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

SOL 8.NS.1 · Covers textbook Chapter 2 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 122 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used throughout: $\sqrt{\ }$ is the positive (principal) square root; $\approx$ means "approximately equal to"; every approximation of an irrational number is given to the nearest hundredth. Where a decimal approximation is shown for a radical, the exact value is the radical, not the decimal.

Lesson 2.1 — Both Square Roots of a Number

Guided practice

1. $12$, since $12^2 = 144$.
2. $-12$. Take the positive root first, then apply the minus sign.
3. $0$. Only $0 \cdot 0 = 0$.
4. $15$ and $-15$, since $15^2 = 225$ and $(-15)^2 = 225$.
5. Rational. $121 = 11^2$ is a perfect square, so $\sqrt{121} = 11$, an integer.
6. Irrational. $122$ lies between the perfect squares $121$ and $144$, so it is not a perfect square and its root has a nonterminating, nonrepeating decimal.

Independent practice

7. a) $13$ b) $-14$ c) $20$ d) $-1$
8. $8$ and $-8$
9. $\sqrt{50}$ and $-\sqrt{80}$ are irrational. $\sqrt{49} = 7$ and $-\sqrt{81} = -9$ are rational, because $49$ and $81$ are perfect squares.
10. True. $\sqrt{36} = 6$, and the minus sign in front makes the value $-6$. Both $6$ and $-6$ square to $36$, and the notation $-\sqrt{36}$ names the negative one.
11. $\sqrt{100} = 10$ and $-\sqrt{100} = -10$. They are opposites: the same distance from zero, $10$ units, but in opposite directions. On a number line, $\sqrt{100}$ sits $10$ units right of zero and $-\sqrt{100}$ sits $10$ units left of zero, so $-\sqrt{100} < \sqrt{100}$.
12. $17$ feet, since $17^2 = 289$. The other square root of $289$ is $-17$, and a side length cannot be negative, so only the positive root answers the question.
13. Every other perfect square has two roots because a positive number and its opposite both square to the same positive value. Zero has no distinct opposite — the opposite of $0$ is $0$ — so there is only one number that squares to $0$.
14. The minus sign is not inside the radical and nothing cancels. Read the expression as "the opposite of $\sqrt{49}$": first $\sqrt{49} = 7$, then apply the minus sign. Correct value: $-7$.

Exit ticket 2.1

15. $16$, since $16^2 = 256$.
16. $-18$, since $18^2 = 324$.
17. $19$ and $-19$, since $19^2 = 361$.
18. $\sqrt{16} = 4$ is the positive square root of $16$; $-\sqrt{16} = -4$ is its opposite, the negative square root. Both square to $16$, but they sit on opposite sides of zero, and $-\sqrt{16} < \sqrt{16}$.

Lesson 2.2 — Between Which Two Natural Numbers?

For every item in this lesson, the two consecutive natural numbers come from the perfect squares that bracket the radicand, and the better approximation is decided by comparing the two gaps: the smaller gap wins.

Guided practice

19. $36 < 40 < 49$, so between $6$ and $7$. Gaps $4$ and $9$, so $6$ is better.
20. $81 < 90 < 100$, so between $9$ and $10$. Gaps $9$ and $10$, so $9$ is better. It is a near tie; $\sqrt{90} = 9.4868\ldots$, just under $9.5$.
21. $144 < 150 < 169$, so between $12$ and $13$. Gaps $6$ and $19$, so $12$ is better.
22. $64 < 75 < 81$, so between $8$ and $9$. Gaps $11$ and $6$, so $9$ is better.
23. $16 < 20 < 25$, so between $4$ and $5$. Gaps $4$ and $5$, so $4$ is better.
24. $289 < 300 < 324$, so between $17$ and $18$. Gaps $11$ and $24$, so $17$ is better.
25. $4 < 8 < 9$, so between $2$ and $3$. Gaps $4$ and $1$, so $3$ is better.
26. $361 < 380 < 400$, so between $19$ and $20$. Gaps $19$ and $20$, so $19$ is better.

Independent practice

27. a) $3$ and $4$ b) $7$ and $8$ c) $9$ and $10$ d) $14$ and $15$
28. a) $3$, since $12 - 9 = 3$ and $16 - 12 = 4$. b) $7$, since $55 - 49 = 6$ and $64 - 55 = 9$. c) $10$, since $99 - 81 = 18$ and $100 - 99 = 1$. d) $14$, since $210 - 196 = 14$ and $225 - 210 = 15$.
29. Between $11$ and $12$, since $121 < 125 < 144$. Gaps $4$ and $19$, so $11$ is better.
30. Between $15$ and $16$, since $225 < 250 < 256$. Gaps $25$ and $6$, so $16$ is better.
31. Between $13$ and $14$, since $169 < 170 < 196$. Gaps $1$ and $26$, so $13$ is better.
32. $400$ is a perfect square: $\sqrt{400} = 20$ exactly. The question assumes the root falls strictly between two whole numbers, and this one is a whole number, so there is no pair of consecutive natural numbers it lies between.
33. The side is $\sqrt{130}$, and $121 < 130 < 144$, so it is between $11$ and $12$ meters. Gaps $9$ and $14$, so $11$ meters is the better estimate.
34. The side is $\sqrt{45}$, between $6$ and $7$ feet. Gaps $9$ and $4$, so $7$ feet is the nearer whole foot. Since $45 < 49$, the true side is less than $7$, so the estimate of $7$ feet is a little too big.
35. Halfway between the bracketing perfect squares $36$ and $49$ is $42.5$. A radicand below $42.5$ puts the root in the lower half of the interval from $6$ to $7$, and a radicand above $42.5$ puts it in the upper half. Since $40 < 42.5$, the root is in the lower half and $6$ is nearer — and none of that required knowing the value of $\sqrt{40}$.
36. "Close to $100$" is not the test; the comparison of both gaps is. Here $90 - 81 = 9$ and $100 - 90 = 10$, so $90$ is actually nearer to $81$. The better approximation is $9$. The gaps between consecutive perfect squares grow as the numbers grow, which is exactly why the eyeball estimate misleads.

Exit ticket 2.2

37. $49 < 60 < 64$, so between $7$ and $8$. Gaps $11$ and $4$, so $8$ is better.
38. $100 < 110 < 121$, so between $10$ and $11$. Gaps $10$ and $11$, so $10$ is better.
39. $36 < 45 < 49$, so between $6$ and $7$. Gaps $9$ and $4$, so $7$ is better.
40. An acceptable response: find the two perfect squares that the radicand sits between; find the number halfway between those two perfect squares; if the radicand is below the halfway number, the smaller root is the better approximation, and if it is above, the larger root is. Comparing the two gaps directly gives the same answer.

Lesson 2.3 — Rational Approximations to the Nearest Hundredth

Each answer below shows the two hundredths whose squares bracket the radicand; the nearer one is the approximation.

Guided practice

41. $1.41^2 = 1.9881$ and $1.42^2 = 2.0164$. Since $2 - 1.9881 = 0.0119 < 0.0164$, $\sqrt{2} \approx 1.41$.
42. $2.23^2 = 4.9729$ and $2.24^2 = 5.0176$. Since $5 - 4.9729 = 0.0271$ and $5.0176 - 5 = 0.0176$, $\sqrt{5} \approx 2.24$.
43. $3.16^2 = 9.9856$ and $3.17^2 = 10.0489$. Since $10 - 9.9856 = 0.0144 < 0.0489$, $\sqrt{10} \approx 3.16$.
44. $5.47^2 = 29.9209$ and $5.48^2 = 30.0304$. Since $30.0304 - 30 = 0.0304 < 0.0791$, $\sqrt{30} \approx 5.48$.
45. $6.32^2 = 39.9424$ and $6.33^2 = 40.0689$. Since $40 - 39.9424 = 0.0576 < 0.0689$, $\sqrt{40} \approx 6.32$.
46. $\sqrt{8} \approx 2.83$, since $2.82^2 = 7.9524$ and $2.83^2 = 8.0089$, and $8.0089 - 8 = 0.0089$ is the smaller gap. So $-\sqrt{8} \approx -2.83$.
47. $14.14^2 = 199.9396$ and $14.15^2 = 200.2225$. Since $200 - 199.9396 = 0.0604 < 0.2225$, $\sqrt{200} \approx 14.14$.
48. $\sqrt{150} \approx 12.25$, since $12.24^2 = 149.8176$ and $12.25^2 = 150.0625$, and $0.0625 < 0.1824$. So $-\sqrt{150} \approx -12.25$.

Independent practice

49. a) $\sqrt{3} \approx 1.73$ b) $\sqrt{7} \approx 2.65$ c) $\sqrt{15} \approx 3.87$ d) $\sqrt{20} \approx 4.47$
50. a) $-\sqrt{12} \approx -3.46$ b) $-\sqrt{50} \approx -7.07$ c) $\sqrt{90} \approx 9.49$ d) $-\sqrt{300} \approx -17.32$
51. $\sqrt{110} \approx 10.49$. The hundredths squared are $10.48^2 = 109.8304$ and $10.49^2 = 110.0401$; since $0.0401 < 0.1696$, the upper one is nearer.
52. $\sqrt{250} \approx 15.81$. The hundredths squared are $15.81^2 = 249.9561$ and $15.82^2 = 250.2724$; since $0.0439 < 0.2724$, the lower one is nearer.
53. $7.1$ is greater. Squaring both: $(\sqrt{50})^2 = 50$ and $7.1^2 = 50.41$. Since $50 < 50.41$ and both numbers are positive, $\sqrt{50} < 7.1$. (Consistent with $\sqrt{50} \approx 7.07$.)
54. $\sqrt{10}$ is greater. $\pi \approx 3.14$ and $\sqrt{10} \approx 3.16$, so $\sqrt{10} > \pi$. Squaring agrees: $\pi^2 \approx 9.87 < 10$.
55. $-\sqrt{26}$ is greater. $\sqrt{26} \approx 5.10$, more precisely $5.099\ldots$, so $-\sqrt{26} \approx -5.099$, which lies to the right of $-5.2$ on the number line. Squaring the sizes: $26 < 27.04 = 5.2^2$, so $\sqrt{26} < 5.2$, and taking opposites reverses the inequality to $-\sqrt{26} > -5.2$.
56. The side is $\sqrt{45}$. Since $6.70^2 = 44.89$ and $6.71^2 = 45.0241$, and $45.0241 - 45 = 0.0241 < 0.11$, the side is about $6.71$ feet.
57. $\pi$ is irrational, so its decimal never terminates and never repeats; $3.14$ terminates, so it is a rational number and cannot be $\pi$. Writing $\pi \approx 3.14$ claims only that $3.14$ is the closest hundredth to $\pi$ — close enough to work with, not equal.
58. The student stopped at tenths and never checked hundredths, and also never compared $8.4$ against the next tenth. Continuing: $8.4^2 = 70.56$ and $8.5^2 = 72.25$, so the root is between $8.4$ and $8.5$; then $8.48^2 = 71.9104$ and $8.49^2 = 72.0801$, and since $72 - 71.9104 = 0.0896$ while $72.0801 - 72 = 0.0801$, the correct approximation is $\sqrt{72} \approx 8.49$.

Exit ticket 2.3

59. $\sqrt{80} \approx 8.94$. ($8.94^2 = 79.9236$ and $8.95^2 = 80.1025$; the lower gap $0.0764$ is smaller.)
60. $-\sqrt{130} \approx -11.40$. ($11.40^2 = 129.96$ and $11.41^2 = 130.1881$; the lower gap $0.04$ is smaller.)
61. $\sqrt{99}$ is greater. Squaring both: $99$ against $9.9^2 = 98.01$. Since $99 > 98.01$, $\sqrt{99} > 9.9$. (Indeed $\sqrt{99} \approx 9.95$.)
62. An irrational number has a decimal expansion that never terminates and never repeats. Any decimal you can finish writing terminates, so it is rational, and it must differ from the irrational number somewhere past the last place you wrote. Adding more places shrinks the difference but never makes it zero.

Lesson 2.4 — Locating Real Numbers on a Number Line

Guided practice

63. Between the ticks $7$ and $8$, since $49 < 55 < 64$. The gaps $6$ and $9$ put it in the lower half, and $\sqrt{55} \approx 7.42$, so it goes a little less than halfway across the gap.
64. $-\sqrt{30} \approx -5.48$, plotted between $-6$ and $-5$, just to the right of the midpoint $-5.5$, since $-5.48 > -5.5$.
65. $\pi$ goes on the tick labeled $3.14$ and $\sqrt{10}$ goes on the tick labeled $3.16$, two hundredths to its right.
66. $\frac{22}{7}$ lies farther right, since $\frac{22}{7} = 3.142857\ldots$ and $\pi = 3.14159\ldots$. The difference is about $0.0013$, so the line has to be marked in thousandths for the two points to separate.
67. $\sqrt{2} \approx 1.41$, plotted a little less than halfway from $1$ to $2$.
68. The tick at $6.3$. $6.3^2 = 39.69$ and $6.4^2 = 40.96$, and $40 - 39.69 = 0.31$ is much less than $40.96 - 40 = 0.96$.

Independent practice

69. $\sqrt{17} \approx 4.12$ (between $4$ and $5$); $\sqrt{45} \approx 6.71$ (between $6$ and $7$); $\sqrt{95} \approx 9.75$ (between $9$ and $10$).
70. $-\sqrt{12} \approx -3.46$ (between $-4$ and $-3$); $-\sqrt{50} \approx -7.07$ (between $-8$ and $-7$, just left of $-7$); $-\sqrt{85} \approx -9.22$ (between $-10$ and $-9$).
71. Between $7.7$ and $7.8$, since $7.7^2 = 59.29$ and $7.8^2 = 60.84$.
72. Between $14.14$ and $14.15$, since $14.14^2 = 199.9396$ and $14.15^2 = 200.2225$.
73. $5.5$ is farther right. $\sqrt{30} \approx 5.48$, which sits just left of the $5.5$ tick. Squaring confirms it: $30 < 30.25 = 5.5^2$.
74. Since $4.8^2 = 23.04$ and $4.9^2 = 24.01$, the mark falls between the tenths $4.8$ and $4.9$. It is very close to the upper one, because $24.01$ misses $24$ by only $0.01$, so the carpenter should use $4.9$ feet. (To the nearest hundredth, $\sqrt{24} \approx 4.90$.)
75. On a whole-number line both $\pi$ and $\frac{22}{7}$ land in the same gap between $3$ and $4$, at points about $0.0013$ apart — far closer together than the width of a pencil mark, so the picture cannot separate them. A line from $3.140$ to $3.145$ marked in thousandths shows $\pi$ near $3.1416$ and $\frac{22}{7}$ near $3.1429$, with $\frac{22}{7}$ clearly to the right.
76. Halfway between $9$ and $10$ is $9.5$, and $9.5^2 = 90.25$, which is more than $90$. So $\sqrt{90}$ is less than $9.5$ and belongs just left of halfway, at about $9.49$.

Exit ticket 2.4

77. $\sqrt{70} \approx 8.37$, plotted between $8$ and $9$. The nearest tenth is $8.4$, since $8.3^2 = 68.89$ and $8.4^2 = 70.56$, and $70$ is nearer to $70.56$.
78. $-\sqrt{45} \approx -6.71$, plotted between $-7$ and $-6$.
79. $5.2$ lies farther right. Squaring: $26 < 27.04 = 5.2^2$, so $\sqrt{26} < 5.2$. (To the nearest hundredth $\sqrt{26} \approx 5.10$, which rounds to the same two places as $5.1$, so the squaring argument is the honest one here.)
80. $\sqrt{50} \approx 7.07$, so $-\sqrt{50} \approx -7.07$, which is farther from zero on the negative side than $-7$ and therefore plots to the left of $-7$. On a number line, anything to the left is less, so $-\sqrt{50} < -7$.

Lesson 2.5 — Comparing and Ordering Five Real Numbers

Decimal values are shown in parentheses; the ordered answer is written in the original forms, as the standard requires.

Guided practice

81. $\sqrt{50} < 7.2$. Squaring both: $50 < 51.84$.
82. $\pi < 3.2$, since $\pi \approx 3.14$.
83. $-\sqrt{30} > -5.5$. $\sqrt{30} \approx 5.48$, so $-\sqrt{30} \approx -5.48$, which is to the right of $-5.5$.
84. $\frac{3}{8} < 40\%$, since $\frac{3}{8} = 0.375$ and $40\% = 0.40$.
85. $2.5 \times 10^{2} > 249$, since $2.5 \times 10^{2} = 250$.
86. Decimals: $\frac{5}{8} = 0.625$; $0.58$; $62\% = 0.62$; $0.65$; $\frac{3}{5} = 0.6$. Ascending: $0.58 < \frac{3}{5} < 62\% < \frac{5}{8} < 0.65$
87. Decimals: $\sqrt{20} \approx 4.47$; $4.5$; $\frac{21}{5} = 4.2$; $430\% = 4.3$; $4\frac{3}{8} = 4.375$. Ascending: $\frac{21}{5} < 430\% < 4\frac{3}{8} < \sqrt{20} < 4.5$
88. Decimals: $-\sqrt{10} \approx -3.16$; $-3.1$; $-\frac{22}{7} \approx -3.14$; $-3\frac{1}{4} = -3.25$; $-305\% = -3.05$. Descending: $-305\% > -3.1 > -\frac{22}{7} > -\sqrt{10} > -3\frac{1}{4}$

Independent practice

89. Decimals: $\sqrt{30} \approx 5.48$; $5.4$; $\frac{26}{5} = 5.2$; $545\% = 5.45$; $5\frac{1}{2} = 5.5$. Ascending: $\frac{26}{5} < 5.4 < 545\% < \sqrt{30} < 5\frac{1}{2}$
90. Decimals: $\sqrt{72} \approx 8.49$; $8.4$; $\frac{43}{5} = 8.6$; $855\% = 8.55$; $8\frac{3}{8} = 8.375$. Descending: $\frac{43}{5} > 855\% > \sqrt{72} > 8.4 > 8\frac{3}{8}$
91. Decimals: $-\sqrt{8} \approx -2.83$; $-2.9$; $-\frac{11}{4} = -2.75$; $-285\% = -2.85$; $-2\frac{4}{5} = -2.8$. Ascending: $-2.9 < -285\% < -\sqrt{8} < -2\frac{4}{5} < -\frac{11}{4}$
92. Decimals: $1.5 \times 10^{1} = 15$; $15.2$; $\sqrt{200} \approx 14.14$; $14\frac{1}{2} = 14.5$; $1460\% = 14.6$. Ascending: $\sqrt{200} < 14\frac{1}{2} < 1460\% < 1.5 \times 10^{1} < 15.2$
93. Decimals: $\pi \approx 3.1416$; $\sqrt{10} \approx 3.1623$; $\frac{22}{7} \approx 3.1429$; $3.15$; $3.1$. These are close enough that hundredths alone would tie $\pi$ and $\frac{22}{7}$ at $3.14$, so carry extra places. Descending: $\sqrt{10} > 3.15 > \frac{22}{7} > \pi > 3.1$
94. $\sqrt{99} > 9.9$. Squaring both: $99 > 98.01$. (And $\sqrt{99} \approx 9.95$.)
95. $-\sqrt{145} < -12$. $\sqrt{145} \approx 12.04$, since $145 > 144 = 12^2$, so $\sqrt{145} > 12$ and taking opposites reverses the inequality.
96. Side lengths are the square roots of the areas: $\sqrt{5} \approx 2.24$, $\sqrt{5.76} = 2.4$, $\sqrt{6} \approx 2.45$, $\sqrt{6.25} = 2.5$, $\sqrt{7} \approx 2.65$. Shortest to longest: the $5$, $5.76$, $6$, $6.25$, and $7$ square-inch tiles, in that order. Because a bigger square has a bigger side, ordering the areas already orders the sides.
97. Strategy 1 — common denominator: $\frac{5}{8} = \frac{25}{40}$ and $62\% = \frac{62}{100} = \frac{31}{50} = \frac{124}{200}$, while $\frac{5}{8} = \frac{125}{200}$; since $125 > 124$, $\frac{5}{8}$ is greater. Strategy 2 — decimals: $\frac{5}{8} = 0.625$ and $62\% = 0.62$, and $0.625 > 0.620$. Either way, $\frac{5}{8} > 62\%$.
98. The student ordered by distance from zero instead of by position on the number line. For negative numbers, the farther from zero, the smaller. Correct ascending order: $-3.5 < -3.2 < -3.05$.

Exit ticket 2.5

99. Decimals: $\sqrt{45} \approx 6.71$; $6.8$; $\frac{33}{5} = 6.6$; $690\% = 6.9$; $6\frac{3}{4} = 6.75$. Ascending: $\frac{33}{5} < \sqrt{45} < 6\frac{3}{4} < 6.8 < 690\%$
100. Decimals: $-\sqrt{20} \approx -4.47$; $-4.4$; $-\frac{21}{5} = -4.2$; $-4\frac{1}{2} = -4.5$; $-455\% = -4.55$. Descending: $-\frac{21}{5} > -4.4 > -\sqrt{20} > -4\frac{1}{2} > -455\%$
101. $7.05 < \sqrt{50}$. Squaring: $7.05^2 = 49.7025 < 50$. (And $\sqrt{50} \approx 7.07$.)
102. Converting to decimals is reliable because it puts every number in one common form, and decimals in the same place value can be compared digit by digit from the left. It is not always fastest: comparing two fractions such as $\frac{5}{8}$ and $\frac{3}{5}$ is quicker with a common denominator, and comparing a radical to a decimal is both quicker and exact if you square both instead of rounding.

Chapter 2 Review

Part A — Estimating between consecutive natural numbers (8.NS.1a)

103. Between $7$ and $8$, since $49 < 55 < 64$. Gaps $6$ and $9$, so $7$ is better.
104. Between $14$ and $15$, since $196 < 210 < 225$. Gaps $14$ and $15$, so $14$ is better.
105. Between $11$ and $12$, since $121 < 130 < 144$. Gaps $9$ and $14$, so $11$ is better.
106. Between $9$ and $10$, since $81 < 85 < 100$. Gaps $4$ and $15$, so $9$ is better.
107. Between $15$ and $16$, since $225 < 240 < 256$. Halfway between $225$ and $256$ is $240.5$, and $240 < 240.5$, so the root falls in the lower half of the interval and $15$ is the better approximation. The gap comparison says the same thing by one unit: $240 - 225 = 15$ and $256 - 240 = 16$. (Check: $\sqrt{240} = 15.4919\ldots$, just under $15.5$.)

Part B — Rational approximations and the number line (8.NS.1b)

108. a) $\sqrt{11} \approx 3.32$ b) $-\sqrt{29} \approx -5.39$ c) $\sqrt{160} \approx 12.65$ d) $-\sqrt{390} \approx -19.75$
109. $\sqrt{150} \approx 12.25$, which lies between the tenths $12.2$ and $12.3$. Since $12.2^2 = 148.84$ and $12.3^2 = 151.29$, and $150 - 148.84 = 1.16$ is less than $151.29 - 150 = 1.29$, the nearest tenth is $12.2$.
110. $-\sqrt{82}$ is greater. $\sqrt{82} \approx 9.06$, so $-\sqrt{82} \approx -9.06$, which lies to the right of $-9.1$. Squaring the sizes: $82 < 82.81 = 9.1^2$, so $\sqrt{82} < 9.1$, and taking opposites reverses it.
111. Between $8.54$ and $8.55$, since $8.54^2 = 72.9316$ and $8.55^2 = 73.1025$. (To the nearest hundredth, $\sqrt{73} \approx 8.54$.)
112. $\sqrt{2} \approx 1.41$ and $\sqrt{3} \approx 1.73$, so on a line from $1$ to $2$ marked in tenths, $\sqrt{2}$ plots just past $1.4$ and $\sqrt{3}$ just past $1.7$. $\sqrt{3}$ is farther right, which makes sense because $3 > 2$ and larger radicands give larger positive roots.

Part C — Comparing and ordering real numbers (8.NS.1c)

113. Decimals: $\sqrt{40} \approx 6.32$; $6.4$; $\frac{31}{5} = 6.2$; $635\% = 6.35$; $6\frac{1}{4} = 6.25$. Ascending: $\frac{31}{5} < 6\frac{1}{4} < \sqrt{40} < 635\% < 6.4$
114. Decimals: $-\sqrt{110} \approx -10.49$; $-10.5$; $-\frac{53}{5} = -10.6$; $-1040\% = -10.4$; $-10\frac{3}{8} = -10.375$. Descending: $-10\frac{3}{8} > -1040\% > -\sqrt{110} > -10.5 > -\frac{53}{5}$
115. Decimals: $2.5 \times 10^{-1} = 0.25$; $0.3$; $\frac{7}{25} = 0.28$; $26\% = 0.26$; $0.245$. Ascending: $0.245 < 2.5 \times 10^{-1} < 26\% < \frac{7}{25} < 0.3$
116. Decimals: $\pi \approx 3.14$; $3.2$; $\sqrt{11} \approx 3.32$; $\frac{17}{5} = 3.4$; $315\% = 3.15$. Descending: $\frac{17}{5} > \sqrt{11} > 3.2 > 315\% > \pi$
117. Decimals: $\sqrt{0} = 0$; $-\sqrt{16} = -4$; $3.5$; $-\frac{7}{2} = -3.5$; $400\% = 4$. Ascending: $-\sqrt{16} < -\frac{7}{2} < \sqrt{0} < 3.5 < 400\%$
118. $\frac{5}{6}$ is greater. As a decimal, $\frac{5}{6} = 0.8\overline{3} = 0.8333\ldots$, while $83\% = 0.83$ exactly. Comparing place by place, the two agree through the hundredths and then $\frac{5}{6}$ has a $3$ in the thousandths place where $83\%$ has a $0$. As fractions: $\frac{5}{6} = \frac{250}{300}$ and $83\% = \frac{83}{100} = \frac{249}{300}$, and $250 > 249$.

Part D — Mixed application and reasoning

119. Side lengths: $\sqrt{120} \approx 10.95$, $\sqrt{144} = 12$, $\sqrt{130} \approx 11.40$, $\sqrt{156} \approx 12.49$, $\sqrt{121} = 11$. Shortest to longest: the $120$, $121$, $130$, $144$, and $156$ square-foot rooms, with sides about $10.95$, $11$, $11.40$, $12$, and $12.49$ feet. The rooms with whole-number sides are the $121$ square-foot room ($11$ feet) and the $144$ square-foot room ($12$ feet), because $121$ and $144$ are perfect squares.
120. Both numbers lie in the same gap between the ticks $14$ and $15$ — $\sqrt{200}$ because $196 < 200 < 225$, and $14.1$ by inspection — so the whole-number picture puts them in the same place and settles nothing. Redraw the line from $14$ to $15$ marked in hundredths, or compare with squares. Squaring: $14.1^2 = 198.81 < 200$, so $14.1 < \sqrt{200}$. $\sqrt{200}$ is greater, and to the nearest hundredth it is $14.14$.
121. The equals sign claims the two numbers are the same, and they are not: $50$ is not a perfect square, so $\sqrt{50}$ is irrational and cannot equal the terminating decimal $7.07$. The computation that shows it: $7.07^2 = 49.9849$, not $50$. A correct statement is $\sqrt{50} \approx 7.07$, or more fully $7.07 < \sqrt{50} < 7.08$, since $7.08^2 = 50.1264$.
122. Values: $-\sqrt{12} \approx -3.46$; $-3.5$; $\pi \approx 3.14$; $\sqrt{2} \approx 1.41$; $2.9$. Ascending: $-3.5 < -\sqrt{12} < \sqrt{2} < 2.9 < \pi$ The picture justifies the list because plotting each value at its approximate position and then reading the points from left to right is the ascending order — the number line is built so that "farther right" and "greater" mean the same thing. The only pair needing care is $-3.5$ and $-\sqrt{12} \approx -3.46$: both sit between $-4$ and $-3$, and $-\sqrt{12}$ is the one nearer zero, so it is the greater of the two.

Workbook-only frames

Page 1, perfect squares table. $n^2$ for $n = 0$ to $10$: $0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100$. For $n = 11$ to $20$: $121, 144, 169, 196, 225, 256, 289, 324, 361, 400$.

Page 2, fill in the blanks. The symbol $\sqrt{\ }$ always means the positive (principal) square root. To name the negative root, write a minus sign in front. Every positive number has two square roots; $0$ has one. If the radicand is a perfect square the root is rational; if it is not, the root is irrational.

Page 5, the method. Step 2: $6 < \sqrt{40} < 7$. Step 3: $40 - 36 = 4$ and $49 - 40 = 9$. Step 4: the better approximation is $6$.

Page 5 and 6, estimating tables. Lower square, upper square, between, lower gap, upper gap, better:

Page 9, the method on $\sqrt{40}$. Tenths: $6.3^2 = 39.69$ and $6.4^2 = 40.96$. Hundredths: $6.32^2 = 39.9424$ and $6.33^2 = 40.0689$. Nearer to $40$: $39.9424$, so $\sqrt{40} \approx 6.32$.

Page 17, conversion table. $\frac{21}{5} = 4.2$; $4\frac{3}{8} = 4.375$; $430\% = 4.3$; $1.5 \times 10^{1} = 15$; $2.5 \times 10^{-1} = 0.25$; $\sqrt{20} \approx 4.47$; $\pi \approx 3.14$.

Page 18, items 86–88 decimal rows. Item 86: $0.625$, $0.58$, $0.62$, $0.65$, $0.6$. Item 87: $4.47$, $4.5$, $4.2$, $4.3$, $4.375$. Item 88: $-3.16$, $-3.1$, $-3.14$, $-3.25$, $-3.05$.

Page 20, item 96 table. Sides in the order the areas are listed: $2.24$, $2.5$, $2.45$, $2.4$, $2.65$ inches.

Page 25, item 119 table. Sides in the order the areas are listed: $10.95$, $12$, $11.40$, $12.49$, $11$ feet.