Appendix A — Answer Key, Chapter 2: Comparing and Ordering Rational Numbers

SOL 7.NS.2 · Covers textbook Chapter 2 and the companion workbook. Item numbers match the textbook and run continuously from 1 to 85; workbook items are the same problems, so this key serves both. Reasoning answers show an acceptable response, not the only wording.

Lesson 2.1 — What Makes a Number Rational

Guided practice

1. a) $\dfrac{7}{1}$ b) $-\dfrac{5}{2}$ (from $-\tfrac{25}{10}$) c) $\dfrac{3}{5}$ (from $\tfrac{6}{10}$) d) $-\dfrac{7}{4}$
2. $\dfrac{3}{8} = 0.375 = 37.5\%$
3. $45\% = \dfrac{45}{100} = \dfrac{9}{20}$, and $0.45$
4. $-2\tfrac{1}{5} = -\dfrac{11}{5} = -2.2$
5. Yes. $0.125$ is one hundred twenty-five thousandths, so $0.125 = \dfrac{125}{1000} = \dfrac{1}{8}$, a ratio of two integers with a nonzero denominator.

Independent practice

6. a) $0.875$ b) $0.15$ c) $0.36$ d) $2.75$
7. a) $\dfrac{2}{5}$ b) $-\dfrac{3}{8}$ c) $\dfrac{1}{8}$ d) $\dfrac{8}{5}$ (or $1\tfrac{3}{5}$)
8. a) $25\%$ b) $7\%$ c) $90\%$ d) $125\%$
9. a) $0.6$ and $\dfrac{3}{5}$ b) $0.08$ and $\dfrac{2}{25}$ c) $1.5$ and $\dfrac{3}{2}$
10. Rational: $-6$, $0.75$, $-2\tfrac{1}{3}$, $\dfrac{12}{5}$. Not rational: $0.121121112\ldots$, because its digits never end and never fall into a repeating block, so it cannot be written as a ratio of two integers.
11. The same thickness. $\dfrac{5}{8} = 5 \div 8 = 0.625$, so the board matches the plans exactly.
12. Every integer $n$ can be written as $\dfrac{n}{1}$, a ratio of two integers with a nonzero denominator; for example $-9 = \dfrac{-9}{1}$. The reverse is not true, because a rational number like $\dfrac{3}{4}$ is not a whole number or the opposite of one — it falls between $0$ and $1$.

Exit ticket 2.1

13. $\dfrac{35}{100} = \dfrac{7}{20}$, and $35\%$
14. $-\dfrac{11}{8}$ and $-1.375$
15. $0.8$ and $80\%$
16. $-7$ is rational because it can be written as $\dfrac{-7}{1}$, a ratio of two integers with a denominator that is not zero.

Lesson 2.2 — Comparing with Benchmarks and Equivalency

Guided practice

17. Greater than $\dfrac{1}{2}$. Half of $9$ is $4.5$, and the numerator $5$ is more than $4.5$, so $\dfrac{5}{9} > \dfrac{1}{2}$.
18. Closest to $\dfrac{1}{2}$. $\dfrac{3}{8} = 0.375$, which is $0.375$ from $0$, only $0.125$ from $0.5$, and $0.625$ from $1$.
19. $\dfrac{2}{3} = \dfrac{16}{24}$ and $\dfrac{5}{8} = \dfrac{15}{24}$, so $\dfrac{2}{3} > \dfrac{5}{8}$.
20. $0.6 = \dfrac{3}{5}$, so $0.6 = \dfrac{3}{5}$.
21. $45\% = 0.45$ and $\dfrac{2}{5} = 0.4$, so $45\% > \dfrac{2}{5}$.

Independent practice

22. a) less ($4 < 4.5$) b) greater ($7 > 6$) c) less ($5 < 5.5$) d) greater ($9 > 8$)
23. a) $\dfrac{15}{20} > \dfrac{14}{20}$, so $\dfrac{3}{4} > \dfrac{7}{10}$ b) $\dfrac{15}{18} < \dfrac{16}{18}$, so $\dfrac{5}{6} < \dfrac{8}{9}$ c) $-\dfrac{10}{15} < -\dfrac{9}{15}$, so $-\dfrac{2}{3} < -\dfrac{3}{5}$
24. a) $0.375 < 0.400$, so $\dfrac{3}{8} < 0.4$ b) $\dfrac{7}{20} = 0.35$, so $\dfrac{7}{20} = 0.35$ c) $-0.25 < -0.20$, so $-\dfrac{1}{4} < -0.2$
25. a) $\dfrac{5}{8} = 62.5\%$, so $65\% > \dfrac{5}{8}$ b) $\dfrac{3}{8} = 37.5\%$, so $30\% < \dfrac{3}{8}$ c) $120\% = \dfrac{6}{5}$, so $120\% = \dfrac{6}{5}$
26. $-0.6$ is greater. $-\dfrac{5}{8} = -0.625$, and $0.6 < 0.625$, so $-0.6$ is closer to zero and therefore lies farther right: $-0.6 > -\dfrac{5}{8}$.
27. Mr. Kim's class. $\dfrac{3}{8} = 0.375$ and $40\% = 0.40$. Padding to thousandths, $0.375 < 0.400$, so a larger share of Mr. Kim's class walks.
28. Half of $12$ is $6$ and $5 < 6$, so $\dfrac{5}{12} < \dfrac{1}{2}$. Half of $13$ is $6.5$ and $7 > 6.5$, so $\dfrac{7}{13} > \dfrac{1}{2}$. One is below the benchmark and the other is above it, so $\dfrac{5}{12} < \dfrac{7}{13}$ without ever building $156$ths.

Exit ticket 2.2

29. Less than $\dfrac{1}{2}$. Half of $15$ is $7.5$, and $7 < 7.5$.
30. $\dfrac{5}{6} > 0.8$, since $\dfrac{5}{6} = 0.8\overline{3}$ and $0.833\ldots > 0.800$.
31. $-\dfrac{3}{4} < -0.7$, since $-\dfrac{3}{4} = -0.75$ and $0.75 > 0.70$, so $-0.75$ is farther from zero.
32. Half of the denominator $20$ is $10$. The numerator $9$ is less than $10$, so the fraction is less than half of the whole: $\dfrac{9}{20} < \dfrac{10}{20} = \dfrac{1}{2}$.

Lesson 2.3 — Comparing on a Number Line with $<$, $>$, and $=$

Guided practice

33. $-0.5$ is greater. On the figure, $-0.5$ sits to the right of $-\dfrac{7}{4} = -1.75$, and anything farther right is greater.
34. $-2.05$ is farther right, because $2.05 < 2.50$ makes $-2.05$ closer to zero.
35. $-1.2 < -1.02$ (pad to $-1.20$ versus $-1.02$; $1.20 > 1.02$, so $-1.20$ is farther left)
36. $\dfrac{3}{4} = 0.75$
37. $0.1 > -\dfrac{2}{5}$

Independent practice

38. a) $-0.8 = -\dfrac{4}{5}$ b) $1\dfrac{1}{2} = 1.50 > 1.45$ c) $-\dfrac{3}{8} = -0.375 > -0.400$, so $-\dfrac{3}{8} > -0.4$ d) $85\% = 0.85 = 0.850$
39. Any value strictly between them works, such as $-0.4$ (or $-\dfrac{3}{8} = -0.375$). It sits to the right of $-0.5$ and to the left of $-0.25$, between the halfway mark and the quarter mark on the negative side.
40. $-3.45$ is greater. $-\dfrac{7}{2} = -3.5$, and on the number line $-3.5$ lies to the left of $-3.45$ because it is farther from zero.
41. a) $-1.7 > -1.75$ b) $0.375 < \dfrac{2}{5}$
42. a) False. $-0.6 < -0.06$, because $0.60 > 0.06$, so $-0.6$ is farther left. b) True. c) False. $-2\dfrac{1}{4} = -2.25$ and $2.25 < 2.40$, so $-2.25$ is closer to zero: $-2\dfrac{1}{4} > -2.4$.
43. $-1.5 < -1\dfrac{1}{4}$ (that is, $-1.5 < -1.25$). Wednesday's change was the greater value, since $-1.25$ is closer to zero — the temperature dropped less than it did on Tuesday.
44. Padding with zeros adds tenths, hundredths, or thousandths worth nothing at all: $0.5 = \dfrac{5}{10} = \dfrac{50}{100} = \dfrac{500}{1000}$. Since all three fractions are equal, all three decimals name the identical point on the number line, so padding never changes a value — it only lines up the place values for comparison.

Exit ticket 2.3

45. $-1.6 < -1.06$
46. $\dfrac{7}{8} = 0.875$
47. $-\dfrac{5}{4}$ is greater, since $-\dfrac{5}{4} = -1.25$ and $1.25 < 1.30$, so $-1.25$ lies to the right of $-1.3$.
48. Write the fraction as a decimal so both numbers are in the same form, then locate both to the left of zero. The one closer to zero is farther right, and farther right always means greater. The number farther from zero is the lesser one.

Lesson 2.4 — Ordering a Set of Rational Numbers

Guided practice

49. Decimals: $0.400$, $0.125$, $0.650$, $0.500$. Ascending: $\dfrac{1}{8}$, $0.4$, $\dfrac{1}{2}$, $0.65$
50. Decimals: $-0.50$, $0.25$, $-1.25$, $0.75$. Descending: $\dfrac{3}{4}$, $0.25$, $-\dfrac{1}{2}$, $-1.25$
51. Decimals: $0.600$, $0.625$, $0.575$, $0.666\ldots$ Ascending: $0.575$, $60\%$, $\dfrac{5}{8}$, $\dfrac{2}{3}$
52. Decimals: $-0.75$, $-0.60$, $-0.80$, $-0.70$. Descending: $-\dfrac{3}{5}$, $-\dfrac{7}{10}$, $-0.75$, $-0.8$
53. On a line from $-2$ to $2$ in fourths: $-1.25$ is one tick left of $-1$; $-\dfrac{1}{2}$ is two ticks left of $0$; $0.25$ is one tick right of $0$; $\dfrac{3}{4}$ is three ticks right of $0$, one tick short of $1$. Left to right: $-1.25$, $-\dfrac{1}{2}$, $0.25$, $\dfrac{3}{4}$ — which is the reverse of the descending answer in item 50, as it should be.

Independent practice

54. $-2$, $-0.3$, $\dfrac{1}{4}$, $0.75$
55. Decimals: $0.375$, $0.400$, $0.350$, $0.333\ldots$ Descending: $0.4$, $\dfrac{3}{8}$, $35\%$, $\dfrac{1}{3}$
56. Decimals: $-1.50$, $-1.05$, $-1.60$, $-1.25$. Ascending: $-1.6$, $-1\dfrac{1}{2}$, $-1\dfrac{1}{4}$, $-1.05$
57. Decimals: $0.102$, $0.120$, $0.125$, $0.090$. Descending: $\dfrac{1}{8}$, $12\%$, $0.102$, $0.09$
58. Coldest to warmest: $-1.5$°C, $-0.25$°C, $0.75$°C, $1\dfrac{1}{4}$°C
59. Decimals: $2.500$, $2.375$, $2.450$, $2.250$. Heaviest to lightest: $2.5$ lb, $2.45$ lb, $2\dfrac{3}{8}$ lb, $2\dfrac{1}{4}$ lb. The tenths place did most of the work: all four weights have the same whole-number part of $2$, so the comparison began at the tenths digit ($5$, $3$, $4$, $2$), and only $2.45$ versus $2.375$ needed a look past it.
60. The student sorted the digits $25$, $3$, $5$ as if the numbers were positive. On the negative side, larger digits mean farther from zero, which means less. The list given is actually descending. Correct ascending order: $-0.5$, $-0.3$, $-0.25$.

Exit ticket 2.4

61. Decimals: $0.60$, $0.50$, $0.55$, $0.48$. Ascending: $0.48$, $\dfrac{1}{2}$, $55\%$, $0.6$
62. Decimals: $-0.25$, $-0.20$, $-0.35$, $-0.50$. Descending: $-0.2$, $-\dfrac{1}{4}$, $-0.35$, $-\dfrac{1}{2}$
63. Decimals: $0.333\ldots$, $0.300$, $0.350$, $0.303$. Ascending: $0.3$, $0.303$, $\dfrac{1}{3}$, $35\%$
64. Sample justification: I converted every value to a decimal and padded to thousandths, because place-value comparison is fastest. That gives $0.300$, $0.303$, $0.333\ldots$, and $0.350$. All four agree in the tenths place with a $3$, so the hundredths place decides the order: $0$, $0$, $3$, $5$. The tie between $0.300$ and $0.303$ breaks in the thousandths place, where $0 < 3$. So the ascending order is $0.3$, $0.303$, $\dfrac{1}{3}$, $35\%$.

Chapter 2 Review

Part A — Forms of a rational number (7.NS.2a)

65. a) $0.625$ b) $-0.75$ c) $2.2$ d) $3.5$
66. a) $9\%$ b) $60\%$ c) $125\%$
67. a) $\dfrac{24}{100} = \dfrac{6}{25}$ b) $\dfrac{5}{100} = \dfrac{1}{20}$ c) $\dfrac{250}{100} = \dfrac{5}{2}$
68. $-\dfrac{875}{1000} = -\dfrac{7}{8}$
69. Rational: $-11$, $0.25$, $\dfrac{4}{9}$. Not rational: $0.303003000\ldots$, since it neither terminates nor repeats.
70. $\dfrac{11}{8}$ and $1.375$

Part B — Comparing with benchmarks, equivalency, and symbols (7.NS.2a)

71. a) $\dfrac{5}{8} = 0.625 > 0.600$, so $>$ b) $-\dfrac{2}{5} = -0.40$ and $0.40 > 0.35$, so $<$ c) $=$ d) $-1\dfrac{1}{8} = -1.125$ and $1.125 < 1.200$, so $>$
72. a) greater ($4 > 3.5$) b) less ($11 < 12$) c) less ($0.499 < 0.500$) d) greater ($51\% > 50\%$)
73. $\dfrac{7}{12} = \dfrac{21}{36}$ and $\dfrac{5}{9} = \dfrac{20}{36}$, so $\dfrac{7}{12} > \dfrac{5}{9}$.
74. $-0.85$ is greater. $-\dfrac{9}{10} = -0.90$, and $0.90 > 0.85$, so $-0.9$ is farther from zero and lies to the left.
75. $\dfrac{1}{3} < 0.35$
76. Any value strictly between $0.25$ and $0.3$ works, such as $0.28$. Since $\dfrac{1}{4} = 0.25$ and $0.250 < 0.280 < 0.300$, the number $0.28$ lies between them. ($\dfrac{9}{32} = 0.28125$ also works.)

Part C — Ordering a set of four (7.NS.2a)

77. Decimals: $0.700$, $0.750$, $0.680$, $0.625$. Ascending: $\dfrac{5}{8}$, $68\%$, $0.7$, $\dfrac{3}{4}$
78. Decimals: $-0.400$, $-0.333\ldots$, $-0.450$, $-0.375$. Descending: $-\dfrac{1}{3}$, $-\dfrac{3}{8}$, $-0.4$, $-0.45$
79. Ascending: $-1.5$, $-0.75$, $\dfrac{1}{2}$, $1.05$
80. Decimals: $1.250$, $1.025$, $1.300$, $1.200$. Descending: $130\%$, $1\dfrac{1}{4}$, $1.2$, $1.025$

Part D — Mixed application and reasoning

81. Decimals: $3.250$, $3.125$, $3.400$, $3.500$. Longest to shortest: $3\dfrac{1}{2}$ m, $3.4$ m, $3.25$ m, $3\dfrac{1}{8}$ m
82. Least to greatest: $-\$12.50$, $-\$12.05$, $-\$3.40$, $\$8.75$. The change that hurt the balance most is $-\$12.50$, which is also the least value. It is the least because it sits farthest to the left of zero, but describing it as "smallest" is misleading: the amount removed, $\$12.50$, is the largest of the three withdrawals. Size measured as distance from zero and value measured as position on the number line point in opposite directions for negatives.
83. The student compared $50$ and $45$ as if both numbers were positive. On the negative side of zero, the larger the digits, the farther from zero the number sits, and farther from zero on the left means less. Since $0.50 > 0.45$, the value $-0.5$ lies to the left of $-0.45$. Correct comparison: $-0.5 < -0.45$.
84. Strategy 1, convert to decimals: $\dfrac{5}{8} = 5 \div 8 = 0.625$; padded, compare $0.625$ and $0.600$. The tenths digits tie at $6$, and the hundredths place gives $2 > 0$, so $\dfrac{5}{8} > 0.6$. Strategy 2, convert to a common fraction form: $0.6 = \dfrac{3}{5}$, and with the common denominator $40$, $\dfrac{5}{8} = \dfrac{25}{40}$ while $\dfrac{3}{5} = \dfrac{24}{40}$, so $\dfrac{5}{8} > \dfrac{3}{5} = 0.6$. Both strategies give the same result, which is the check.
85. Mark the line from $-2$ to $2$ with a tick every one fourth, labeling the integers. Then $-1\dfrac{1}{4}$ falls one tick to the left of $-1$; $-0.5 = -\dfrac{2}{4}$ falls two ticks to the left of $0$; $0.25$ falls one tick to the right of $0$; and $1\dfrac{1}{2} = 1\dfrac{2}{4}$ falls two ticks to the right of $1$. Ascending order: $-1\dfrac{1}{4}$, $-0.5$, $0.25$, $1\dfrac{1}{2}$. The picture proves it because values increase from left to right on a number line, so reading the plotted points left to right is the least-to-greatest ordering — no further arithmetic is needed.

Workbook-only items

Page 2, fill in the blanks. Every integer is rational because you can write it over 1. A proper fraction has a numerator smaller than its denominator. An improper fraction has a numerator at least as large as its denominator. The word percent means per hundred.

Page 6, benchmark blanks (item 17). Half of $9$ is 4.5, and $5$ is more than that.

Page 10, padding practice.

Page 12, plot it. Left to right: $-\dfrac{7}{4}$ (one tick left of $-1.5$, that is three ticks left of $-1$), $-0.5$, $\dfrac{3}{4}$, $1\dfrac{1}{4}$. In order: $-\dfrac{7}{4}$, $-0.5$, $\dfrac{3}{4}$, $1\dfrac{1}{4}$.

Page 14, the four steps. 1. Convert everything to decimals and pad to the same number of places. 2. Sort the negatives first, then zero, then the positives. 3. Write the answer back in the original forms. 4. Check that the first value is the least and the last is the greatest.

Page 20, item 85 blank number line. Ticks every one fourth from $-2$ to $2$; the four plotted points are at $-1\dfrac{1}{4}$, $-0.5$, $0.25$, and $1\dfrac{1}{2}$, matching the answer to item 85 above.