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3B · Chapter 8 · Solutions

Complete worked answer key

Answers and explanations for every original lesson practice set and chapter assessment.

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The Multiplication Table of 6

Prerequisite

  1. 20 — Five equal groups of four have 4 + 4 + 4 + 4 + 4 = 20 objects. You can also picture five rows with four dots each.
  2. 18 — Three groups of six have 18 objects.
  3. 4 — Because five groups of four make 20.

Understanding checks

  1. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  2. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.
  3. 30 — 6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.

Guided

  1. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  2. 36 — 6 groups of 6 make 6 × 6 = 36. Or find 5 × 6 and add 6.
  3. 18 — 6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
  4. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  5. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.

Independent

  1. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.
  2. 36 — 6 groups of 6 make 6 × 6 = 36. Or find 5 × 6 and add 6.
  3. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  4. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  5. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  6. 18 — 6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
  7. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  8. 30 — 6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.

Challenge

  1. 42 — One decomposition is (5 × 7) + (1 × 7) = 42. Another is (3 × 7) + (3 × 7) = 42. The partitions use different group splits but preserve the same product.
  2. 24 — Turning an array swaps rows and columns but adds no dots. Both 6 × 4 and 4 × 6 equal 24.

Exit

  1. 18 — 6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
  2. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.
  3. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.

The Multiplication Table of 7

Prerequisite

  1. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  2. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  3. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.

Understanding checks

  1. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
  2. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  3. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.

Guided

  1. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
  2. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  3. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
  4. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
  5. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.

Independent

  1. 21 — 7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
  2. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  3. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  4. 28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
  5. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
  6. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
  7. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
  8. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.

Challenge

  1. 21 — One decomposition is (6 × 3) + (1 × 3) = 21. Another is (3 × 3) + (4 × 3) = 21. The partitions use different group splits but preserve the same product.
  2. 28 — Turning an array swaps rows and columns but adds no dots. Both 7 × 4 and 4 × 7 equal 28.

Exit

  1. 28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
  2. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
  3. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.

Multiplying by 6 and 7

Prerequisite

  1. 21 — 7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
  2. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
  3. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.

Understanding checks

  1. 28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
  2. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  3. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.

Guided

  1. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
  2. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  3. 28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
  4. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  5. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.

Independent

  1. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  2. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.
  3. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
  4. 18 — 6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
  5. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
  6. 30 — 6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.
  7. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  8. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.

Challenge

  1. 28 — One decomposition is (6 × 4) + (1 × 4) = 28. Another is (3 × 4) + (4 × 4) = 28. The partitions use different group splits but preserve the same product.
  2. 35 — Turning an array swaps rows and columns but adds no dots. Both 7 × 5 and 5 × 7 equal 35.

Exit

  1. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
  2. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  3. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.

Dividing by 6 and 7

Prerequisite

  1. 21 — 7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
  2. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  3. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.

Understanding checks

  1. 3 — Find how many groups of 7 fit into 21: 21 ÷ 7 = 3.
  2. 9 — Use inverse multiplication: 6 × 9 = 54; therefore 54 ÷ 6 = 9.
  3. 8 — Find how many groups of 7 fit into 56: 56 ÷ 7 = 8.

Guided

  1. 10 — Use inverse multiplication: 7 × 10 = 70; therefore 70 ÷ 7 = 10.
  2. 3 — Use inverse multiplication: 7 × 3 = 21; therefore 21 ÷ 7 = 3.
  3. 10 — Use inverse multiplication: 6 × 10 = 60; therefore 60 ÷ 6 = 10.
  4. 6 — Find how many groups of 7 fit into 42: 42 ÷ 7 = 6.
  5. 7 — Use inverse multiplication: 7 × 7 = 49; therefore 49 ÷ 7 = 7.

Independent

  1. 3 — Use inverse multiplication: 6 × 3 = 18; therefore 18 ÷ 6 = 3.
  2. 9 — Use inverse multiplication: 7 × 9 = 63; therefore 63 ÷ 7 = 9.
  3. 4 — Find how many groups of 7 fit into 28: 28 ÷ 7 = 4.
  4. 7 — Use inverse multiplication: 6 × 7 = 42; therefore 42 ÷ 6 = 7.
  5. 5 — Find how many groups of 6 fit into 30: 30 ÷ 6 = 5.
  6. 4 — Use inverse multiplication: 7 × 4 = 28; therefore 28 ÷ 7 = 4.
  7. 3 — Find how many groups of 6 fit into 18: 18 ÷ 6 = 3.
  8. 6 — Find how many groups of 6 fit into 36: 36 ÷ 6 = 6.

Challenge

  1. 5 — 7 × 6 = 42 is more than 35. The correct share is 35 ÷ 7 = 5.
  2. 5 bags and 2 extras — 37 = 7 × 5 + 2, with 2 less than the group size 7.

Exit

  1. 8 — Use inverse multiplication: 6 × 8 = 48; therefore 48 ÷ 6 = 8.
  2. 7 — Find how many groups of 6 fit into 42: 42 ÷ 6 = 7.
  3. 10 — Find how many groups of 7 fit into 70: 70 ÷ 7 = 10.

The Multiplication Table of 8

Prerequisite

  1. 3 — Use inverse multiplication: 7 × 3 = 21; therefore 21 ÷ 7 = 3.
  2. 5 — Find how many groups of 6 fit into 30: 30 ÷ 6 = 5.
  3. 5 — Use inverse multiplication: 6 × 5 = 30; therefore 30 ÷ 6 = 5.

Understanding checks

  1. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.
  2. 32 — 8 groups of 4 make 8 × 4 = 32. Or find 7 × 4 and add 4.
  3. 80 — 8 groups of 10 make 8 × 10 = 80. Or find 7 × 10 and add 10.

Guided

  1. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5.
  2. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5.
  3. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.
  4. 48 — 8 groups of 6 make 8 × 6 = 48. Or find 7 × 6 and add 6.
  5. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.

Independent

  1. 48 — 8 groups of 6 make 8 × 6 = 48. Or find 7 × 6 and add 6.
  2. 32 — 8 groups of 4 make 8 × 4 = 32. Or find 7 × 4 and add 4.
  3. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.
  4. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.
  5. 32 — 8 groups of 4 make 8 × 4 = 32. Or find 7 × 4 and add 4.
  6. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.
  7. 80 — 8 groups of 10 make 8 × 10 = 80. Or find 7 × 10 and add 10.
  8. 24 — 8 groups of 3 make 8 × 3 = 24. Or find 7 × 3 and add 3.

Challenge

  1. 48 — One decomposition is (7 × 6) + (1 × 6) = 48. Another is (4 × 6) + (4 × 6) = 48. The partitions use different group splits but preserve the same product.
  2. 40 — Turning an array swaps rows and columns but adds no dots. Both 8 × 5 and 5 × 8 equal 40.

Exit

  1. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  2. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  3. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.

The Multiplication Table of 9

Prerequisite

  1. 48 — 8 groups of 6 make 8 × 6 = 48. Or find 7 × 6 and add 6.
  2. 24 — 8 groups of 3 make 8 × 3 = 24. Or find 7 × 3 and add 3.
  3. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.

Understanding checks

  1. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  2. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  3. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8.

Guided

  1. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.
  2. 36 — 9 groups of 4 make 9 × 4 = 36. Or find 8 × 4 and add 4.
  3. 36 — 9 groups of 4 make 9 × 4 = 36. Or find 8 × 4 and add 4.
  4. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7.
  5. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.

Independent

  1. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8.
  2. 81 — 9 groups of 9 make 9 × 9 = 81. Or find 8 × 9 and add 9.
  3. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8.
  4. 81 — 9 groups of 9 make 9 × 9 = 81. Or find 8 × 9 and add 9.
  5. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7.
  6. 81 — 9 groups of 9 make 9 × 9 = 81. Or find 8 × 9 and add 9.
  7. 90 — 9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10.
  8. 90 — 9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10.

Challenge

  1. 63 — One decomposition is (8 × 7) + (1 × 7) = 63. Another is (4 × 7) + (5 × 7) = 63. The partitions use different group splits but preserve the same product.
  2. 54 — Turning an array swaps rows and columns but adds no dots. Both 9 × 6 and 6 × 9 equal 54.

Exit

  1. 54 — 9 groups of 6 make 9 × 6 = 54. Or find 8 × 6 and add 6.
  2. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.
  3. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.

Multiplying by 8 and 9

Prerequisite

  1. 54 — 9 groups of 6 make 9 × 6 = 54. Or find 8 × 6 and add 6.
  2. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.
  3. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.

Understanding checks

  1. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  2. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7.
  3. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5.

Guided

  1. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.
  2. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.
  3. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  4. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7.
  5. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8.

Independent

  1. 48 — 8 groups of 6 make 8 × 6 = 48. Or find 7 × 6 and add 6.
  2. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  3. 36 — 9 groups of 4 make 9 × 4 = 36. Or find 8 × 4 and add 4.
  4. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  5. 48 — 8 groups of 6 make 8 × 6 = 48. Or find 7 × 6 and add 6.
  6. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5.
  7. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  8. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.

Challenge

  1. 27 — One decomposition is (8 × 3) + (1 × 3) = 27. Another is (4 × 3) + (5 × 3) = 27. The partitions use different group splits but preserve the same product.
  2. 54 — Turning an array swaps rows and columns but adds no dots. Both 9 × 6 and 6 × 9 equal 54.

Exit

  1. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.
  2. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9.
  3. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.

Dividing by 8 and 9

Prerequisite

  1. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  2. 90 — 9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10.
  3. 32 — 8 groups of 4 make 8 × 4 = 32. Or find 7 × 4 and add 4.

Understanding checks

  1. 4 — Find how many groups of 9 fit into 36: 36 ÷ 9 = 4.
  2. 9 groups, 8 left — 9 × 9 = 81, leaving 8. The remainder is smaller than 9.
  3. 3 — Find how many groups of 8 fit into 24: 24 ÷ 8 = 3.

Guided

  1. 4 — Use inverse multiplication: 9 × 4 = 36; therefore 36 ÷ 9 = 4.
  2. 10 — Find how many groups of 8 fit into 80: 80 ÷ 8 = 10.
  3. 8 groups, 3 left — 8 × 8 = 64, leaving 3. The remainder is smaller than 8.
  4. 5 — Find how many groups of 8 fit into 40: 40 ÷ 8 = 5.
  5. 3 — Use inverse multiplication: 9 × 3 = 27; therefore 27 ÷ 9 = 3.

Independent

  1. 5 — Use inverse multiplication: 8 × 5 = 40; therefore 40 ÷ 8 = 5.
  2. 9 groups, 1 left — 8 × 9 = 72, leaving 1. The remainder is smaller than 8.
  3. 6 — Find how many groups of 8 fit into 48: 48 ÷ 8 = 6.
  4. 9 — Use inverse multiplication: 9 × 9 = 81; therefore 81 ÷ 9 = 9.
  5. 7 groups, 2 left — 8 × 7 = 56, leaving 2. The remainder is smaller than 8.
  6. 7 — Use inverse multiplication: 9 × 7 = 63; therefore 63 ÷ 9 = 7.
  7. 8 groups, 1 left — 8 × 8 = 64, leaving 1. The remainder is smaller than 8.
  8. 6 — Use inverse multiplication: 8 × 6 = 48; therefore 48 ÷ 8 = 6.

Challenge

  1. 4 — 9 × 5 = 45 is more than 36. The correct share is 36 ÷ 9 = 4.
  2. 4 bags and 2 extras — 38 = 9 × 4 + 2, with 2 less than the group size 9.

Exit

  1. 6 groups, 4 left — 8 × 6 = 48, leaving 4. The remainder is smaller than 8.
  2. 4 — Find how many groups of 8 fit into 32: 32 ÷ 8 = 4.
  3. 8 — Use inverse multiplication: 8 × 8 = 64; therefore 64 ÷ 8 = 8.

Practice A

  1. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  2. 21 — 7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
  3. 21 — 7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
  4. 4 — Find how many groups of 7 fit into 28: 28 ÷ 7 = 4.
  5. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  6. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  7. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  8. 10 — Use inverse multiplication: 7 × 10 = 70; therefore 70 ÷ 7 = 10.
  9. 36 — 6 groups of 6 make 6 × 6 = 36. Or find 5 × 6 and add 6.
  10. 63 — 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
  11. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
  12. 7 — Use inverse multiplication: 6 × 7 = 42; therefore 42 ÷ 6 = 7.
  13. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.
  14. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  15. 48 — 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
  16. 3 — Find how many groups of 7 fit into 21: 21 ÷ 7 = 3.
  17. 42 — Turning an array swaps rows and columns but adds no dots. Both 7 × 6 and 6 × 7 equal 42.
  18. 30 — One decomposition is (5 × 5) + (1 × 5) = 30. Another is (3 × 5) + (3 × 5) = 30. The partitions use different group splits but preserve the same product.

Practice B

  1. 30 — 6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.
  2. 49 — 7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
  3. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
  4. 8 — Find how many groups of 6 fit into 48: 48 ÷ 6 = 8.
  5. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5.
  6. 36 — 9 groups of 4 make 9 × 4 = 36. Or find 8 × 4 and add 4.
  7. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  8. 6 — Find how many groups of 8 fit into 48: 48 ÷ 8 = 6.
  9. 18 — 6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
  10. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  11. 30 — 6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.
  12. 5 — Use inverse multiplication: 6 × 5 = 30; therefore 30 ÷ 6 = 5.
  13. 24 — 8 groups of 3 make 8 × 3 = 24. Or find 7 × 3 and add 3.
  14. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.
  15. 27 — 9 groups of 3 make 9 × 3 = 27. Or find 8 × 3 and add 3.
  16. 3 groups, 3 left — 8 × 3 = 24, leaving 3. The remainder is smaller than 8.
  17. 28 — Turning an array swaps rows and columns but adds no dots. Both 7 × 4 and 4 × 7 equal 28.
  18. 36 — One decomposition is (5 × 6) + (1 × 6) = 36. Another is (3 × 6) + (3 × 6) = 36. The partitions use different group splits but preserve the same product.

Chapter Review

  1. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.
  2. 70 — 7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
  3. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  4. 4 — Use inverse multiplication: 6 × 4 = 24; therefore 24 ÷ 6 = 4.
  5. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8.
  6. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7.
  7. 81 — 9 groups of 9 make 9 × 9 = 81. Or find 8 × 9 and add 9.
  8. 8 — Use inverse multiplication: 8 × 8 = 64; therefore 64 ÷ 8 = 8.
  9. 36 — 6 groups of 6 make 6 × 6 = 36. Or find 5 × 6 and add 6.
  10. 35 — 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
  11. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  12. 9 — Find how many groups of 6 fit into 54: 54 ÷ 6 = 9.
  13. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  14. 90 — 9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10.
  15. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8.
  16. 10 groups, 8 left — 9 × 10 = 90, leaving 8. The remainder is smaller than 9.
  17. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.
  18. 42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
  19. 28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
  20. 10 — Use inverse multiplication: 6 × 10 = 60; therefore 60 ÷ 6 = 10.
  21. 56 — 8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
  22. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  23. 45 — 9 groups of 5 make 9 × 5 = 45. Or find 8 × 5 and add 5.
  24. 4 — Find how many groups of 9 fit into 36: 36 ÷ 9 = 4.
  25. 35 — Turning an array swaps rows and columns but adds no dots. Both 7 × 5 and 5 × 7 equal 35.
  26. 42 — One decomposition is (5 × 7) + (1 × 7) = 42. Another is (3 × 7) + (3 × 7) = 42. The partitions use different group splits but preserve the same product.

Practice Exam

  1. 8 — The suggested answer is not correct. Use inverse multiplication: 7 × 8 = 56; therefore 56 ÷ 7 = 8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  2. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  3. 72 — 8 groups of 9 make 8 × 9 = 72. Or find 7 × 9 and add 9. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  4. 42 — 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  5. 48 — The suggested answer is not correct. 6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  6. 7 — Find how many groups of 6 fit into 42: 42 ÷ 6 = 7. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  7. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  8. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  9. 56 — The suggested answer is not correct. 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  10. 5 — Find how many groups of 8 fit into 40: 40 ÷ 8 = 5. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  11. 90 — 9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  12. 40 — 8 groups of 5 make 8 × 5 = 40. Or find 7 × 5 and add 5. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  13. 63 — The suggested answer is not correct. 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  14. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  15. 9 — Use inverse multiplication: 9 × 9 = 81; therefore 81 ÷ 9 = 9. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  16. 63 — 9 groups of 7 make 9 × 7 = 63. Or find 8 × 7 and add 7. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  17. 35 — The suggested answer is not correct. 7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  18. 64 — 8 groups of 8 make 8 × 8 = 64. Or find 7 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  19. 72 — 9 groups of 8 make 9 × 8 = 72. Or find 8 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  20. 56 — 7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  21. 10 — The suggested answer is not correct. Find how many groups of 8 fit into 80: 80 ÷ 8 = 10. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  22. 54 — 6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  23. 21 — Turning an array swaps rows and columns but adds no dots. Both 7 × 3 and 3 × 7 equal 21. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  24. 24 — One decomposition is (5 × 4) + (1 × 4) = 24. Another is (3 × 4) + (3 × 4) = 24. The partitions use different group splits but preserve the same product. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)