78, -93
negative numbers
Understand Vocabulary
Complete the sentences using the preview words.
2. A
perfect square
Review Words
integers (enteros)
✔negative numbers
(números negativos)
✔positive numbers
(números positivos)
✔whole number (número
entero)
Preview Words
whole numbers
positive numbers
1. One of the two equal factors of a number is a
Vocabulary
square root .
has integers as its square roots.
cube root (raiz cúbica)
irrational numbers (número
irracional)
perfect cube (cubo
perfecto)
perfect square (cuadrado
perfecto)
principal square root (raíz
cuadrada principal)
rational number (número
racional)
real numbers (número real)
repeating decimal (decimal
periódico)
square root (raíz cuadrada)
terminating decimal
(decimal finito)
3. The principal square root is the nonnegative square root
of a number.
Integrating Language Arts
Students can use these reading and note-taking strategies to help
them organize and understand new concepts and vocabulary.
Additional Resources
Differentiated Instruction
• Reading Strategies EL
© Houghton Mifflin Harcourt Publishing Company
Active Reading
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
8_MCAAESE206984_U1MO01.indd 5
18/05/13 10:45 AM
Focus | Coherence | Rigor
Tracking Your Learning Progression
Before
Students understand:
• write rational numbers as decimals
• describe relationships between
sets and subsets of rational
numbers
• compare rational numbers
5
Module 1
In this module
Students will learn how to:
• express a rational number as a decimal
• approximate the value of an irrational number
• describe the relationship between sets of real numbers
• order a set of real numbers arising from mathematical
and real-world contexts
5
After
Students will connect that:
• the rational numbers are those
with decimal expansions that
terminate in 0s or eventually
repeat
• non-rational numbers are called
irrational numbers
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GETTING READY FOR
GETTING READY FOR
Real Numbers
Real Numbers
CA Common Core
Standards
Content Areas
The Number System—8.NS
Cluster Know that there are numbers that are not rational, and approximate them by
rational numbers.
Go online to
see a complete
unpacking of the
CA Common Core
Standards.
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8.NS.1
Know that numbers that are
not rational are called irrational.
Understand informally that
every number has a decimal
expansion; for rational numbers
show that the decimal expansion
repeats eventually, and convert a
decimal expansion which repeats
eventually into a rational number.
Key Vocabulary
rational number (número
racional)
A number that can be
expressed as a ratio of two
integers.
irrational number (número
irracional)
A number that cannot be
expressed as a ratio of two
integers or as a repeating or
terminating decimal.
8.NS.2
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).
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to see all CA
Common Core
Standards
explained.
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6
What It Means to You
You will recognize a number as rational or
irrational by looking at its fraction or decimal form.
EXAMPLE 8.NS.1
Classify each number as rational or irrational.
_
0.3 = _13
0.25 = _14
These numbers are rational because they can be written as ratios
of integers or as repeating or terminating decimals.
_
√ 5 ≈ 2.236067977…
π ≈ 3.141592654…
These numbers are irrational because they cannot be written as
ratios of integers or as repeating or terminating decimals.
What It Means to You
You will learn to estimate the values of irrational numbers.
EXAMPLE 8.NS.2
_
Estimate the value of √8.
8 is not a perfect square. Find the two perfect squares closest to 8.
8 is between
the perfect
squares
4 and 9.
_
_
_
So √_8 is between √4 and √9.
√ 8 is between 2 and 3.
_
8 is closer to 9, so √8 is closer to 3.
2
2.8
2.92 = 8.41
_ = 7.84
√ 8 is between 2.8 and 2.9
_
A good estimate for √8 is 2.85.
Unit 1
8_MCABESE206984_U1MO01.indd 6
California Common Core Standards
© Houghton Mifflin Harcourt Publishing Company
Use the examples on the page to help students know exactly what
they are expected to learn in this module.
Understanding the standards and the vocabulary terms in the standards
will help you know exactly what you are expected to learn in this module.
Lesson
1.1
Lesson
1.2
10/29/13 11:23 PM
Lesson
1.3
8.NS.1 Know that numbers that are not rational are called irrational. Understand
informally that every number has a decimal expansion; for rational numbers show that the decimal
expansion repeats eventually, and convert a decimal expansion which repeats eventually into a
rational number.
8.NS.2 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).
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the
form x 2 = p and x 3 = p, where p is a positive rational number.
_ Evaluate square roots of small perfect
squares and cube roots of small perfect cubes. Know that √ 2 is irrational.
Real Numbers
6
LESSON
1.1 Rational and Irrational Numbers
Lesson Support
Content Objective
Students will learn to rewrite rational numbers and decimals, take square roots and
cube roots, and approximate irrational numbers.
Language Objective
Students will show and explain how to rewrite rational numbers and decimals, take
square roots and cube roots, and approximate irrational numbers.
California Common Core Standards
8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal
expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually
into a rational number.
8.NS.2 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., π 2).
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p,_where p is a
positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √ 2 is irrational.
MP.6 Attend to precision.
Building Background
Eliciting Prior Knowledge Have students work with a
partner to review the relationship between fractions and
decimals. Ask students to provide an example of writing a
fraction or mixed number as a decimal and vice versa.
Discuss how students chose and wrote their examples.
2 = 1.6
1_
3
7 = 0.7
_
10
1
4.5 = 4_
2
Learning Progressions
Cluster Connections
In this lesson, students work with positive rational and irrational
numbers. They make connections among the real numbers by
converting fractions and decimals and approximating irrational
numbers. Important understandings for students include the
following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Know that there are numbers that are not
rational, and approximate them by rational numbers. Tell
students, “A square garden has an area of 20 square feet.”
• Understand that every number has a decimal expansion.
• Convert a repeating decimal to a rational number.
• Evaluate square roots of perfect squares and cube roots of
perfect cubes.
• Estimate an irrational number.
Work with the real number system will continue in this unit as
students extend the positive rational and irrational numbers to
include negative numbers and compare and order real
numbers.
7A
_
3
_
= 0.75
4
20 ft 2
Have students explain why the side length cannot be rational.
Then have them approximate the length of each side of the
garden to the nearest tenth and hundredth.
Sample answer:
_
The length is the solution to s 2 = 20, √ 20 , which is not a rational
number. 4.5 ft; 4.47 ft; The length is between 4 and 5 feet. 20 is
closer to 4.5 2 than to 4.4 2 or 4.6 2. It is also closer to 4.47 2 than to
4.46 2 or 4.48 2.
PROFESSIONAL DEVELOPMENT
Language Support
EL
California ELD Standards
basic ways.
Emerging 2.I.12b. Selecting language resources – Use knowledge of morphology to appropriately select affixes in
Expanding 2.I.12b. Selecting language resources – Use knowledge of morphology to appropriately select affixes in a
growing number of ways to manipulate language.
Bridging 2.I.12b. Selecting language resources – Use knowledge of morphology to appropriately select affixes in a variety
of ways to manipulate language.
Linguistic Support
EL
Academic/Content Vocabulary
Background Knowledge
square – In this lesson, the word square has multiple
meanings, which can cause confusion. For example,
to square as in to take the square root of a number is a
verb. It is different from the nouns square or square of
a number. The text also refers to perfect square and
principal square root of a number, and the square root
symbol is used. These different usages of square as a
mathematical term need to be clarified. Sentence
frames can be used to help define the meaning.
suffixes – When added to a root word, the suffix -th is
used in math to indicate one of a specified number of
parts, such as tenth, hundredth, or thousandth. Remind
students that the suffix -th also indicates place value.
Note that Spanish, Vietnamese, Mandarin, and other
languages do not have the ending /th/ sound, so
teachers need to enunciate carefully.
To square a number means to _______.
The perfect square of a number means _______.
Leveled Strategies for English Learners
cognates – The words terminating and terminal used
in this lesson are cognates in Spanish: terminar,
meaning “to end” or “to finish.” A Spanish cognate for
approximate is aproximar.
EL
Emerging Use cards with root words ten, hundred, and thousand and a card with the -th suffix.
Have students place them together to show place value. Then complete a sentence. Use the
same procedure to identify decimals.
Expanding Support students at this level of English proficiency by providing sentence frames for
them to use to describe their mathematical reasoning.
To write the fraction _______ as a decimal, I _______.
Bridging Have students identify different meanings of the term square by matching examples of
math problems with a written out sentence frame that defines the usage of the term square: to
square a number; perfect square; square root. Use this procedure also with the term cube.
Math Talk
Be sure to clarify the different uses of the term square when referring to square
roots, perfect squares, and so on.
Rational and Irrational Numbers
7B
LESSON
1.1 Rational and Irrational Numbers
CA Common Core
Standards
The student is expected to:
The Number System—8.NS.1
Know that numbers that are not rational are called
irrational. Understand informally that every number has a
decimal expansion; for rational numbers show that the
decimal expansion repeats eventually, and convert a
decimal expansion which repeats eventually into a
rational number.
Engage
ESSENTIAL QUESTION
How do you rewrite rational numbers and decimals, take square roots and cube roots,
and approximate irrational numbers? To express as a decimal, divide the numerator by
the denominator. To take a square root or cube root of a number, find the number that
when squared or cubed equals the original number. To approximate an irrational number,
estimate a number between two consecutive perfect squares.
Motivate the Lesson
Ask: Which type of rational number do you see more often, fractions or decimals? Which
do you prefer to use? Why?
The Number System—8.NS.2
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. π 2)
Expressions and
Equations—8.EE.2
Use square root and cube root symbols to represent
solutions to equations of the form x 2 = p and x 3 = p,
where p is a positive rational number. Evaluate square
roots of small perfect squares
_ and cube roots of small
perfect cubes. Know that √2 is irrational.
Explain
Questioning Strategies
MP.6 Precision
ADDITIONAL EXAMPLE 1
Write each fraction as a decimal.
5 0._5
2 0.4
A _
B _
5
9
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Interactive example available online
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ADDITIONAL EXAMPLE 2
Write each decimal as a fraction in
simplest form.
71
A 0.355 ___
200
Mathematical Practices
• How does the denominator of a fraction in simplest form tell whether the decimal
equivalent of the fraction is a terminating decimal? The decimal will terminate if
the denominator is an even number, a multiple of 5, or a multiple of 10.
Avoid Common Errors
To avoid interpreting __14 as 4 divided by 1, tell students to start at the top of the fraction and
read the bar as “divided by.”
YOUR TURN
Talk About It
Check for Understanding
Ask: Can an improper fraction be written as a decimal? Give an example to support
your answer. Yes; __54 = 1.25.
EXAMPLE 2
Questioning Strategies
_
43
B 0.43 __
99
Interactive Whiteboard
Interactive example available online
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Lesson 1.1
Have students write examples of ratios, and then share with the class the various notations
for ratios that they used (for example 2:5, 2 to 5, __25 ). Point out the connection between the
word ratio and the meaning of rational number. See also Explore Activity in student text.
EXAMPLE 1
Mathematical Practices
7
Explore
Mathematical Practices
• How can you use place value to write a terminating decimal as a fraction with a power of
ten in the denominator? Start by identifying the place value of the decimal's last digit, and
then use the corresponding power of 10 as the denominator of the fraction.
• How can you tell if a decimal can be written as a rational number? If the decimal is a
terminating or repeating decimal, then it can be written as a rational number.
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1.1
?
Rational and
Irrational Numbers
ESSENTIAL QUESTION
8.NS.1
Know that numbers that are not
rational are called irrational.
Understand informally that every
number has a decimal expansion;
for rational numbers show that the
decimal expansion repeats eventually,
and convert a decimal expansion
which repeats eventually into a
relation number. Also 8.NS.2, 8.EE.2
YOUR TURN
Personal
Math Trainer
Online Practice
and Help
You can express terminating and repeating decimals as rational numbers.
Math On the Spot
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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
6 and 0.5.
Math On the Spot
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My Notes
0.5 can be written as _12 .
© Houghton Mifflin Harcourt Publishing Company
1
_
= 0.25
4
_
1
_
= 0.3
3
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
_
Remember that the fraction bar means “divided by.”
Divide the numerator by the denominator.
_
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.
_
Divide until the remainder is zero, adding zeros after
the decimal point in the dividend as needed.
x = 0.37
_
(100)x = 100(0.37)
_
1
— = 0.3333333333333...
3
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.
1
B _3
0.333
⎯
3⟌ 1.000
−9
10
−9
10
−9
1
A 0.825
825
____
1000
My Notes
A _14
8.NS.1
The decimal 0.825 means “825 thousandths.” Write this as a fraction.
8.NS.1
Write each fraction as a decimal.
EXAMPLE 2
Write each decimal as a fraction in simplest form.
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.
0.25
⎯
4⟌ 1.00
-8
20
-20
0
2.3
3. 2_13
Expressing Decimals as
Rational Numbers
Expressing Rational Numbers
as Decimals
EXAMPL 1
EXAMPLE
_
0.125
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How do you rewrite rational numbers and decimals, take square
roots and cube roots, and approximate irrational numbers?
6 can be written as _61 .
Write each fraction as a decimal.
_
5
0.45
1. __
2. _18
11
_
100x = 37.37
−x
Divide until the remainder is zero or until the digits in
the quotient begin to repeat.
99x = 37
Add zeros after the decimal point in the dividend as
needed.
_
_
37.37 minus 0.37 is 37.
Now solve the equation for x. Simplify if necessary.
99x __
___
= 37
99
99
When a decimal has one or more digits that repeat
indefinitely, write the decimal with a bar over the
repeating digit(s).
Divide both sides of the equation by 99.
37
x = __
99
Lesson 1.1
8_MCABESE206984_U1M01L1.indd 7
_
−0.37
© Houghton Mifflin Harcourt Publishing Company
LESSON
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7
11/1/13 1:28 AM
8
Unit 1
8_MCAAESE206984_U1M01L1.indd 8
12/04/13 8:38 PM
PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.6
This lesson provides an opportunity to address
this Mathematical Practices standard. It calls for
students to attend to precision. Students learn
to express rational numbers accurately and
precisely in both fractional and decimal forms,
and learn to translate from one form to the other.
They also learn how to precisely represent and
communicate ideas about irrational numbers,
square roots, and cube roots.
Math Background
Some decimals may have a pattern but still not
be a repeating decimal that is rational. For
example, in 3.12112111211112…, you can
predict the next digit, and describe the pattern.
(There is one more 1 each time before the 2.)
However, this is not a terminating decimal, nor is
it a repeating decimal, and it is therefore NOT a
rational number.
Rational and Irrational Numbers
8
Focus on Technology
Mathematical Practices
Point out the importance of entering a repeating decimal correctly
when using a graphing
_
calculator to convert the decimal to a fraction. The decimal 0.59 must be entered as
0.595959595959, not 0.59.
YOUR TURN
Focus on Math Connections
Make sure students understand that the place value of the last digit in Exercises 4 and 6
determines the denominator of the corresponding fraction or mixed number. So, for
Exercise 4, the place value hundredths gives a denominator of 100, and for Exercise 6, the
place value tenths gives a denominator of 10.
ADDITIONAL EXAMPLE 3
Solve each equation for x.
A x 2 = 324
18, -18
25
5
5
__
B x 2 = ___
, -__
144
12
12
C 343 = x 3 7
EXAMPLE 3
Questioning Strategies
Mathematical Practices
• How can a solution of an equation of the form x 2 = p be negative if p is a positive number?
Since the square of a negative number is positive, a negative number is also a solution of
x 2 equals a positive number.
• When is a solution of an equation of the form x 3 = p larger than p? The solution is larger
than p if p is a number between 0 and 1.
125 __
5
D x 3 = ___
512
8
Interactive Whiteboard
Interactive example available online
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Focus on Math Connections
_
Make sure_students understand the difference in finding √ 121 and solving x 2 = 121. The
symbol √ indicates the positive or principal square root only, while the equation x 2 = 121
has two roots, the principal square root and its opposite.
YOUR TURN
Avoid Common Errors
To avoid sign errors in Exercise 9, make sure that students understand that the cube of a
negative number is not a positive number. Therefore, -8 is not a solution of x 3 = 512.
Talk About It
Check for Understanding
Ask: Kris predicts that there are two real solutions for Exercises 7 and 8 and that
there are three real solutions for Exercises 9 and 10. Is his prediction correct?
Explain. His prediction is correct for Exercises 7 and 8 because there are two numbers
whose squares are the same positive number given in the exercises. His prediction is not
correct for Exercises 9 and 10, however, because there is only one real number whose cube
is the same positive number given in the exercises.
EXPLORE ACTIVITY
Questioning Strategies
Mathematical Practices
• Compare the values for 13 2 and 1.3 2. The digits are the same, but 1.3 2 has two decimal
places (1.69), while 13 2 has none (169).
_
• How do you know whether √ 2 will be closer to 1 or closer to 2? It will be closer to
1 because 2 is between the perfect squares of 1 and 4, but closer to 1 than it is to 4.
Connect Vocabulary
EL
Explain to students that the word irrational, when used as an ordinary word in English,
means without logic or reason. In mathematics, when we say that a number is irrational it
means only that the number cannot be written as the quotient of two integers.
Engage with the Whiteboard
9
Have students extend the number line in both directions and label the locations of
the whole numbers _1 and 2. These are the roots of the consecutive perfect squares
1 and 4 used to estimate √ 7.
Lesson 1.1
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3
__
25
_
5. 0.57
19
__
33
_
_
3
√ 729 = √ x3
3
Write each decimal as a fraction in simplest form.
4. 0.12
729 = x3
C
YOUR TURN
6. 1.4
Personal
Math Trainer
1_25
Solve for x by taking the cube root of both sides.
_
3
√
729 = x
Online Practice
and Help
Apply the definition of cube root.
9=x
my.hrw.com
Think: What number cubed equals 729?
The solution is 9.
Finding Square Roots and Cube Roots
8
x3 = ___
125
D
The square root of a positive number p is x if x = p. There are two square
roots for every positive number. For example, the square roots of 36 are
1
__
_1
_1
6 and −6 because 62 = 36 and (−6)2 = 36. The square roots
_ of 25 are 5 and − 5.
1
You can write the square roots of __
as ±_15. The symbol √5 indicates the positive,
25
or principal square root.
√
Math On the Spot
my.hrw.com
()
Apply the definition of square root.
x = ±11
Think: What numbers squared equal 121?
The solutions are 11 and −11.
B
16
x = ___
169
EXPLORE ACTIVITY
_
16
x = ±√ ___
169
4
x = ±__
13
Solve for x by taking the square root of both sides.
Apply the definition of square root.
16
Think: What numbers squared equal ____
?
169
4
4
The solutions are __
and −__
.
13
13
Mathematical Practices
Can you square an integer
and get a negative number?
What does this indicate
about whether negative
numbers have square
roots?
No; the square of a
positive integer is
positive, the square
of a negative integer
is positive, and the
square of 0 is 0. So
negative numbers do
not have (real) square
roots.
Lesson 1.1
8_MCAAESE206984_U1M01L1.indd 9
x = _47
64
10. x3 = ___
343
8.NS.2, 8.EE.2
Estimating Irrational Numbers
Math Talk
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.
Square roots of perfect squares are rational numbers. Square roots of numbers
that are not perfect squares are irrational. Some equations like those in
Example 3 involve square roots of numbers that are not perfect squares.
9
4/19/13 2:40 PM
_
x2 = 2
2
16
x2 = ___
169
x=8
9. x3 = 512
my.hrw.com
3
x = ±__
16
9
8. x2 = ___
256
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
_
x = ±√121
x = ±14
7. x2 = 196
Online Practice
and Help
8.EE.2
Solve for x by taking the square root of both sides.
Solve each equation for x.
Personal
Math Trainer
Solve each equation for x.
x2 = 121
8
Think: What number cubed equals ____
?
125
The solution is _25.
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.
x2 = 121
Apply the definition of cube root.
x = _25
YOUR TURN
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
_
3
1
1
is _13 because _13 = __
. The symbol √
The cube root of __
1 indicates the
27
27
cube root.
A
Solve for x by taking the cube root of both sides.
_
8
x = ___
125
3
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.
EXAMPL 3
EXAMPLE
_
_
3 3
8
3 ___
√
x = √
125
2
_
√ 2 is irrational.
x = ± √2
_
Estimate the value of √2.
A Find two consecutive perfect squares that 2 is between. Complete
the inequality by writing these perfect squares in the boxes.
B Now take the square root of each number.
C Simplify the square roots of perfect squares.
_
√ 2 is between
10
1
and
2
1
_
.
√
< 2 <
_
√2
1
<
1
< √2 <
_
<
4
_
√
4
2
Unit 1
8_MCAAESE206984_U1M01L1.indd 10
4/16/13 12:11 AM
DIFFERENTIATE INSTRUCTION
Critical Thinking
Modeling
Additional Resources
In the Explore_Activity, students estimated the
location of √2 on a number line. Ask students
whether they think that it is possible to locate
_
more precisely the point that represents √ 2. In
other words, can you graph irrational numbers
exactly on a number line, along with rational _
numbers? Students should understand that √2
is a real number, and all real numbers can be
located on a real number line. A more precise
estimate will allow more precise placement on a
number line.
Have students use a ruler to represent a number
line with a unit that is one inch long. Have them
draw a square with a side of one inch, and draw
the diagonal to make two isosceles triangles.
Lead students to understand that_the length of
the diagonal (or hypotenuse) is √2 .
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
Have them copy the length of their diagonal
onto their ruler, or number line, starting at zero.
The end point of the diagonal represents
_ the
exact point for the irrational number √ 2 on a
number line.
The Modeling note tells one way to do this.
Rational and Irrational Numbers
10
Elaborate
Talk About It
Summarize the Lesson
Ask: If someone claims that a certain number is irrational but you know it is
actually rational, how could you prove to that person that the number is rational?
You could find a fraction equal to the number, such that the number is the ratio of two
integers, with the denominator not equal to zero.
GUIDED PRACTICE
Engage with the Whiteboard
Have students plot each number in Exercises 16–18 on a number line. Students
should label each point with the irrational number written as a radical and as a
decimal.
Avoid Common Errors
Exercises 1–6 To avoid reversing the order of the dividend and divisor, tell students to start
at the top of the fraction and read the bar as “divided by.”
Focus on Technology
Have students use a calculator to investigate the decimal equivalents of such fractions as
10
1 __
__1 , __2 , ..., __8 and __
. Ask them to describe the patterns they find as a result of these
, 2 , ..., __
11 11
11
9 9
9
investigations.
11
Lesson 1.1
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√2 ≈ 1.5
0
1
2
3
Guided Practice
4
E 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.
1.69
1.32 =
1.42 =
1.96
Write each fraction or mixed number as a decimal.
_ (Example 1)
No; 2 is not between 1.69 and 1.96.
2.25
0.8
3. 3 _34
5. 2 _38
2.375
6. _56
_
1.4
-x
.
9
_
4.4
-
x=
√2 ≈ 1.4
_
0.4
_
1.41 =
© Houghton Mifflin Harcourt Publishing Company
1.9881
1.42 =
than to
1.43 =
_
x=
_
1000x =
-x
325.325
-
_
0.325
_______________________
x=
999
26
26
__
99
325
325
___
999
x=
25
14. x2 = ___
289
√
17
√
≈±
4.1
__
25
5
289
17
x=
x = ± __________ = ± _____
2.0449
2.0164 , so √_2 ≈ 1.41
15. x3 = 216
__
12. Find a better estimate of √2.
2 is closer to
0.26
x=
__
x= ±
2
-x -
4
_
9
13. x2 = 17
Test the squares of numbers between 1.4 and 1.5.
2.0164
26.26
Solve each equation for x. (Example 3 and Explore Activity)
_
2
_
100x =
99
11
__
25
12. 0.325
___________________
4
x=
11. How could you find an even better estimate of √2?
1.9881
_
0.83
_
11. 0.26
_______________
1.1 1.2 1.3 1.4 1.5
2
3.75
9. 0.44
_
10x =
, so √ 2 ≈
5 _35
8. 5.6
_
10. 0.4
F Locate and label this value on the number line.
Reflect
27
__
40
7. 0.675
Yes; 2 is between 1.96 and 2.25.
than to
2. _89
Write each decimal as a fraction or mixed number in simplest form. (Example 2)
_
Is √2 between 1.4 and 1.5? How do you know?
1.96
0.7
7
4. __
10
_
Is √2 between 1.3 and 1.4? How do you know?
2 is closer to
0.4
1. _25
2.25
1.52 =
√
3
216
=
6
Approximate each irrational number to one decimal place without a calculator.
(Explore Activity)
.
Draw a number line and locate and label your estimate.
16.
√2 ≈ 1.41
?
1.41 1.42 1.43 1.44 1.45
13. Solve x = 7. Write your answer as a radical expression. Then estimate
to one decimal place.
_
x = ±√7 ; x ≈ ±2.6
2
_
√5 ≈
2.2
17.
_
√3 ≈
1.7
18.
_
√ 10 ≈
3.2
ESSENTIAL QUESTION CHECK-IN
19. What is the difference between rational and irrational numbers?
Rational numbers can be written in the form __ba , where
© Houghton Mifflin Harcourt Publishing Company
_
D Estimate that √2 ≈ 1.5.
a and b are integers and b ≠ 0. Irrational numbers cannot
be written in this form.
Lesson 1.1
8_MCAAESE206984_U1M01L1.indd 11
11
4/16/13 12:11 AM
12
Unit 1
8_MCAAESE206984_U1M01L1.indd 12
4/16/13 12:11 AM
Rational and Irrational Numbers
12
Personal
Math Trainer
Online Assessment
and Intervention
Online homework
assignment available
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1.1 LESSON QUIZ
8.NS.1, 8.NS.2, 8.EE.2
7
1. Write as a decimal: 2__58 , 1__
12
_
2. Write as a fraction: 0.34, 1.24
9
3. Solve x 2 = __
for x.
49
4. Solve x 3 = 216 for x.
5.
_
Estimate the value of √ 13 to one
Evaluate
GUIDED AND INDEPENDENT PRACTICE
8.NS.1, 8.NS.2, 8.EE.2
Concepts & Skills
Practice
Example 1
Expressing Rational Numbers as Decimals
Exercises 1–6, 20–21, 24–25
Example 2
Expressing Decimals as Rational Numbers
Exercises 7–12, 22–23, 26–27
Example 3
Finding Square Roots and Cube Roots
Exercises 13–15, 28, 30–31, 35
Explore Activity
Estimating Irrational Numbers
Exercises 13, 16–18, 29, 32–34
decimal place without using a
calculator.
Lesson Quiz available online
Exercise
my.hrw.com
Answers
_
1. 2.625, 1.583
17 __
2. __
,18
50 33
3. x = ±__3
4. x = 6
7
5. 3.6
Depth of Knowledge (D.O.K.)
MP.4 Modeling
3 Strategic Thinking
MP.4 Modeling
2 Skills/Concepts
MP.6 Precision
33
3 Strategic Thinking
MP.7 Using Structure
34
2 Skills/Concepts
MP.3 Logic
35
2 Skills/Concepts
MP.4 Modeling
36
3 Strategic Thinking
MP.3 Logic
37
3 Strategic Thinking
MP.7 Using Structure
38
3 Strategic Thinking
MP.2 Reasoning
28
29–32
Differentiated Instruction includes:
• Leveled Practice worksheets
Lesson 1.1
Mathematical Practices
2 Skills/Concepts
20–27
Additional Resources
13
Focus | Coherence | Rigor
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Class
_
Date
1.1 Independent Practice
Personal
Math Trainer
8.NS.1, 8.NS.2, 8.EE.2
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7
-inch-long bolt is used in a machine.
20. A __
16
What is this length written as a decimal?
Online Practice
and Help
_
0.16
24. A heartbeat takes 0.8 second. How many
seconds is this written as a fraction?
25. There are 26.2 miles in a marathon. Write
the number of miles using a fraction.
26. The average
score on a biology test
_
was 72.1. Write the average score using a
fraction.
27. The metal in a penny is worth about
0.505 cent. How many cents is this written
as a fraction?
4
_
second
5
26 _15 mi
101
___
cent
200
34. Justify Reasoning What is a good estimate for the solution to the
equation x3 = 95? How did you come up with your estimate?
Sample answer: about 4.5; 43 = 64 and 53 = 125.
Because 95 is about halfway between 64 and 125, try 4.5.
4.53 = 91.125, which is a good estimate.
35. 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.
3 feet
FOCUS ON HIGHER ORDER THINKING
Yes; the cube root of a negative number is negative,
because a negative number cubed is always negative,
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? How many solutions
does the equation have?
and a nonnegative number cubed is always nonnegative.
x = 400; x = ± 20; the equation has 2 solutions.
2
37. Make a Conjecture Evaluate and compare the following expressions.
_
painting cannot have a side length of -20 inches.
_
_
√ 16
16
_
and ____
√__
81
√ 81
16
4
16
4
_
_
= _25 = ____
= _4 = ____
√__
√___
25
81
√ 25
√ 81
_ _
_
_ 9
a
a
___
_ = __ ; √ a · √ b = √ a · b
√b
√b
√
c. What is the length of the wood trim needed to go around the painting?
√
4 × 20 = 80 inches
_
_
√ 36
36
_
and ____
√__
49
√ 49
Lesson 1.1
√
√ 36
36
_;
= _67 = ____
√__
49
√ 49
38. 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.
Solve each equation for x. Write your answers as radical expressions. Then
estimate to one decimal place, if necessary.
_
_
3
29. x2 = 14 x = ±√ 14 ≈ ±3.7
30. x3 = 1331 x = √ 1331 = 11
_
_
x = ±√ 29 ≈ ±5.4
31. x2 = 144 x = ±√ 144 = ±12
32. x2 = 29
225; the solutions to x2 = a are x = ±15, and
15 - (-15) = 30.
13
4/16/13 12:11 AM
PRE-AP
√
Use your results to make a conjecture about a division rule for square
roots. Since division is multiplication by the reciprocal, make a conjecture
_
_
_square roots._
about
rule for
_ a multiplication
_
x = 20 makes sense, but x = -20 doesn’t, because a
EXTEND THE MATH
_
4
4
_
and ____
√__
25
√ 25
b. Do all of the solutions that you found make sense in the
context of the problem? Explain.
8_MCAAESE206984_U1M01L1.indd 13
Work Area
36. Draw Conclusions Can you find the cube root of a negative number? If
so, is it positive or negative? Explain your reasoning.
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
wood trim that is needed to go around the painting.
© Houghton Mifflin Harcourt Publishing Company • ©Photodisc/Getty Images
would be 3.8 or 3.9.
6
23. A baseball pitcher has pitched 98 _32 innings.
What is the number of innings written as a
decimal?
_
_
_
so √15 is very close to √16 , or 4. A better estimate
98.6 innings
2.8 km
72 _19
His estimate is low because 15 is very close to 16,
21. The weight of an object on the moon is _16
its weight on Earth. Write _1 as a decimal.
0.4375 in.
22. The distance to the nearest gas station is
2 _54 kilometers. What is this distance written
as a decimal?
2
2
33. Analyze Relationships To find √15, Beau
_found 3 = 9 and 4 = 16. He
said that since 15 is between_
9 and 16, √15 must be between 3 and 4. He
3+4
thinks a good estimate for √15 is ____
= 3.5. Is Beau’s estimate high, low,
2
or correct? Explain.
© Houghton Mifflin Harcourt Publishing Company • © Ilene MacDonald/Alamy Images
Name
Activity available online
14
Unit 1
8_MCABESE206984_U1M01L1.indd 14
10/29/13 11:42 PM
my.hrw.com
_
Activity Write √0.9 on the board and invite students to conjecture what the value
might be. Have them
_check their conjectures by squaring. Invite them to suggest
ways to estimate √0.9. As a hint, point out that 0.9 is close to 1.0, and so they might
use that to help guide _
their estimates. Lead them to see that, since 0.92 is 0.81 and
2
√
1.0 is 1, the value of 0.9 is greater
_ than 0.9 and less than 1.0. Try squaring 0.95 to get
0.9025. A good estimate for √ 0.9 is 0.95.
Rational and Irrational Numbers
14
LESSON
1.2
Sets of Real Numbers
Lesson Support
Content Objective
Language Objective
Students will learn to describe relationships between sets of numbers.
Students will explain how to describe relationships between sets of real numbers.
California Common Core Standards
8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal
expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually
into a rational number.
MP.7 Look for and make use of structure.
Focus | Coherence | Rigor
Building Background
Eliciting Prior Knowledge Have students draw a number
line from -5 to 5. Ask them to plot points on the number line to
approximate the location_of rational and irrational numbers,
such as -1, __34 , 2.5, -4__23 , √2 , and -π.
√2
-1
3
4
2.5
-5-4 -3-2-1 0 1 2 3 4 5
Learning Progressions
Cluster Connections
In this lesson, students clarify their understanding of the real
number system. They characterize sets and subsets of the real
numbers. They also identify sets for real-world situations.
Important understandings for students include the following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Know that there are numbers that are not
rational, and approximate them by rational numbers. Have
students copy this diagram, which relates the sets of real
numbers.
• Identify all of the possible subsets of the real numbers for
a given number.
• Decide whether a statement about a subset of the real
numbers is true or false.
• Identify the set of numbers that best describes a realworld situation.
Understanding the relationships among the sets of numbers
that make up the real numbers is essential as students are
introduced to different forms of numbers throughout the school
year. This lesson provides a foundation for the comparing and
ordering of real numbers in the next lesson.
15A
-4 23 -π
Real Numbers
Rational Numbers
Irrational
Numbers
Integers
Whole
Numbers
Ask students to complete the diagram by writing three
examples for each set of numbers. Have students share
examples and explain how they knew each number they
selected belonged in the appropriate set. Answers may vary.
Check students’ work.
PROFESSIONAL DEVELOPMENT
Language Support
EL
California ELD Standards
Emerging 2.II.5. Modifying to add details – Expand sentences with simple adverbials to provide details about a familiar
activity or process.
Expanding 2.II.5. Modifying to add details – Expand sentences with adverbials to provide details about a familiar or new
activity or process.
Bridging 2.II.5. Modifying to add details – Expand sentences with increasingly complex adverbials to provide details about a
variety of familiar and new activities and processes.
Linguistic Support
EL
Academic/Content Vocabulary
Rules and Patterns
Venn diagrams – Students need descriptive
language to describe the categories that the different
areas and colors of a Venn diagram represent, the
concept of a set, and how sets are distinct or can
overlap. Use sentence frames, such as:
Abbreviations – In this lesson, the abbreviation mph
is used. Be sure to point out that mph stands for miles
per hour and is used to give units in a rate of speed.
Students may also have seen mpg (miles per gallon),
which gives the units in a rate of fuel efficiency.
The big oval represents __________.
The dark/light blue color in the middle of the
big ovals represents __________.
These sets overlap because __________.
Borrowed Words – Terminology used in baseball,
such as inning and pitcher, may require some
explanation. Spanish, as well as some other
languages, have borrowed these terms from English,
so some students may be familiar with these words
already. Despite this, whenever a word is critical to
students understanding the word problem, it is best
to explain the meaning.
In this way, students have the language and structure
to identify the criteria that distinguish a set and to
explain the abstract representation. Also point out the
use of the prefix sub-, meaning “under,” in the term
subset.
Leveled Strategies for English Learners
EL
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10.
Expanding Have students use sentence frames to describe the meaning of regions and colors
used in a Venn diagram. Then give them similar sentence frames orally and have them draw and
shade a Venn diagram based on the oral prompts.
Bridging Have students work in groups to draw a Venn diagram to represent sets based on
real-world examples in the lesson.
Math Talk
To help students answer the question posed in Math Talk, provide a sentence
frame for their answer.
The numbers between 3.1 and 3.9 on a number line
are __________ because __________.
Sets of Real Numbers
15B
LESSON
1.2 Sets of Real Numbers
CA Common Core
Standards
The student is expected to:
The Number System—8.NS.1
Know that numbers that are not rational are called
irrational. Understand informally that every number has
a decimal expansion; for rational numbers show that
the decimal expansion repeats eventually, and convert a
decimal expansion which repeats eventually into a
rational number.
Mathematical Practices
MP.7 Using Structure
ADDITIONAL EXAMPLE 1
Write all names that apply to each
number.
A -10
integer, rational, real
12
B _
3
whole, integer, rational, real
Interactive Whiteboard
Interactive example available online
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Animated Math
Classifying Numbers
Students build fluency in classifying
numbers in this engaging, fast-paced
game.
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ADDITIONAL EXAMPLE 2
Tell whether the given statement is
true or false. Explain your choice.
Engage
ESSENTIAL QUESTION
How can you describe relationships between sets of real numbers? Sample answer: Describe
them as two different sets, or one set as being a subset of another.
Motivate the Lesson
Ask: How many different types of tigers can you name? How does the set of Bengal tigers
relate to the set of tigers?
Explore
Point to different locations in the Animals diagram and ask for examples for that
classification. Do the same for the Real Numbers diagram. Students should understand that
everything within a region is part of the set, for example both -3 and 2 are integers.
Explain
EXAMPLE 1
Questioning Strategies
Mathematical Practices
• In A, why is 5 not a perfect square? It does not have rational numbers as its square roots.
• Can the number in B be written as a fraction? Why or why not? Yes; it is a terminating
decimal, so it is a rational number.
Engage with the Whiteboard
Have students place the numbers in Example 1 and Additional Example 1 in the
Venn diagram for numbers.
YOUR TURN
Avoid Common Errors
Be sure that students read Exercise 2 carefully before answering. The number given in the
problem, 10, is the area, not the side length.
EXAMPLE 2
Questioning Strategies
Mathematical Practices
• What two major sets are the real numbers composed of? rational and irrational numbers
No integers are whole numbers.
• What is the location of the set of whole numbers in the Venn diagram in relation to the set
of rational numbers? Explain. Inside it; whole numbers are rational numbers.
False; every whole number is also an
integer.
Focus on Reasoning
Interactive Whiteboard
Interactive example available online
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15
Lesson 1.2
Mathematical Practices
Remind students that it takes only one counterexample to show that a statement is false.
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Sets of Real
Numbers
1.2
?
8.NS.1
YOUR TURN
Know that numbers that are not rational
are called irrational. Understand informally
that every number has a decimal expansion;
for rational numbers show that the decimal
expansion repeats eventually, and convert a
decimal expansion which repeats eventually
into a relation number.
ESSENTIAL QUESTION
Personal
Math Trainer
Write all names that apply to each number.
1. A baseball pitcher has pitched 12_23 innings.
Online Practice
and Help
rational, real
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2. The length of the side of a square that has an
How can you describe relationships between sets of real numbers?
area of 10 square yards.
Classifying Real Numbers
Animals
Biologists classify animals based on shared
characteristics. A cardinal is an animal, a vertebrate,
a bird, and a passerine.
Vertebrates
Understanding Sets and Subsets
of Real Numbers
Birds
Math On the Spot
Passerines
You already know that the set of rational numbers
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.
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By understanding which sets are subsets of types of numbers, you can verify
whether statements about the relationships between sets are true or false.
Math On the Spot
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EXAMPLE 2
Real Numbers
Rational Numbers
27
4
7
Whole
Numbers
-1
0
1
√4
A All irrational numbers are real numbers.
Sample answer:
8; 8 = _81
√17
-2
Passerines, such
as the cardinal,
are also called
“perching birds.”
- √11
√2
3
B No rational numbers are whole numbers.
Give an example of a
rational number that is a
whole number. Show that
the number is both whole
and rational.
EXAMPL 1
EXAMPLE
_
√5
8.NS.1
YOUR TURN
Tell whether the given statement is true or false. Explain your choice.
5 is a whole number that is not a perfect square.
irrational, real
B –17.84
rational, real
_
√ 81
C ____
9
Animated
Math
3. All rational numbers are integers.
False. Every integer is a rational number, but not every
rational number is an integer. Rational numbers such as _35
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–17.84 is a terminating decimal.
_
√ 81
9
_____
= __
=1
9
9
whole, integer, rational, real
False. A whole number can be written as a fraction with a denominator
of 1, so every whole number is included in the set of rational numbers.
The whole numbers are a subset of the rational numbers.
Mathematical Practices
Write all names that apply to each number.
A
True. Every irrational number is included in the set of real numbers.
The irrational numbers are a subset of the real numbers.
Math Talk
π
4.5
8.NS.1
Tell whether the given statement is true or false. Explain your choice.
Irrational
Numbers
-6
Integers
-3
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Wikimedia Commons
0.3
irrational, real
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Digital Image copyright ©2004 Eyewire
LESSON
rational,
irrational, real
and - _52 are not integers.
Math Talk
Mathematical Practices
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What types of numbers are
between 3.1 and 3.9 on a
number line?
Online Practice
and Help
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Lesson 1.2
8_MCABESE206984_U1M01L2.indd 15
16
15
06/11/13 11:44 AM
4. Some irrational numbers are integers.
False. Real numbers are either rational or irrational numbers.
Integers are rational numbers, so no integers are irrational numbers.
Unit 1
8_MCAAESE206984_U1M01L2.indd 16
4/16/13 1:36 AM
PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.7
This lesson provides an opportunity to address
this Mathematical Practices standard. It calls for
students to discern structure to connect and
communicate mathematical ideas.
Students use a Venn diagram to structure
relationships between sets of numbers. They
connect and communicate mathematical ideas
when they make logical statements about the
sets and describe which set best describes
numbers applied to real-life situations.
Math Background
The relationships between sets of numbers
extend to include complex numbers. A complex
number can be written as a sum of a real
number, a, and an imaginary number, bi.
a + bi
An imaginary number is a special number that,
when squared gives a negative value. When you
square a real number, you get a nonnegative
number. When you square an imaginary number,
you get a negative value. The imaginary unit is i.
i=
_
√ -1
Sets of Real Numbers
16
YOUR TURN
Avoid Common Errors
Students may see the word “All“ or ”No” in Exercises 3 and 4 and immediately assume that
any absolute statements like these are false. Remind them that there are true statements
that begin with these words, and encourage them to provide examples.
ADDITIONAL EXAMPLE 3
Identify the set of numbers that
best describes the situation.
Explain your choice.
A the amount of time that has passed
since midnight
The set of real numbers; time is
continuous, so the amount of time
can be rational or irrational.
B the number of tickets sold to a
basketball game
The set of whole numbers; the
number of tickets sold may be 0 or
a counting number
Interactive Whiteboard
Interactive example available online
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EXAMPLE 3
Questioning Strategies
Mathematical Practices
• In A, how does the phrase “number of ” give you a clue about the number classification?
It indicates a counting number.
• What is the relationship between the circumference of a circle and the diameter?
The circumference is diameter times π.
Focus on Critical Thinking
Mathematical Practices
25 __
31
28 __
In B, suppose the diameters in inches were __
π , π , π , and so on. What set of numbers would
best describe the circumferences? Explain. Whole numbers; the circumferences would be
the whole numbers 25, 28, 31, and so on.
YOUR TURN
Focus on Critical Thinking
Mathematical Practices
Have students compare and contrast the classification of numbers in the answers in
Exercises 5 and 6.
Elaborate
Talk About It
Summarize the Lesson
Ask: What are some ways that number sets can be related? Sets may be subsets of
other sets or they may be separate from other sets.
GUIDED PRACTICE
Engage with the Whiteboard
Have students place the numbers in Exercises 1– 8 in the Venn diagram for numbers
at the beginning of the lesson.
Integrating Language Arts
EL
Encourage English learners to ask for clarification on any terms or phrases that they do not
understand.
Avoid Common Errors
Exercise 7 Remind students that a repeating decimal is a rational number.
Exercises 9–10 Remind students that it only takes one counterexample to show that a
statement is false.
17
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Guided Practice
Identifying Sets for Real-World
Situations
Write all names that apply to each number. (Example 1)
Real numbers can be used to represent real-world quantities. Highways have
posted speed limit signs that are represented by natural numbers such as
55 mph. Integers appear on thermometers. Rational numbers are used in many
daily activities, including cooking. For example, ingredients in a recipe are often
given in fractional amounts such as _23 cup flour.
EXAMPL 3
EXAMPLE
Math On the Spot
2.
3.
_
√ 24
4. 0.75
rational, real
irrational, real
_
5. 0
whole, integer, rational, real
My Notes
_
integer, rational, real
integer, rational, real
rational, real
The set of whole numbers best describes the situation. The number of
people wearing glasses may be 0 or a counting number.
6. - √ 100
18
8. - __
6
7. 5.45
A the number of people wearing glasses in a room
_
√ 36
whole, integer, rational, real
rational, real
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8.NS.1
Identify the set of numbers that best describes each situation. Explain
your choice.
1. _78
Tell whether the given statement is true or false. Explain your choice.
(Example 2)
B the circumference of a flying disk has a diameter of 8, 9, 10, 11, or
14 inches
9. All whole numbers are rational numbers.
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.
True. Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1.
10. No irrational numbers are whole numbers.
True. Whole numbers are rational numbers.
Identify the set of numbers that best describes each situation. Explain your
choice. (Example 3)
YOUR TURN
11. the change in the value of an account when given to the nearest dollar
© Houghton Mifflin Harcourt Publishing Company
and can be positive, negative, or zero.
5. the amount of water in a glass as it evaporates
Real numbers; the amount can be any number greater
12. the markings on a standard ruler
1
th inch.
Rational numbers; the ruler is marked every __
16
than 0.
?
6. the weight of a person in pounds
IN.
1
ESSENTIAL QUESTION CHECK-IN
13. What are some ways to describe the relationships between sets of
numbers?
Rational numbers; a person’s weight can be a decimal
such as 83.5 pounds.
1
inch
16
© Houghton Mifflin Harcourt Publishing Company
Integers; the change can be a whole dollar amount
Identify the set of numbers that best describes the situation. Explain
your choice.
Sample answer: Describe one set as being a subset of
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another, or show their relationships in a Venn diagram.
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8_MCAAESE206984_U1M01L2.indd 17
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4/16/13 5:20 AM
18
Unit 1
8_MCAAESE206984_U1M01L2.indd 18
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DIFFERENTIATE INSTRUCTION
Graphic Organizers
Number Sense
Additional Resources
Give students a list of numbers (including
terminating and repeating decimals, fractions,
integers, and rational and irrational square roots)
and a graphic organizer as shown below.
Point out to students that knowing the types of
numbers to expect in different situations can
alert them to incorrect math as well as to
impossible situations. For example, 13.5 shots
made in basketballs is not possible, but an
average number of shots can equal 13.5.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
Real Numbers
Rational numbers
Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in
the correct section of the organizer.
Sets of Real Numbers
18
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Online Assessment
and Intervention
Online homework
assignment available
Evaluate
Focus | Coherence | Rigor
GUIDED AND INDEPENDENT PRACTICE
8.NS.1
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1.2 LESSON QUIZ
8.NS.1
1. Write all the names
_ that apply to
the number -1.5.
2. Tell whether the given statement is
true or false. Explain your choice.
All numbers between 1 and 2 are
rational numbers.
3. Identify the set of numbers that
best describes the situation. Explain
your choice.
The choices on a survey question
change the total points for the survey
by -2, -1, 0, 1, or 2 points.
Lesson Quiz available online
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Answers
1. rational, real
_
2. False; √2 is an example of an irrational
number between 1 and 2
3. Integers; each number is an integer, but
only three are whole numbers.
Concepts & Skills
Practice
Example 1
Classifying Real Numbers
Exercises 1–8, 14–19, 22–24
Example 2
Understanding Sets and Subsets of Real
Numbers
Exercises 9–10
Example 3
Identifying Sets for Real-World Situations
Exercises 11–12, 20–21, 25
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
14–19
2 Skills/Concepts
MP.7 Using Structure
20–21
2 Skills/Concepts
MP.6 Precision
22–23
2 Skills/Concepts
MP.3 Logic
24
1 Recall of Information
MP.7 Using Structure
25
2 Skills/Concepts
MP.2 Reasoning
26–27
3 Strategic Thinking
MP.3 Logic
28
3 Strategic Thinking
MP.8 Patterns
29
3 Strategic Thinking
MP.3 Logic
Additional Resources
Differentiated Instruction includes:
• Leveled Practice worksheets
Exercise 29 combines concepts from the California Common Core
cluster “Know that there are numbers that are not rational, and
approximate them by rational numbers.”
19
Lesson 1.2
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Name
Class
Date
1.2 Independent Practice
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8.NS.1
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Online Practice
and Help
Write all names that apply to each number. Then place the numbers in the
correct location on the Venn diagram.
14.
_
-√9
integer, rational, real
16.
_
√ 50
18.
16.6 rational, real
15.
irrational, real
19.
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?
rational, real
_
√ 16
rational number
whole, integer, rational, real
Rational Numbers
Integers
1
26. Explain the Error Katie said, “Negative numbers are integers.” What was
her error?
Irrational Numbers
82
√50
The set of negative numbers also includes non-integer
rational numbers and irrational numbers.
Whole Numbers
√16
Work Area
FOCUS ON HIGHER ORDER THINKING
Real Numbers
16.6
π mi
It can be a rational number that is not an integer, or an irrational number.
257 whole, integer, rational, real
17. 8 _12
23. Critique Reasoning The circumference of a circular region is shown.
What type of number best describes the diameter of the circle? Explain
π
_
your answer. Whole; the diameter is π = 1 mile.
27. Justify Reasoning Can you ever use a calculator to determine if a
number is rational or irrational? Explain.
√9
Sample answer: If the calculator shows a decimal that
257
terminates in fewer digits than what the calculator screen
allows, then you can tell that the number is rational. If not,
you cannot tell from the calculator display whether the
Identify the set of numbers that best describes each situation. Explain
your choice.
number terminates because you see a limited number
© Houghton Mifflin Harcourt Publishing Company
Real numbers; the height can be any number greater than zero.
non-terminating non-repeating decimal (irrational).
_
21. the score with respect to par of several golfers: 2, – 3, 5, 0, – 1
_1
28. Draw Conclusions
_ The decimal _0.3 represents 3 . What type of number
best describes 0._9, which is 3 · 0.3? Explain.
_
Whole; 3 · 0.3 represents 3 · _13 = 1, so 0.9 is exactly 1.
Integers; the scores are counting numbers, their
opposites, and zero.
29. Communicate Mathematical Ideas Irrational numbers can never be
precisely represented in decimal form. Why is this?
1
22. Critique Reasoning Ronald states that the number __
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.
© Houghton Mifflin Harcourt Publishing Company
of digits. It may be a repeating decimal (rational), or
20. the height of an airplane as it descends to an airport runway
Sample answer: In decimal form, irrational numbers never
Nathaniel is correct. A rational number is a number that
1
can be written as a fraction, and __
11 is a fraction.
terminate and never repeat. Therefore, no matter how
many decimal places you include, the number will never
be precisely represented. There are always more digits.
Lesson 1.2
8_MCAAESE206984_U1M01L2.indd 19
EXTEND THE MATH
19
4/16/13 1:36 AM
PRE-AP
Activity available online
20
Unit 1
8_MCAAESE206984_U1M01L2.indd 20
12/04/13 9:09 PM
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Activity Have students consider the concept of restricted domain for the sets of
numbers that describe situations. For example, the number of sisters a person has
can best be described by whole numbers, but no one has ever had 1,500 sisters. An
area code is an integer or whole number between 200 and 999.
Have students use a source, such as the Guinness Book of World Records, and give
examples of sets of numbers that describe situations where the domain is restricted.
Ask whether the restriction may be changed in the future.
Sets of Real Numbers
20
LESSON
1.3
Ordering Real Numbers
Lesson Support
Content Objective
Language Objective
Students will learn to order a set of real numbers.
Students will show and describe how to order a set of real numbers.
California Common Core Standards
8.NS.2 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., π 2).
MP.4 Model with mathematics.
Focus | Coherence | Rigor
Building Background
Eliciting Prior Knowledge Have students draw a
number line to compare
a rational number and an irrational
_
number, such as -√5 and -4__12. Ask them to explain how
they approximated the irrational number on the number
line. Then have them identify the greater and the lesser real
number. Repeat with several other pairs of real numbers in
different forms.
-5-4 -3-2-1 0 1 2 3 4 5
Learning Progressions
Cluster Connections
In this lesson, students order a set of real numbers. They use
rational approximations to compare the sizes of irrational
numbers. They also order numbers for real-world situations.
Important understandings for students include the following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Know that there are numbers that are not
rational, and approximate them by rational numbers. Tell
students that there is a special number, called the golden
ratio, with applications in mathematics, geometry, art, and
architecture. The golden ratio is called phi and is represented by
the Greek letter ϕ. It includes an irrational number in its
definition.
•
•
•
•
Compare irrational numbers.
Estimate the value of expressions with irrational numbers.
Order a set of real numbers.
Order real numbers in a real-world context.
Work with real numbers continues throughout Grade 8 and into
high school. This lesson provides students with a foundation for
understanding the relative sizes of numbers in different forms in
the real number system.
21A
-4 12 -√5
_
1 + √5
ϕ=_
2
Have students explain why the golden ratio is irrational. Ask
them to find the two whole numbers the golden ratio lies
between. Then challenge them to approximate the golden ratio
to the nearest tenth. It is irrational because it includes an
irrational number in its definition. It lies between 1 and 2. To the
nearest tenth, ϕ = 1.6.
PROFESSIONAL DEVELOPMENT
Language Support
EL
California ELD Standards
Emerging 2.I.8. Analyzing language choices – Explain how phrasing or different common words with similar meanings
produce different effects on the audience.
Expanding 2.I.8. Analyzing language choices – Explain how phrasing or different words with similar meanings or figurative
language produce shades of meaning and different effects on the audience.
Bridging 2.I.8. Analyzing language choices – Explain how phrasing or different words with similar meanings or figurative
language produce shades of meaning, nuances, and different effects on the audience.
Linguistic Support
EL
Academic/Content Vocabulary
Background Knowledge
Post a chart like this to remind students of the regular
comparative forms of adjectives that use the -er and
-est suffixes. Add to the chart for terms that appear in
examples and exercises in each lesson. Include any
irregular verb forms.
Go On – the title of the module review or quiz is
Ready to Go On. This title uses an idiomatic expression.
In this context, to go on means “to move ahead” or
“to proceed.” It is different from the use of go on that
means having enough facts to use meaningfully,
as in having enough to go on. Also, the intonation
used in pronouncing an expression can give it
different meanings. For example, when the speaker
emphasizes the word on, he or she might be
expressing disbelief as in, “Go ON! You’re kidding,
right?” Discuss with students other ways that the
phrase go on may be used.
Adjective
Comparative
Superlative
Far
Farther
Farthest
Large
Larger
Largest
Great
Greater
Greatest
Some
Less
Least
Some
More
Most
Leveled Strategies for English Learners
EL
Emerging Label points on a number line with the terms used in ordering: greater, greatest, less,
lesser, least. Use sentence frames to insert the correct terms.
Expanding Have students give two or three complete sentences to compare the placement of
numbers on a number line using the correct forms of the comparative and superlative adjectives.
Bridging Have students work in pairs, with one student giving directions to the other in
complete sentences to order numbers on a number line.
Math Talk
To help students answer the question posed in Math Talk, make sure that
students have a command of the forms for making comparisons and the
superlative and the concept of opposite order so that the focus is on the math
concept instead of the language skills needed to describe and explain order.
Ordering Real Numbers
21B
LESSON
1.3 Ordering Real Numbers
CA Common Core
Standards
The student is expected to:
The Number System—8.NS.2
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. π 2)
Engage
ESSENTIAL QUESTION
How do you order a set of real numbers? Sample answer: Find their approximate decimal
values and order them.
Motivate the Lesson
Ask: What kind of numbers are you comparing when you compare the price of gasoline at
two different gas stations?
Mathematical Practices
Explore
MP.4 Modeling
Give students two rational numbers and ask them to name a number between them.
Repeat a few times and then give them two irrational numbers and ask them to name a
number between them.
Explain
ADDITIONAL EXAMPLE 1
Compare. Write <, >, or =.
A
B
_
√8 - 2
_
√ 20 + 1
_
√8
_
3 + √2
4-
<
>
Interactive Whiteboard
Interactive example available online
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EXAMPLE 1
Questioning Strategies
Mathematical Practices
_
_
√ 5 and √ 3?
• Which is greater, the difference between 5 and 3, or the difference
_
_ between
The difference between 5 and 3 is 2; the difference between √ 5 and √ 3 is approximately
1. So the difference between 5 and 3 is greater.
Avoid Common Errors
Caution students to read the problem carefully and think about what the radical sign means
so that they do not misread the problem and answer that the two sides are equal.
YOUR TURN
Focus on Technology
Calculators should not be used at this point, because developing number sense is the goal.
ADDITIONAL
_ EXAMPLE 2
Order 3π, √10, and 3.25 from greatest
to least.
_
3π, 3.25, √10
Interactive Whiteboard
Interactive example available online
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EXAMPLE 2
Questioning Strategies
Mathematical Practices
_
• How do you determine whether √22 is less than or greater than 4.5? The square of 4.5 is
20.25, which is less than 22, so the square root of 22 must be greater than 4.5.
Engage with the Whiteboard
Have students graph and label various real numbers between 4.2 and 4.4 and
between 4.7 and 5.
YOUR TURN
Focus on Modeling
Mathematical Practices
Have students label the integers on the number line with their equivalent
square
root.
_
_ _
For example, 1, 2, and 3 on the number line would be labeled √1, √4, and √ 9.
21
Lesson 1.3
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Ordering Real
Numbers
1.3
8.NS.2
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).
Ordering Real Numbers
You can compare and order real numbers and list them from least to greatest.
Math On the Spot
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EXAMPLE 2
STEP 1
How do you order a set of real numbers?
Between any two real numbers is another real number. To compare and order
real numbers, you can approximate irrational numbers as decimals.
STEP 1
STEP 2
8.NS.2
4.52 = 20.25
_
My Notes
STEP 2
+ 5 is between 6 and 7.
© Houghton Mifflin Harcourt Publishing Company
_
√3√5
0.5
1
1.5
2
Compare. Write <, >, or =.
4.
_
√ 12
+6
<
_
12 + √6
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8_MCAAESE206984_U1M01L3.indd 21
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6. π2, 10, √75
√75
8
8.5
9
2.5
3
3.5
4
Math Talk
Mathematical Practices
If real numbers a, b, and c
are in order from least to
greatest, what is the order
of their opposites from
least to greatest?
Explain.
_
√ 75 , π2, 10
_
YOUR TURN
>
5
Order the numbers from least to greatest. Then graph them on the
number line.
_ _
_
_
√ 3 , √ 5 , 2.5
5. √5 , 2.5, √3
0
3.
4.8
YOUR TURN
17 and 18
_
2 + √4
√22
4.6
Math Talk answer: -c, -b, -a; -c is farthest to the left on a
number line, -b is in the middle, and -a is farthest to the right.
_
_
_
√2 + 4
4.4
From least to greatest, the numbers are π + 1, 4 _12 , and √22 .
What are the closest two integers that √300 is between?
2.
4.2
_
_
_
1
42
Read the numbers from left to right to place them in order from
least to greatest.
If 7 + √5 is equal to √5 plus a number, what do you know about the
number? Why?
_
The number is 7; both expressions must equal 7 + √5 .
1.
4.82 = 23.04
_
4
So, √3 + 5 > 3 + √5 .
Reflect
_
Plot √22 , π + 1, and 4 _21 on a number line.
π+1
3 + √5 is between 5 and 6.
_
4.72 = 22.09
An approximate value of π is 3.14. So an approximate value
of π +1 is 4.14.
3 + √5 . Write <, >, or =.
Use perfect squares to estimate
square roots.
12 = 1 22 = 4 32 = 9
4.62 = 21.16
Since 4.72 = 22.09, an approximate value for √22 is 4.7.
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Then use your approximations to simplify the expressions.
_
√3
_
√ 22
Since 22 is closer to 25 than 16, use squares
_ of numbers between
4.5 and 5 to find a better estimate of √22 .
Math On the Spot
EXAMPL 1
EXAMPLE
_
First approximate √3 .
_
√ 3 is between 1 and 2.
_
Next approximate √5 .
_
√ 5 is between 2 and 3.
_
First approximate √22 .
is 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
1
_
you can compare it to 4 2 .
Comparing Irrational Numbers
_
from least to greatest.
2
ESSENTIAL QUESTION
Compare √3 + 5
8.NS.2
_
Order √22 , π + 1, and 4 _1
π2
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9.5 10 10.5 11 11.5 12
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Unit 1
8_MCAAESE206984_U1M01L3.indd 22
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PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.4
This lesson provides an opportunity to address
this Mathematical Practices standard. It calls for
students to model relationships using multiple
representations, including diagrams, graphs, and
language as appropriate. Students use multiple
representations when they use number lines to
estimate the locations of and order rational and
irrational numbers given as symbols.
Math Background
In this lesson, students estimate irrational
numbers in the form of square roots of nonperfect squares by finding two perfect squares
between which the number falls. A more precise
method involves
repeated division. For example,
_
to find √28 , find a whole number whose perfect
square is close to 28, such as 5. Divide 28 by that
number: 28 ÷ 5 = 5.6. Find the average of the
5 + 5.6
= 5.3. Continue
quotient and divisor: _____
2
dividing 28 by each result and averaging until
you get the desired accuracy.
Ordering Real Numbers
22
ADDITIONAL EXAMPLE 3
The diameter of a meteorite in
millimeters is calculated by four
different methods. Order the results
from least to greatest.
_
13
mm,
Joe: √18 mm, Lisa: __
3
4π
Pablo: 4.6 mm, Julien: __
mm
3
4π
13
Julien: __
mm, Lisa: __
mm,
3
3
_
Joe: √18 mm, Pablo: 4.6 mm
Interactive Whiteboard
Interactive example available online
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EXAMPLE 3
Questioning Strategies_
Mathematical Practices
• How can you verify that √ 28 is between 5.2 and 5.3? 5.2 2 = 27.04 and 5.3 2 = 28.09
_
• Explain how to determine which number is greater: 5.5 or 5.5. When the repeating decimal
is rounded to the nearest tenth or hundredth, you can see that it is greater.
Connect to Daily Life
Discuss how measuring across a canyon might involve different methods than measuring
along a road. Explain that measurements like these are often done using calculations that
approximate the distance.
YOUR TURN
Focus on Critical Thinking
Mathematical Practices
_
_
Discuss_with students which number is greater, 3.45 or 3.450? 3.45 or 3.455_
and why. Explain
that 3.45 can be written out as 3.4545…Make sure they understand that 3.45 is greater than
3.45, but less than 3.455.
Elaborate
Talk About It
Summarize the Lesson
Ask: How can you order two numbers in different forms whose decimal approximations appear to be equal? Approximate one or both numbers to an additional
number of decimal places.
GUIDED PRACTICE
Engage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9.
Allow the points to be in any format other than decimal.
Avoid Common Errors
Exercises 3–4 Caution students to read the problem carefully so that they do not misread
the problem as the same numbers combined by addition on each side of the circle.
Exercise 10 Remind students that the calculations have units.
23
Lesson 1.3
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Guided Practice
Ordering Real Numbers in
a Real-World Context
Compare. Write <, >, or =. (Example 1)
EXAMPL 3
EXAMPLE
Math On the Spot
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8.NS.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.
My Notes
Distance Across Quarry Canyon (km)
Juana
Lee Ann
_
√ 28
Ryne
Jackson
5.5
5_12
_
23
__
4
1.
_
√3
+2
<
_
√3 + 3
2.
_
√ 8 + 17
3.
_
√6
+5
<
6+
_
√5
4.
_
√9 + 3
5.
_
√ 17 - 3
7.
_
√7 + 2
23
__
= 5.75
4
_
5
5.2
1
5 2 5.5
0.5
5.4
5.6
5.8
6.28
1.5
2
3 + √ 11
_
1.75
.
.
2π
2.5
3
3.5
4
4.5
5
5.5
1.5
6
6.5
7
_
√3
,
,
.
2
5
Forest Perimeter (km)
Leon
_
√ 17
-2
Mika
Jason
Ashley
π
1 + __
2
12
___
2.5
5
6
?
_
23 km, 5.5 km, _
__
5 12 km, √28 km.
4
ESSENTIAL QUESTION CHECK-IN
11. Explain how to order a set of real numbers.
Sample answer: Convert each number to a decimal
YOUR TURN
equivalent, using estimation to find equivalents for
Four people have found the distance in miles across a crater using
different methods. Their results are given below.
_
irrational numbers. Graph each number on a number line.
_
10
√ 10
3_1
Jonathan: __
3 , Elaine: 3.45, José: 2 , Lashonda:
Order the distances
from greatest
_ to least.
_
10
mi, √10 mi
3_1 mi, 3.45 mi, __
2
>
, so √3 ≈
10. Four people have found the perimeter of a forest
using different methods. Their results are given
in the table. Order their calculations from
greatest to least. (Example 3)
_
π
12 km, √ 17 - 2 km
1+_
km, 2.5 km, __
From greatest to least, the distances are:
_
1
2π
23
4
_
14 - √8
_
1.8
and
From least to greatest, the numbers are
2
_
√3
<
_
√ 17 + 3
√3
0
_
23 _
, 5.5, and 5 _1 on a number line.
Plot √28 , __
√28
8.
1.7
π ≈ 3.14, so 2π ≈
_
4
6. 12 - √2
9+
_
5.5 is 5.555…, so 5.5 to the nearest hundredth is 5.56.
STEP 2
_
_
√5
_
√ 10 - 1
_
√ 3 is between
5 _12 = 5.5
<
9. Order √ 3 , 2π, and 1.5 from least to greatest. Then graph them on the
number line. (Example 2)
_
√28 is between 5.2 and 5.3. Since 5.32 = 28.09, an approximate
_
value for √28 is 5.3.
© Houghton Mifflin Harcourt Publishing Company
>
-2 +
Write each value as a decimal.
STEP 1
7.
>
_
√ 11 + 15
>
3
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Elena Elisseeva/Alamy Images
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
least values, depending upon the context.
Read the numbers from left to right for least to greatest.
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Read the numbers from right to left for greatest to least.
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Lesson 1.3
8_MCAAESE206984_U1M01L3.indd 23
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24
Unit 1
8_MCAAESE206984_U1M01L3.indd 24
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DIFFERENTIATE INSTRUCTION
Modeling
Multiple Representations
Additional Resources
Place papers around the room with the numbers
from 1 to 5, one per sheet. Give each student a
card showing a number between 1 and 5 in
different forms. Have students place his or her
card between the correct integers, and decide
where the number goes in relation to any
numbers already placed.
Give students a vertical number line, which
some students might find easier to use than a
horizontal one. Have them decide whether to
place points for rational and irrational numbers
above or below existing points.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
Ordering Real Numbers
24
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Online Assessment
and Intervention
Online homework
assignment available
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GUIDED AND INDEPENDENT PRACTICE
8.NS.2
Practice
Example 1
Comparing Irrational Numbers
Exercises 1–8
1. Compare. Write <, >, or =.
Example 2
Ordering Real Numbers
Exercises 9, 12–15, 18–21
2.
Example 3
Ordering Real Numbers in a Real-World Context
Exercises 10, 16–17
8.NS.2
_
√ 95 - 5
_
√ 62 - 2
_
Order 10.5, √105, and 3π + 1 from
greatest to least.
3. A length in centimeters is calculated
differently by four different people.
Order their calculations from least
to greatest.
5 π cm,
11 cm, Silvio: __
K.D.: __
2 _
3 _
Paula: 5.4 cm, Luis: √33 cm
Lesson Quiz available online
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Answers
_
2.
Focus | Coherence | Rigor
Concepts & Skills
1.3 LESSON QUIZ
1.
Evaluate
_
√ 95 - 5 < √ 62 - 2
_
√ 105, 3π + 1, 10.5
Lesson 1.3
Depth of Knowledge (D.O.K.)
Mathematical Practices
1 Recall of Information
MP.5 Using Tools
16
2 Skills/Concepts
MP.2 Reasoning
17
2 Skills/Concepts
MP.6 Precision
18–21
2 Skills/Concepts
MP.2 Reasoning
22
3 Strategic Thinking
MP.4 Modeling
23–24
3 Strategic Thinking
MP.3 Logic
12–15
Additional Resources
_
3. Silvio: __53 π cm, Paula: 5.4 cm,
_
11
K.D.: __
cm, Luis: √33 cm
2
25
Exercise
Differentiated Instruction includes:
• Leveled Practice worksheets
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Name
Class
Date
1.3 Independent Practice
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?
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8.NS.2
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Online Practice
and Help
Order the numbers from least to greatest.
14.
_
_
√8
___
√7
_, 2,
13.
2
_
√8
___
√
2 , 2, 7
11
_
√ 10 , π, 3.5
_
π, √10 , 3.5
_
_
√ 220 , -10, √ 100 , 11.5
_
15.
_
-10, √100 , 11.5, √220
_
9
√ 8 , -3.75, 3, _
4
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.
_
-3.75, _9 , √8 , 3
a. Between which two square roots of integers could you find this
number?
_
_
between √7 ≈ 2.65 and √8 ≈ 2.83
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.
b. Between which two square roots of integers can you find π?
_
_
between √9 = 3 and √10 ≈ 3.16
12.25 m2
4π m2, or approximately 12.6 m2
a. Find the area of the square.
b. Find the area of the circle.
22. Analyze Relationships There are several approximations used for π,
22
including 3.14 and __
7 . π is approximately 3.14159265358979 . . .
The circle would give her more space to plant because it has a
a. Label π and the two approximations on the number line.
larger area.
17. Winnie measured the length of her father’s ranch
four times and got four different distances.
Her measurements are shown in the table.
© Houghton Mifflin Harcourt Publishing Company
Work Area
FOCUS ON HIGHER ORDER THINKING
c. Compare your answers from parts a and b. Which garden would give
your sister the most space to plant?
1
2
3
_
√ 60
58
__
8
7.3
a. To estimate the actual length, Winnie first
approximated each distance to the nearest
hundredth. Then she averaged the four
numbers.
Using a calculator, find_Winnie’s estimate.
_
58
3
√ 60 ≈ 7.75, __ = 7.25, 7.3 ≈ 7.33, 7 _ =
5
8
3.14
Distance Across Father’s Ranch (km)
_
3.140
4
7 _35
x
than the two given approximations.
2; rational numbers can have the same location, and
.
irrational numbers can have the same location, but they
cannot share a location.
Sample answer: 3.7
_
_
Sample answer: √31
24. Critique Reasoning Jill says that 12.6 is less than 12.63. Explain her error.
She did not consider the repeating digit. 12.66. . .
Lesson 1.3
8_MCAAESE206984_U1M01L3.indd 25
EXTEND THE MATH
25
4/16/13 4:48 AM
PRE-AP
355
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
the fewest number of distinct points that need to be graphed? Explain.
Give an example of each type of number.
19. an irrational number between 5 and 7
3.143
c. Find a whole number x so that the ratio ___
113 is a better estimate for π
_
_
3.142
22; it is closer to π on the number line.
__
7
7.60, so the average
b. Winnie’s father estimated the distance across his ranch to be √56 km.
How does this distance compare to Winnie’s estimate?
_
They are nearly identical. √56 is approximately 7.4833…
_
22
7
b. Which of the two approximations is a better estimate for π? Explain.
is 7.4825 km.
18. a real number between √13 and √14
3.141
π
© Houghton Mifflin Harcourt Publishing Company Image Credits: ©3DStock/iStockPhoto.com
12.
_
115
√ 115 , ___
11 , and 10.5624
Neither student is correct. The answer
_
115
should be ___, 10.5624, √115 .
26
Unit 1
8_MCAAESE206984_U1M01L3.indd 26
Activity available online
21/05/13 8:01 AM
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Activity Have students investigate whether there are infinitely many numbers
between two numbers by giving examples for each of the following.
• Between any two rational numbers there is at least one other rational number.
Sample answer: 4.5 is between 4.1 and 4.8
• Between any two irrational numbers
is at least one rational number.
_ there_
Sample answer: 4.5 is between √11 and √ 29
• Between any two _
rational numbers there is at least one irrational number.
Sample answer: √ 11 is between 3.1 and 3.6
• Between any two _
irrational numbers
at least one irrational number.
_ there is_
Sample answer: √ 17 is between √11 and √29
Ordering Real Numbers
26
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Module Quiz
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Assess Mastery
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1.1 Rational and Irrational Numbers
Online Practice
and Help
Write each fraction as a decimal or each decimal as a fraction.
Use the assessment on this page to determine if students have
mastered the concepts and standards covered in this module.
7
​​​​ ​ 0.35
​ ​ ​ ​
1. ​__
20
2.
___
1.​27​​​​​
14​
__
​ ​ 11
​ ​ ​
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3. 1​_​78​ ​​ ​ 1.875
​ ​ ​ ​ ​ ​
Solve each equation for x.
3
4. x2​=​81​​ 9,​-9
​ ​ ​ ​ ​
Response to
Intervention
2
1
1​​
__
​ 1​​,​-__
5. x3​=​343​​ ​ ​ 7​ ​ ​
1 10
​ ​​​​ ​ ​ ​ ​10
​ ​
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​tenth?​​ ​ ​ ​ ​ ​ ​ ​ 14.1​ft
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
Intervention
Enrichment
1.2 Sets of Real Numbers
Write all names that apply to each number.
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instant scoring, feedback, and customized intervention
or enrichment.
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​√121​​​
π irrational,​real
​​​​​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​​ ​ ​ ​ ​ ​ ​ ​ ​
9. ​__
2
10. Tell​whether​the​statement​“All​integers​are​rational​numbers”​is​true​​
or​false.​Explain​your​choice.​
Online and Print Resources
Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies
• Success for English
Learners EL
EL
True;​integers​can​be​written​as​the​quotient​of​two​integers.
PRE-AP
1.3 Ordering Real Numbers
Compare. Write <, >, or =.
Extend the Math PRE-AP
Lesson Activities in TE
__
__
11. ​√8​​+​3​​​<​ ​8​+​​√3​​
© Houghton Mifflin Harcourt Publishing Company
Online Assessment
and Intervention
121
____​​​
​​whole,​integer,​rational,​real​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
8. ​____
Additional Resources
Assessment Resources includes
• Leveled Module Quizzes
Order the numbers from least to greatest.
__ ___
___
__
2
13. ​√99​​,​​π2,​9.​8​​​​​ π​ ,​9.​
​ ​8​​,​​​√​99​
​ ​​
__
___
12. ​√5​​+​11​ >
5​+​​√11​​​
___
√__1
__
1
_
​ ​ ​ ​​​​,​​0.​​ 2​​​,​​​4​​
1 _
1
14. ​ ​__
​​​,​​​ ​,​0.​2​​​​ ​ 25
25 4
___
√
__
ESSENTIAL QUESTION
15. How​are​real​numbers​used​to​describe​real-world​situations?
​
Sample​answer:​Real​numbers,​such​as​the​rational​​
number​​​_14​,​can​describe​amounts​used​in​cooking.
Module​1
8_MCAAESE206984_U1M01RT.indd 27
California Common Core Standards
27
Common Core Standards
Lesson
Exercises
1.1
1–7
8.NS.1, 8.NS.2, 8.EE.2
1.2
8–10
8.NS.1
1.3
11–14
8.NS.2
Unit 1 Module 1
27
4/15/13 11:13 PM
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Module 1
MIXed ReVIeW
Assessment Readiness
Scoring Guide
Item 3 Award the student 1 point for finding the edge length of
the cube and 1 point for correctly explaining how to use a cube
root to solve the problem.
Item 4 Award the student 1 point for determining that the
student is incorrect and 1 point for correctly justifying the
reasoning for this conclusion.
Assessment Readiness
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Online Practice
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___
1. Look at each number. Is the number between 2π and √52 ?
Select Yes or No for expressions A–C.
A. 6_23
Yes
No
5π
 
B. __
2
Yes
No
C. 3√5
Yes
No
__
11
.
2. Consider the number -  __
15
Choose True or False for each statement.
Additional Resources
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A. The number is rational.
B. The number can be written as
a repeating decimal.
C. The number is less than –0.8.
True
True
False
False
True
False
3. The volume of a cube is given by V = x , where x is the length of an edge
of the cube. A cube-shaped end table has a volume of 3_83 cubic feet. What
is the length of an edge of the end table? Explain how you solved this
problem.
3
1_12 ft; Sample answer: The equation x3 = 3_38 can be used
to find the edge length in feet. To solve the equation,
write the mixed number as a fraction greater than 1:
___
29
4. A student says that √83 is greater than __
. Is the student correct? Justify your
3
reasoning.
___
__
29
= 9.6.
No; Sample answer: √83 ≈ 9.1, and __
3
__
___
29
.
Because 9.1 < 9.6, √83 < __
3
28
© Houghton Mifflin Harcourt Publishing Company
27
. Then take the cube root of both sides: x = _32 = 1_12.
x3 = __
8
Unit 1
8_MCAAESE206984_U1M01RT.indd 28
24/04/13 9:46 AM
California Common Core Standards
Items
Grade 8 Standards
Mathematical Practices
1
8.NS.2
MP.7
2*
7.NS.2b, 7.NS.2d, 8.NS.1
MP.7
3
8.EE.2
MP.1, MP.4
4
8.NS.1, 8.NS.2
MP.3
Item 4 combines concepts from the California
Common Core cluster “Know that there are
numbers that are not rational, and approximate
them by rational numbers.”
* Item integrates mixed review concepts from previous modules or a previous course.
Real Numbers
28