MathBored

Virginia SOL Mathematics Textbook

Appendix A — Answer Key, Chapter 2: Adding and Subtracting Integers

SOL 6.CE.2 a, b, d · Covers textbook Chapter 2 and the companion workbook. Item numbers match the textbook; workbook items that repeat textbook problems share those answers, and every workbook-only item is keyed in the final section. Reasoning answers show an acceptable response, not the only wording.

Lesson 2.1 — Modeling Integer Addition

Guided practice

  1. Seven red chips are left. −2+(−5)=−7-2 + (-5) = -7
  2. Four zero pairs form, leaving 2 yellow chips. 6+(−4)=26 + (-4) = 2
  3. Three zero pairs form, leaving 4 red chips. −7+3=−4-7 + 3 = -4
  4. −1+5=4-1 + 5 = 4
  5. 2+(−6)=−42 + (-6) = -4

Independent practice

  1. a) −7-7 b) 33 c) −5-5 d) 00
  2. a) 44 b) −4-4 c) −8-8
  3. Three zero pairs form. Four yellow chips are left over, so 7+(−3)=47 + (-3) = 4.
  4. The pile is worth 00. All 5 yellow chips pair with all 5 red chips, making 5 zero pairs and leaving nothing.
  5. 3+(−8)=−53 + (-8) = -5
  6. −4+9=5-4 + 9 = 5. On a number line, start at −4-4 and jump 9 units right, landing on 55. The temperature at noon was 55°F.
  7. Both piles contain the same chips: 6 red and 2 yellow. Order of placement does not change how many zero pairs form or what is left over, so both give −4-4. Addition is commutative.

Exit ticket 2.1

  1. −7-7
  2. 33; four zero pairs formed.
  3. −6+4=−2-6 + 4 = -2
  4. A zero pair is one yellow chip and one red chip together, worth 1+(−1)=01 + (-1) = 0. Adding zero to a pile does not change its value, so you may add zero pairs whenever you need more chips of a particular color.

Lesson 2.2 — Adding Integers

Guided practice

  1. −15-15
  2. −7-7
  3. 88
  4. 00
  5. 1111

Independent practice

  1. a) −38-38 b) −14-14 c) −16-16 d) 00
  2. a) −18-18 b) 1515 c) 1818
  3. −14+6=−8-14 + 6 = -8; then −8+(−5)=−13-8 + (-5) = -13.
  4. Estimate: −200+90=−110-200 + 90 = -110. Exact: 197−88=109197 - 88 = 109, and −197-197 has the larger absolute value, so the sum is −109-109. The estimate is within 1 of the exact answer and has the same sign.
  5. a) 1111 b) −8-8
  6. 7+(−12)+3=−5+3=−27 + (-12) + 3 = -5 + 3 = -2. The team lost 2 yards overall.
  7. Negative. The signs differ, so the sign of the sum comes from the addend with the larger absolute value, and ∣−56∣=56|-56| = 56 is greater than ∣43∣=43|43| = 43.

Exit ticket 2.2

  1. −15-15
  2. −8-8
  3. −17-17
  4. Compare the absolute values. The addend that is farther from zero decides the sign of the sum, because the smaller amount cancels away against it.

Lesson 2.3 — Subtraction as Adding the Opposite

Guided practice

  1. −3-3
  2. −8-8
  3. −6-6
  4. 1515
  5. −4+(−11)=−15-4 + (-11) = -15

Independent practice

  1. a) −12-12 b) −20-20 c) −12-12 d) 3636
  2. a) 1515 b) −16-16 c) −26-26
  3. −7−(−7)=−7+7=0-7 - (-7) = -7 + 7 = 0. Subtracting a number is the same as adding its opposite, and a number plus its opposite is always zero.
  4. 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8; then 8−8=08 - 8 = 0.
  5. −4-4, since −6−(−4)=−6+4=−2-6 - (-4) = -6 + 4 = -2.
  6. 8−(−6)=8+6=148 - (-6) = 8 + 6 = 14. The temperature fell 14 degrees.
  7. Counterexample: 3−(−5)=3+5=83 - (-5) = 3 + 5 = 8, which is greater than 33. Subtraction increases a value whenever the number being subtracted is negative, because subtracting a negative means adding a positive.

Exit ticket 2.3

  1. −7-7
  2. −15-15
  3. 77
  4. To subtract an integer, add its opposite: a−b=a+(−b)a - b = a + (-b). Sample: −2−(−9)=−2+9=7-2 - (-9) = -2 + 9 = 7.

Lesson 2.4 — Adding and Subtracting in Context

Guided practice

  1. −24+40=16-24 + 40 = 16, a balance of $16.
  2. −7+12=5-7 + 12 = 5, or 55°F.
  3. 85+(−110)=−2585 + (-110) = -25, or 25 feet below sea level.
  4. 6−(−14)=206 - (-14) = 20 degrees apart.
  5. −8+3=−5-8 + 3 = -5, a net loss of 5 yards.

Independent practice

  1. −3+9=6-3 + 9 = 6; then 6−4=26 - 4 = 2. It ends on level 2.
  2. −58+120=62-58 + 120 = 62; then 62−35=2762 - 35 = 27. The final balance is $27.
  3. −46+19=−27-46 + 19 = -27; then −27−33=−60-27 - 33 = -60. The final depth is −60-60 feet.
  4. Estimate: −20+40=20-20 + 40 = 20°C. Exact: −18+41=23-18 + 41 = 23°C.
  5. Greatest is 99, least is −11-11. 9−(−11)=209 - (-11) = 20 degrees.
  6. 14,505−(−282)=14,505+282=14,78714{,}505 - (-282) = 14{,}505 + 282 = 14{,}787 feet.
  7. Subtraction answers "how far apart," which is what a difference in elevation asks. Subtracting a negative elevation adds its absolute value, and that is correct here because the distance from the summit down to sea level and the distance from sea level down to Death Valley must be added together.

Exit ticket 2.4

  1. −30+18=−12-30 + 18 = -12, a balance of −$12-\$12, meaning the account is overdrawn by $12.
  2. 11−(−4)=1511 - (-4) = 15 degrees.
  3. −25−15=−40-25 - 15 = -40 feet, or 40 feet below the surface.
  4. Estimate: −200+205=5-200 + 205 = 5. Exact: −198+205=7-198 + 205 = 7. The estimate told you to expect a small positive number, so an answer such as −7-7 or 403403 would have been caught immediately.

Chapter 2 Review

Part A — Modeling addition and subtraction (6.CE.2a)

  1. a) −8-8 b) 55 c) 00
  2. a) 44 b) −5-5
  3. 2+(−7)=−52 + (-7) = -5
  4. −4−(−6)=2-4 - (-6) = 2. You start with only 4 red chips but need to remove 6, so you add 2 zero pairs. That leaves the pile still worth −4-4 while giving you 6 red chips to remove. After removing them, 2 yellow chips remain.

Part B — Adding and subtracting two integers (6.CE.2b)

  1. a) −41-41 b) 1414 c) −17-17 d) 00
  2. a) −11-11 b) −29-29 c) 99 d) 2828
  3. −9+14=5-9 + 14 = 5; then 5+(−20)=−155 + (-20) = -15.
  4. −6−(−11)=−6+11=5-6 - (-11) = -6 + 11 = 5; then 5+(−8)=−35 + (-8) = -3.
  5. a) 88 b) −5-5

Part C — Problems in context (6.CE.2d)

  1. −62+95=33-62 + 95 = 33; then 33−18=1533 - 18 = 15. The final balance is $15.
  2. 7−(−21)=287 - (-21) = 28 degrees apart.
  3. −38+12=−26-38 + 12 = -26; then −26−27=−53-26 - 27 = -53. The final depth is −53-53 feet.

Part D — Mixed application and reasoning

  1. Estimate: −400+300=−100-400 + 300 = -100. Exact: 406−289=117406 - 289 = 117, and −406-406 has the larger absolute value, so the sum is −117-117.
  2. Both expressions describe the same trip on a number line. Starting at aa and removing bb moves you ∣b∣|b| units in the direction opposite to bb, which is exactly the jump you make when you add −b-b. For example, 3−83 - 8 and 3+(−8)3 + (-8) both start at 33 and move 8 units left to −5-5.
  3. a) Always true. Two negatives are both left of zero, so the jumps combine in the same direction and the sum stays negative. b) False. Counterexample: 4+(−9)=−54 + (-9) = -5. The sum takes the sign of the addend with the larger absolute value, which may be either one.
  4. Sample: a diver 15 feet below the surface rises 40 feet. The 2525 means the diver ends 25 feet above the surface. (Any situation with a starting value of −15-15, an increase of 4040, and a correct interpretation of 2525 is acceptable.)

Workbook-only items

Page 2, fill in the blanks. A yellow chip is worth +1+1. A red chip is worth −1-1. Together they make a zero pair, worth 00. Adding a zero pair to a pile does not change what the pile is worth.

Page 2, table.

Addition sentence Zero pairs formed Sum
−4+(−3)-4 + (-3) 00 −7-7
8+(−5)8 + (-5) 55 33
−9+4-9 + 4 44 −5-5
−6+6-6 + 6 66 00
−2+(−5)-2 + (-5) 00 −7-7
6+(−4)6 + (-4) 44 22

Page 3, jumps. 1. 44 2. −4-4 3. −8-8 4. −1+5=4-1 + 5 = 4 5. 2+(−6)=−42 + (-6) = -4

Page 3, blank number line. −6+4=−2-6 + 4 = -2

Page 3, chip sentence. 3+(−8)=−53 + (-8) = -5

Page 3, apply it. −4+9=5-4 + 9 = 5; the noon temperature was 55°F.

Page 5, fill in the rules. Same signs: add the absolute values, then keep the sign they share. Different signs: subtract the smaller absolute value from the larger, then use the sign of the addend with the larger absolute value.

Page 5, find each sum. a) −15-15 b) −7-7 c) 88 d) 00 e) 1111 f) −38-38 g) −14-14 h) −16-16 i) 00

Page 5, sort. Negative sums: a, b, f, g, h. Sums equal to zero: d, i.

Page 6, estimate table.

Problem Estimate Exact sum
−197+88-197 + 88 −200+90=−110-200 + 90 = -110 −109-109
−403+150-403 + 150 −400+150=−250-400 + 150 = -250 −253-253
88+(−312)88 + (-312) 90+(−300)=−21090 + (-300) = -210 −224-224

Page 6, three integers. 1. −14+6=−8-14 + 6 = -8, then −8+(−5)=−13-8 + (-5) = -13 2. −18-18 3. 1515 4. 1818

Page 6, missing addend. 1111; −8-8; 1515

Page 6, apply it. 7+(−12)+3=−27 + (-12) + 3 = -2; the team lost 2 yards overall.

Page 6, think. Negative, because ∣−56∣>∣43∣|-56| > |43| and the sign of the sum follows the addend farther from zero.

Page 8, rewrite table.

Subtraction Rewrite as addition Value
4−94 - 9 4+(−9)4 + (-9) −5-5
−3−5-3 - 5 −3+(−5)-3 + (-5) −8-8
−6−(−10)-6 - (-10) −6+10-6 + 10 44
12−(−7)12 - (-7) 12+712 + 7 1919
−4−11-4 - 11 −4+(−11)-4 + (-11) −15-15
−9−(−4)-9 - (-4) −9+4-9 + 4 −5-5

Page 8, explain. You need to remove 6 red chips but only 4 are on the table. Adding 2 zero pairs keeps the value at −4-4 while supplying the 2 extra red chips you need.

Page 9, differences. a) −12-12 b) −20-20 c) −12-12 d) 3636 e) 1515 f) −16-16 g) −26-26 h) 00

Page 9, error hunt. 5−8=35 - 8 = 3 false, correct value −3-3. −2−6=−8-2 - 6 = -8 true. −5−(−9)=−14-5 - (-9) = -14 false, correct value 44. 10−(−4)=1410 - (-4) = 14 true. −6−(−6)=−12-6 - (-6) = -12 false, correct value 00.

Page 9, missing number. −4-4

Page 9, two steps. 5−(−3)−8=8−8=05 - (-3) - 8 = 8 - 8 = 0

Page 9, challenge. Sample: 3−(−5)=83 - (-5) = 8, which is greater than 33. Subtracting a negative increases the value.

Page 11, choosing the operation table.

Situation Add or subtract? Number sentence Answer
Balance −$24-\$24, deposit $40 add −24+40-24 + 40 1616, a balance of $16
Temperature −7-7°F rises 12 degrees add −7+12-7 + 12 55°F
Hiker at 8585 ft descends 110 ft add 85+(−110)85 + (-110) −25-25 ft
How far apart are 66°F and −14-14°F? subtract 6−(−14)6 - (-14) 2020 degrees
Team loses 8 yards, then gains 3 add −8+3-8 + 3 −5-5, a 5-yard loss

Page 12, two-step problems.

  1. −3+9=6-3 + 9 = 6; 6−4=26 - 4 = 2. Ends on level 2.
  2. −58+120=62-58 + 120 = 62; 62−35=2762 - 35 = 27. Final balance $27.
  3. −46+19=−27-46 + 19 = -27; −27−33=−60-27 - 33 = -60. Final depth −60-60 ft.
  4. 9−(−11)=209 - (-11) = 20 degrees.
  5. 14,505−(−282)=14,78714{,}505 - (-282) = 14{,}787 feet.
  6. Estimate −20+40=20-20 + 40 = 20; exact −18+41=23-18 + 41 = 23°C.

Page 13, exit ticket 2.4. Same as the textbook exit ticket 2.4 above: 1. −$12-\$12 2. 1515 degrees 3. −40-40 ft 4. Estimate 55, exact 77; the estimate confirms a small positive answer.

Pages 14–15, Chapter 2 review. Same items and answers as the textbook Chapter 2 Review above.