Appendix A — Answer Key, Chapter 4: Operations with Rational Numbers in Context

SOL 7.CE.1 · Covers textbook Chapter 4 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 through 96. Reasoning and justification answers show an acceptable response, not the only wording. For multistep problems, the intermediate steps are shown, because those steps are what the standard asks students to justify.

Lesson 4.1 — Adding and Subtracting Rational Numbers

Guided practice

1. Positive, because $12.9 > 7.4$. Size: $12.9 - 7.4 = 5.5$. Answer: $5.5$
2. $\tfrac{9}{4} + \left(-\tfrac{23}{4}\right) = -\tfrac{14}{4} = -\tfrac{7}{2} = -3\tfrac{1}{2}$
3. $6.2 - (-3.85) = 6.2 + 3.85 = 10.05$
4. $-2.5 - 4.75 = -7.25$, so $-7.25$°C
5. $34.20 - 52.75 = -18.55$, so $-\$18.55$. The negative sign means the account is overdrawn by $18.55 — the check was larger than the balance.

Independent practice

6. a) $-9.6 + 4.25 = -5.35$ b) $-\tfrac{11}{3} + \tfrac{5}{4} = -\tfrac{44}{12} + \tfrac{15}{12} = -\tfrac{29}{12} = -2\tfrac{5}{12}$ c) $7.5 - 11.375 = -3.875$ d) $-\tfrac{5}{6} + \tfrac{1}{3} = -\tfrac{5}{6} + \tfrac{2}{6} = -\tfrac{3}{6} = -\tfrac{1}{2}$
7. $1240.5 - 386.75 = 853.75$ ft; $853.75 + 152.25 = 1006$ ft. Answer: $1{,}006$ ft
8. $-42.15 + 120.00 = 77.85$; $77.85 - 18.60 = 59.25$; $59.25 - 35.75 = 23.50$. Answer: $23.50
9. $2\tfrac{3}{4} + 1\tfrac{5}{8} = \tfrac{22}{8} + \tfrac{13}{8} = \tfrac{35}{8} = 4\tfrac{3}{8}$ qt poured. $7\tfrac{1}{2} = \tfrac{60}{8}$, so $\tfrac{60}{8} - \tfrac{35}{8} = \tfrac{25}{8} = 3\tfrac{1}{8}$. Answer: $3\tfrac{1}{8}$ qt
10. $-1.25 + 0.875 = -0.375$; $-0.375 - 2.5 = -2.875$; $-2.875 + 3.125 = 0.25$. Answer: a total change of $+\$0.25$; the price ended higher than it started.
11. Noon: $-6.5 + 13.25 = 6.75$°F. 9 p.m.: $6.75 - 8.75 = -2$°F. Answer: $-2$°F
12. The correct answer is $-4.6 - 2.8 = -4.6 + (-2.8) = -7.4$. Jordan read the problem as though the $2.8$ were being removed from a distance of $4.6$, but both numbers pull in the same direction here: starting at $-4.6$ and subtracting means moving $2.8$ further left, not back toward zero. Subtraction only moves you toward zero when the number being subtracted is negative.

Exit ticket 4.1

13. $-8.25 + 3.5 = -4.75$
14. $\tfrac{11}{6} - \tfrac{26}{6} = -\tfrac{15}{6} = -\tfrac{5}{2} = -2\tfrac{1}{2}$
15. $-16.40 + 25.00 = 8.60$. Answer: $8.60
16. Sample: a $5 fee is charged to an account, then reversed. Removing a charge of $-5$ is subtracting a negative, and the balance goes up by $5. Symbolically, $a - (-5) = a + 5$: taking away a loss leaves you better off than before.

Lesson 4.2 — Multiplying and Dividing Rational Numbers

Guided practice

17. One negative factor, so the product is negative. Size: $7 \times 2.5 = 17.5$. Answer: $-17.5$
18. Two negatives, so the product is positive. $\tfrac{3}{5} \times \tfrac{10}{9} = \tfrac{30}{45} = \tfrac{2}{3}$. Answer: $\tfrac{2}{3}$
19. $-20.4 \div 6 = -3.4$
20. $\tfrac{21}{4} \div \left(-\tfrac{3}{2}\right) = \tfrac{21}{4} \times \left(-\tfrac{2}{3}\right) = -\tfrac{42}{12} = -\tfrac{7}{2} = -3\tfrac{1}{2}$
21. $(-0.375)(8) = -3$. Answer: $-3$ ft, a drop of 3 feet

Independent practice

22. a) $(-1.2)(-4.5) = 5.4$ b) $-\tfrac{7}{8} \times \tfrac{2}{3} = -\tfrac{14}{24} = -\tfrac{7}{12}$ c) $-15.75 \div (-2.5) = 6.3$ d) $-\tfrac{8}{3} \div \tfrac{4}{5} = -\tfrac{8}{3} \times \tfrac{5}{4} = -\tfrac{40}{12} = -\tfrac{10}{3} = -3\tfrac{1}{3}$
23. $146.20 \div 4 = 36.55$. Answer: $36.55 each
24. $\tfrac{7}{3} \times \tfrac{9}{4} = \tfrac{63}{12} = \tfrac{21}{4} = 5\tfrac{1}{4}$. Answer: $5\tfrac{1}{4}$ cups
25. $(-8.75)(7.2) = -63$. Answer: $-63$ psi, a loss of 63 psi
26. $(-12.5)(6.4) = -80$. Answer: $-80$ ft
27. $\tfrac{21}{2} \div \tfrac{3}{2} = \tfrac{21}{2} \times \tfrac{2}{3} = \tfrac{42}{6} = 7$. Answer: 7 portions. Nothing is left over because the quotient is a whole number; checking backward, $7 \times 1\tfrac{1}{2} = 10\tfrac{1}{2}$ cups exactly.
28. Sample: five roommates share a $73.50 overcharge, so each balance changes by $-73.50 \div 5 = -14.70$. Now ask the reverse question — how many people would have to absorb a $-\$14.70$ change to account for the full $-\$73.50$? That count is $-73.50 \div (-14.70) = 5$, a count of people, which cannot be negative. Dividing a negative by a negative asks "how many of these fit into that," and both being negative means they point the same direction, so the count is positive.

Exit ticket 4.2

29. $(-6)(-3.5) = 21$
30. $-\tfrac{3}{4} \times \tfrac{8}{9} = -\tfrac{24}{36} = -\tfrac{2}{3}$
31. $-22.5 \div 4.5 = -5$
32. Negative. Division is multiplication in disguise: if $a \div b = q$, then $q \times b = a$. With exactly one negative between $a$ and $b$, the only way for $q \times b$ to have the right sign is for $q$ to be negative.

Lesson 4.3 — Estimating to Judge Whether an Answer Is Reasonable

Guided practice

33. Estimate $9 \times 3 = 27$. Exact: $8.9 \times 3.1 = 27.59$
34. Compatible numbers: $-20 \div 5 = -4$. Exact: $-19.6 \div 4.9 = -4$
35. Estimate $5 + 3 = 8$. Exact: $\tfrac{41}{8} + \tfrac{23}{8} = \tfrac{64}{8} = 8$
36. Estimate $6 \times 0.5 = 3$. Not reasonable — the report is about 10 times too large, which points to a misplaced decimal point. Exact: $6.4 \times 0.52 = 3.328$
37. Low: $3 \times 4 = 12$. High: $4 \times 5 = 20$. Exact: $3.7 \times 4.6 = 17.02$, which lies between 12 and 20.

Independent practice

38. a) Estimate $12 \times 5 = 60$; exact $12.3 \times 4.8 = 59.04$ b) Estimate $-45 \div 5 = -9$; exact $-44.8 \div 5.6 = -8$ c) Estimate $10 + 4 = 14$; exact $\tfrac{78}{8} + \tfrac{33}{8} = \tfrac{111}{8} = 13\tfrac{7}{8}$ d) Estimate $8 \times 2 = 16$; exact $\tfrac{39}{5} \times 2 = \tfrac{78}{5} = 15\tfrac{3}{5}$
39. Low: $8 \times 6 = 48$. High: $9 \times 7 = 63$. Exact: $8.3 \times 6.7 = 55.61$, which lies between 48 and 63.
40. a) Reasonable. Estimate $5 \times 5 = 25$, and $24.99$ is right there. (It is also exactly correct.) b) Unreasonable. Estimate $63 \div 7 = 9$, so 90 is ten times too large. Correct: $63.9 \div 7.1 = 9$. c) Unreasonable. A negative times a positive must be negative. Correct: $-3.2 \times 6.5 = -20.8$. d) Reasonable. Estimate $20 - 5 = 15$. Exact: $19.6 + (-4.85) = 14.75$.
41. Estimate $11 \times 3 = 33$ (or $11.5 \times 3.2 = 36.80$). Exact: $11.4 \times 3.199 = 36.4686$, which rounds to $36.47.
42. Estimate $100 \div 5 = 20$. Exact: $97.35 \div 5 = 19.47$, so $19.47 each. The estimate confirms it: the true bill is slightly under $100, so each share should be slightly under $20, and $19.47 is.
43. Estimate $4 \times 14 = 56$, plus about $9, is about $65 — more than $60, so $60 is not enough. Exact: $4 \times 13.75 = 55.00$; $55.00 + 8.50 = 63.50$. Answer: $60 is not enough; the total is $63.50.
44. For two positive numbers, making either factor larger makes the product larger, so rounding both up produces a product larger than the true one. A single estimate tells you only a nearby value, and you still have to judge how far off is "too far." A low-and-high pair converts that judgment into a test with a definite answer: the exact product must fall between them, so any result outside the bracket is certainly wrong.

Exit ticket 4.3

45. Estimate $8 \times 4 = 32$. Exact: $7.8 \times 4.2 = 32.76$
46. Not reasonable — the size is about right but the sign is wrong, since a negative divided by a positive must be negative. Estimate $-56 \div 7 = -8$. Correct: $-56.4 \div 7.05 = -8$.
47. Low: $6 \times 3 = 18$. High: $7 \times 4 = 28$. Exact: $6.4 \times 3.7 = 23.68$
48. A misplaced decimal point changes an answer by a factor of 10, 100, or more, while a calculator reports whatever was typed without complaint. An estimate gives you the right order of magnitude in advance, so a result ten times too large or too small stands out immediately — as in $48.6 \div 0.6$, where an estimate of 80 exposes a reported answer of $8.1$.

Lesson 4.4 — Multistep Problems in Context

Guided practice

49. $44.80 - 4.80 = 40.00$; $40.00 \div 4 = 10.00$. Answer: $10.00 each
50. $(2.25)(4) = 9$; $-3.5 + 9 = 5.5$. Answer: $5.5$°C
51. $\tfrac{11}{4} \times 6.40 = 17.60$. Answer: $17.60
52. $(1.375)(6) = 8.25$ gal lost; $20 - 8.25 = 11.75$. Answer: $11.75$ gallons remain
53. $\tfrac{25}{2} \div \tfrac{5}{8} = \tfrac{25}{2} \times \tfrac{8}{5} = \tfrac{200}{10} = 20$. Answer: 20 pieces

Independent practice

54. $60 \times 0.045 = 2.70$; $24.99 + 2.70 = 27.69$. Answer: $27.69
55. $(128.25)(6) = 769.5$ ft descended; $2150.5 - 769.5 = 1381$. Answer: $1{,}381$ ft
56. $3 \times \tfrac{7}{4} = \tfrac{21}{4} = 5\tfrac{1}{4}$ gal needed; $\tfrac{21}{4} - \tfrac{18}{4} = \tfrac{3}{4}$. Answer: $\tfrac{3}{4}$ gallon more
57. $-28.35 + 150.00 = 121.65$; $4 \times 22.50 = 90.00$; $121.65 - 90.00 = 31.65$. Answer: $31.65
58. $6 \times 0.625 = 3.75$ drop per share; $18.75 - 3.75 = 15.00$ new price; $20 \times 3.75 = 75.00$ total loss. Answer: $15.00 per share; a total loss of $75.00
59. $27 \times 8.75 = 236.25$ admission; $236.25 + 148.50 = 384.75$ total; $384.75 \div 27 = 14.25$. Answer: $14.25 per student
60. The student would get $44.80 \div 4 = 11.20$, then $11.20 - 4.80 = 6.40$, or $6.40 each. That is wrong because it gives every one of the four people the full $4.80 discount, applying $19.20 of discount to a bill that only had $4.80 taken off. The discount comes off the bill one time, so it must be subtracted before the division, not after. Checking backward confirms the correct order: $4 \times 10.00 + 4.80 = 44.80$.

Exit ticket 4.4

61. $56.40 - 6.40 = 50.00$; $50.00 \div 5 = 10.00$. Answer: $10.00 each
62. $(-1.25)(6) = -7.5$; $2.5 - 7.5 = -5$. Answer: $-5$°F
63. $\tfrac{26}{3} \div \tfrac{2}{3} = \tfrac{26}{3} \times \tfrac{3}{2} = 13$. Answer: 13 bowls
64. Sample: estimate first — a fall of about $1$ degree per hour for 6 hours is a fall of about 6 degrees, from a starting point only $2.5$ above zero, so the answer must be a few degrees below zero. Then work backward from the answer: starting at $-5$ and rising $1.25$°F per hour for 6 hours gives $-5 + 7.5 = 2.5$°F, the original temperature.

Lesson 4.5 — Justifying a Solution

Guided practice

65. Sample: $9.8$ is close to 10 and $5.2$ is close to 5, and $10 \times 5 = 50$, so a product just under 51 is exactly what should be expected.
66. Division, because the question asks how many $\tfrac{5}{8}$-pound portions fit inside $6\tfrac{1}{4}$ pounds — a measuring question, not a sharing one. $\tfrac{25}{4} \div \tfrac{5}{8} = \tfrac{25}{4} \times \tfrac{8}{5} = \tfrac{200}{20} = 10$. Answer: 10 bags. Check: $10 \times \tfrac{5}{8} = \tfrac{50}{8} = 6\tfrac{1}{4}$.
67. The student subtracted instead of adding the opposite. Subtracting $-2.4$ means adding $2.4$: $-5.6 + 2.4 = -3.2$. Answer: $-3.2$
68. The temperature starts only $1.5$ degrees above zero, and the total fall is $2.5 \times 3 = 7.5$ degrees, which is much larger than $1.5$. Falling farther than your distance above zero puts you below zero, so the answer must be negative. $1.5 - 7.5 = -6$. Answer: $-6$°F
69. Estimate: $81.25 is close to $80, and $80 \div 5 = 16$, so each share is about $16. Operation: division, because one total is being shared into 5 equal parts. Computation: $81.25 \div 5 = 16.25$. Answer sentence: each friend pays $16.25, which matches the estimate of about $16. Check: $5 \times 16.25 = 81.25$.

Independent practice

70. The student added instead of multiplying: $\tfrac{2}{3} + \left(-\tfrac{3}{4}\right) = \tfrac{8}{12} - \tfrac{9}{12} = -\tfrac{1}{12}$. Multiplying gives $\tfrac{2}{3} \times \left(-\tfrac{3}{4}\right) = -\tfrac{6}{12} = -\tfrac{1}{2}$. Answer: $-\tfrac{1}{2}$
71. Dividing by $0.3$, a number less than 1, must make the result larger than $18.6$, not smaller — so an answer under 1 cannot be right. A compatible estimate is $18 \div 0.3 = 60$. Exact: $18.6 \div 0.3 = 62$. The reported answer was off by a factor of 100.
72. Estimate: about $\$104 - \$5 = \$99$, and $99 \div 7$ is about $14. Operations: subtract the coupon once from the whole bill, because it reduces the bill itself and not each person's share; then divide by 7, because the remainder is shared into 7 equal parts. Computation: $103.60 - 5.25 = 98.35$; $98.35 \div 7 = 14.05$. Answer sentence: each friend pays $14.05. Check: $7 \times 14.05 = 98.35$, and $98.35 + 5.25 = 103.60$, the original bill.
73. $(-0.375)(12) = -4.5$. Answer: $-4.5$ inches. The sign must be negative because evaporation removes water, so every day's change is a loss. The size is reasonable because a loss of a bit more than $\tfrac{1}{3}$ inch per day for 12 days should total a bit more than 4 inches, and $4.5$ is.
74. The scaling factor is $\tfrac{20}{8} = \tfrac{5}{2}$, because 20 servings is $2\tfrac{1}{2}$ times as many as 8 servings and every ingredient scales by that same factor. $\tfrac{4}{3} \times \tfrac{5}{2} = \tfrac{20}{6} = \tfrac{10}{3} = 3\tfrac{1}{3}$. Answer: $3\tfrac{1}{3}$ cups of sugar. Check: $3\tfrac{1}{3} \div \tfrac{5}{2} = \tfrac{10}{3} \times \tfrac{2}{5} = \tfrac{4}{3} = 1\tfrac{1}{3}$ cups, the original amount.
75. $14.05 is right. After the coupon the bill is about $98, and $98 \div 7 = 14$ exactly, so each share must be just over $14. Seven shares of $14.50 would total $101.50, which is more than the $98.35 actually owed — so $14.50 is too large without doing the division at all.
76. Working backward uses the answer to rebuild a number the problem gave you, so it tests the whole chain — the operations chosen, their order, and the arithmetic. Saying "I used a calculator" tests nothing, because a calculator computes exactly what was entered, including a mistyped digit, a dropped negative sign, or the wrong operation entirely. Evidence has to be independent of the step it is checking.

Exit ticket 4.5

77. Sample: $6.1$ is close to 6 and $3.9$ is close to 4, so the size should be near $6 \times 4 = 24$, and one negative factor makes the product negative. An answer of $-23.79$ matches both the size and the sign.
78. The student divided by 2 instead of dividing by $\tfrac{1}{2}$ — that is, they halved the number rather than multiplying by the reciprocal. $\tfrac{7}{2} \div \tfrac{1}{2} = \tfrac{7}{2} \times \tfrac{2}{1} = 7$. Answer: 7. (It makes sense: seven halves fit inside $3\tfrac{1}{2}$.)
79. Estimate: $63.80 is close to $64, and $64 \div 4 = 16$, so each share is about $16. Operation: division, because one bill is shared into 4 equal parts. Computation: $63.80 \div 4 = 15.95$. Check: $4 \times 15.95 = 63.80$, the original bill. Answer: $15.95 each.
80. (1) An estimate showing about where the answer must land, including its sign. (2) The operations chosen, with a reason tying each to the situation. (3) An answer sentence with units that answers the question asked, plus a check when one is available.

Chapter 4 Review

Part A — Adding and subtracting rational numbers (7.CE.1a)

81. a) $-14.75 + 6.5 = -8.25$ b) $-\tfrac{19}{8} - \tfrac{11}{6} = -\tfrac{57}{24} - \tfrac{44}{24} = -\tfrac{101}{24} = -4\tfrac{5}{24}$ c) $9.6 - (-3.45) = 9.6 + 3.45 = 13.05$ d) $\tfrac{21}{4} + \left(-\tfrac{30}{4}\right) = -\tfrac{9}{4} = -2\tfrac{1}{4}$
82. $-52.80 + 140.00 = 87.20$; $87.20 - 36.45 = 50.75$. Answer: $50.75
83. $1.5 - 9.25 = -7.75$; $-7.75 + 3.75 = -4$. Answer: $-4$°F

Part B — Multiplying and dividing rational numbers (7.CE.1a)

84. a) $(-8)(2.75) = -22$ b) $-\tfrac{3}{5} \times -\tfrac{5}{6} = \tfrac{15}{30} = \tfrac{1}{2}$ c) $-33.6 \div (-4.2) = 8$ d) $\tfrac{20}{3} \div \left(-\tfrac{4}{3}\right) = \tfrac{20}{3} \times \left(-\tfrac{3}{4}\right) = -\tfrac{60}{12} = -5$
85. $\tfrac{39}{4} \div \tfrac{3}{4} = \tfrac{39}{4} \times \tfrac{4}{3} = 13$. Answer: 13 pieces
86. $(-2.125)(8) = -17$. Answer: $-17$ gallons

Part C — Estimating and judging reasonableness (7.CE.1a)

87. a) Estimate $12 \times 5 = 60$; exact $11.8 \times 5.1 = 60.18$ b) Estimate $-72 \div 8 = -9$; exact $-71.4 \div 8.4 = -8.5$ c) Estimate $7 + 3 = 10$; exact $\tfrac{55}{8} + \tfrac{25}{8} = \tfrac{80}{8} = 10$
88. Low: $5 \times 7 = 35$. High: $6 \times 8 = 48$. Exact: $5.6 \times 7.3 = 40.88$, which lies between 35 and 48.
89. a) Reasonable. Estimate $4 \times 20 = 80$, and the report is $80.19$. (It is also exactly correct.) b) Unreasonable. Dividing by $0.3$ must make the result larger than $12.6$. Correct: $12.6 \div 0.3 = 42$. c) Unreasonable. Adding $2.25$ to $-7.5$ cannot reach a positive number, since $2.25 < 7.5$. Correct: $-7.5 + 2.25 = -5.25$.

Part D — Multistep problems in context (7.CE.1a)

90. $96.40 - 14.40 = 82.00$; $82.00 \div 4 = 20.50$. Answer: $20.50 each
91. $24 \times 6.25 = 150.00$ admission; $150.00 + 114.00 = 264.00$ total; $264.00 \div 24 = 11.00$. Answer: $11.00 per student
92. Scaling factor $\tfrac{15}{6} = \tfrac{5}{2}$; $\tfrac{7}{2} \times \tfrac{5}{2} = \tfrac{35}{4} = 8\tfrac{3}{4}$. Answer: $8\tfrac{3}{4}$ cups
93. $(-14.25)(6) = -85.5$; $96.5 - 85.5 = 11$; $11 + 8.5 = 19.5$. Answer: $19.5$ ft

Part E — Mixed reasoning and justification (7.CE.1a)

94. The student added before multiplying, computing $-18.4 + 6.2 = -12.2$ and then doubling it. The order of operations requires the multiplication first: $6.2 \times 2 = 12.4$, then $-18.4 + 12.4 = -6$. Answer: $-6$
95. Estimate: $77.70 is close to $78, and $78 \div 6 = 13$, so each share is about $13. Operation: division, because one bill is shared into 6 equal parts. Computation: $77.70 \div 6 = 12.95$. Answer sentence: each friend pays $12.95, which matches the estimate of about $13. Check: $6 \times 12.95 = 77.70$, the original bill.
96. A calculator computes exactly what it is given, so it cannot tell you that what you gave it was wrong. An estimate is an independent prediction made before the keystrokes, so it catches errors the calculator will happily repeat. One specific example: a misplaced decimal point. If you enter $48.6 \div 0.6$ as $4.86 \div 0.6$, the calculator returns $8.1$ with full confidence, but an estimate of $48 \div 0.6 = 80$ shows immediately that the true answer must be near 80, not near 8. A dropped negative sign is caught the same way.

Workbook-only items

Page 2, fill in the blanks. Adding a positive moves you to the right. Adding a negative moves you to the left. To subtract, add the opposite: $a - b = a + (-b)$. Decimals in this chapter go no further than the thousandths place.

Page 2, signed-number table. $+45.50$; $-12.25$; $-6.25$; $-18.60$; $+9.75$

Page 3, fraction blanks. $3\tfrac{1}{4} = \tfrac{26}{8}$; $1\tfrac{5}{8} = \tfrac{13}{8}$; difference $= \tfrac{13}{8} = 1\tfrac{5}{8}$

Page 6, sign table. positive; negative; negative; positive

Page 6, pattern. $-9$; $-4.5$; $0$; $4.5$. Each time the second factor drops by 1, the product goes up by $4.5$.

Page 6, sharing or measuring. $73.50 among 5 people — sharing. Ribbon into $\tfrac{5}{8}$-ft pieces — measuring. Cups into $\tfrac{3}{4}$-cup bowls — measuring. $146.20 among 4 friends — sharing.

Page 7, reciprocal blanks. $\tfrac{51}{8} \times \tfrac{4}{3} = \tfrac{204}{24} = 8\tfrac{1}{2}$

Page 10, strategy table. Rounding: $7 \times 2.40 = 16.80$. Low: $6.5 \times 2 = 13$. High: $7 \times 2.5 = 17.5$. Exact product: $16.44$. Yes, it lands inside the bracket.

Page 10, two errors. A misplaced decimal point; a dropped negative sign.

Page 10, warning. Dividing by a number less than 1 makes the result larger.

Page 14, the plan. Step 1: in what unit. Step 4: work backward.

Page 14, order matters. $78.60 - 12.60 = 66.00$; $66.00 \div 3 = 22.00$. You cannot divide first because the coupon reduces the bill one time. Dividing first and then subtracting $12.60 from each share would apply the coupon three times, once per person.

Page 14, rate over time. $4.5 + (-1.75)(6) = 4.5 - 10.5 = -6$

Page 18, three parts. (1) An estimate, including its sign. (2) The operations you chose and why. (3) An answer sentence with units, plus a check.

Page 18, model table. Estimate: about $\$47 - \$5 = \$42$, and $42 \div 3 = 14$, so about $14 each. Operations and why: subtract the coupon once, because it comes off the bill as a whole; then divide by 3, because the remainder is shared equally. Computation: $47.25 - 5.25 = 42.00$; $42.00 \div 3 = 14.00$. Answer sentence: each friend pays $14.00. Check: $3 \times 14.00 + 5.25 = 47.25$.

Page 18, error analysis. The student added first, getting $-12.6 + 4.2 = -8.4$ before multiplying by 3. Correct value: $4.2 \times 3 = 12.6$, then $-12.6 + 12.6 = 0$.

Pages 20 and 24, justification tables. See the keys for items 72 and 95 above; the five rows correspond to estimate, operations and why, computation, answer sentence, and check.