Grade 7 Workbook — Chapter 2: Comparing and Ordering Rational Numbers

SOL 7.NS.2 · 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/. Item numbers match the textbook exactly, so one answer key serves both books.

PAGE 1 — Chapter opener

Chapter 2 · Comparing and Ordering Rational Numbers

Standard 7.NS.2

In this chapter you will:

• Recognize a rational number in all of its forms
• Compare rational numbers using benchmarks and equivalency
• Compare on a number line using $<$, $>$, and $=$
• Order a set of up to four rational numbers, least to greatest or greatest to least
• Justify every answer in writing or with a model

Words to know: rational number · proper fraction · improper fraction · mixed number · percent · repeating decimal · equivalent · benchmark · ascending order · descending order

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 — What makes a number rational

2.1 What Makes a Number Rational

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

FIGURE: fig2-three-forms.png (full width)

Fill in the blanks.

Every integer is rational because you can write it over ________.

A ________ fraction has a numerator smaller than its denominator.

An ________ fraction has a numerator at least as large as its denominator.

The word percent means per ____________.

1. Write each number as a fraction of two integers.

5. Is $0.125$ a rational number? Explain how you know.

PAGE 3 — Moving between forms

Fraction · Decimal · Percent

FIGURE: fig3-fraction-decimal-percent-line.png (full width)

2. Write $\dfrac{3}{8}$ as a decimal ________ and as a percent ________

3. Write $45\%$ as a fraction in simplest form ________ and as a decimal ________

4. Write $-2\tfrac{1}{5}$ as an improper fraction ________ and as a decimal ________

6. Write each fraction as a decimal.

7. Write each decimal as a fraction in simplest form.

PAGE 4 — Percents and reasoning

Percents, Both Directions

8. Write each as a percent.

9. Write each percent as a decimal and as a fraction in simplest form.

10. Circle the rational numbers. Cross out anything that is not rational.

$$-6 \qquad 0.75 \qquad -2\tfrac{1}{3} \qquad 0.121121112\ldots \qquad \tfrac{12}{5}$$

11. Apply it. Plans call for a board $\tfrac{5}{8}$ in thick. The store label says $0.625$ in. Same or different?

Conversion: _______________ Answer: _______________

12. Explain. Why is every integer a rational number? Use $-9$. Why is the reverse not true?

PAGE 5 — Exit ticket 2.1

Exit Ticket · Lesson 2.1

Name: ________________________ Date: ____________

13. Write $0.35$ as a fraction in simplest form ________ and as a percent ________

14. Write $-1\tfrac{3}{8}$ as an improper fraction ________ and as a decimal ________

15. Write $\dfrac{4}{5}$ as a decimal ________ and as a percent ________

16. Explain in a sentence why $-7$ is a rational number.

PAGE 6 — Benchmarks

2.2 Comparing with Benchmarks

FIGURE: fig1-benchmark-line.png (full width)

A fraction is greater than $\tfrac{1}{2}$ when the numerator is more than half the denominator.

17. Is $\dfrac{5}{9}$ less than, greater than, or equal to $\dfrac{1}{2}$? ________

Half of $9$ is ________, and $5$ is ________ than that.

18. Is $\dfrac{3}{8}$ closest to $0$, $\dfrac{1}{2}$, or $1$? ________

22. Greater than or less than $\dfrac{1}{2}$? Write G or L.

28. Explain. How does the benchmark $\dfrac{1}{2}$ let you compare $\dfrac{5}{12}$ and $\dfrac{7}{13}$ with no common denominator?

PAGE 7 — Equivalency

Comparing by Equivalency

FIGURE: fig5-zoom-compare.png (full width)

19. Compare $\dfrac{2}{3}$ and $\dfrac{5}{8}$ using the common denominator $24$.

$\dfrac{2}{3} = \dfrac{\ \ \ }{24}$ $\dfrac{5}{8} = \dfrac{\ \ \ }{24}$ So $\dfrac{2}{3} \ \square \ \dfrac{5}{8}$

20. Compare $0.6$ and $\dfrac{3}{5}$: $\ 0.6 \ \square \ \dfrac{3}{5}$

21. Compare $45\%$ and $\dfrac{2}{5}$: $\ 45\% \ \square \ \dfrac{2}{5}$

23. Common denominator. Fill in $<$, $>$, or $=$.

24. Convert to decimals. Fill in $<$, $>$, or $=$.

25. Percent against fraction. Fill in $<$, $>$, or $=$.

PAGE 8 — Negatives and an application

Comparing Negative Rational Numbers

On the negative side of zero, the number CLOSER to zero is GREATER.

26. Which is greater, $-\dfrac{5}{8}$ or $-0.6$? ________

Show the work that proves it:

27. Apply it. In Ms. Reyes' class $\dfrac{3}{8}$ of the students walk to school. In Mr. Kim's class $40\%$ walk. Which class has the greater share of walkers?

Conversions: $\dfrac{3}{8} =$ ________ $40\% =$ ________

Answer: ________________

Written justification: _______________________________________________

PAGE 9 — Exit ticket 2.2

Exit Ticket · Lesson 2.2

Name: ________________________ Date: ____________

29. Is $\dfrac{7}{15}$ greater than or less than $\dfrac{1}{2}$? ________

30. Fill in: $\dfrac{5}{6} \ \square \ 0.8$

31. Fill in: $-\dfrac{3}{4} \ \square \ -0.7$

32. Explain how you know $\dfrac{9}{20} < \dfrac{1}{2}$ without dividing.

PAGE 10 — The number line rule

2.3 Comparing on a Number Line

The number farther to the RIGHT is GREATER.

FIGURE: fig4-negatives-on-the-line.png (full width)

33. Using the figure, which is greater, $-\dfrac{7}{4}$ or $-0.5$? ________

How does the picture show it? _______________________________________________

34. Which is farther to the right, $-2.5$ or $-2.05$? ________

Fill in $<$, $>$, or $=$.

35. $-1.2 \ \square \ -1.02$

36. $\dfrac{3}{4} \ \square \ 0.75$

37. Rewrite $-\dfrac{2}{5} < 0.1$ using $>$: _______________

Padding practice. Rewrite each pair with the same number of decimal places before comparing.

PAGE 11 — Symbols and error hunt

Symbols and Corrections

38. Fill in $<$, $>$, or $=$.

39. Name a rational number between $-0.5$ and $-0.25$: ________

Where does it sit on the number line? _______________________________________________

40. Which is greater, $-\dfrac{7}{2}$ or $-3.45$? ________ Explain: ____________________

41. Rewrite each statement the other way.

a) $-1.75 < -1.7$ becomes _______________

b) $\dfrac{2}{5} > 0.375$ becomes _______________

42. True or false? Correct every false one.

PAGE 12 — Apply and explain

Apply It · Lesson 2.3

43. Apply it. Tuesday's temperature change was $-1.5$°C. Wednesday's was $-1\dfrac{1}{4}$°C.

Comparison with a symbol: _______________

Which day's change was greater? _______________

Why? _______________________________________________

44. Explain. Why do $0.5$, $0.50$, and $0.500$ all sit at the same point on a number line? Why is padding with zeros safe?

Plot it. Place $-\dfrac{7}{4}$, $-0.5$, $\dfrac{3}{4}$, and $1\dfrac{1}{4}$ on the line below, then write the four values in order left to right.

BLANK NUMBER LINE: -2 to 2, tick marks every one fourth, only integers labeled

PAGE 13 — Exit ticket 2.3

Exit Ticket · Lesson 2.3

Name: ________________________ Date: ____________

45. Fill in: $-1.6 \ \square \ -1.06$

46. Fill in: $\dfrac{7}{8} \ \square \ 0.875$

47. Which is greater, $-\dfrac{5}{4}$ or $-1.3$? ________

48. Explain how a number line decides a comparison between a negative fraction and a negative decimal.

PAGE 14 — Ordering: the procedure

2.4 Ordering a Set of Rational Numbers

Ascending = least to greatest. Descending = greatest to least.

FIGURE: fig6-ordering-four.png (full width)

The four steps.

1. Convert everything to ____________ and pad to the same number of places.
2. Sort the ____________ first, then zero, then the positives.
3. Write the answer back in the ____________ forms.
4. Check that the first value is the ____________ and the last is the ____________.

49. Least to greatest: $0.4$, $\dfrac{1}{8}$, $0.65$, $\dfrac{1}{2}$

Decimals: ________ ________ ________ ________

Answer: _______________________________________________

50. Greatest to least: $-\dfrac{1}{2}$, $0.25$, $-1.25$, $\dfrac{3}{4}$

Decimals: ________ ________ ________ ________

Answer: _______________________________________________

PAGE 15 — Ordering practice

Ordering Practice

51. Least to greatest: $60\%$, $\dfrac{5}{8}$, $0.575$, $\dfrac{2}{3}$

52. Greatest to least: $-0.75$, $-\dfrac{3}{5}$, $-0.8$, $-\dfrac{7}{10}$

53. Describe where each number from item 50 sits on a number line from $-2$ to $2$ marked in fourths, then read them off left to right.

54. Least to greatest: $-2$, $\dfrac{1}{4}$, $-0.3$, $0.75$

55. Greatest to least: $\dfrac{3}{8}$, $0.4$, $35\%$, $\dfrac{1}{3}$

56. Least to greatest: $-1\dfrac{1}{2}$, $-1.05$, $-1.6$, $-1\dfrac{1}{4}$

57. Greatest to least: $0.102$, $12\%$, $\dfrac{1}{8}$, $0.09$

PAGE 16 — Ordering in the real world

Apply It · Lesson 2.4

58. Apply it. Four days' temperature change from normal: $-1.5$°C, $0.75$°C, $-0.25$°C, $1\dfrac{1}{4}$°C.

Coldest to warmest: _______________________________________________

59. Apply it. Four bags of flour weigh $2.5$ lb, $2\dfrac{3}{8}$ lb, $2.45$ lb, $2\dfrac{1}{4}$ lb.

Heaviest to lightest: _______________________________________________

Which single place value did the most work in your comparison, and why?

60. Find the error. A student ordered $-0.3$, $-0.25$, $-0.5$ as $-0.25$, $-0.3$, $-0.5$ and called it ascending.

What went wrong? _______________________________________________

Correct ascending order: _______________________________________________

PAGE 17 — Exit ticket 2.4

Exit Ticket · Lesson 2.4

Name: ________________________ Date: ____________

61. Least to greatest: $0.6$, $\dfrac{1}{2}$, $55\%$, $0.48$

62. Greatest to least: $-\dfrac{1}{4}$, $-0.2$, $-0.35$, $-\dfrac{1}{2}$

63. Least to greatest: $\dfrac{1}{3}$, $0.3$, $35\%$, $0.303$

64. Justify your ordering in item 63. Name the strategy and show the evidence.

PAGE 18 — Chapter 2 review, part 1

Chapter 2 Review

Part A · Forms of a rational number

65. Write as a decimal: a) $\dfrac{5}{8}$ ______ b) $-\dfrac{3}{4}$ ______ c) $2\dfrac{1}{5}$ ______ d) $\dfrac{7}{2}$ ______

66. Write as a percent: a) $0.09$ ______ b) $\dfrac{3}{5}$ ______ c) $1\dfrac{1}{4}$ ______

67. Percent to fraction in simplest form: a) $24\%$ ______ b) $5\%$ ______ c) $250\%$ ______

68. Write $-0.875$ as a fraction in simplest form. ______

69. Circle the rational numbers: $-11$ · $0.25$ · $\dfrac{4}{9}$ · $0.303003000\ldots$ (never ends, never repeats)

70. Write $1\dfrac{3}{8}$ as an improper fraction ______ and as a decimal ______

PAGE 19 — Chapter 2 review, part 2

Chapter 2 Review (continued)

Part B · Comparing

71. Fill in $<$, $>$, or $=$:

72. Greater than or less than $\dfrac{1}{2}$? a) $\dfrac{4}{7}$ ______ b) $\dfrac{11}{24}$ ______ c) $0.499$ ______ d) $51\%$ ______

73. Compare $\dfrac{7}{12}$ and $\dfrac{5}{9}$ using a common denominator.

$\dfrac{7}{12} = \dfrac{\ \ \ }{36}$ $\dfrac{5}{9} = \dfrac{\ \ \ }{36}$ Symbol: ______

74. Which is greater, $-\dfrac{9}{10}$ or $-0.85$? ______

75. Rewrite $0.35 > \dfrac{1}{3}$ using $<$: _______________

76. Name a rational number between $\dfrac{1}{4}$ and $0.3$: ______ Show why: ____________________

Part C · Ordering a set of four

77. Least to greatest: $0.7$, $\dfrac{3}{4}$, $68\%$, $\dfrac{5}{8}$

78. Greatest to least: $-0.4$, $-\dfrac{1}{3}$, $-0.45$, $-\dfrac{3}{8}$

79. Least to greatest: $-1.5$, $\dfrac{1}{2}$, $-0.75$, $1.05$

80. Greatest to least: $1\dfrac{1}{4}$, $1.025$, $130\%$, $1.2$

PAGE 20 — Chapter 2 review, part 3

Chapter 2 Review (continued)

Part D · Application and reasoning

81. Four long jumps: $3.25$ m, $3\dfrac{1}{8}$ m, $3.4$ m, $3\dfrac{1}{2}$ m. Longest to shortest:

82. Four account changes: $-\$12.50$, $\$8.75$, $-\$12.05$, $-\$3.40$. Least to greatest:

Which change hurt the balance most? ______ Why is that not the same as "smallest in size"?

83. A student says $-0.5 > -0.45$ because $50 > 45$. Explain the error and write the correct comparison.

Correct comparison: _______________

84. Describe TWO strategies for comparing $\dfrac{5}{8}$ and $0.6$. Carry out both and confirm they agree.

Strategy 1: _______________________________________________

Strategy 2: _______________________________________________

Result: _______________

85. Describe a number line from $-2$ to $2$ marked in fourths. Place $-1\dfrac{1}{4}$, $-0.5$, $0.25$, $1\dfrac{1}{2}$, state the ascending order, and explain how the picture proves it.

BLANK NUMBER LINE: -2 to 2, tick marks every one fourth, only integers labeled

Ascending order: _______________________________________________

Explanation: _______________________________________________

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
• Blank number lines: pages 12 and 20 need an unlabeled line from $-2$ to $2$ with ticks every one fourth and only the integers labeled; generate a blank variant with figures/make_grade7_ch02.py if one is added
• Figure widths: all six chapter figures are landscape number lines; place them full width
• Comparison boxes: the $\square$ symbol marks where the student writes $<$, $>$, or $=$; keep it at least 24 pt so there is room to write
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow