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Multiply Smarter: Break Apart, Regroup, Estimate, Check

Build reliable multi-digit multiplication with distributive thinking, friendly-number regrouping, estimation, partial products and 40 original Grade 4–5 practice questions.

Grades 4–5 · 40 practice questions · Original material
Learning objectiveMultiply one- and two-digit factors accurately by breaking numbers apart, regrouping useful factors, estimating before calculating, and checking whether an answer is reasonable.

The idea

A multiplication expression can often be rewritten without changing its value. The safest shortcuts come from place value and the distributive and associative properties, not from changing one part of the expression and hoping the answer stays the same.

Break apart when place value helps: 47 × 6 = (40 × 6) + (7 × 6). Regroup when friendly factors help: 25 × 28 = 25 × 4 × 7 = 100 × 7. For two-digit factors, partial products keep every piece visible: 34 × 12 = 34 × 10 + 34 × 2.

Estimate before or after you calculate. If 58 × 23 is a little more than 60 × 20, an answer such as 133 or 13,340 should look suspicious. Estimation is a reasonableness check, not a replacement for the exact answer.

Worked examples

Example 1 · Break apart by place value

Find 47 × 6.

  1. Split 47 into 40 + 7.
  2. 40 × 6 = 240 and 7 × 6 = 42.
  3. Add the partial products: 240 + 42 = 282.

47 × 6 = 282.

Example 2 · Regroup friendly factors

Find 25 × 28 without long multiplication.

  1. Write 28 as 4 × 7.
  2. Regroup: 25 × 4 × 7 = 100 × 7.
  3. 100 × 7 = 700.

25 × 28 = 700.

Example 3 · Use partial products

Find 34 × 12.

  1. Split 12 into 10 + 2.
  2. 34 × 10 = 340 and 34 × 2 = 68.
  3. 340 + 68 = 408.

34 × 12 = 408.

Example 4 · Use a nearby friendly number

Find 58 × 19.

  1. Think of 19 as 20 − 1.
  2. 58 × 20 = 1,160.
  3. Subtract one group of 58: 1,160 − 58 = 1,102.

58 × 19 = 1,102.

Example 5 · Estimate first

Estimate, then calculate 72 × 39.

  1. Estimate with 70 × 40 = 2,800.
  2. Exact: 72 × 39 = 72 × (40 − 1) = 2,880 − 72.
  3. 2,880 − 72 = 2,808, which is close to the estimate.

Estimate ≈ 2,800; exact answer = 2,808.

Example 6 · Spot a fake shortcut

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

  1. 25 × 9 is nine groups of 25.
  2. To make ten groups of 25, add another 25, not 9.
  3. 25 × 9 + 9 = 234, while 25 × 10 = 250.

No. The missing addend is 25, not 9.

Common mistakes

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

Breaking apart a factor but forgetting one partial product.

Why this fails: For 43 × 6, both 40 × 6 and 3 × 6 are required. Leaving out 3 × 6 changes the value.

Changing 19 to 20 without compensating.

Why this fails: 58 × 20 has one extra group of 58 compared with 58 × 19, so you must subtract 58.

Using an attractive regrouping that is not equivalent.

Why this fails: 25 × 24 can become 25 × 4 × 6 because 24 = 4 × 6. It cannot become 25 × 4 + 6.

Ignoring place value in two-digit multiplication.

Why this fails: In 34 × 12, the 1 in 12 means one ten, so the first partial product is 34 × 10, not 34 × 1.

Trusting an exact-looking answer without estimating.

Why this fails: An estimate catches many misplaced zeros and missing partial products before they become final answers.

Foundation

Foundation practice

Use place value, basic facts and friendly factors. Say what you changed and why the value stayed the same.

Calculate

1. 32 × 4

Calculate

2. 46 × 5

Choose an equivalent expression

3. Which expression is exactly equal to 57 × 6?

  1. 50 × 6 + 7 × 6
  2. 50 × 6 + 7
  3. 57 × 5 + 6
  4. 60 × 6 + 7 × 6
Friendly factors

4. 25 × 12

Friendly factors

5. 125 × 8

Calculate

6. 63 × 7

Calculate

7. 38 × 4

Estimate

8. What is a sensible estimate for 49 × 8?

Place value

9. 34 × 10

Break apart

10. 34 × 11

Practice

Practice practice

Multiply two-digit numbers with partial products, nearby friendly numbers and a reasonableness check.

Partial products

11. 27 × 13

Partial products

12. 42 × 16

Regroup

13. 35 × 24

Nearby number

14. 48 × 19

Partial products

15. 62 × 17

Friendly factors

16. 75 × 28

Friendly factors

17. 125 × 24

Nearby number

18. 99 × 37

Word problem

19. A warehouse packs 24 boxes with 18 notebooks in each box. How many notebooks are packed?

Word problem

20. A theatre has 36 rows with 27 seats in each row. How many seats are there?

Think

Think practice

Test equivalence, diagnose errors and work backwards instead of following one memorized procedure.

Equivalent expressions

21. Which expression equals 25 × 36?

  1. 100 × 9
  2. 25 × 30 + 6
  3. 50 × 36
  4. 100 × 36
Missing addend

22. 25 × 9 + ___ = 25 × 10

Compare

23. Without doing two full multiplications, how much larger is 50 × 21 than 49 × 21?

Explain the mistake

24. A student says 38 × 14 = 52 because 38 + 14 = 52. What should the student do instead?

Friendly factors

25. 64 × 25

Reverse problem

26. 32 × ? = 32 × (10 + 7). What is the missing factor?

True or false

27. 18 × 24 and 9 × 48 have the same product.

  1. True
  2. False
Reasoning

28. Explain why 16 × 25 = 8 × 50.

Break apart

29. 72 × 15

Estimate and judge

30. Which is the only reasonable answer for 58 × 23?

  1. 133
  2. 1,334
  3. 5,823
  4. 13,340

Challenge

Challenge practice

Use structure, reverse reasoning and a first step toward algebraic thinking.

Friendly factors

31. 125 × 48

Friendly factors

32. 250 × 36

Nearby number

33. 48 × 49

Distributive property

34. 101 × 67

Application

35. A rectangle is 36 m long and 24 m wide. What is its area?

Word problem

36. A school sells 128 tickets each day for 24 days. How many tickets are sold?

Reverse problem

37. 25 × n = 2,300. Find n.

Equivalent expression

38. Complete the decomposition: 37 × 26 = 37 × 20 + ______.

Can this shortcut work?

39. A student rewrites 48 × 25 as 12 × 100. Is the rewrite valid? Explain.

Prealgebra preview

40. If x = 6, evaluate 23 × (x + 4).

Answer key

  1. 1. 128
    30 × 4 = 120 and 2 × 4 = 8; 120 + 8 = 128.
  2. 2. 230
    40 × 5 = 200 and 6 × 5 = 30; total 230.
  3. 3. 50 × 6 + 7 × 6
    57 = 50 + 7, so the distributive property gives both partial products.
  4. 4. 300
    12 = 4 × 3, so 25 × 4 × 3 = 100 × 3 = 300.
  5. 5. 1,000
    Eight groups of 125 make 1,000.
  6. 6. 441
    60 × 7 = 420 and 3 × 7 = 21; total 441.
  7. 7. 152
    30 × 4 = 120 and 8 × 4 = 32; total 152.
  8. 8. About 400
    49 is close to 50, and 50 × 8 = 400.
  9. 9. 340
    Multiplying by 10 makes 34 tens, which is 340.
  10. 10. 374
    34 × (10 + 1) = 340 + 34 = 374.
  11. 11. 351
    27 × 10 + 27 × 3 = 270 + 81 = 351.
  12. 12. 672
    42 × 10 + 42 × 6 = 420 + 252 = 672.
  13. 13. 840
    35 × (20 + 4) = 700 + 140 = 840.
  14. 14. 912
    48 × 20 − 48 = 960 − 48 = 912.
  15. 15. 1,054
    62 × 10 + 62 × 7 = 620 + 434 = 1,054.
  16. 16. 2,100
    75 × (4 × 7) = 300 × 7 = 2,100.
  17. 17. 3,000
    24 = 8 × 3, so 125 × 8 × 3 = 1,000 × 3 = 3,000.
  18. 18. 3,663
    100 × 37 − 37 = 3,700 − 37 = 3,663.
  19. 19. 432 notebooks
    24 × 18 = 24 × (20 − 2) = 480 − 48 = 432.
  20. 20. 972 seats
    36 × 27 = 36 × 20 + 36 × 7 = 720 + 252 = 972.
  21. 21. 100 × 9
    36 = 4 × 9, so 25 × 36 = 25 × 4 × 9 = 100 × 9 = 900.
  22. 22. 25
    One more group of 25 turns nine groups into ten groups.
  23. 23. 21
    The first factor is larger by 1, so the products differ by 1 × 21 = 21.
  24. 24. Multiply, for example 38 × 10 + 38 × 4 = 532.
    The problem asks for 14 groups of 38, not the sum of the two factors.
  25. 25. 1,600
    25 × 4 = 100 and 64 ÷ 4 = 16, so 64 × 25 = 16 × 100 = 1,600.
  26. 26. 17
    10 + 7 = 17, so both sides equal 32 × 17.
  27. 27. True
    Halving 18 to 9 while doubling 24 to 48 keeps the product the same: both equal 432.
  28. 28. One factor was halved and the other doubled, so the product stayed 400.
    16 × 25 = 400 and 8 × 50 = 400; the ×2 and ÷2 changes cancel.
  29. 29. 1,080
    72 × (10 + 5) = 720 + 360 = 1,080.
  30. 30. 1,334
    60 × 20 is about 1,200, so 1,334 is plausible. Exact partial products give 1,160 + 174 = 1,334.
  31. 31. 6,000
    48 = 8 × 6, so 125 × 8 × 6 = 1,000 × 6 = 6,000.
  32. 32. 9,000
    250 × 4 = 1,000 and 36 ÷ 4 = 9, so the product is 1,000 × 9 = 9,000.
  33. 33. 2,352
    48 × 50 − 48 = 2,400 − 48 = 2,352.
  34. 34. 6,767
    100 × 67 + 1 × 67 = 6,700 + 67 = 6,767.
  35. 35. 864 m²
    36 × 24 = 36 × 20 + 36 × 4 = 720 + 144 = 864 square metres.
  36. 36. 3,072 tickets
    128 × 24 = 128 × 20 + 128 × 4 = 2,560 + 512 = 3,072.
  37. 37. 92
    2,300 ÷ 25 = 92. Check: 25 × 92 = 25 × (100 − 8) = 2,500 − 200 = 2,300.
  38. 38. 37 × 6
    26 = 20 + 6, so both partial products must use 37.
  39. 39. Yes.
    Dividing 48 by 4 and multiplying 25 by 4 keeps the product unchanged: 12 × 100 = 1,200.
  40. 40. 230
    Substitute 6 for x: 23 × (6 + 4) = 23 × 10 = 230. This previews how a letter can stand for a number.

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