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Math · Number Sense

Multiply a 2-Digit Number by a 1-Digit Number: Area Model and Partial Products

Break a two-digit number into tens and ones, use an area model and partial products, regroup correctly, estimate to check, and solve multiplication word problems.

Grades 4–5 · 28 practice questions · Original material
Learning objectiveMultiply a two-digit whole number by a one-digit number using an area model, partial products and the standard method, estimate to check that the answer is reasonable, and solve one- and two-step word problems.

The idea

Multiplying 47 × 6 is easier when 47 is broken into 40 and 7. Each part is multiplied by 6, and the two partial products are added: 40 × 6 = 240 and 7 × 6 = 42, so 47 × 6 = 240 + 42 = 282.

An area model draws this idea. A rectangle that is 47 long and 6 tall is cut into a 40-by-6 piece and a 7-by-6 piece. The area of the whole rectangle is the sum of the areas of the pieces.

The vertical method records the same two products. You multiply the ones first (7 × 6 = 42), write 2 and regroup 4 tens, then multiply the tens (4 × 6 = 24 tens) and add the regrouped 4 tens: 24 + 4 = 28 tens, so the answer is 282.

Always estimate before you trust an answer. Round 47 to 50, and 50 × 6 = 300. Since 47 is a little less than 50, the exact answer should be a little less than 300. 282 is reasonable; 242 is not.

In a word problem, look for equal groups: 9 boxes with 68 markers each means 9 × 68. Name what the answer counts (markers), and check it against an estimate.

Worked examples

An area model and partial products for 47 times 6Left panel: a rectangle that is 47 long and 6 tall is cut into two parts. The 40 part has area 40 times 6 equals 240. The 7 part has area 7 times 6 equals 42. Adding the parts gives 240 plus 42 equals 282. Right panel: the same product in a vertical layout. 7 times 6 equals 42, then 40 times 6 equals 240, then 42 plus 240 equals 282.Break the two-digit number apart, then add the partsOriginal schematic, drawn to the labelled values (not to scale).1 · Area model: 47 × 6407640 × 6= 2407 × 6= 42240 + 42 = 282So 47 × 6 = 2822 · Partial products47× 6427 × 624040 × 6282Multiply the ones, then the tens,then add the partial products.
Original schematic, drawn to the labelled values. Break the two-digit number into tens and ones, multiply each part by the single digit, then add the partial products: 40 × 6 = 240 and 7 × 6 = 42, so 47 × 6 = 282. The vertical method records the same two products; the carried digit is a shortcut for adding them.

Example 1 · Area model

Find 53 × 4 with an area model.

  1. Break 53 into 50 + 3.
  2. 50 × 4 = 200 and 3 × 4 = 12.
  3. Add the partial products: 200 + 12 = 212.

212

Example 2 · Vertical method with regrouping

Find 36 × 4 with the vertical method.

  1. Ones: 6 × 4 = 24. Write 4 and regroup 2 tens.
  2. Tens: 3 × 4 = 12 tens, and 12 + 2 = 14 tens.
  3. Write 14 tens in front of the 4: 144.

144

Example 3 · Estimate to check

A student says 29 × 5 = 125. Is that reasonable?

  1. Round 29 to 30. 30 × 5 = 150.
  2. 29 is 1 less than 30, so the exact product is 5 less than 150.
  3. 150 − 5 = 145. The answer 125 is too small.

No. 29 × 5 = 145.

Example 4 · Word problem

A school has 9 boxes with 68 markers in each. How many markers is that?

  1. Equal groups: 9 groups of 68, so multiply 68 × 9.
  2. 60 × 9 = 540 and 8 × 9 = 72.
  3. 540 + 72 = 612. Estimate: 70 × 9 = 630, so 612 is reasonable.

612 markers

Common mistakes

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

Forgetting to add the regrouped tens (47 × 6 = 242).

Why this fails: After 7 × 6 = 42 you regroup 4 tens. Add them to the tens product: 24 + 4 = 28 tens. Estimating with 50 × 6 = 300 shows that 242 is too small.

Multiplying the tens digit as if it were ones (40 × 6 = 24).

Why this fails: The 4 in 47 means 4 tens. 4 tens × 6 = 24 tens = 240, so the zero must come back.

Multiplying only the ones, or only the tens.

Why this fails: Both parts of the number are multiplied by the single digit. In an area model, both rectangles are part of the total area.

Adding the partial products in the wrong places.

Why this fails: 240 + 42 = 282, not 24 + 42 = 66. Keep place value: 240 is 24 tens, not 24 ones.

Choosing the wrong operation in a word problem.

Why this fails: Equal groups mean multiply. If each of 4 friends has 75 stickers, the total is 4 × 75, not 4 + 75.

Foundation

Foundation practice

Break numbers apart, multiply tens, and use an area model.

Break apart

1. Which expression shows 47 × 6 broken into tens and ones?

  1. (40 + 6) × 7
  2. (40 × 6) + (7 × 6)
  3. (4 × 6) + (7 × 6)
  4. (40 × 7) + (6 × 6)
Tens times a digit

2. What is 40 × 6?

  1. 24
  2. 46
  3. 2,400
  4. 240
Area model

3. An area model cuts 53 × 4 into 50 × 4 and 3 × 4. What are the two partial products?

  1. 200 and 12
  2. 20 and 12
  3. 200 and 7
  4. 54 and 7
Ones first

4. In 36 × 4, what is the first partial product, the ones times 4?

  1. 12
  2. 144
  3. 24
  4. 10
Estimate

5. About how much is 48 × 5 if you round 48 to the nearest ten?

  1. 200
  2. 250
  3. 25
  4. 500
Product

6. What is 24 × 3?

  1. 62
  2. 60
  3. 18
  4. 72
Product

7. What is 15 × 5?

  1. 75
  2. 55
  3. 50
  4. 30

Practice

Practice practice

Multiply two-digit numbers by one digit, including word problems.

Product

8. What is 36 × 4?

  1. 124
  2. 120
  3. 144
  4. 36
Product

9. What is 47 × 6?

  1. 242
  2. 282
  3. 240
  4. 66
Product

10. What is 58 × 3?

  1. 154
  2. 150
  3. 39
  4. 174
Product

11. What is 29 × 5?

  1. 145
  2. 105
  3. 100
  4. 55
Product

12. What is 63 × 7?

  1. 421
  2. 420
  3. 441
  4. 63
Product

13. What is 84 × 6?

  1. 484
  2. 504
  3. 480
  4. 72
Product

14. What is 92 × 8?

  1. 726
  2. 720
  3. 88
  4. 736
Word problem

15. A school has 9 boxes of markers, and each box holds 68 markers. How many markers are there in all?

  1. 612
  2. 542
  3. 540
  4. 126

Think

Think practice

Find mistakes, estimate, compare and solve two-step problems.

Word problem

16. Four friends each collect 75 stickers. How many stickers do they have together?

  1. 280
  2. 48
  3. 300
  4. 285
Word problem

17. A week has 7 days. A library lends 39 books each day. How many books does it lend in one week?

  1. 213
  2. 273
  3. 210
  4. 84
Error analysis

18. Ravi wrote 47 × 6 = 242. What mistake did he most likely make?

  1. He multiplied 4 × 6 and 7 × 6 but added them
  2. He used 40 × 6 = 24
  3. He divided instead of multiplying
  4. He forgot to add the 4 tens carried from 7 × 6 = 42
Reasonable?

19. Mia says 47 × 6 = 242. Use 50 × 6 to decide if 242 is reasonable.

  1. No. 50 × 6 = 300, and 47 is only a little less than 50, so the answer should be close to 280
  2. Yes, because 242 is less than 300
  3. Yes, because 4 × 6 = 24
  4. No, because the answer must end in 6
Compare

20. Which is greater, 26 × 4 or 25 × 5?

  1. 26 × 4
  2. They are equal
  3. 25 × 5
  4. Not enough information
Equal expressions

21. Which expression has the same value as 38 × 5?

  1. (30 + 5) × (8 + 5)
  2. (30 × 5) + (8 × 5)
  3. (3 × 5) + (8 × 5)
  4. 30 × 8 × 5

Challenge

Challenge practice

Missing digits and reasoning about products.

Missing digit

22. In 4 ■ × 5 = 235, what is the missing digit?

  1. 6
  2. 8
  3. 5
  4. 7
Two-step

23. Jo buys 4 packs of 56 hockey cards. How many cards is that?

  1. 224
  2. 204
  3. 200
  4. 44
Two-step

24. A bus holds 48 people. Six buses are full. Then 15 more people come on a seventh bus. How many people ride in all?

  1. 288
  2. 333
  3. 303
  4. 63
Create

25. Which multiplication has a product closest to 400?

  1. 62 × 6
  2. 79 × 5
  3. 48 × 7
  4. 91 × 4
Missing digit

26. In 6 ■ × 4 = 2■■, the product is 272 when the missing tens digit is…

  1. 7
  2. 6
  3. 9
  4. 8
Reasoning

27. A student says 90 × 6 and 9 × 60 give the same answer. Is the student right?

  1. Yes. Both equal 540 because each is 9 × 6 with one zero
  2. No, because 90 is greater than 9
  3. No, because 6 is less than 60
  4. Only if you add the zeros
Create

28. Which two numbers could be multiplied to make 336 (use 6 as the one-digit factor)?

  1. 46
  2. 66
  3. 56
  4. 96

Answer key

  1. 1. (40 × 6) + (7 × 6)
    47 is 40 + 7, so multiply each part by 6 and add: (40 × 6) + (7 × 6). Using 4 × 6 loses the zero because 4 here means 4 tens. The other choices change the numbers being multiplied.
  2. 2. 240
    4 tens × 6 = 24 tens = 240. 24 forgets that the 4 is tens. 46 adds instead of multiplying. 2,400 gives one zero too many.
  3. 3. 200 and 12
    50 × 4 = 200 and 3 × 4 = 12, and 200 + 12 = 212. Using 20 mixes up 5 × 4 and 50 × 4. 7 and 54 come from adding.
  4. 4. 24
    6 × 4 = 24 for the ones. 12 is 3 × 4, the tens digit without its place value. 144 is the final answer. 10 is 6 + 4.
  5. 5. 250
    48 rounds to 50 and 50 × 5 = 250. 200 would round down to 40. 25 forgets the zero. 500 multiplies by 10 by mistake.
  6. 6. 72
    20 × 3 = 60 and 4 × 3 = 12, so 60 + 12 = 72. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 6 and 12 without placing the tens correctly.
  7. 7. 75
    10 × 5 = 50 and 5 × 5 = 25, so 50 + 25 = 75. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 5 and 25 without placing the tens correctly.
  8. 8. 144
    30 × 4 = 120 and 6 × 4 = 24, so 120 + 24 = 144. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 12 and 24 without placing the tens correctly.
  9. 9. 282
    40 × 6 = 240 and 7 × 6 = 42, so 240 + 42 = 282. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 24 and 42 without placing the tens correctly.
  10. 10. 174
    50 × 3 = 150 and 8 × 3 = 24, so 150 + 24 = 174. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 15 and 24 without placing the tens correctly.
  11. 11. 145
    20 × 5 = 100 and 9 × 5 = 45, so 100 + 45 = 145. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 10 and 45 without placing the tens correctly.
  12. 12. 441
    60 × 7 = 420 and 3 × 7 = 21, so 420 + 21 = 441. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 42 and 21 without placing the tens correctly.
  13. 13. 504
    80 × 6 = 480 and 4 × 6 = 24, so 480 + 24 = 504. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 48 and 24 without placing the tens correctly.
  14. 14. 736
    90 × 8 = 720 and 2 × 8 = 16, so 720 + 16 = 736. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 72 and 16 without placing the tens correctly.
  15. 15. 612
    60 × 9 = 540 and 8 × 9 = 72, so 540 + 72 = 612. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 54 and 72 without placing the tens correctly.
  16. 16. 300
    70 × 4 = 280 and 5 × 4 = 20, so 280 + 20 = 300. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 28 and 20 without placing the tens correctly.
  17. 17. 273
    30 × 7 = 210 and 9 × 7 = 63, so 210 + 63 = 273. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 21 and 63 without placing the tens correctly.
  18. 18. He forgot to add the 4 tens carried from 7 × 6 = 42
    7 × 6 = 42, so write 2 and carry 4 tens. Ravi then wrote 4 × 6 = 24 without adding the carried 4 (24 + 4 = 28), giving 242 instead of 282. Adding 24 and 42 would give 66.
  19. 19. No. 50 × 6 = 300, and 47 is only a little less than 50, so the answer should be close to 280
    47 is 3 less than 50, so the product is 3 × 6 = 18 less than 300, which is 282. 242 is about 40 too small. A product does not have to end in 6.
  20. 20. 25 × 5
    26 × 4 = 104 and 25 × 5 = 125. The larger first number does not decide it; the extra group of 25 matters more than the extra 1.
  21. 21. (30 × 5) + (8 × 5)
    38 = 30 + 8, so 38 × 5 = (30 × 5) + (8 × 5) = 150 + 40 = 190. Using 3 × 5 drops the place value, and the other choices change the operation.
  22. 22. 7
    235 ÷ 5 = 47, so the missing digit is 7. Check: 47 × 5 = 200 + 35 = 235. 6 gives 230 and 8 gives 240.
  23. 23. 224
    50 × 4 = 200 and 6 × 4 = 24, so 200 + 24 = 224. Wrong answers come from forgetting the regrouped tens, multiplying only the tens, or adding 20 and 24 without placing the tens correctly.
  24. 24. 303
    6 × 48 = 288, and 288 + 15 = 303. 288 forgets the extra 15. 333 adds 15 twice by mistake. 63 adds 48 and 15.
  25. 25. 79 × 5
    79 × 5 = 395. The others are 372, 336 and 364. Closest means the smallest difference from 400.
  26. 26. 8
    68 × 4 = 272 (60 × 4 = 240 and 8 × 4 = 32). 67 × 4 = 268 is too small. 69 × 4 = 276 is too big.
  27. 27. Yes. Both equal 540 because each is 9 × 6 with one zero
    9 × 6 = 54, and each expression has one zero in total, so both are 540. Multiplication can be turned around, and the zero stays with the tens.
  28. 28. 56
    56 × 6 = 336 (300 + 36). 46 × 6 = 276, 66 × 6 = 396 and 96 × 6 = 576.

Related practice

Multiplication and division facts: arrays and fact familiesPlace value to 100,000Estimate sums and differencesFactors and multiples