Grade 7 Workbook — Chapter 8: Simplifying Numerical Expressions

SOL 7.PFA.2 (a) · Companion to Textbook Chapter 8

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 and run continuously from 1 to 103, so one answer key serves both books.

PAGE 1 — Chapter opener

Chapter 8 · Simplifying Numerical Expressions

Standard 7.PFA.2 (a)

In this chapter you will:

• Use the order of operations to simplify an expression to one value
• Work with exponents of 1, 2, 3, and 4 on positive integer bases
• Take square roots of perfect squares inside expressions
• Peel nested brackets and absolute value bars from the inside out
• Use the properties of real numbers as shortcuts

Words to know: numerical expression · simplify · value · order of operations · exponent · base · square root · radical sign · radicand · perfect square · brackets · absolute value · commutative property · associative property · distributive property · identity property · inverse property

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

Notation note: this chapter uses parentheses ( ), brackets [ ], and absolute value bars | |. It never uses braces { }.

PAGE 2 — Why we need an agreed order

8.1 The Order of Operations

FIGURE: fig1-order-of-operations-ladder.png (full width)

Two students simplify $7 + 3 \times 4$.

Student A adds first: $10 \times 4 =$ ______

Student B multiplies first: $7 + 12 =$ ______

The order of operations says student ______ is correct, so $7 + 3 \times 4 =$ ______

Fill in the blanks.

Tier 1 is ______________ symbols. Simplify the ______________ group first.

Tier 2 is ______________ and perfect-square ______________.

Tier 3 is multiplication and ______________. Work ______ to ______.

Tier 4 is addition and ______________. Work ______ to ______.

Watch the traps.

$24 \div 6 \times 2 =$ ______ (NOT $24 \div 12$)

$18 - 5 + 2 =$ ______ (NOT $18 - 7$)

PAGE 3 — Guided practice 8.1

One Tier at a Time

Fill in each frame. Rewrite the whole expression on every line.

1. $9 + 6 \times 2$

$= 9 +$ ________ $=$ ________ First tier used: ______________

2. $(9 + 6) \times 2$

$=$ ________ $\times 2$ $=$ ________ What did the parentheses change? ____________________

3. $20 - 12 \div 4$

$= 20 -$ ________ $=$ ________

4. $36 \div 9 \times 2$

$=$ ________ $\times 2$ $=$ ________ Leftmost operation: ______________

5. $5 \times (11 - 4) + 8$

$= 5 \times$ ________ $+ 8$ $=$ ________ $+ 8$ $=$ ________

6. $48 \div (6 + 2) - 3$

$= 48 \div$ ________ $- 3$ $=$ ________ $- 3$ $=$ ________

PAGE 4 — Independent practice 8.1

Simplify Each Expression

7. Simplify each.

8. Simplify each.

9. $60 \div 5 \times 3 =$ ________

10. $18 - 4 + 7 =$ ________

11. $7 \times 4 - 15 \div 5 =$ ________

12. $(25 - 9) \div 4 + 6 \times 2 =$ ________

13. $100 - (36 \div 4 + 11) \times 3$

Line 1: _______________________________

Line 2: _______________________________

Line 3: _______________________________

Value: ________

PAGE 5 — Apply, reason, and repair

Thinking It Through

14. Application. A class buys 4 packs of markers that cost $3 each and 2 notebooks that cost $5 each.

Expression: _______________________________

Total cost: $________

15. Reasoning. Explain why $18 - 5 + 2$ does not equal $18 - 7$.

Correct value: ________

Rule that settles it: _______________________________________________

16. Error analysis. Riley simplified $6 + 2 \times 9$ like this:

$6 + 2 = 8$, then $8 \times 9 = 72$.

The error is: _______________________________________________

Correct value: ________

PAGE 6 — Exit ticket 8.1

Exit Ticket · Lesson 8.1

Name: ________________________ Date: ____________

17. $12 + 8 \div 2 =$ ________

18. $(12 + 8) \div 2 =$ ________

19. $9 \times 3 - 20 \div 4 =$ ________

20. Why did mathematicians need to agree on an order of operations at all?

PAGE 7 — Powers and roots

8.2 Exponents and Perfect-Square Roots

Complete the power table.

Fill in the blanks.

In $2^3$, the base is ______ and the exponent is ______, so $2^3 = 2 \times 2 \times 2 =$ ______.

$2^3$ is NOT $2 \times$ ______.

$2^3 =$ ______ but $3^2 =$ ______, so the base and exponent are not interchangeable.

The radical sign is also a ______________ symbol, so $\sqrt{25 + 11} = \sqrt{\underline{\ \ \ \ }} =$ ______.

Guided practice

21. a) $2^3 =$ ____ b) $3^4 =$ ____ c) $10^2 =$ ____ d) $7^1 =$ ____

22. $4 + 5^2 =$ ________

23. $6 \times 2^3 =$ ________

24. $\sqrt{64} + 12 =$ ________

25. $3^2 + \sqrt{16} \times 2 =$ ________

26. $(2 + 3)^2 - 15 =$ ________ Do this before the exponent: ______________

PAGE 8 — Independent practice 8.2

Powers and Roots in Longer Expressions

27. a) $2^2 =$ ____ b) $2^4 =$ ____ c) $5^3 =$ ____ d) $1^4 =$ ____

28. a) $\sqrt{36} =$ ____ b) $\sqrt{100} =$ ____ c) $\sqrt{144} =$ ____ d) $\sqrt{225} =$ ____

Use the three-line frame for each.

29. $50 - 4^2 \times 2$ → ________ → ________ → ________

30. $\sqrt{81} \times 3 + 2^3$ → ________ → ________ → ________

31. $(6 - 2)^3 \div 8$ → ________ → ________ → ________

32. $\sqrt{9 + 16} + 5^2$ → ________ → ________ → ________

33. $3^3 - \sqrt{49} \times 2 + 1$ → ________ → ________ → ________

34. $100 \div (2 + 3)^2 \times 3$ → ________ → ________ → ________

PAGE 9 — Apply, reason, and repair

Thinking It Through

35. Application. A square garden covers 144 square feet. Write an expression using a square root for the distance all the way around it.

Expression: _______________________________

Value: ________ feet

36. Reasoning. Why are $2^3$ and $3^2$ not equal?

$2^3 =$ ______ $3^2 =$ ______

Explanation: _______________________________________________

37. Error analysis. Tomas simplified $3 + 2^3$ like this:

$3 + 2 = 5$, then $5^3 = 125$.

The error is: _______________________________________________

Correct value: ________

PAGE 10 — Exit ticket 8.2

Exit Ticket · Lesson 8.2

Name: ________________________ Date: ____________

38. $7 + 2^4 =$ ________

39. $\sqrt{100} - 3^2 =$ ________

40. $(4 + 5)^2 \div 3 =$ ________

41. Why is $\sqrt{9 + 16}$ not the same as $\sqrt{9} + \sqrt{16}$?

$\sqrt{9 + 16} =$ ______ $\sqrt{9} + \sqrt{16} =$ ______

PAGE 11 — Peeling the layers

8.3 Brackets and Absolute Value Bars

FIGURE: fig2-peeling-the-layers.png (full width)

Fill in the blanks.

Brackets [ ] do exactly what ______________ do. They are used so the layers are easy to ______.

With nested groups, always simplify the ______________ group first.

In $2[8]$, the 2 against the bracket means ______________, so $2[8] =$ ______.

This book never uses ______________, written { }.

Guided practice

42. $2[9 + (4 - 1)]$ → $2[$ ________ $]$ → ________ First group simplified: ______________

43. $[30 - (6 \times 4)] + 5$ → $[$ ________ $] + 5$ → ________

44. $|-9| =$ ________

45. $|5 - 12| = |$ ________ $| =$ ________

46. $|2 - 8| + 3^2$ → ________ $+$ ________ → ________

47. $4[|7 - 7| + 5]$ → $4[$ ________ $+ 5]$ → $4[$ ________ $]$ → ________

PAGE 12 — Absolute value is a distance

Distance from Zero

FIGURE: fig3-absolute-value-distance.png (full width)

Two moves, always in this order. 1. Simplify everything between the bars. 2. Take that number's distance from zero.

48. a) $|-15| =$ ____ b) $|20| =$ ____ c) $|0| =$ ____ d) $|-4| + |4| =$ ____

Simplify. Show the value inside each grouping symbol before you clear it.

49. $3[14 - (2 + 5)] =$ ________

50. $[45 \div (2 + 7)] \times 4 =$ ________

51. $50 - 3|4 - 9| =$ ________

52. $[2^3 + |1 - 5|] \div 3 =$ ________

53. $\sqrt{|9 - 25|} =$ ________

54. $2[(3 + 2)^2 - |10 - 14|] =$ ________

55. $[100 - 4^2] \div |0 - 7| =$ ________

PAGE 13 — Apply, reason, and repair

Thinking It Through

56. Application. At 6 a.m. the temperature was $-5$°F. By noon it was $9$°F.

Expression using absolute value bars: _______________________________

Change: ________ degrees

57. Reasoning. Why do $|3 - 10|$ and $|10 - 3|$ have the same value?

$|3 - 10| =$ ______ $|10 - 3| =$ ______

Explanation using distance: _______________________________________________

58. Error analysis. Priya dropped the bars and wrote $|4 - 9| + 6 = 4 - 9 + 6 = 1$.

The error is: _______________________________________________

Correct value: ________

PAGE 14 — Exit ticket 8.3

Exit Ticket · Lesson 8.3

Name: ________________________ Date: ____________

59. $|8 - 15| =$ ________

60. $4[6 - (1 + 3)] =$ ________

61. $[3^2 + |2 - 6|] \div 13 =$ ________

62. Why are the bars in $|4 - 9|$ called grouping symbols? What must you finish before taking the absolute value?

PAGE 15 — The properties

8.4 Properties of Real Numbers as Shortcuts

Complete the table.

The boundary line. Commutative and associative work for addition and multiplication only.

$10 - 4 =$ ______ but $4 - 10 =$ ______, so subtraction is ______________ commutative.

Guided practice

63. $6 + 9 = 9 + 6$ → property: _______________________________

64. $(3 \times 5) \times 2 = 3 \times (5 \times 2)$ → property: _______________________________

65. $28 + 15 + 2 =$ ________ Numbers paired: ______ and ______, because they make ______

66. $5 \times 17 \times 2 =$ ________

67. $7(20 + 3) = 7($ ____ $) + 7($ ____ $) =$ ______ $+$ ______ $=$ ________

68. $14 \times 1 = 14$ → property: _______________________________

PAGE 16 — Independent practice 8.4

Shortcuts That Keep the Value

FIGURE: fig4-distributive-area-model.png (full width)

69. Name the property.

70. $47 + 68 + 53 = ($ ____ $+$ ____ $) +$ ____ $=$ ________

71. $4 \times 19 \times 25 = ($ ____ $\times$ ____ $) \times$ ____ $=$ ________

72. $6(50 + 7) =$ ______ $+$ ______ $=$ ________ Check by adding inside first: $6($ ____ $) =$ ________

73. $9(30 - 2) =$ ______ $-$ ______ $=$ ________

74. $58 + 27 + (-58) =$ ________ Three properties used: ______________, ______________, ______________

75. $\left(\tfrac{1}{4} \times 4\right) \times 23 =$ ________ Two properties used: ______________, ______________

76. $5(20 + 4) - 3^2 =$ ________

PAGE 17 — Apply, reason, and repair

Thinking It Through

77. Application. A teacher buys 8 boxes of pencils at $12 each and 8 boxes of erasers at $3 each.

As a sum of two products: $8(12) + 8(3) =$ ________

As a single product: $8($ ____ $) =$ ________

Total: $________

78. Reasoning. Show that subtraction is not commutative.

Example: ______ $-$ ______ $=$ ______, but ______ $-$ ______ $=$ ______

Why are these properties shortcuts and not exceptions to the order of operations?

79. Error analysis. Devon wrote $3(8 + 5) = 3(8) + 5 = 29$.

The error is: _______________________________________________

Correct value: ________ Check: $3($ ____ $) =$ ________

PAGE 18 — Exit ticket 8.4

Exit Ticket · Lesson 8.4

Name: ________________________ Date: ____________

80. $(7 + 2) + 8 = 7 + (2 + 8)$ → property: _______________________________

81. $25 \times 13 \times 4 =$ ________

82. $6(40 + 3) =$ ______ $+$ ______ $=$ ________

83. Why can a property shortcut never change the value of an expression?

PAGE 19 — Chapter 8 review, part 1

Chapter 8 Review

Part A · Order of operations

84. $15 + 6 \times 3 =$ ________

85. $(15 + 6) \times 3 =$ ________

86. $72 \div 9 \times 2 =$ ________

87. $40 - 24 \div 6 + 5 =$ ________

88. $9 \times (12 - 5) - 18 \div 3 =$ ________

Part B · Exponents and perfect-square roots

89. a) $3^3 =$ ____ b) $2^4 =$ ____ c) $5^2 =$ ____ d) $10^3 =$ ____

90. a) $\sqrt{16} =$ ____ b) $\sqrt{121} =$ ____ c) $\sqrt{196} =$ ____ d) $\sqrt{400} =$ ____

91. $4^2 + \sqrt{81} \times 2 =$ ________

92. $(7 - 3)^3 \div \sqrt{64} =$ ________

PAGE 20 — Chapter 8 review, part 2

Chapter 8 Review (continued)

Part C · Brackets and absolute value bars

93. $5[13 - (4 + 6)] =$ ________

94. $|7 - 16| + 2^3 =$ ________

95. $[6^2 - |3 - 11|] \div 7 =$ ________

96. $2[|5 - 12| + \sqrt{100}] =$ ________

Part D · Properties of real numbers

97. Name the property.

a) $5 + (6 + 4) = (5 + 6) + 4$ _______________________________

b) $12 \times 1 = 12$ _______________________________

c) $7(10 + 2) = 7(10) + 7(2)$ _______________________________

98. $36 + 89 + 64 =$ ________

99. $8(60 + 4) =$ ______ $+$ ______ $=$ ________

PAGE 21 — Chapter 8 review, part 3

Chapter 8 Review (continued)

Part E · Mixed application and reasoning

100. $3[\sqrt{49} + (2 + 6)] - 4^2$

Line 1: _______________________________

Line 2: _______________________________

Line 3: _______________________________

Value: ________

101. Application. A square courtyard covers 225 square feet. Fencing goes all the way around, except for a 6-foot gate.

Expression: _______________________________

Fencing needed: ________ feet

102. Reasoning. $|2 - 9| \times 3 =$ ________ and $2 - 9 \times 3 =$ ________

Why are the values different?

103. Error analysis. Sam wrote $20 - 2[3 + (4 \times 2)] = 18 \times 11 = 198$.

The error is: _______________________________________________

Correct work, line by line:

Correct value: ________

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: all four Chapter 8 figures are landscape; place fig1-order-of-operations-ladder.png and fig4-distributive-area-model.png at full width, and fig2-peeling-the-layers.png and fig3-absolute-value-distance.png at full width above the practice frames
• Arrow frames: the → chains on pages 8 and 11 should be one text box per line so the blanks do not reflow
• Bars and brackets: set absolute value bars and brackets in the same weight as parentheses; do not substitute braces { } anywhere in this chapter
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow