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Complete Chapter 4 answer key

Full answers and reasoning for every chapter lesson and assessment.

Print this separately when students work on assessment pages. These are publicly available explanatory solutions, not secure school answers.

Lesson 1: Looking Back at Multiplication

Entry

  1. 12 — 4 equal groups with 3 in each group make 4 × 3 = 12. This is prerequisite equal-group reasoning.
  2. 30 — 5 equal groups with 6 in each group make 5 × 6 = 30. This is prerequisite equal-group reasoning.

Worked checks

  1. 21 — There are 7 groups of 3; multiply 7×3=21. Turning the array does not change the total.
  2. 80 — There are 8 groups of 10; multiply 8×10=80. Turning the array does not change the total.
  3. 18 — There are 2 groups of 9; multiply 2×9=18. Turning the array does not change the total.

Guided

  1. 33 — There are 3 groups of 11; multiply 3×11=33. Turning the array does not change the total.
  2. 36 — There are 4 groups of 9; multiply 4×9=36. Turning the array does not change the total.
  3. 35 — There are 5 groups of 7; multiply 5×7=35. Turning the array does not change the total.
  4. 30 — There are 6 groups of 5; multiply 6×5=30. Turning the array does not change the total.

Independent

  1. 32 — There are 8 groups of 4; multiply 8×4=32. Turning the array does not change the total.
  2. 6 — There are 2 groups of 3; multiply 2×3=6. Turning the array does not change the total.
  3. 30 — There are 3 groups of 10; multiply 3×10=30. Turning the array does not change the total.
  4. 32 — There are 4 groups of 8; multiply 4×8=32. Turning the array does not change the total.
  5. 30 — There are 5 groups of 6; multiply 5×6=30. Turning the array does not change the total.
  6. 24 — There are 6 groups of 4; multiply 6×4=24. Turning the array does not change the total.
  7. 77 — There are 7 groups of 11; multiply 7×11=77. Turning the array does not change the total.
  8. 72 — There are 8 groups of 9; multiply 8×9=72. Turning the array does not change the total.

Challenge

  1. 24 objects — Rotating the rectangular array changes which side is called rows, but the set of 24 positions is unchanged.
  2. 3×8 and 4×6 — Three rows of eight and four rows of six each count 24 equally arranged counters; factor pairs can differ without changing total.

Exit

  1. 40 — There are 4 groups of 10; multiply 4×10=40. Turning the array does not change the total.
  2. 40 — There are 5 groups of 8; multiply 5×8=40. Turning the array does not change the total.
  3. 36 — There are 6 groups of 6; multiply 6×6=36. Turning the array does not change the total.

Lesson 2: Strategies for Finding the Product

Entry

  1. 20 — There are 4 groups of 5; multiply 4×5=20. Turning the array does not change the total.
  2. 15 — There are 5 groups of 3; multiply 5×3=15. Turning the array does not change the total.

Worked checks

  1. 132 — (20+2)×6=(20×6)+(2×6)=132.
  2. 203 — (20+9)×7=(20×7)+(9×7)=203.
  3. 288 — (30+6)×8=(30×8)+(6×8)=288.

Guided

  1. 34 — (10+7)×2=(10×2)+(7×2)=34.
  2. 72 — (20+4)×3=(20×3)+(4×3)=72.
  3. 124 — (30+1)×4=(30×4)+(1×4)=124.
  4. 190 — (30+8)×5=(30×5)+(8×5)=190.

Independent

  1. 469 — (60+7)×7=(60×7)+(7×7)=469.
  2. 592 — (70+4)×8=(70×8)+(4×8)=592.
  3. 162 — (80+1)×2=(80×2)+(1×2)=162.
  4. 39 — (10+3)×3=(10×3)+(3×3)=39.
  5. 80 — (20+0)×4=(20×4)+(0×4)=80.
  6. 135 — (20+7)×5=(20×5)+(7×5)=135.
  7. 204 — (30+4)×6=(30×6)+(4×6)=204.
  8. 287 — (40+1)×7=(40×7)+(1×7)=287.

Challenge

  1. 114 — Think (20−1)×6 = 20×6 − 1×6 = 120−6=114.
  2. 90 — Both parts are multiplied by five: (10×5)+(8×5)=50+40=90.

Exit

  1. 51 — (10+7)×3=(10×3)+(7×3)=51.
  2. 96 — (20+4)×4=(20×4)+(4×4)=96.
  3. 155 — (30+1)×5=(30×5)+(1×5)=155.

Lesson 3: Looking Back at Division

Entry

  1. 28 — There are 7 groups of 4; multiply 7×4=28. Turning the array does not change the total.
  2. 88 — There are 8 groups of 11; multiply 8×11=88. Turning the array does not change the total.

Worked checks

  1. 7 — 4 equal groups contain 28 counters: 28÷4=7; verify 7×4=28.
  2. 10 — 5 equal groups contain 50 counters: 50÷5=10; verify 10×5=50.
  3. 13 — 6 equal groups contain 78 counters: 78÷6=13; verify 13×6=78.

Guided

  1. 11 — 6 equal groups contain 66 counters: 66÷6=11; verify 11×6=66.
  2. 14 — 7 equal groups contain 98 counters: 98÷7=14; verify 14×7=98.
  3. 4 — 8 equal groups contain 32 counters: 32÷8=4; verify 4×8=32.
  4. 7 — 9 equal groups contain 63 counters: 63÷9=7; verify 7×9=63.

Independent

  1. 5 — 6 equal groups contain 30 counters: 30÷6=5; verify 5×6=30.
  2. 8 — 7 equal groups contain 56 counters: 56÷7=8; verify 8×7=56.
  3. 11 — 8 equal groups contain 88 counters: 88÷8=11; verify 11×8=88.
  4. 14 — 9 equal groups contain 126 counters: 126÷9=14; verify 14×9=126.
  5. 4 — 2 equal groups contain 8 counters: 8÷2=4; verify 4×2=8.
  6. 7 — 3 equal groups contain 21 counters: 21÷3=7; verify 7×3=21.
  7. 10 — 4 equal groups contain 40 counters: 40÷4=10; verify 10×4=40.
  8. 13 — 5 equal groups contain 65 counters: 65÷5=13; verify 13×5=65.

Challenge

  1. Six objects per box, four boxes — 24 objects arranged in bags of 6 form 4 bags. Division may count groups rather than their sizes.
  2. How many boxes if there are 4 crayons in each? 6 boxes — Originally 24÷6=4 boxes. Change the group size to 4 crayons per box: 24÷4=6 boxes.

Exit

  1. 6 — 6 equal groups contain 36 counters: 36÷6=6; verify 6×6=36.
  2. 9 — 7 equal groups contain 63 counters: 63÷7=9; verify 9×7=63.
  3. 12 — 8 equal groups contain 96 counters: 96÷8=12; verify 12×8=96.

Lesson 4: Multiplying and Dividing with 0 and 1

Entry

  1. 6 — 2 equal groups contain 12 counters: 12÷2=6; verify 6×2=12.
  2. 9 — 3 equal groups contain 27 counters: 27÷3=9; verify 9×3=27.

Worked checks

  1. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  2. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  3. 57 — The meaning of equal groups gives 57. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.

Guided

  1. 31 — The meaning of equal groups gives 31. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  2. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  3. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  4. 52 — The meaning of equal groups gives 52. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.

Independent

  1. 58 — The meaning of equal groups gives 58. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  2. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  3. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  4. 79 — The meaning of equal groups gives 79. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  5. 86 — The meaning of equal groups gives 86. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  6. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  7. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  8. 18 — The meaning of equal groups gives 18. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.

Challenge

  1. Undefined — If 8÷0=q, then 0×q would have to equal 8. No number q satisfies that.
  2. 0, undefined, 0 — 0÷7=0 because seven equal groups can each receive 0; 7÷0 has no defined quotient; seven groups of 0 contain 0.

Exit

  1. 23 — The meaning of equal groups gives 23. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  2. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  3. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.

Lesson 5: Division with Remainders

Entry

  1. 10 — 4 equal groups contain 40 counters: 40÷4=10; verify 10×4=40.
  2. 13 — 5 equal groups contain 65 counters: 65÷5=13; verify 13×5=65.

Worked checks

  1. 18 bags, 1 left — 37=2×18+1; since 1<2, 18 bags are full and 1 oranges remain.
  2. 25 bags, 1 left — 76=3×25+1; since 1<3, 25 bags are full and 1 oranges remain.
  3. 32 bags, 2 left — 130=4×32+2; since 2<4, 32 bags are full and 2 oranges remain.

Guided

  1. 34 bags, 2 left — 206=6×34+2; since 2<6, 34 bags are full and 2 oranges remain.
  2. 41 bags, 1 left — 288=7×41+1; since 1<7, 41 bags are full and 1 oranges remain.
  3. 11 bags, 5 left — 93=8×11+5; since 5<8, 11 bags are full and 5 oranges remain.
  4. 18 bags, 5 left — 167=9×18+5; since 5<9, 18 bags are full and 5 oranges remain.

Independent

  1. 7 bags, 1 left — 43=6×7+1; since 1<6, 7 bags are full and 1 oranges remain.
  2. 15 bags, 2 left — 107=7×15+2; since 2<7, 15 bags are full and 2 oranges remain.
  3. 22 bags, 1 left — 177=8×22+1; since 1<8, 22 bags are full and 1 oranges remain.
  4. 29 bags, 7 left — 268=9×29+7; since 7<9, 29 bags are full and 7 oranges remain.
  5. 36 bags, 1 left — 73=2×36+1; since 1<2, 36 bags are full and 1 oranges remain.
  6. 6 bags, 1 left — 19=3×6+1; since 1<3, 6 bags are full and 1 oranges remain.
  7. 13 bags, 1 left — 53=4×13+1; since 1<4, 13 bags are full and 1 oranges remain.
  8. 20 bags, 3 left — 103=5×20+3; since 3<5, 20 bags are full and 3 oranges remain.

Challenge

  1. 5 vans — Four vans hold only 32. Since three students remain, a fifth van is required, even though 35÷8 is 4 remainder 3.
  2. 7 teams; 5 on last — Six full teams hold 48, leaving 5 children. A seventh smaller team is necessary.

Exit

  1. 29 bags, 4 left — 178=6×29+4; since 4<6, 29 bags are full and 4 oranges remain.
  2. 36 bags, 6 left — 258=7×36+6; since 6<7, 36 bags are full and 6 oranges remain.
  3. 6 bags, 2 left — 50=8×6+2; since 2<8, 6 bags are full and 2 oranges remain.

Lesson 6: Odd and Even Numbers

Entry

  1. 6 — 2 equal groups with 3 in each group make 2 × 3 = 6. This is prerequisite equal-group reasoning.
  2. 18 — 3 equal groups with 6 in each group make 3 × 6 = 18. This is prerequisite equal-group reasoning.

Worked checks

  1. odd — 145 is odd, 88 is even; their sum 233 is odd because 233 leaves 1 unpaired item.
  2. odd — 162 is even, 19 is odd; their sum 181 is odd because 181 leaves 1 unpaired item.
  3. even — 34 is even, 30 is even; their sum 64 is even because 64 leaves 0 unpaired item.

Guided

  1. odd — 34 is even, 15 is odd; their sum 49 is odd because 49 leaves 1 unpaired item.
  2. odd — 51 is odd, 26 is even; their sum 77 is odd because 77 leaves 1 unpaired item.
  3. odd — 68 is even, 37 is odd; their sum 105 is odd because 105 leaves 1 unpaired item.
  4. odd — 85 is odd, 48 is even; their sum 133 is odd because 133 leaves 1 unpaired item.

Independent

  1. odd — 104 is even, 15 is odd; their sum 119 is odd because 119 leaves 1 unpaired item.
  2. odd — 121 is odd, 26 is even; their sum 147 is odd because 147 leaves 1 unpaired item.
  3. odd — 138 is even, 37 is odd; their sum 175 is odd because 175 leaves 1 unpaired item.
  4. odd — 155 is odd, 48 is even; their sum 203 is odd because 203 leaves 1 unpaired item.
  5. odd — 172 is even, 59 is odd; their sum 231 is odd because 231 leaves 1 unpaired item.
  6. even — 44 is even, 70 is even; their sum 114 is even because 114 leaves 0 unpaired item.
  7. even — 61 is odd, 81 is odd; their sum 142 is even because 142 leaves 0 unpaired item.
  8. even — 78 is even, 92 is even; their sum 170 is even because 170 leaves 0 unpaired item.

Challenge

  1. Even — Each odd collection leaves one unpaired counter; the two leftovers make a complete pair.
  2. Yes — Each odd number contributes one unpaired counter. Three leftover counters form a pair and one leftover: odd.

Exit

  1. even — 99 is odd, 15 is odd; their sum 114 is even because 114 leaves 0 unpaired item.
  2. even — 116 is even, 26 is even; their sum 142 is even because 142 leaves 0 unpaired item.
  3. even — 133 is odd, 37 is odd; their sum 170 is even because 170 leaves 0 unpaired item.

Lesson 7: Word Problems — Part 1

Entry

  1. 55 — There are 5 groups of 11; multiply 5×11=55. Turning the array does not change the total.
  2. 54 — There are 6 groups of 9; multiply 6×9=54. Turning the array does not change the total.

Worked checks

  1. 60 — The total is 4 equal groups of 15: 4 × 15 = 60.
  2. 132 — The total is 6 equal groups of 22: 6 × 22 = 132.
  3. 48 — The total is 8 equal groups of 6: 8 × 6 = 48.

Guided

  1. 56 — The total is 4 equal groups of 14: 4 × 14 = 56.
  2. 126 — The total is 6 equal groups of 21: 6 × 21 = 126.
  3. 40 — The total is 8 equal groups of 5: 8 × 5 = 40.
  4. 130 — The total is 10 equal groups of 13: 10 × 13 = 130.

Independent

  1. 65 — The total is 13 equal groups of 5: 13 × 5 = 65.
  2. 48 — The total is 4 equal groups of 12: 4 × 12 = 48.
  3. 114 — The total is 6 equal groups of 19: 6 × 19 = 114.
  4. 208 — The total is 8 equal groups of 26: 8 × 26 = 208.
  5. 100 — The total is 10 equal groups of 10: 10 × 10 = 100.
  6. 216 — The total is 12 equal groups of 18: 12 × 18 = 216.
  7. 75 — The total is 3 equal groups of 25: 3 × 25 = 75.
  8. 54 — The total is 6 equal groups of 9: 6 × 9 = 54.

Challenge

  1. The groups are not equal — Multiplication as equal groups requires the same amount per group. Otherwise count or add the actual different sizes.
  2. 59 — 9×7=63 planted; 63−4=59 sprout. The plants that failed do not change the original group structure.

Exit

  1. 90 — The total is 9 equal groups of 10: 9 × 10 = 90.
  2. 187 — The total is 11 equal groups of 17: 11 × 17 = 187.
  3. 312 — The total is 13 equal groups of 24: 13 × 24 = 312.

Lesson 8: Word Problems — Part 2

Entry

  1. 190 — The total is 10 equal groups of 19: 10 × 19 = 190.
  2. 312 — The total is 12 equal groups of 26: 12 × 26 = 312.

Worked checks

  1. 64 — Draw one unit 16 for Sam and 4 matching units for Nora: 16 × 4 = 64.
  2. 95 — Draw one unit 19 for Sam and 5 matching units for Nora: 19 × 5 = 95.
  3. 132 — Draw one unit 22 for Sam and 6 matching units for Nora: 22 × 6 = 132.

Guided

  1. 18 — Draw one unit 9 for Sam and 2 matching units for Nora: 9 × 2 = 18.
  2. 36 — Draw one unit 12 for Sam and 3 matching units for Nora: 12 × 3 = 36.
  3. 60 — Draw one unit 15 for Sam and 4 matching units for Nora: 15 × 4 = 60.
  4. 90 — Draw one unit 18 for Sam and 5 matching units for Nora: 18 × 5 = 90.

Independent

  1. 12 — Draw one unit 6 for Sam and 2 matching units for Nora: 6 × 2 = 12.
  2. 27 — Draw one unit 9 for Sam and 3 matching units for Nora: 9 × 3 = 27.
  3. 48 — Draw one unit 12 for Sam and 4 matching units for Nora: 12 × 4 = 48.
  4. 75 — Draw one unit 15 for Sam and 5 matching units for Nora: 15 × 5 = 75.
  5. 108 — Draw one unit 18 for Sam and 6 matching units for Nora: 18 × 6 = 108.
  6. 42 — Draw one unit 21 for Sam and 2 matching units for Nora: 21 × 2 = 42.
  7. 72 — Draw one unit 24 for Sam and 3 matching units for Nora: 24 × 3 = 72.
  8. 108 — Draw one unit 27 for Sam and 4 matching units for Nora: 27 × 4 = 108.

Challenge

  1. 11 versus 24 — Three more gives 8+3=11; three times gives 3 copies of 8, or 24.
  2. 28 beads — Eva has 5×7=35. Difference is 35−7=28. The difference is four equal seven-bead units.

Exit

  1. 58 — Draw one unit 29 for Sam and 2 matching units for Nora: 29 × 2 = 58.
  2. 96 — Draw one unit 32 for Sam and 3 matching units for Nora: 32 × 3 = 96.
  3. 24 — Draw one unit 6 for Sam and 4 matching units for Nora: 6 × 4 = 24.

Lesson 9: 2-Step Word Problems

Entry

  1. 189 — The total is 9 equal groups of 21: 9 × 21 = 189.
  2. 55 — The total is 11 equal groups of 5: 11 × 5 = 55.

Worked checks

  1. 102 — First find total 7×18=126, then subtract 24 to get 102.
  2. 27 — First find total 8×7=56, then subtract 29 to get 27.
  3. 86 — First find total 9×10=90, then subtract 4 to get 86.

Guided

  1. 18 — First find total 7×6=42, then subtract 24 to get 18.
  2. 43 — First find total 8×9=72, then subtract 29 to get 43.
  3. 104 — First find total 9×12=108, then subtract 4 to get 104.
  4. 141 — First find total 10×15=150, then subtract 9 to get 141.

Independent

  1. 94 — First find total 7×14=98, then subtract 4 to get 94.
  2. 127 — First find total 8×17=136, then subtract 9 to get 127.
  3. 40 — First find total 9×6=54, then subtract 14 to get 40.
  4. 71 — First find total 10×9=90, then subtract 19 to get 71.
  5. 12 — First find total 3×12=36, then subtract 24 to get 12.
  6. 31 — First find total 4×15=60, then subtract 29 to get 31.
  7. 86 — First find total 5×18=90, then subtract 4 to get 86.
  8. 33 — First find total 6×7=42, then subtract 9 to get 33.

Challenge

  1. 15 pens — Before selling there were 33+12=45 pens. Three equal packs held 45÷3=15 each.
  2. 77 pencils — Filled pencils: 5×12=60. Add 17 leftovers to obtain 77.

Exit

  1. 88 — First find total 7×16=112, then subtract 24 to get 88.
  2. 123 — First find total 8×19=152, then subtract 29 to get 123.
  3. 68 — First find total 9×8=72, then subtract 4 to get 68.

Practice

Solutions

  1. 35 — There are 5 groups of 7; multiply 5×7=35. Turning the array does not change the total.
  2. 30 — There are 6 groups of 5; multiply 6×5=30. Turning the array does not change the total.
  3. 21 — There are 7 groups of 3; multiply 7×3=21. Turning the array does not change the total.
  4. 268 — (60+7)×4=(60×4)+(7×4)=268.
  5. 370 — (70+4)×5=(70×5)+(4×5)=370.
  6. 486 — (80+1)×6=(80×6)+(1×6)=486.
  7. 12 — 6 equal groups contain 72 counters: 72÷6=12; verify 12×6=72.
  8. 15 — 7 equal groups contain 105 counters: 105÷7=15; verify 15×7=105.
  9. 5 — 8 equal groups contain 40 counters: 40÷8=5; verify 5×8=40.
  10. 84 — The meaning of equal groups gives 84. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  11. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  12. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  13. 39 bags, 5 left — 239=6×39+5; since 5<6, 39 bags are full and 5 oranges remain.
  14. 9 bags, 4 left — 67=7×9+4; since 4<7, 9 bags are full and 4 oranges remain.
  15. 16 bags, 1 left — 129=8×16+1; since 1<8, 16 bags are full and 1 oranges remain.
  16. odd — 124 is even, 15 is odd; their sum 139 is odd because 139 leaves 1 unpaired item.
  17. odd — 141 is odd, 26 is even; their sum 167 is odd because 167 leaves 1 unpaired item.
  18. odd — 158 is even, 37 is odd; their sum 195 is odd because 195 leaves 1 unpaired item.
  19. 253 — The total is 11 equal groups of 23: 11 × 23 = 253.
  20. 91 — The total is 13 equal groups of 7: 13 × 7 = 91.
  21. 56 — The total is 4 equal groups of 14: 4 × 14 = 56.
  22. 60 — Draw one unit 30 for Sam and 2 matching units for Nora: 30 × 2 = 60.
  23. 99 — Draw one unit 33 for Sam and 3 matching units for Nora: 33 × 3 = 99.
  24. 28 — Draw one unit 7 for Sam and 4 matching units for Nora: 7 × 4 = 28.
  25. 80 — First find total 7×12=84, then subtract 4 to get 80.
  26. 111 — First find total 8×15=120, then subtract 9 to get 111.
  27. 148 — First find total 9×18=162, then subtract 14 to get 148.

Chapter Review

Solutions

  1. 75 — The total is 3 equal groups of 25: 3 × 25 = 75.
  2. 138 — First find total 9×18=162, then subtract 24 to get 138.
  3. 64 — Draw one unit 32 for Sam and 2 matching units for Nora: 32 × 2 = 64.
  4. 216 — The total is 12 equal groups of 18: 12 × 18 = 216.
  5. 98 — (10+4)×7=(10×7)+(4×7)=98.
  6. odd — 158 is even, 57 is odd; their sum 215 is odd because 215 leaves 1 unpaired item.
  7. 5 — 6 equal groups contain 30 counters: 30÷6=5; verify 5×6=30.
  8. 7 teams, 5 left — 6×7=42, with 47−42=5 children left.
  9. 8 bags, 1 left — 65=8×8+1; since 1<8, 8 bags are full and 1 oranges remain.
  10. 168 — (20+1)×8=(20×8)+(1×8)=168.
  11. 50 — There are 5 groups of 10; multiply 5×10=50. Turning the array does not change the total.
  12. 56 — Eight copies of the two-part sum contain 40 of the first part and 16 of the second.
  13. 6×5 and 5×6 — Both rectangles have 30 cells; rotating swaps rows and columns without changing count.
  14. 18 — Draw one unit 6 for Sam and 3 matching units for Nora: 6 × 3 = 18.
  15. 41 — First find total 10×7=70, then subtract 29 to get 41.
  16. 45 — The total is 5 equal groups of 9: 5 × 9 = 45.
  17. odd — 124 is even, 35 is odd; their sum 159 is odd because 159 leaves 1 unpaired item.
  18. 41 — 5×9=45; 45−4=41.
  19. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  20. 62 — The meaning of equal groups gives 62. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  21. 12 — 4 equal groups contain 48 counters: 48÷4=12; verify 12×4=48.
  22. 492 — (80+2)×6=(80×6)+(2×6)=492.
  23. 48 — There are 6 groups of 8; multiply 6×8=48. Turning the array does not change the total.
  24. 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  25. 22 bags, 1 left — 45=2×22+1; since 1<2, 22 bags are full and 1 oranges remain.
  26. 15 bags, 8 left — 143=9×15+8; since 8<9, 15 bags are full and 8 oranges remain.
  27. 12 — There are 4 groups of 3; multiply 4×3=12. Turning the array does not change the total.
  28. 36 — Draw one unit 9 for Sam and 4 matching units for Nora: 9 × 4 = 36.
  29. odd — 141 is odd, 46 is even; their sum 187 is odd because 187 leaves 1 unpaired item.
  30. 24 — First find total 3×10=30, then subtract 6 to get 24.
  31. 15 — 5 equal groups contain 75 counters: 75÷5=15; verify 15×5=75.

Practice Exam

Solutions

  1. 56 — There are 7 groups of 8; multiply 7×8=56. Turning the array does not change the total.
  2. 28 — 9 teams total; two contain 8 children, leaving 28.
  3. 36 and 15 — 3×12=36 whereas 12+3=15; the relationships are distinct.
  4. 84 — Multiply both parts by 6: 6×10+6×4=60+24=84.
  5. 40 — First find total 9×6=54, then subtract 14 to get 40.
  6. 54 — 6×10−6×1=60−6=54.
  7. 52 — Draw one unit 26 for Sam and 2 matching units for Nora: 26 × 2 = 52.
  8. Zero shares versus zero divisor — Zero items can be divided into 5 empty groups; division by 0 cannot satisfy inverse multiplication for nonzero 5.
  9. Yes — 17+19=36, and 36÷4=9 equal groups.
  10. even — 119 is odd, 35 is odd; their sum 154 is even because 154 leaves 0 unpaired item.
  11. Even — Eight equal groups of seven can be paired by joining two groups at a time; 8 is even.
  12. 7 groups, 1 left — 6×7=42 and 43−42=1, which is smaller than 6.
  13. 176 — The total is 8 equal groups of 22: 8 × 22 = 176.
  14. 12 — 7×6=42; 42−30=12.
  15. 13 — 4 equal groups contain 52 counters: 52÷4=13; verify 13×4=52.
  16. 36 — Each row has 45÷5=9; four rows remain, 4×9=36.
  17. 64 — (30+2)×2=(30×2)+(2×2)=64.
  18. 27 — The meaning of equal groups gives 27. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
  19. 30 bags, 1 left — 241=8×30+1; since 1<8, 30 bags are full and 1 oranges remain.
  20. 38 — 5×7+3=35+3=38.
  21. 8 trays — Seven trays hold 56, leaving two plants for an eighth tray.