Study CommonsRead · reason · practise

Home / Math / Multiplication Shortcuts: Valid Distributive Steps or Traps?

Math · Multiplication & Mathematical Reasoning

Multiplication Shortcuts: Valid Distributive Steps or Traps?

Tell a valid distributive calculation from a convincing but incorrect shortcut; explain what must be multiplied and why.

Grades 3–5 · 28 practice questions · Original material
Learning objectiveUse a(b+c)=ab+ac correctly, spot incorrect omitted terms, and choose direct multiplication when a shortcut does not help.

The idea

Breaking apart a factor works because every part is multiplied by the same number. For example, 6 × 14 = 6 × (10 + 4) = 60 + 24 = 84.

A common trap is to multiply only the first part: 6 × (10 + 4) is not 6 × 10 + 4. The final 4 must also be multiplied by 6.

Another trap is to change a factor to a friendlier number without adjusting the product correctly. Replacing 19 by 20 in 7 × 19 adds one extra group of 7, not one extra unit.

Not every expression has a useful trick. A clear direct calculation and an independent check are better than an invented shortcut.

Worked examples

Nine rows of nineteen, each shown as a row of twenty minus one extra unit.9 × 19: subtract ONE from EACH of 9 groupsBegin with 9 rows of 20; remove the orange unit in every row.12345678919 green units in every row+19 × 19 = 9 × 20 − 9 = 171Not 9 × 20 − 1 = 179
Nine groups of nineteen are nine groups of twenty minus nine single units, not minus one. The rectangles are a conceptual model.

Distribute correctly

Calculate 7 × 14.

  1. Split 14 into 10 + 4.
  2. 7 × 10 + 7 × 4 = 70 + 28.
  3. The total is 98.

98

The omitted-factor trap

Is 8 × 13 = 8 × 10 + 3?

  1. The expression 8 × (10 + 3) has eight groups of each part.
  2. Correct: 8 × 10 + 8 × 3 = 80 + 24.
  3. The incorrect shortcut gives 83; correct answer 104.

No; 104

Compensate by a whole group

Calculate 9 × 19.

  1. Use 9 × 20 = 180.
  2. One extra group of 9 was counted.
  3. 180 − 9 = 171.

171

Two independent routes

Calculate 12 × 15.

  1. First: 12 × (10 + 5) = 120 + 60 = 180.
  2. Check: (10 + 2) × 15 = 150 + 30 = 180.
  3. Both decompositions agree.

180

When the trick is unhelpful

Calculate 6 × 7.

  1. Multiplication facts or repeated addition give 42 directly.
  2. Changing 7 to 10 would require 6 × 3 = 18 adjustment.
  3. A complicated shortcut is unnecessary when a fact is known.

42

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Forgetting to distribute to every addend.

Why this fails: 4 × (20 + 3) = 4 × 20 + 4 × 3, not 4 × 20 + 3.

Correcting a rounded factor by only 1.

Why this fails: In 8 × 19, replacing 19 with 20 changes the product by 8.

Distributing across addition in the wrong direction.

Why this fails: (4 + 2) × 9 = 4 × 9 + 2 × 9; multiply both addends by 9.

Assuming every calculation requires a trick.

Why this fails: Simple direct multiplication can be faster and less error-prone.

Foundation · Break the factor

Foundation · Break the factor practice

Show the split and both products.

Explain

1. Calculate 4 × 13.

Explain

2. Calculate 6 × 12.

Explain

3. Calculate 8 × 15.

Explain

4. Calculate 3 × 24.

Explain

5. Calculate 5 × 18.

Explain

6. Calculate 7 × 11.

Explain

7. Calculate 9 × 12.

Practice · Round then compensate

Practice · Round then compensate practice

Compensate for the entire group you added or removed.

Explain

8. Calculate 6 × 19.

Explain

9. Calculate 7 × 29.

Explain

10. Calculate 8 × 21.

Explain

11. Calculate 4 × 39.

Explain

12. Calculate 9 × 31.

Explain

13. Calculate 5 × 49.

Explain

14. Calculate 3 × 28.

Think · Find the missing part

Think · Find the missing part practice

Explain why plausible-looking equalities fail.

Check equality

15. Is 6 × 14 = 6 × 10 + 4 correct?

Check equality

16. A classmate writes 7 × 19 = 7 × 20 − 1. Correct?

Check equality

17. Is (5 + 3) × 11 = 5 × 11 + 3 × 11 correct?

Check equality

18. Is 4 × (8 + 2) = 4 × 8 + 2 × 4 correct?

Check equality

19. Does 9 × 17 equal 9 × (20 − 3)?

Check equality

20. Is 3 × 26 = 3 × 20 + 6 × 3?

Check equality

21. Is 8 × (10 + 5) = 8 × 10 + 5 × 10 correct?

Challenge · Which method is actually sound?

Challenge · Which method is actually sound? practice

Choose a sound method, reject a shortcut or compare methods.

Explain

22. Find 12 × 13 using distribution.

Explain

23. Find 11 × 19 using compensation.

Check equality

24. A student claims 25 × 9 + 9 = 25 × 10. Is that true?

Check equality

25. Is 15 × 8 = (10 × 8) + (5 × 8) true?

Explain

26. Find 13 × 17 and check by a different split.

Check equality

27. A student writes 6 × (30 − 2) = 180 − 2. Explain.

Explain

28. Is 7 × 8 easier by a direct fact or by rounding 8 to 10? Explain.

Answer key

  1. 1. 52
    4 × 10 + 4 × 3 = 40 + 12 = 52.
  2. 2. 72
    6 × 10 + 6 × 2 = 60 + 12 = 72.
  3. 3. 120
    8 × 10 + 8 × 5 = 80 + 40 = 120.
  4. 4. 72
    3 × 20 + 3 × 4 = 60 + 12 = 72.
  5. 5. 90
    5 × 10 + 5 × 8 = 50 + 40 = 90.
  6. 6. 77
    7 × 10 + 7 × 1 = 70 + 7 = 77.
  7. 7. 108
    9 × 10 + 9 × 2 = 90 + 18 = 108.
  8. 8. 114
    6 × 20 − 6 = 120 − 6 = 114.
  9. 9. 203
    7 × 30 − 7 = 210 − 7 = 203.
  10. 10. 168
    8 × 20 + 8 = 168.
  11. 11. 156
    4 × 40 − 4 = 156.
  12. 12. 279
    9 × 30 + 9 = 279.
  13. 13. 245
    5 × 50 − 5 = 245.
  14. 14. 84
    3 × 30 − 3 × 2 = 90 − 6 = 84.
  15. 15. No
    The right side is 64, but the correct distribution is 60 + 24 = 84.
  16. 16. No
    Subtract the extra group of 7: 140 − 7 = 133.
  17. 17. Yes
    Each addend receives one factor of 11, giving 55 + 33 = 88.
  18. 18. Yes
    This is distribution to both addends: 32 + 8 = 40.
  19. 19. Yes
    9 × 20 − 9 × 3 = 180 − 27 = 153.
  20. 20. Yes
    Multiplication order can switch: 60 + 18 = 78.
  21. 21. No
    The second term is 8 × 5, not 5 × 10; answer 120.
  22. 22. 156
    12 × 10 + 12 × 3 = 120 + 36 = 156.
  23. 23. 209
    11 × 20 − 11 = 220 − 11 = 209.
  24. 24. No
    25 × 9 + 9 = 234, whereas 25 × 10 = 250. The extra term would need to be 25.
  25. 25. Yes
    80 + 40 = 120.
  26. 26. 221
    13 × (10 + 7) = 130 + 91 = 221; (10 + 3) × 17 = 170 + 51 = 221.
  27. 27. Incorrect; 168
    The removed 2 must be multiplied by 6: 180 − 12 = 168.
  28. 28. 56; either valid
    Direct 7 × 8 = 56 is short, while 7 × 10 − 7 × 2 = 56 is valid but longer.

Related practice

Mental Addition and CompensationMultiply Smarter