Algebra 1 Workbook — Chapter 1: Translating and Evaluating Algebraic Expressions

SOL A.EO.1 · 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 122.

PAGE 1 — Chapter opener

Chapter 1 · Translating and Evaluating Algebraic Expressions

Standard A.EO.1

In this chapter you will:

• Translate a verbal phrase into an algebraic expression, and back into words
• Spot the two phrases that reverse the order of the quantities they name
• Define a variable for a real situation and write an expression that models it
• Say what each number and each variable in a contextual expression means
• Evaluate expressions for given replacement values, including fractions and decimals
• Evaluate expressions containing absolute value, square roots, and cube roots

Words to know: algebraic expression · numerical expression · variable · term · coefficient · constant · verbal quantitative situation · contextual situation · translate · evaluate · replacement value · absolute value · square root · cube root

Three rules this chapter fixes. A radical left in a denominator, such as $\dfrac{6}{\sqrt{2}}$, is a finished answer here — do NOT rationalize it. $\sqrt{25}$ means $5$, never $\pm 5$. And $\sqrt[3]{-8} = -2$, because a cube root of a negative number is real.

Nothing in this chapter is solved, because nothing in this chapter is an equation. Solving starts in Chapter 2.

PAGE 2 — Words into symbols

1.1 The Language of Algebra

FIGURE: fig2-operation-word-bank.png (full width)

Fill in the blanks.

An expression built from numbers, variables, and operations is an ____________________ expression.

The pieces of an expression separated by $+$ and $-$ signs are its ____________.

The number multiplying a variable is the ____________________.

A term with no variable is a ____________________.

In $5x^2 - 8x + 3$ there are ______ terms, the coefficients are ______ and ______, and the constant is ______.

FIGURE: fig1-phrase-to-expression.png (full width)

The two phrases that reverse the order of what they name are ____________________ and ____________________.

$$\text{five less than twice a number} \ \longrightarrow \ \underline{\hspace{2.5cm}}$$

Translate. Let $n$ be the number.

1. the sum of a number and $15$: ______________

2. $9$ less than a number: ______________

3. the product of $6$ and a number: ______________

4. the quotient of a number and $4$: ______________

5. twice a number, decreased by $7$: ______________

6. $5$ more than three times a number: ______________

PAGE 3 — Practice · both directions

Practice · Translating

7. A verbal phrase for $n + 10$: _______________________________________

8. A verbal phrase for $4n - 1$: _______________________________________

9. Translate each. Only two different expressions appear among the five.

10. Translate each.

11. Translate each, using parentheses only where the phrase requires them.

a) ______________  b) ______________  c) ______________  d) ______________

12. Write a verbal phrase for each.

    a) $7n$: _______________________________________

    b) $n - 11$: _______________________________________

    c) $\dfrac{n + 5}{2}$: _______________________________________

    d) $2n + 9$: _______________________________________

    e) $6(n - 4)$: _______________________________________

PAGE 4 — Parts of an expression

Naming the Parts

13. For $5x^2 - 8x + 3$:

    a) number of terms: ______

    b) coefficient of the first term: ______

    c) coefficient of the second term: ______

    d) constant: ______

14. Complete the table.

15. Circle the expression that matches "ten decreased by the product of $4$ and a number."

    $4n - 10$ $10 - 4n$ $4(10 - n)$ $\dfrac{10}{4n}$

    How did you eliminate the other three? _______________________________________

PAGE 5 — Apply, reason, correct

Show What You Know · 1.1

16. Apply it. A parking garage holds $n$ cars at the start of the hour, and $12$ cars leave during that hour.

    Cars remaining: ______________

    Why is the expression not $12 - n$? _______________________________________

17. Explain. Why is "$7$ less than a number" written $n - 7$ rather than $7 - n$?

    Replace "a number" with $20$: $\ n - 7 =$ ______ and $7 - n =$ ______

    ---

18. Find the error. A student translates "$5$ less than the product of $3$ and a number" as $5 - 3n$.

    What went wrong? _______________________________________________

    Correct expression: ______________

    A one-line check the student could have done: _______________________________

PAGE 6 — Exit ticket 1.1

Exit Ticket · Lesson 1.1

Name: ________________________ Date: ____________

19. $10$ less than twice a number: ______________

20. the quotient of a number and $6$, increased by $5$: ______________

21. A verbal phrase for $3(n - 2)$: _______________________________________

22. In words, the difference between $4n - 9$ and $9 - 4n$:

    ---

    ---

PAGE 7 — Real situations into expressions

1.2 Expressions from Contextual Situations

FIGURE: fig3-context-bar-model.png (full width)

Fill in the blanks.

Step one is always to ____________ the variable in a full sentence.

A quantity charged once is a ____________________ in the expression.

A quantity charged once per unit is ____________________ by the variable.

$$\text{a } \$25 \text{ fee plus } \$15 \text{ per hour} \ \longrightarrow \ \underline{\hspace{2.5cm}}$$

Define the variable, then write the expression.

23. A movie ticket costs $\$12$. Cost of $t$ tickets.

    Let ______ = _______________________ Expression: ______________

24. A pizza costs $\$14$ plus $\$2$ per topping.

    Let ______ = _______________________ Expression: ______________

25. Sara has $\$50$ and spends some of it. Amount left.

    Let ______ = _______________________ Expression: ______________

26. A class of $s$ students is divided into $5$ equal groups. Size of each group: ______________

27. A rectangle's length is $4$ cm more than its width $w$. Length: ______________

28. Perimeter of that rectangle, simplified: ______________

29. A gym charges a $\$30$ joining fee plus $\$22$ per month. Total after $m$ months: ______________

PAGE 8 — Practice · modeling

Practice · Writing Expressions

30. Notebooks cost $\$3.25$ each.

    a) cost of $n$ notebooks: ______________

    b) cost of $n$ notebooks plus one $\$5$ bag: ______________

31. A car travels $55$ miles per hour.

    a) distance in $h$ hours: ______________

    b) total distance if $20$ miles were already covered: ______________

32. A jar holds $q$ quarters and $d$ dimes.

    a) total value in cents: ______________

    b) total value in dollars: ______________

33. A tank holds $80$ gallons and drains $6$ gallons per minute.

    Gallons left after $m$ minutes: ______________

    Why is the rate subtracted rather than added? _______________________________

PAGE 9 — Practice · reading an expression

Practice · What the Parts Mean

34. A phone bill in dollars is $40 + 0.25g$, where $g$ is gigabytes used beyond the plan limit.

35. Match each situation to its expression. Write the expression on the line.

    Choices: $8n$ $8 + n$ $n - 8$ $\dfrac{n}{8}$ $8 - n$

    a) $n$ boxes of $8$ pencils: ______

    b) $n$ dollars after spending $\$8$: ______

    c) $n$ cookies shared equally by $8$ people: ______

    d) $8$ dollars plus $n$ dollars: ______

    e) $8$ feet of ribbon with $n$ feet cut off: ______

36. Define the variable, then write the expression.

    a) the number of days in $w$ weeks: ______________

    b) the number of inches in $f$ feet: ______________

    c) the age of someone $5$ years younger than a person aged $a$: ______________

PAGE 10 — Apply, reason, correct

Show What You Know · 1.2

37. Apply it. A theater has some number of rows with $18$ seats each, plus $12$ balcony seats.

    Let ______ = _______________________

    Total seats: ______________

    If the balcony closed, the expression becomes: ______________

38. Explain. Why do $5 + 3n$ and $3n + 5$ describe the same situation, while $5 - 3n$ and $3n - 5$ do not?

    ---

    ---

39. Find the error. A jacket costs $c$ dollars and a shirt costs $\$20$ less than the jacket. A student writes the shirt's cost as $20 - c$.

    What went wrong? _______________________________________________

    Correct expression: ______________

    What does $20 - c$ describe instead? _______________________________

PAGE 11 — Exit ticket 1.2

Exit Ticket · Lesson 1.2

Name: ________________________ Date: ____________

40. A taxi charges $\$3.50$ plus $\$2$ per mile.

    Let ______ = _______________________ Fare: ______________

41. A bag of $s$ marbles shared equally among $6$ friends: ______________

42. Describe a situation modeled by $8n + 15$.

    $n$ counts _______________ the $8$ means _______________ the $15$ means _______________

43. How do you decide which quantity gets the variable?

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PAGE 12 — Substitute, then simplify

1.3 Evaluating Expressions

FIGURE: fig5-substitution-steps.png (full width)

Fill in the blanks.

Every replacement value is substituted inside ____________________.

After substituting, follow the ____________ of ____________________.

$-6^2 =$ ______ but $(-6)^2 =$ ______, because the exponent is attached to ____________________.

FIGURE: fig4-absolute-value-distance.png (full width)

Absolute value reports a ____________________ from zero, so it is never ____________________.

$|-6| =$ ______ and $-|-6| =$ ______

Evaluate.

44. $3x + 5$ for $x = 4$: ______

45. $3x + 5$ for $x = -4$: ______

46. $7 - 2n$ for $n = -3$: ______

47. $x^2$ for $x = -6$: ______

48. $-x^2$ for $x = -6$: ______

49. $|a|$ for $a = -9$: ______

50. $-|a|$ for $a = -9$: ______

51. $|a - 4|$ for $a = -9$: ______

52. $4c - 1$ for $c = \tfrac{1}{2}$: ______

PAGE 13 — Practice · substitution

Practice · Evaluating

53. Evaluate for $x = -2$ and $y = 5$.

54. Evaluate for $a = -3$ and $b = 4$.

55. Evaluate for $m = \tfrac{2}{3}$.

56. Evaluate for $t = -1.5$.

PAGE 14 — Practice · absolute value

Practice · Absolute Value

57. Evaluate.

Which two parts above show that where the bars fall changes the answer? ______ and ______

58. Evaluate $|2x - 7|$.

59. Evaluate $\dfrac{x + y}{x - y}$.

    a) $x = 5$, $y = 3$: ______

    b) $x = \tfrac{1}{2}$, $y = \tfrac{1}{4}$: ______

60. Evaluate $5 - 2(x - 3)^2$ for $x = 1$: ______

    WORK SPACE: 1.5 in tall, full width — blank

PAGE 15 — Apply, reason, correct

Show What You Know · 1.3

61. Apply it. The number of degrees between two temperature readings is $|T_2 - T_1|$. Morning: $-4\,^\circ\text{F}$. Afternoon: $11\,^\circ\text{F}$.

    Value: ______

    Why is absolute value the right tool here? _______________________________

62. Apply it. A repair shop charges $25 + 15h$ dollars for $h$ hours of work.

    For $h = 3.5$: ______ As money: $______

63. Explain. Evaluate $(-6)^2 =$ ______ and $-6^2 =$ ______

    Why do they differ? _______________________________________________

64. Find the error. A student evaluates $5 - 2x$ for $x = -3$ and writes $5 - 6 = -1$.

    Where did it go wrong? _______________________________________________

    Correct value: ______ Rule missed: _______________________________

PAGE 16 — Exit ticket 1.3

Exit Ticket · Lesson 1.3

Name: ________________________ Date: ____________

65. For $x = -5$: a) $2x - 9 =$ ______ b) $x^2 + x =$ ______

66. $6n + 1$ for $n = \tfrac{1}{3}$: ______

67. For $x = 2.5$: $|x - 10| =$ ______ and $-|x - 10| =$ ______

68. Why does substituting a negative value in parentheses matter? Give an example.

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PAGE 17 — Roots inside evaluation

1.4 Absolute Value, Square Roots, and Cube Roots

FIGURE: fig6-roots-on-the-number-line.png (full width)

Fill in the blanks.

$\sqrt{a}$ asks for the ____________________ number whose ____________ is $a$.

$\sqrt[3]{a}$ asks for the number whose ____________ is $a$.

$\sqrt[3]{-27} =$ ______, because $(\underline{\hspace{1cm}})^3 = -27$.

$\sqrt{-64}$ is ____________________, because no real number squares to a negative.

$-\sqrt{16} =$ ______, and the minus sign is applied ____________________ the root is taken.

A radical is a ____________________ symbol, so $\sqrt{9 + 16} =$ ______ while $\sqrt{9} + \sqrt{16} =$ ______.

A radical left in a denominator is a finished answer in this chapter. Do NOT rationalize.

Evaluate.

69. $\sqrt{49} =$ ______

70. $\sqrt{\tfrac{1}{4}} =$ ______

71. $\sqrt[3]{27} =$ ______

72. $\sqrt[3]{-8} =$ ______

73. $\sqrt{x}$ for $x = 121$: ______

74. $\sqrt[3]{x}$ for $x = -125$: ______

75. $3\sqrt{x} - 2$ for $x = 16$: ______

76. $\sqrt[3]{x} + |x|$ for $x = -27$: ______

77. $\dfrac{10}{\sqrt{x}}$ for $x = 5$ (leave the denominator as it is): ______

PAGE 18 — Practice · roots

Practice · Square Roots and Cube Roots

78. Evaluate.

79. Evaluate.

80. Evaluate $\sqrt{a + b}$.

    a) $a = 7$, $b = 9$: ______ b) $a = -11$, $b = 47$: ______ c) $a = \tfrac{1}{2}$, $b = \tfrac{1}{2}$: ______

81. Evaluate $\sqrt[3]{x - 1}$.

    a) $x = 9$: ______ b) $x = -7$: ______ c) $x = 1$: ______

82. Evaluate $\sqrt{b^2 - 4ac}$.

    a) $a = 2$, $b = 7$, $c = 3$: ______

    b) $a = 1$, $b = 6$, $c = 5$: ______

    WORK SPACE: 1.5 in tall, full width — blank

PAGE 19 — Practice · mixed roots

Practice · Roots in Larger Expressions

83. Evaluate $|x| + \sqrt{x}$.

    a) $x = 9$: ______ b) $x = \tfrac{1}{4}$: ______

84. Evaluate $\dfrac{12}{\sqrt{x}}$.

    a) $x = 3$: ______ b) $x = 4$: ______

    Which one finishes as a rational number, and why? _______________________________

85. Evaluate $2\sqrt{x} + \sqrt[3]{y}$ for $x = 25$ and $y = -8$: ______

86. Apply it. A square patio of area $A$ square feet has sides $\sqrt{A}$ feet. A cubical bin of volume $V$ cubic inches has edges $\sqrt[3]{V}$ inches.

    $A = 169$: side ______ ft $V = 216$: edge ______ in

87. Apply it. An object dropped from $d$ feet falls for $\sqrt{\dfrac{d}{16}}$ seconds.

    For $d = 144$: ______ seconds

PAGE 20 — Reason and correct

Show What You Know · 1.4

88. Explain. Why is $\sqrt[3]{-64}$ a real number while $\sqrt{-64}$ is not?

    $(\underline{\hspace{1cm}})^3 = -64$, so the cube root is ______

    Squaring a positive gives a ____________ and squaring a negative gives a ____________, so nothing squares to $-64$.

    ---

89. Find the error. A student claims $\sqrt{x^2 + y^2} = x + y$ and evaluates at $x = 3$, $y = 4$ to get $7$.

    Correct value of $\sqrt{x^2 + y^2}$: ______

    What false assumption does the shortcut make? _______________________________

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PAGE 21 — Exit ticket 1.4

Exit Ticket · Lesson 1.4

Name: ________________________ Date: ____________

90. a) $\sqrt{x}$ for $x = 100$: ______ b) $\sqrt[3]{x}$ for $x = -216$: ______

91. $5\sqrt[3]{x} - \sqrt{y}$ for $x = 8$, $y = 49$: ______

92. $\dfrac{9}{\sqrt{x}}$ for $x = 6$: ______

    Why are you finished? _______________________________________

93. Why does $\sqrt{-25}$ have no real value while $\sqrt[3]{-25}$ does?

    ---

PAGE 22 — Chapter 1 review, part 1

Chapter 1 Review

Part A · Translating verbal situations into expressions

94. Translate. Let $n$ be the number.

95. Write a verbal phrase for each.

    a) $5n - 2$: _______________________________________

    b) $\dfrac{n + 4}{3}$: _______________________________________

    c) $4(n - 6)$: _______________________________________

96. "the difference of $6$ and a number" is ______________

    "$6$ less than a number" is ______________

    The difference between them: _______________________________________

97. Apply it. A cell plan costs $\$30$ per month plus $\$0.10$ per text.

    Let ______ = _______________________ Monthly bill: ______________

98. Apply it. A rectangle's length is $3$ more than twice its width.

    Let ______ = _______________________

    a) length: ______________ b) perimeter, simplified: ______________

PAGE 23 — Chapter 1 review, part 2

Chapter 1 Review (continued)

99. Apply it. Marcus has $d$ dollars and spends $\$18.75$. What he has left: ______________

100. Apply it. A cash drawer holds $n$ nickels and $d$ dimes. Total value in cents: ______________

101. Apply it. An auditorium has $r$ rows of $24$ seats plus $8$ floor seats.

     Let ______ = _______________________ Total seats: ______________

102. For $7x^2 - 3x + 10$:

     a) number of terms: ______ b) coefficient of the second term: ______ c) constant: ______

103. Complete the table.

PAGE 24 — Chapter 1 review, part 3

Chapter 1 Review (continued)

104. Explain. Why is "$5$ less than $x$" written $x - 5$ and not $5 - x$?

     For $x = 12$: $\ x - 5 =$ ______ and $5 - x =$ ______

     ---

105. Find the error. A student writes "the quotient of $8$ and a number" as $\dfrac{n}{8}$.

     What went wrong? _______________________________________________

     Correct expression: ______________

Part B · Evaluating expressions

106. Evaluate for $x = -4$ and $y = 2$.

107. Evaluate for $a = \tfrac{1}{2}$ and $b = -\tfrac{3}{4}$.

PAGE 25 — Chapter 1 review, part 4

Chapter 1 Review (continued)

108. Evaluate for $p = -2.5$.

109. Evaluate.

110. Evaluate $|3x - 5|$.

111. Evaluate.

112. Evaluate.

PAGE 26 — Chapter 1 review, part 5

Chapter 1 Review (continued)

113. Evaluate $\sqrt{x + y}$.

     a) $x = 52$, $y = 12$: ______ b) $x = -9$, $y = 13$: ______ c) $x = 0.75$, $y = 0.25$: ______

114. Evaluate $2\sqrt[3]{x} + \sqrt{y}$ for $x = -27$, $y = 36$: ______

115. Evaluate $\dfrac{14}{\sqrt{x}}$. Do not rationalize.

     a) $x = 7$: ______ b) $x = 49$: ______

116. Evaluate $\sqrt{b^2 - 4ac}$ for $a = 3$, $b = 11$, $c = 6$: ______

     WORK SPACE: 1.5 in tall, full width — blank

PAGE 27 — Chapter 1 review, part 6

Chapter 1 Review (continued)

117. Apply it. A repair shop charges $25 + 15h$ dollars.

     For $h = 2.5$: ______ As money: $______

118. Apply it. A square rug covers $6.25$ square meters, so each side is $\sqrt{6.25}$ meters: ______ m

119. Apply it. A cubical box holds $343$ cubic centimeters, so each edge is $\sqrt[3]{343}$ centimeters: ______ cm

120. Apply it. Overnight the temperature was $-13\,^\circ\text{F}$; by noon it was $9\,^\circ\text{F}$. The degrees between the readings is $|T_2 - T_1|$.

     Value: ______

     Why is the answer the same if the readings are named in the other order? _______________________________

PAGE 28 — Chapter 1 review, part 7

Chapter 1 Review (continued)

121. Explain. For $x = -5$: $x^2 =$ ______ and $-x^2 =$ ______

     Why do they differ? What is the exponent attached to in each?

     ---

     ---

122. Find the error. A student evaluates $\dfrac{20}{\sqrt{x}}$ for $x = 5$, writes $\dfrac{20}{\sqrt{5}}$, then says the problem is unfinished because a radical is still in the denominator.

     What does this chapter's convention say? _______________________________________

     ---

     Which chapter teaches the technique the student is thinking of? ______

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 six figures are full width. fig2-operation-word-bank.png and fig1-phrase-to-expression.png both sit on page 2; if the page overflows, move fig1 to the top of page 3 rather than shrinking either one, since the word bank is unreadable below full width. The same applies to fig5 and fig4 on page 12.
• Work space (items 60, 82, 116): a plain white area, 1.5 in tall, no border and no grid. These items need a substitution written out over two or three lines, and a single blank invites a guess.
• Evaluation tables (items 53–58, 78–79, 106–112): make every answer cell at least 0.45 in tall. Students write intermediate steps inside the cell when the value is negative or fractional.
• Fraction and radical prompts: items 55, 70, 78c, 78e, 79e, 107, 111b, and 112c contain stacked fractions inside radicals. Set these as images or use a math font at 16 pt; a 12 pt stacked fraction inside a radical is not legible in print.
• Absolute value bars: keep the bars in items 49–51, 57, 58, 67, 76, 83, 108–110, and 120 tall enough to be distinguished from the digit $1$ and from a table's cell divider. If a bar lands adjacent to a table border, add 4 pt of cell padding.
• The "do not rationalize" note: repeat the callout from page 1 in the margin of pages 17 and 28. Items 77, 84a, 92, 115a, and 122 all end with a radical in a denominator, and a student who has been taught otherwise elsewhere will assume the page is wrong.
• Answer blanks: keep every blank on the same line as its prompt. Items 23–29, 36, 37, 40, 97, 98, and 101 each need two blanks — one for the variable definition sentence and one for the expression — so give those items a full line each.
• Exit tickets (pages 6, 11, 16, 21): each is a half page. Print two per sheet or leave the lower half blank for teacher notes.