Grade 8 Workbook — Chapter 1: Subsets of the Real Number System

SOL 8.NS.2 · Companion to Textbook Chapter 1

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 102.

PAGE 1 — Chapter opener

Chapter 1 · Subsets of the Real Number System

Standard 8.NS.2

In this chapter you will:

Describe the natural numbers, whole numbers, and integers
Describe the rational numbers in fraction, decimal, and radical form
Describe the irrational numbers and what makes a decimal irrational
Draw the nested diagram of the real number system
Classify any real number into every subset it belongs to
Give examples and non-examples of each subset

Words to know: subset · natural numbers (counting numbers) · whole numbers · opposite · integers · rational number · terminating decimal · repeating decimal · perfect square · irrational number · real numbers

Calculator note: a calculator cannot tell you which subset a number belongs to, so work these by hand. Know these perfect squares on sight: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$ .

PAGE 2 — Counting, then zero, then the opposites

1.1 Natural, Whole, and Integer

FIGURE: fig1-counting-to-integers.png (full width)

Fill in the blanks.

The natural numbers, also called the ____________ numbers, are $1, 2, 3, \dots$

The whole numbers are the natural numbers together with ______.

The integers are the whole numbers together with their ____________.

The symbol $\subset$ is read "____________________."

$\text{natural} \subset \underline{\hspace{2.5cm}} \subset \underline{\hspace{2.5cm}}$

Circle every number in each row that belongs to the named subset.

Subset	$3$	$0$	$-5$
1. Natural numbers
2. Whole numbers
3. Integers

A whole number that is NOT a natural number: ______
An integer that is NOT a whole number: ______
Why is every natural number also an integer?

PAGE 3 — Practice · naming subsets

Which Subsets Contain It?

List every subset from Lesson 1.1 that contains each number.

a) $7$	b) $0$	c) $-13$	d) $100$

Circle TRUE or FALSE, then give a reason.

a) Every whole number is a natural number.	TRUE / FALSE
b) Every integer is a whole number.	TRUE / FALSE
c) Every natural number is a whole number.	TRUE / FALSE
d) $0$ is an integer.	TRUE / FALSE

Reasons: _______________________________________________

Give a non-example.

a) an integer that is not a natural number: ______

b) a number on the number line that is not an integer: ______
$\text{natural numbers} \subset \underline{\hspace{2.5cm}} \subset \text{integers}$
A thermometer reads $-6\,^\circ\text{F}$ .

Which subset contains $-6$ ? _______________

Which do not, and why? _______________________________________________

PAGE 4 — Apply, reason, correct

Show What You Know · 1.1

Apply it. An elevator panel is labeled $-2$ for the second basement, $0$ for the lobby, and $1$ through $5$ for the upper floors.

Smallest subset containing every label: _______________

Why are the whole numbers not enough? _______________________________________________
Explain. Why do the whole numbers need $0$ but the counting numbers do not?
Find the error. A student says $-8$ is a whole number "because it has no fraction part."

What went wrong? _______________________________________________

A subset that does contain $-8$ : _______________

PAGE 5 — Exit ticket 1.1

Exit Ticket · Lesson 1.1

Name: ________________________ Date: ____________

Is $0$ a natural number? ______ A whole number? ______ An integer? ______
An integer that is not a whole number: ______
Every subset from this lesson that contains $25$ : _______________________________
In your own words, the difference between the whole numbers and the integers:

PAGE 6 — Three costumes of one definition

1.2 Rational Numbers

A rational number can be written as $\dfrac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

FIGURE: fig4-decimal-forms.png (full width)

Fill in the frames.

$7 = \frac{7}{\underline{\hspace{1cm}}} \qquad -4 = \frac{\underline{\hspace{1cm}}}{1} \qquad 0 = \frac{0}{\underline{\hspace{1cm}}}$

A decimal that ends is a ____________________ decimal.

A decimal with a block of digits repeating forever is a ____________________ decimal.

Before classifying a radical, always ____________________ it first.

Write as a ratio of two integers.

$6 =$ ______
$-4 =$ ______
$0.25 =$ ______ (simplest form)

Write as a decimal. Circle STOPS or REPEATS.

$\tfrac{3}{8} =$ ______ STOPS / REPEATS
$\tfrac{2}{3} =$ ______ STOPS / REPEATS
Is $\sqrt{49}$ rational? ______ Why? _______________________________________________
Is $2\tfrac{1}{2}$ rational? ______ As a ratio of two integers: ______
Is $-\tfrac{5}{6}$ rational? ______ How does the negative sign fit the definition?

PAGE 7 — Practice · rational forms

Practice · Rational Numbers

Each number is rational. List every OTHER subset from Lesson 1.1 that contains it.

a) $9$	b) $0$	c) $-6$	d) $\tfrac{3}{4}$	e) $-2.5$

Write as a ratio of two integers.

a) $12$	b) $-7$	c) $0.4$	d) $1.75$	e) $0$

Write as a decimal, then circle its kind.

Fraction	Decimal	Kind
a) $\tfrac{5}{8}$		terminates / repeats
b) $\tfrac{1}{9}$		terminates / repeats
c) $\tfrac{7}{20}$		terminates / repeats
d) $\tfrac{5}{6}$		terminates / repeats

Circle the rational numbers, then give a reason for each choice.

$\sqrt{36}$ $\sqrt{10}$ $0.75$ $\pi$ $-11$

Reasons: _______________________________________________

PAGE 8 — Apply, reason, correct

Show What You Know · 1.2

Explain. Why is every integer a rational number? Use the definition.
Apply it. A carpenter cuts a board $3.5$ feet long.

Rational? ______ As a ratio of two integers: ______ Is it an integer? ______
Apply it. A player's batting average is $0.325$ .

As a fraction in simplest form: ______

Every subset that contains it: _______________________________
Find the error. A student says $0.\overline{6}$ must be irrational "because the digits never end."

What went wrong? _______________________________________________

Correct classification: _______________

PAGE 9 — Exit ticket 1.2

Exit Ticket · Lesson 1.2

Name: ________________________ Date: ____________

$-9$ as a ratio of two integers: ______
Is $0.45$ rational? ______ As a fraction in simplest form: ______
Is $\sqrt{25}$ rational? ______ Why? _______________________________________
Why is a decimal that repeats forever still rational?

PAGE 10 — The numbers the fractions miss

1.3 Irrational Numbers

An irrational number cannot be written as a ratio of two integers. Its decimal never terminates AND never repeats.

FIGURE: fig5-irrationals-on-the-number-line.png (full width)

Fill in the frames.

$0.375$ ____________, so it is rational.

$0.\overline{36}$ ____________, so it is rational.

$1.41421356\dots$ never ____________ and never ____________, so it is ____________.

$\underline{\hspace{1cm}} = \sqrt{9} < \sqrt{11} < \sqrt{16} = \underline{\hspace{1cm}}$

$\frac{22}{7} = 3.142857\dots \qquad \pi = 3.1415 \underline{\hspace{1cm}} \dots$

Rational or irrational?

$\sqrt{9}$ : ______________ Why? _______________________________________
$\sqrt{11}$ : ______________ Why? _______________________________________
$\pi$ : ______________
$0.121121112\dots$ (forever, no repeating block): ______________
$\sqrt{11}$ lies between ______ and ______
An irrational number: ______ A non-example: ______

PAGE 11 — Practice · irrational numbers

Practice · Irrational Numbers

Circle R for rational or I for irrational.

a) $\sqrt{16}$	R / I
b) $\sqrt{17}$	R / I
c) $-\sqrt{81}$	R / I
d) $\sqrt{50}$	R / I
e) $3.14$	R / I

Perfect squares closest to $50$ : ______ and ______

So $\sqrt{50}$ lies between ______ and ______
TRUE / FALSE: every square root is irrational.

Example that justifies your answer: ______
Terminates, repeats, or neither?

a) $\tfrac{1}{8}$	b) $\sqrt{3}$	c) $\tfrac{4}{11}$	d) $2.7$

PAGE 12 — Apply, reason, correct

Show What You Know · 1.3

Explain. How does the decimal form of $\sqrt{2}$ show it cannot be a ratio of two integers?
Apply it. A square garden has an area of $20$ square feet, so each side is $\sqrt{20}$ feet.

Rational or irrational? ______________ Between ______ and ______ feet
Apply it. A circle has diameter $5$ cm, so its exact circumference is $5\pi$ cm.

Why is the exact circumference irrational? _______________________________________

Why is $15.7$ cm only an approximation? _______________________________________
Find the error. A student writes " $\pi = \tfrac{22}{7}$ , so $\pi$ is rational."

Error 1: _______________________________________________

Error 2: _______________________________________________

PAGE 13 — Exit ticket 1.3

Exit Ticket · Lesson 1.3

Name: ________________________ Date: ____________

$\sqrt{36}$ is ______________ because _______________________________________
$\sqrt{37}$ is ______________, and it lies between ______ and ______
Example of an irrational number: ______ Non-example: ______
How does the decimal form tell you whether a number is rational or irrational?

PAGE 14 — The whole system

1.4 Nesting and Classifying

FIGURE: fig2-subset-diagram.png (full width)

Label the diagram above, then fill in the chain.

$\text{natural} \subset \underline{\hspace{2cm}} \subset \underline{\hspace{2cm}} \subset \underline{\hspace{2cm}} \subset \text{real}$

The irrational region is drawn ____________ the rational region, never ____________ it, because no number is both.

FIGURE: fig6-classification-flowchart.png (full width)

Name EVERY subset that contains the number.

$8$ : _______________________________________
$0$ : _______________________________________
$-15$ : _______________________________________
$\tfrac{3}{5}$ : _______________________________________
$\sqrt{49}$ : _______________________________________
$\sqrt{13}$ : _______________________________________
$-2.25$ : _______________________________________
$\pi$ : _______________________________________

PAGE 15 — Practice · the full classification table

Practice · Classifying Real Numbers

FIGURE: fig3-subset-diagram-examples.png (full width)

Write "yes" or "no" in every cell.

Number	Natural	Whole	Integer	Rational	Irrational	Real
$4$
$-4$
$0.4$
$\sqrt{4}$
$\sqrt{40}$

Name a number that is:

a) an integer but not a whole number: ______

b) rational but not an integer: ______

c) real but not rational: ______

d) whole but not natural: ______

e) natural: ______
Circle TRUE or FALSE, then justify each.

a) Every rational number is real.	TRUE / FALSE
b) Every real number is rational.	TRUE / FALSE
c) No irrational number is rational.	TRUE / FALSE
d) Some integers are irrational.	TRUE / FALSE
e) Every whole number is rational.	TRUE / FALSE

Justifications: _______________________________________________

Write each number in the correct region of the diagram above: $-6$ , $\tfrac{5}{8}$ , $\sqrt{64}$ , $\sqrt{65}$ , $0$

PAGE 16 — Apply, reason, correct

Show What You Know · 1.4

Explain. Why can a number not be both rational and irrational?
Apply it. Classify each measurement into every subset that contains it.

Measurement	Subsets
length $12$ in
temperature $-3\,^\circ\text{C}$
volume $2.5$ L
diagonal $\sqrt{2}$ m

Which are real numbers? _______________

Apply it. Two square tiles have areas $36$ cm $^2$ and $30$ cm $^2$ , so their sides are $\sqrt{36}$ cm and $\sqrt{30}$ cm.

$\sqrt{36}$ is _______________ $\sqrt{30}$ is _______________

Which side can be measured exactly with a ruler marked in whole centimeters? ______
Find the error. A student draws the subset diagram with the irrational numbers inside the rational rectangle.

What does that drawing claim? _______________________________________________

Why is it false? _______________________________________________

PAGE 17 — Exit ticket 1.4

Exit Ticket · Lesson 1.4

Name: ________________________ Date: ____________

Every subset containing $-11$ : _______________________________________
Every subset containing $\sqrt{81}$ : _______________________________________
Every subset containing $\sqrt{83}$ : _______________________________________
Why is every natural number a real number, but not every real number a natural number?

PAGE 18 — Chapter 1 review, part 1

Chapter 1 Review

Part A · Relationships among the subsets

Sketch the subset diagram. Label all five subset regions and the real numbers.

DIAGRAM SPACE: 4 in tall, full width — blank box with a hairline border
Complete each sentence with a subset name.

a) Every natural number is a $\underline{\hspace{2cm}}$ number.

b) Every whole number is an $\underline{\hspace{2cm}}$ .

c) Every integer is a $\underline{\hspace{2cm}}$ number.

d) Every rational number is a $\underline{\hspace{2cm}}$ number.
Why is the irrational region drawn beside the rational region rather than inside it?
Place each number in the correct region of your diagram: $6$ , $0$ , $-6$ , $6.5$ , $\sqrt{6}$

PAGE 19 — Chapter 1 review, part 2

Chapter 1 Review (continued)

TRUE / FALSE: the whole numbers are a subset of the integers.

Justification: _______________________________________________
Draw a number line from $-4$ to $4$ and mark every integer. Then mark one rational number that is not an integer, and one irrational number between $1$ and $2$ .

NUMBER LINE SPACE: 1.5 in tall, full width — pre-drawn blank line with arrowheads
Using the three number lines from Lesson 1.1, explain what "subset" means.

Part B · Classifying numbers and explaining why

Name every subset containing each number.

a) $14$	b) $-14$	c) $0$	d) $\tfrac{7}{8}$	e) $-3.75$

Name every subset containing each number.

a) $\sqrt{100}$	b) $\sqrt{101}$	c) $-\sqrt{16}$	d) $0.\overline{5}$	e) $\pi$

PAGE 20 — Chapter 1 review, part 3

Chapter 1 Review (continued)

Why is $\sqrt{121}$ a natural number? _______________________________________
Why is $\sqrt{120}$ irrational? _______________________________________
Is $-7$ a whole number? ______ Why not? _______________________________________

Smallest subset in the diagram that contains it: _______________
A number's decimal form is exactly $0.375$ .

Every subset that contains it: _______________________________

As a ratio of two integers: ______
A number's decimal begins $2.6457513\dots$ and never repeats.

Every subset that contains it: _______________________________
Find the error. A student classifies $0$ as a natural number.

What went wrong? _______________________________________________

Complete correct classification: _______________________________

PAGE 21 — Chapter 1 review, part 4

Chapter 1 Review (continued)

Part C · Describing each subset with examples and non-examples

The natural numbers are _______________________________________________

Examples: ______ , ______ Non-examples (with reasons): ______________________
The whole numbers are _______________________________________________

Examples: ______ , ______ Non-examples: ______ , ______
The integers are _______________________________________________

Examples: ______ , ______ Non-examples: ______ , ______
The rational numbers are _______________________________________________

Examples — fraction: ______ decimal: ______ radical: ______

Non-examples: ______ , ______
The irrational numbers are _______________________________________________

Examples: ______ , ______ Non-examples: ______ , ______

PAGE 22 — Chapter 1 review, part 5

Chapter 1 Review (continued)

Complete the table.

Subset	Example	Non-example
Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers

Part D · Application and reasoning

Apply it. A square patio will cover $45$ square feet, so each side must be $\sqrt{45}$ feet.

Classification: _______________ Between ______ and ______ feet

Why does the builder write $6.7$ feet on the plan? _______________________________
Apply it. A bank balance is $-\$45.50$ .

Every subset that contains $-45.50$ : _______________________________

Why is it not an integer? _______________________________________

PAGE 23 — Chapter 1 review, part 6

Chapter 1 Review (continued)

Explain. Can a number be both irrational and an integer?
Explain. A student concludes that a square root sign always means the number is irrational. Use $\sqrt{144}$ and $\sqrt{145}$ to show why that is false.

$\sqrt{144} =$ ______ , which is _______________

$\sqrt{145}$ is _______________ , because _______________________________________
Apply it. A circle has radius $4$ inches, so its exact area is $16\pi$ square inches.

Why is the exact area irrational? _______________________________________

Why is $50.24$ square inches rational? _______________________________________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Figure widths: every figure in this chapter is full width. fig6-classification-flowchart.png is the tallest; give it its own block on page 14 with no text wrapping beside it, or move it to the top of page 15 if page 14 overflows.
Diagram space (item 77): a plain white box with a hairline border, 4 in tall. Students draw the five nested regions themselves; do not pre-print the rectangles, since drawing the nesting is the skill being assessed.
Number line space (item 82): pre-print a horizontal line with arrowheads and nine unlabeled tick marks so students can label $-4$ through $4$ themselves.
Circle-the-answer items: keep the TRUE / FALSE and R / I choices on the same line as the prompt so the Canva text box does not reflow.
Answer blanks: keep every blank on the same line as its prompt. Items 57–64 and 84–85 need long blanks, since a complete answer names five subsets.