Grade 8 Workbook — Chapter 2: Comparing and Ordering Real Numbers

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

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 122.

PAGE 1 — Chapter opener

Chapter 2 · Comparing and Ordering Real Numbers

Standard 8.NS.1

In this chapter you will:

Name both square roots of a number, positive and negative
Trap a square root between two consecutive natural numbers and say which is closer
Write rational approximations of irrational numbers to the nearest hundredth
Locate irrational values on a number line at the right scale
Compare and order up to five real numbers written in any form

Words to know: perfect square · radicand · positive (principal) square root · negative square root · consecutive natural numbers · better approximation · rational number · irrational number · rational approximation · nearest hundredth · $\pi$ · scientific notation · ascending order · descending order

Calculator note: work these by hand. The standard asks you to justify which approximation is better, and a calculator decimal gives you no justification. Every estimate on these pages is done by squaring a candidate by hand. Keep the perfect squares $0^2$ through $20^2$ where you can see them.

Perfect squares you need. Fill in the missing values.

$n$	0	1	2	3	4	5	6	7	8	9	10
$n^2$	0	1		9		25		49		81

$n$	11	12	13	14	15	16	17	18	19	20
$n^2$	121		169		225		289		361

PAGE 2 — Two square roots

2.1 Both Square Roots of a Number

FIGURE: fig1-two-square-roots.png (full width)

Fill in the blanks.

The symbol $\sqrt{\ }$ always means the ____________ square root.

To name the negative root, write a ____________ in front of the radical.

Every positive number has ______ square roots. The number $0$ has ______.

If the radicand is a perfect square, the root is ____________. If it is not, the root is ____________.

Evaluate.

$\sqrt{144} = \underline{\hspace{2cm}}$
$-\sqrt{144} = \underline{\hspace{2cm}}$
$\sqrt{0} = \underline{\hspace{2cm}}$
Both square roots of $225$ : $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
Is $\sqrt{121}$ rational or irrational? ____________

Why? _______________________________________________
Is $\sqrt{122}$ rational or irrational? ____________

Why? _______________________________________________

PAGE 3 — Roots practice

Rational or Irrational?

FIGURE: fig2-perfect-square-landmarks.png (full width)

Evaluate.

a) $\sqrt{169}$	b) $-\sqrt{196}$	c) $\sqrt{400}$	d) $-\sqrt{1}$

Both square roots of $64$ : $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
Circle the irrational values.

$\sqrt{49}$	$\sqrt{50}$	$-\sqrt{81}$	$-\sqrt{80}$

True or false: $-\sqrt{36} = -6$ . ____________

Justify: _______________________________________________
Explain. Why are $\sqrt{100}$ and $-\sqrt{100}$ different numbers?

Where does each one sit on a number line? _______________________________________________
Apply it. A square patio covers $289$ square feet.

Side length: $\underline{\hspace{2cm}}$ feet

Why is only one square root an answer? _______________________________________________
Explain. Why does $0$ have only one square root?
Find the error. A student writes $-\sqrt{49} = 7$ because "the negative signs cancel."

What went wrong? _______________________________________________

Correct value: $\underline{\hspace{2cm}}$

PAGE 4 — Exit ticket 2.1

Exit Ticket · Lesson 2.1

Name: ________________________ Date: ____________

$\sqrt{256} = \underline{\hspace{2cm}}$
$-\sqrt{324} = \underline{\hspace{2cm}}$
Both square roots of $361$ : $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
What is the difference between $\sqrt{16}$ and $-\sqrt{16}$ ?

PAGE 5 — Trapping a root

2.2 Between Which Two Natural Numbers?

FIGURE: fig3-sqrt40-between-6-and-7.png (full width)

The method.

Step 1. Find the perfect square just below and just above the radicand: $36 < 40 < 49$

Step 2. Take the positive root of each: $\underline{\hspace{2cm}} < \sqrt{40} < \underline{\hspace{2cm}}$

FIGURE: fig4-which-neighbor-is-closer.png (full width)

Step 3. Compare the gaps: $40 - 36 = \underline{\hspace{2cm}}$ and $49 - 40 = \underline{\hspace{2cm}}$

Step 4. The smaller gap wins, so the better approximation is $\underline{\hspace{2cm}}$

Complete the table.

Root	Lower square	Upper square	Between	Lower gap	Upper gap	Better
19. $\sqrt{40}$
20. $\sqrt{90}$
21. $\sqrt{150}$
22. $\sqrt{75}$

PAGE 6 — More estimating

Practice · Trapping Roots

Complete the table.

Root	Lower square	Upper square	Between	Lower gap	Upper gap	Better
23. $\sqrt{20}$
24. $\sqrt{300}$
25. $\sqrt{8}$
26. $\sqrt{380}$

Name the two consecutive natural numbers each root lies between.

a) $\sqrt{12}$	b) $\sqrt{55}$	c) $\sqrt{99}$	d) $\sqrt{210}$

For each root in item 27, name the better approximation and the gap comparison that proves it.

	Better	Gap comparison
a)
b)
c)
d)

PAGE 7 — Estimating in context

Estimate and Justify

$\sqrt{125}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{250}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{170}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
Explain. Why does the question "between which two consecutive natural numbers does $\sqrt{400}$ lie?" have no answer?
Apply it. A square patch of lawn covers $130$ square meters.

One side is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ meters. Better estimate: $\underline{\hspace{2cm}}$
Apply it. A square window covers $45$ square feet.

Nearest whole foot: $\underline{\hspace{2cm}}$ Too big or too small? ____________
Explain. Why is comparing $40$ to $42.5$ enough to decide whether $\sqrt{40}$ is closer to $6$ or $7$ ?
Find the error. A student says $\sqrt{90}$ is closer to $10$ "because $90$ is close to $100$ ."

What went wrong? _______________________________________________

Correct better approximation: $\underline{\hspace{2cm}}$

PAGE 8 — Exit ticket 2.2

Exit Ticket · Lesson 2.2

Name: ________________________ Date: ____________

$\sqrt{60}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{110}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{45}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
Write the halfway test as instructions someone else could follow.

PAGE 9 — Squaring your way in

2.3 Approximating to the Nearest Hundredth

FIGURE: fig5-sqrt200-hundredth-zoom.png (full width)

The method, on $\sqrt{40}$ .

Tenths: $6.3^2 = \underline{\hspace{2cm}}$ and $6.4^2 = \underline{\hspace{2cm}}$ , so $\sqrt{40}$ is between $6.3$ and $6.4$ .

Hundredths: $6.32^2 = \underline{\hspace{2cm}}$ and $6.33^2 = \underline{\hspace{2cm}}$

Which is nearer to $40$ ? $\underline{\hspace{2cm}}$ So $\sqrt{40} \approx \underline{\hspace{2cm}}$

Approximate to the nearest hundredth. Show the two hundredths you squared.

$\sqrt{2} \approx \underline{\hspace{2cm}}$
$\sqrt{5} \approx \underline{\hspace{2cm}}$
$\sqrt{10} \approx \underline{\hspace{2cm}}$
$\sqrt{30} \approx \underline{\hspace{2cm}}$
$\sqrt{40} \approx \underline{\hspace{2cm}}$
$-\sqrt{8} \approx \underline{\hspace{2cm}}$
$\sqrt{200} \approx \underline{\hspace{2cm}}$
$-\sqrt{150} \approx \underline{\hspace{2cm}}$

WORK SPACE: 2 in tall, full width — grid for squaring candidates

PAGE 10 — Approximation practice

Practice · Nearest Hundredth

Approximate.

a) $\sqrt{3}$	b) $\sqrt{7}$	c) $\sqrt{15}$	d) $\sqrt{20}$

Approximate.

a) $-\sqrt{12}$	b) $-\sqrt{50}$	c) $\sqrt{90}$	d) $-\sqrt{300}$

$\sqrt{110} \approx \underline{\hspace{2cm}}$

Hundredths squared: $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
$\sqrt{250} \approx \underline{\hspace{2cm}}$

Hundredths squared: $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$

WORK SPACE: 2 in tall, full width

PAGE 11 — Comparing with approximations

Which Is Greater?

FIGURE: fig6-pi-zoom.png (full width)

$\sqrt{50}$ or $7.1$ ? ____________

Squaring argument: _______________________________________________
$\pi$ or $\sqrt{10}$ ? ____________

Justify: _______________________________________________
$-\sqrt{26}$ or $-5.2$ ? ____________

Justify: _______________________________________________
Apply it. A square rug covers $45$ square feet. Side length to the nearest hundredth: $\underline{\hspace{2cm}}$ feet
Explain. Why is $3.14$ not equal to $\pi$ ? What does $\approx$ claim?
Find the error. A student computes $8.4^2 = 70.56$ , sees $70.56 < 72$ , and answers $\sqrt{72} \approx 8.4$ .

What step is missing? _______________________________________________

Correct approximation: $\underline{\hspace{2cm}}$

PAGE 12 — Exit ticket 2.3

Exit Ticket · Lesson 2.3

Name: ________________________ Date: ____________

$\sqrt{80} \approx \underline{\hspace{2cm}}$
$-\sqrt{130} \approx \underline{\hspace{2cm}}$
Which is greater, $\sqrt{99}$ or $9.9$ ? ____________

Justify: _______________________________________________
Why can a rational approximation never equal an irrational number exactly?

PAGE 13 — Plotting on a number line

2.4 Locating Real Numbers

Three steps: bracket it, approximate it, choose a scale fine enough to show the answer.

$\sqrt{55}$ sits between the ticks $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ , at about $\underline{\hspace{2cm}}$

NUMBER LINE: 0 to 10, whole-number ticks, full width
Plot $-\sqrt{30}$ . Value to the nearest hundredth: $\underline{\hspace{2cm}}$

NUMBER LINE: -6 to 0, whole-number ticks, full width
Plot $\pi$ and $\sqrt{10}$ . $\pi \approx \underline{\hspace{2cm}}$ $\sqrt{10} \approx \underline{\hspace{2cm}}$

NUMBER LINE: 3 to 3.2, hundredth ticks, full width
Which lies farther right, $\pi$ or $\frac{22}{7}$ ? ____________

What scale do you need to see the difference? _______________________________________________
Plot $\sqrt{2}$ . Value to the nearest hundredth: $\underline{\hspace{2cm}}$

NUMBER LINE: 1 to 2, tenth ticks, full width
On a line marked in tenths, $\sqrt{40}$ is nearest the tick $\underline{\hspace{2cm}}$

PAGE 14 — Plotting practice

Practice · Plot and Read

Plot $\sqrt{17}$ , $\sqrt{45}$ , and $\sqrt{95}$ .

$\sqrt{17} \approx \underline{\hspace{2cm}}$ $\sqrt{45} \approx \underline{\hspace{2cm}}$ $\sqrt{95} \approx \underline{\hspace{2cm}}$

NUMBER LINE: 0 to 10, whole-number ticks, full width
Plot $-\sqrt{12}$ , $-\sqrt{50}$ , and $-\sqrt{85}$ .

$-\sqrt{12} \approx \underline{\hspace{2cm}}$ $-\sqrt{50} \approx \underline{\hspace{2cm}}$ $-\sqrt{85} \approx \underline{\hspace{2cm}}$

NUMBER LINE: -10 to 0, whole-number ticks, full width
$\sqrt{60}$ lies between the tenths $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
$\sqrt{200}$ lies between the hundredths $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
Plot $\sqrt{30}$ and $5.5$ . Which is farther right? ____________

NUMBER LINE: 5 to 6, tenth ticks, full width

PAGE 15 — Scale, reasoning, error hunt

Choose the Right Scale

Apply it. A carpenter marks $\sqrt{24}$ feet on a tape showing tenths of a foot.

Between the tenths $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ Tenth to use: $\underline{\hspace{2cm}}$
Explain. Why can a whole-number number line not show whether $\pi$ or $\frac{22}{7}$ is greater?

A scale that can: _______________________________________________
Find the error. A student plots $\sqrt{90}$ exactly halfway between $9$ and $10$ .

What went wrong? _______________________________________________

Which side of halfway? ____________

PAGE 16 — Exit ticket 2.4

Exit Ticket · Lesson 2.4

Name: ________________________ Date: ____________

Plot $\sqrt{70}$ . Nearest tenth: $\underline{\hspace{2cm}}$

NUMBER LINE: 0 to 10, whole-number ticks, full width
Plot $-\sqrt{45}$ . Value to the nearest hundredth: $\underline{\hspace{2cm}}$

NUMBER LINE: -10 to 0, whole-number ticks, full width
Which lies farther right, $\sqrt{26}$ or $5.2$ ? ____________ Justify: ____________________________
How does a number line show that $-\sqrt{50} < -7$ ?

PAGE 17 — Converting every form

2.5 Comparing and Ordering Five Real Numbers

Complete the conversion table.

Form	Example	Decimal
Improper fraction	$\frac{21}{5}$
Mixed number	$4\frac{3}{8}$
Percent	$430\%$
Scientific notation	$1.5 \times 10^{1}$
Scientific notation	$2.5 \times 10^{-1}$
Radical	$\sqrt{20}$
$\pi$	$\pi$

Compare using $<$ or $>$ . Justify each in one sentence.

$\sqrt{50} \ \underline{\hspace{2cm}} \ 7.2$
$\pi \ \underline{\hspace{2cm}} \ 3.2$
$-\sqrt{30} \ \underline{\hspace{2cm}} \ -5.5$
$\frac{3}{8} \ \underline{\hspace{2cm}} \ 40\%$
$2.5 \times 10^{2} \ \underline{\hspace{2cm}} \ 249$

PAGE 18 — Ordering with a table

Order Five Numbers

FIGURE: fig7-order-five-mixed.png (full width)

Order ascending: $\frac{5}{8}$ , $0.58$ , $62\%$ , $0.65$ , $\frac{3}{5}$

Number
Decimal

Answer:  $\underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}}$

Order ascending: $\sqrt{20}$ , $4.5$ , $\frac{21}{5}$ , $430\%$ , $4\frac{3}{8}$

Number
Decimal

Answer:  $\underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}} < \underline{\hspace{2cm}}$

Order descending: $-\sqrt{10}$ , $-3.1$ , $-\frac{22}{7}$ , $-3\frac{1}{4}$ , $-305\%$

Number
Decimal

Answer:  $\underline{\hspace{2cm}} > \underline{\hspace{2cm}} > \underline{\hspace{2cm}} > \underline{\hspace{2cm}} > \underline{\hspace{2cm}}$

PAGE 19 — Ordering practice

Practice · Ascending and Descending

FIGURE: fig8-order-negatives-zoom.png (full width)

Ascending: $\sqrt{30}$ , $5.4$ , $\frac{26}{5}$ , $545\%$ , $5\frac{1}{2}$
Descending: $\sqrt{72}$ , $8.4$ , $\frac{43}{5}$ , $855\%$ , $8\frac{3}{8}$
Ascending: $-\sqrt{8}$ , $-2.9$ , $-\frac{11}{4}$ , $-285\%$ , $-2\frac{4}{5}$
Ascending: $1.5 \times 10^{1}$ , $15.2$ , $\sqrt{200}$ , $14\frac{1}{2}$ , $1460\%$
Descending: $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3.15$ , $3.1$

WORK SPACE: 2.5 in tall, full width — conversion scratch

PAGE 20 — Justify, apply, correct

Show Your Reasoning

$\sqrt{99}$ or $9.9$ ? ____________ Squaring argument: ____________________________
$-\sqrt{145}$ or $-12$ ? ____________ Justify: ____________________________
Apply it. Five square tiles have areas $5$ , $6.25$ , $6$ , $5.76$ , and $7$ square inches.

Area (sq in)	$5$	$6.25$	$6$	$5.76$	$7$
Side (nearest hundredth)

Shortest to longest side: _______________________________________________

Explain. Two strategies for deciding whether $\frac{5}{8}$ or $62\%$ is greater.

Strategy 1: _______________________________________________

Strategy 2: _______________________________________________
Find the error. A student orders $-3.5$ , $-3.05$ , $-3.2$ ascending as $-3.05, -3.2, -3.5$ .

What went wrong? _______________________________________________

Correct order: _______________________________________________

PAGE 21 — Exit ticket 2.5

Exit Ticket · Lesson 2.5

Name: ________________________ Date: ____________

Ascending: $\sqrt{45}$ , $6.8$ , $\frac{33}{5}$ , $690\%$ , $6\frac{3}{4}$
Descending: $-\sqrt{20}$ , $-4.4$ , $-\frac{21}{5}$ , $-4\frac{1}{2}$ , $-455\%$
$7.05 \ \underline{\hspace{2cm}} \ \sqrt{50}$
Why is converting everything to a decimal reliable? Name one situation where another strategy is faster.

PAGE 22 — Chapter 2 review, part 1

Chapter 2 Review

Part A · Estimating between consecutive natural numbers

$\sqrt{55}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{210}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{130}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
$\sqrt{85}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$
Explain. $\sqrt{240}$ is between $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$ . Better: $\underline{\hspace{2cm}}$

Halfway comparison: _______________________________________________

PAGE 23 — Chapter 2 review, part 2

Chapter 2 Review (continued)

Part B · Rational approximations and the number line

Approximate to the nearest hundredth.

a) $\sqrt{11}$	b) $-\sqrt{29}$	c) $\sqrt{160}$	d) $-\sqrt{390}$

$\sqrt{150}$ is nearest the tenth $\underline{\hspace{2cm}}$

NUMBER LINE: 12 to 13, tenth ticks, full width
Which is greater, $-\sqrt{82}$ or $-9.1$ ? ____________ Justify: ____________________________
$\sqrt{73}$ lies between the hundredths $\underline{\hspace{2cm}}$ and $\underline{\hspace{2cm}}$
Explain. Place $\sqrt{2}$ and $\sqrt{3}$ between $1$ and $2$ .

$\sqrt{2} \approx \underline{\hspace{2cm}}$ $\sqrt{3} \approx \underline{\hspace{2cm}}$ Farther right: $\underline{\hspace{2cm}}$

NUMBER LINE: 1 to 2, tenth ticks, full width

PAGE 24 — Chapter 2 review, part 3

Chapter 2 Review (continued)

Part C · Comparing and ordering real numbers

Ascending: $\sqrt{40}$ , $6.4$ , $\frac{31}{5}$ , $635\%$ , $6\frac{1}{4}$
Descending: $-\sqrt{110}$ , $-10.5$ , $-\frac{53}{5}$ , $-1040\%$ , $-10\frac{3}{8}$
Ascending: $2.5 \times 10^{-1}$ , $0.3$ , $\frac{7}{25}$ , $26\%$ , $0.245$
Descending: $\pi$ , $3.2$ , $\sqrt{11}$ , $\frac{17}{5}$ , $315\%$
Ascending: $\sqrt{0}$ , $-\sqrt{16}$ , $3.5$ , $-\frac{7}{2}$ , $400\%$
Which is greater, $\frac{5}{6}$ or $83\%$ ? ____________

Justify in writing: _______________________________________________

PAGE 25 — Chapter 2 review, part 4

Chapter 2 Review (continued)

Part D · Application and reasoning

Apply it. Five square rooms have areas $120$ , $144$ , $130$ , $156$ , and $121$ square feet.

Area (sq ft)	$120$	$144$	$130$	$156$	$121$
Side (nearest hundredth)

 Shortest to longest: _______________________________________________

 Rooms with whole-number sides: _______________________________________________

Explain. A number line is marked in whole numbers from $0$ to $20$ .

Why can it not settle whether $\sqrt{200}$ or $14.1$ is greater? _______________________________________________

What should you do instead? _______________________________________________

Which is greater? ____________
Find the error. A student writes $\sqrt{50} = 7.07$ .

What is wrong with the equals sign? _______________________________________________

Supporting computation: _______________________________________________

A correct statement: _______________________________________________
Plot $-\sqrt{12}$ , $-3.5$ , $\pi$ , $\sqrt{2}$ , and $2.9$ , then list them ascending.

NUMBER LINE: -4 to 4, whole-number ticks with tenth marks, full width

Ascending: _______________________________________________

How does the picture justify the list? _______________________________________________

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
Number line spaces: draw as a plain horizontal rule with tick marks at the spacing called out on each page, labels beneath, and 0.75 in of clear space above for students' point labels. The scale named in each callout is part of the problem — do not substitute a generic line.
Work space blocks: plain white boxes with a hairline border, for squaring candidates by hand. Every estimate in this chapter is done by hand, so every approximation page needs room to multiply.
Figure widths: every figure in this chapter is full width. The zoomed lines (fig5, fig6, fig8) carry small tick labels; do not scale them below 90 percent of the text column or the labels stop being legible in print.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow. Ordering answers need a full line each, with the comparison symbols pre-set.
Perfect-square table on Page 1: print this on the inside cover as well, since every lesson in the chapter depends on it.