Grade 7 Workbook — Chapter 3: Square Roots and Perfect Squares

SOL 7.NS.3 · Companion to Textbook Chapter 3

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/. Item numbers match the textbook exactly, so one answer key serves both books.

PAGE 1 — Chapter opener

Chapter 3 · Square Roots and Perfect Squares

Standard 7.NS.3

In this chapter you will:

• Know every perfect square from $0^2$ to $20^2$
• Find the positive square root of a perfect square from 0 to 400
• Use perfect squares as neighbors to reason about other numbers
• Describe how squaring and taking a square root undo each other

Words to know: exponent · base · perfect square · area · square root · radical sign · radicand · positive square root · consecutive perfect squares · inverse operations

Calculator note: this chapter is fully mental-math and pencil work. No calculator needed, and none allowed on the state test for this standard.

PAGE 2 — Why we say "squared"

3.1 Perfect Squares from 0 to 400

FIGURE: fig1-square-array-16.png (full width)

Fill in the blanks.

The small raised 2 in $4^2$ is called an ______________.

$4^2$ means $4 \times$ ______, which equals ______.

$4^2$ does NOT mean $4 \times$ ______.

A square with a side of 4 units has an area of ______ square units.

Complete the rule.

For any square: area $=$ ( ______________ )$^2$

Count and record. Use the figure above.

PAGE 3 — Watching squares grow

Growing Squares

FIGURE: fig2-growing-squares.png (full width)

Complete the table.

Find the gaps. Write the difference between each pair of neighbors above.

$1 \to 4$: ______ $4 \to 9$: ______ $9 \to 16$: ______ $16 \to 25$: ______

What pattern do the gaps follow? _______________________________________________

11. Use the pattern to predict the next perfect square after 25: ______

Guided practice.

1. A square is built from unit squares with 6 tiles along each side. How many unit squares is that? Write it with an exponent.

   Total: ______ With an exponent: ______

PAGE 4 — The full list, 0 to 400

The Perfect Squares You Must Know

FIGURE: fig3-perfect-squares-0-to-400.png (full width)

Fill in the whole table from memory, then check against the figure.

Guided practice. Evaluate.

2. a) $3^2 =$ ______ b) $8^2 =$ ______ c) $12^2 =$ ______

3. a) $15^2 =$ ______ b) $20^2 =$ ______ c) $0^2 =$ ______

4. Fill in the missing whole number: $121 = \underline{\ \ \ \ }^2$ and $169 = \underline{\ \ \ \ }^2$

5. Explain why 400 is the largest perfect square used in this chapter.

PAGE 5 — Is it a perfect square?

Testing a Number

Move 1 — Neighbors. If a number lands strictly between two consecutive perfect squares, it is not a perfect square. Move 2 — Last digit. Perfect squares only ever end in 0, 1, 4, 5, 6, or 9. A number ending in 2, 3, 7, or 8 is ruled out at once.

Guided practice.

4. Is 50 a perfect square? ______ It lies between ______ and ______.

5. Between which two consecutive perfect squares does 200 lie? ______ and ______

Independent practice.

7. a) $7^2 =$ ______ b) $11^2 =$ ______ c) $14^2 =$ ______ d) $17^2 =$ ______

8. a) $9^2 =$ ______ b) $13^2 =$ ______ c) $16^2 =$ ______ d) $19^2 =$ ______

9. List every perfect square between 100 and 200: _______________________________________________

10. Circle the perfect squares: $24$ · $25$ · $60$ · $64$ · $90$ · $100$

11. A number ends in the digit 7. How do you know right away it is not a perfect square?

13. Between which two consecutive perfect squares does 250 lie? ______ and ______

PAGE 6 — Apply and explain

Lesson 3.1 · Apply It

15. Application. A community garden is a square measuring 18 feet on each side. How many square feet of soil does it cover?

    Work: _______________________________________________

    Area: ______ square feet

16. Reasoning. Jo says 20 is a perfect square because $20 = 4 \times 5$ and both 4 and 5 are whole numbers. What is wrong with Jo's thinking?

The two perfect squares 20 falls between: ______ and ______

PAGE 7 — Exit ticket 3.1

Exit Ticket · Lesson 3.1

Name: ________________________ Date: ____________

17. Evaluate $13^2 =$ ______ and $16^2 =$ ______

18. Is 144 a perfect square? ______ If so, what whole number is squared to get it? ______

19. Between which two consecutive perfect squares does 130 lie? ______ and ______

20. Explain, using unit squares, what makes a number a perfect square.

PAGE 8 — Running the question backward

3.2 The Positive Square Root

FIGURE: fig4-square-root-as-side-length.png (full width)

Label the parts.

In $\sqrt{49} = 7$:

The symbol $\sqrt{\phantom{x}}$ is called the ______________ sign.

The number 49 under it is called the ______________.

The number 7 is the positive ______________ ______________ of 49.

Fill in the blanks.

$\sqrt{49} = 7$ because ______ $\times$ ______ $=$ 49.

A square root answers the question: here is the ______________ of a square, how long is one ______________?

We take the POSITIVE root because a side length is a ______________, and those are never negative.

Watch out. $\sqrt{36}$ is NOT 18. Halving undoes multiplying by 2 — it does not undo squaring.

$\sqrt{36} =$ ______ because ______ $\times$ ______ $= 36$.

PAGE 9 — Square root practice

Finding Square Roots

Guided practice.

21. $\sqrt{25} =$ ______

22. $\sqrt{81} =$ ______

23. $\sqrt{0} =$ ______ $\sqrt{1} =$ ______

24. $\sqrt{196} =$ ______

25. A square has an area of 100 square centimeters. Each side is ______ cm.

Independent practice. Evaluate.

30. Order from least to greatest: $\sqrt{81}$, $\sqrt{36}$, $\sqrt{121}$, $\sqrt{4}$

    ---

31. True or false: $\sqrt{36} = 18$. ______ Why?

34. Explain why $\sqrt{0} = 0$ and $\sqrt{1} = 1$.

PAGE 10 — Trapped between neighbors

When It Is Not a Perfect Square

FIGURE: fig5-between-perfect-squares.png (full width)

Follow the reasoning.

$25 < 30 < 36$, so $\sqrt{25} < \sqrt{30} < \sqrt{36}$, so ______ $< \sqrt{30} <$ ______

Use the same three steps for each.

26. $\sqrt{30}$ lies between ______ and ______

27. $\sqrt{150}$: perfect squares used ______ and ______ , so $\sqrt{150}$ lies between ______ and ______

28. $\sqrt{200}$: perfect squares used ______ and ______ , so $\sqrt{200}$ lies between ______ and ______

29. Reasoning. Without finding any decimal, explain how you know $\sqrt{170}$ is between 13 and 14.

35. Application. A square rug covers 225 square feet. How long is one side? ______ ft

    How much braided trim goes all the way around it? ______ ft

PAGE 11 — Exit ticket 3.2

Exit Ticket · Lesson 3.2

Name: ________________________ Date: ____________

37. $\sqrt{64} =$ ______ $\sqrt{289} =$ ______

38. A square patio has an area of 196 square feet. Each side is ______ ft.

39. $\sqrt{55}$ lies between the whole numbers ______ and ______

40. In your own words, what does the symbol $\sqrt{\phantom{x}}$ ask you to find?

PAGE 12 — Two directions along one road

3.3 Undoing Each Other

FIGURE: fig6-inverse-operations.png (full width)

Fill in the blanks.

Squaring takes a ______________ length and gives back an ______________.

Taking the positive square root takes an ______________ and gives back a ______________ length.

Operations that undo each other are called ______________ operations.

$\sqrt{n^2} =$ ______ and $\left(\sqrt{n^2}\right)^2 =$ ______

Guided practice.

41. $6^2 =$ ______ and $\sqrt{36} =$ ______

42. $14^2 =$ ______ and $\sqrt{196} =$ ______

43. $\left(\sqrt{81}\right)^2 =$ ______

44. $\sqrt{15^2} =$ ______

45. Squaring a positive number and taking its positive square root are ______________ operations.

PAGE 13 — Check it by squaring

Every Answer Checks Itself

Square your answer. If you get the radicand back, you are right.

45. A square has an area of 64 square inches. Side length: ______ in

    Check: ______ $\times$ ______ $=$ ______

46. A square has an area of 289 square meters. Side length: ______ m

    Check: ______ $\times$ ______ $=$ ______

Independent practice.

47. a) $\left(\sqrt{49}\right)^2 =$ ______ b) $\sqrt{12^2} =$ ______ c) $\left(\sqrt{100}\right)^2 =$ ______ d) $\sqrt{17^2} =$ ______

48. Complete the table. What do you notice about the last column?

I notice: _______________________________________________

50. $\sqrt{\underline{\ \ \ \ }} = 13$ and $\underline{\ \ \ \ }^2 = 169$

51. $\sqrt{\underline{\ \ \ \ }} = 20$ and $\underline{\ \ \ \ }^2 = 400$

52. If $s^2 = 121$ and $s$ is a positive side length, then $s =$ ______

PAGE 14 — Apply and explain

Lesson 3.3 · Apply It

53. Explain how you can evaluate $\sqrt{9^2}$ without ever multiplying $9 \times 9$.

54. Order from least to greatest: $5^2$, $\sqrt{144}$, $\sqrt{64}$, $3^2$

    Values: ______, ______, ______, ______

    Order: _______________________________________________

55. Application. A square patch of turf covers 324 square feet. The groundskeeper runs edging all the way around it. How many feet of edging is needed?

    Side: ______ ft Edging: ______ ft

56. Reasoning. Dana says that since $4^2 = 16$, you undo squaring by taking half, so $\sqrt{16} = 8$. What is wrong?

Correct value: $\sqrt{16} =$ ______   Check: ______ $\times$ ______ $= 16$

PAGE 15 — Exit ticket 3.3

Exit Ticket · Lesson 3.3

Name: ________________________ Date: ____________

57. $11^2 =$ ______ $\sqrt{121} =$ ______

58. $\sqrt{16^2} =$ ______

59. A square has an area of 400 square centimeters. Each side is ______ cm.

60. Describe how square roots and perfect squares undo each other. Use an example.

PAGE 16 — Chapter 3 review, part 1

Chapter 3 Review

Part A · Perfect squares from 0 to 400

61. a) $2^2 =$ ______ b) $10^2 =$ ______ c) $15^2 =$ ______ d) $18^2 =$ ______

62. Every perfect square between 200 and 400: _______________________________________________

63. Circle the perfect squares: $45$ · $49$ · $81$ · $96$ · $196$ · $300$

64. 275 lies between the perfect squares ______ and ______

65. A square tile measures 13 cm on each side. Area: ______ square cm

Part B · Positive square roots

66. a) $\sqrt{4} =$ ______ b) $\sqrt{36} =$ ______ c) $\sqrt{100} =$ ______ d) $\sqrt{256} =$ ______

67. a) $\sqrt{144} =$ ______ b) $\sqrt{324} =$ ______ c) $\sqrt{361} =$ ______ d) $\sqrt{0} =$ ______

68. Order least to greatest: $\sqrt{196}$, $\sqrt{25}$, $\sqrt{289}$, $\sqrt{81}$

    ---

69. A square garden has an area of 144 square feet. Each side is ______ ft.

70. $\sqrt{90}$ lies between ______ and ______

71. $\sqrt{380}$ lies between ______ and ______

PAGE 17 — Chapter 3 review, part 2

Chapter 3 Review (continued)

Part C · The inverse relationship

72. $\left(\sqrt{225}\right)^2 =$ ______

73. $\sqrt{14^2} =$ ______

74. $\sqrt{\underline{\ \ \ \ }} = 15$ and $\underline{\ \ \ \ }^2 = 225$

75. If $s^2 = 256$ and $s$ is a positive side length, then $s =$ ______

76. How do you check that $\sqrt{169} = 13$ without a calculator?

Part D · Application and reasoning

77. A square patio covers 289 square feet. Side: ______ ft Distance around: ______ ft

78. A square mural with area 100 square feet is repainted with area 400 square feet.

    First side: ______ ft New side: ______ ft How the side changed: _______________

79. A square has an area of 49 square units. Why is there exactly one acceptable side length?

80. How do you know $\sqrt{260}$ is not a whole number? Which two whole numbers does it lie between?

Between ______ and ______

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: fig1, fig4, and fig6 are near-square or wide line art — place at 60–70 percent width, centered. fig2, fig3, and fig5 are wide strips — place at full width.
• Page 4 table: leave the $n^2$ row at least 0.4 in tall so students can write three-digit answers legibly
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
• No calculator icon: repeat the mental-math note from page 1 as a small footer on every exit ticket page