Study CommonsRead · reason · practise

Home / Math / Multiply 2-Digit Numbers: Area Model and Four Partial Products

Math · Number Sense

Multiply 2-Digit Numbers: Area Model and Four Partial Products

Multiply a two-digit number by a two-digit number with an area model, add the four partial products, estimate to check, and solve word problems.

Grades 4–5 · 36 practice questions · Original material
Learning objectiveMultiply two two-digit whole numbers by breaking both into tens and ones, finding the four partial products in an area model, adding them, checking with an estimate, and solving word problems.

The idea

On November 9 you multiplied a two-digit number by one digit with two boxes. When both numbers have two digits, each number splits into tens and ones, so the rectangle has four boxes.

To find 23 × 14, write 23 = 20 + 3 and 14 = 10 + 4. The four boxes are 20 × 10 = 200, 20 × 4 = 80, 3 × 10 = 30 and 3 × 4 = 12. Every box is part of the rectangle, so you add all four: 200 + 80 + 30 + 12 = 322.

Multiplying tens is a place-value idea: 2 tens × 1 ten = 2 hundreds = 200. A quick estimate, 20 × 10 = 200 or better 20 × 15 = 300, tells you whether 322 is the right size.

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 · Four boxes

Find 23 × 14 with an area model.

  1. Break both numbers apart: 23 = 20 + 3 and 14 = 10 + 4.
  2. Multiply to fill the boxes: 20 × 10 = 200, 20 × 4 = 80, 3 × 10 = 30, 3 × 4 = 12.
  3. Add the partial products: 200 + 80 = 280, 280 + 30 = 310, 310 + 12 = 322.

322

Example 2 · Tens times tens

Find 30 × 40, then 35 × 12.

  1. 3 tens × 4 tens = 12 hundreds, so 30 × 40 = 1,200.
  2. For 35 × 12: 30 × 10 = 300, 30 × 2 = 60, 5 × 10 = 50, 5 × 2 = 10.
  3. 300 + 60 + 50 + 10 = 420.

30 × 40 = 1,200 and 35 × 12 = 420.

Example 3 · Estimate to check

Estimate 38 × 21, then find the exact product.

  1. 38 is about 40 and 21 is about 20, so the estimate is 40 × 20 = 800.
  2. Exact: 30 × 20 = 600, 30 × 1 = 30, 8 × 20 = 160, 8 × 1 = 8.
  3. 600 + 30 + 160 + 8 = 798, which is close to 800.

798, close to the estimate of 800.

Example 4 · Money problem

Tickets cost $22 each and a class buys 15. What is the total in dollars?

  1. Equal groups of $22 mean multiply: 22 × 15.
  2. 20 × 10 = 200, 20 × 5 = 100, 2 × 10 = 20, 2 × 5 = 10.
  3. 200 + 100 + 20 + 10 = 330.

$330

Example 5 · A tempting wrong answer

A student says 24 × 13 = 200 + 12 = 212. What went wrong?

  1. 200 is 20 × 10 and 12 is 4 × 3, the two corner boxes.
  2. The middle boxes 20 × 3 = 60 and 4 × 10 = 40 were left out.
  3. Correct total: 200 + 60 + 40 + 12 = 312. The estimate 25 × 12 = 300 also shows 212 is too small.

24 × 13 = 312; the student skipped both middle boxes.

Common mistakes

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

Adding only the two corner boxes (tens × tens and ones × ones).

Why this fails: Each part of one number must multiply each part of the other. That makes four boxes, and the two middle boxes cannot be skipped.

Losing zeros when multiplying tens, such as 30 × 40 = 120.

Why this fails: 3 × 4 = 12 counts hundreds here, because tens × tens is hundreds. 30 × 40 = 1,200.

Writing 20 × 3 as 6 instead of 60.

Why this fails: The 2 in 23 and 24 means 2 tens, so 20 × 3 is 6 tens, or 60.

Multiplying the numbers only in the tens places and ones places and adding the result.

Why this fails: That ignores the cross products. Check with an estimate: a product far from 20 × 10 = 200 or 30 × 20 = 600 is not reasonable.

Using the wrong operation in a word problem, such as adding 22 + 15 for 15 tickets at $22.

Why this fails: Equal groups call for multiplication. 15 groups of $22 is 22 × 15 = $330.

Foundation

Foundation practice

Break two-digit numbers into tens and ones and find the four partial products.

Break apart

1. Which list shows all four partial products for 23 × 14, using 23 = 20 + 3 and 14 = 10 + 4?

  1. 20 × 10, 20 × 4, 3 × 10, 3 × 4
  2. 20 × 10 and 3 × 4
  3. 20 + 10, 20 + 4, 3 + 10, 3 + 4
  4. 23 × 10 and 14 × 4
Tens times tens

2. What is 20 × 10?

  1. 20
  2. 200
  3. 2,000
  4. 30
Tens times tens

3. What is 30 × 40?

  1. 120
  2. 12,000
  3. 1,200
  4. 70
Partial product

4. In the area model for 23 × 14, what is the 20 × 4 box?

  1. 8
  2. 800
  3. 24
  4. 80
Add the boxes

5. An area model for 12 × 13 has boxes 100, 30, 20 and 6. What is 12 × 13?

  1. 156
  2. 126
  3. 136
  4. 150
Estimate

6. Which estimate is closest to 28 × 31?

  1. 600
  2. 900
  3. 90
  4. 1,200
Product

7. What is 21 × 13?

  1. 203
  2. 219
  3. 273
  4. 213
True or false

8. True or false: 14 × 12 = (14 × 10) + (14 × 2).

  1. True
  2. False
Multiples of ten

9. What is 15 × 20?

  1. 300
  2. 30
  3. 3,000
  4. 35

Practice

Practice practice

Multiply two-digit numbers with an area model and add the partial products.

Product

10. What is 24 × 13?

  1. 212
  2. 312
  3. 252
  4. 272
Product

11. What is 35 × 12?

  1. 310
  2. 360
  3. 420
  4. 370
Product

12. What is 53 × 14?

  1. 512
  2. 542
  3. 712
  4. 742
Product

13. What is 18 × 15?

  1. 270
  2. 140
  3. 220
  4. 190
Partial product

14. In 46 × 12, what is the partial product 6 × 10?

  1. 6
  2. 60
  3. 600
  4. 16
Multiples of ten

15. What is 50 × 20?

  1. 100
  2. 10,000
  3. 1,000
  4. 70
Missing box

16. The area model for 32 × 14 has boxes 300, 120 and 8. Which box is missing?

  1. 2
  2. 200
  3. 12
  4. 20
Product

17. What is 62 × 11?

  1. 682
  2. 602
  3. 622
  4. 662
Product

18. What is 27 × 30?

  1. 81
  2. 810
  3. 8,100
  4. 57
Area

19. A rectangle is 24 cm long and 15 cm wide. What is its area?

  1. 220 cm²
  2. 260 cm²
  3. 360 cm²
  4. 320 cm²

Think

Think practice

Word problems, error analysis, estimates and comparisons.

Word problem

20. A theatre has 24 rows with 18 seats in each row. How many seats are there?

  1. 42
  2. 332
  3. 392
  4. 432
Word problem

21. Tickets for a school play cost $22 each. A class buys 15 tickets. How much do they pay?

  1. $330
  2. $37
  3. $300
  4. $320
Error analysis

22. A student writes 23 × 14 = 200 + 12 = 212. Which boxes did the student forget?

  1. 20 × 3 = 60 and 4 × 10 = 40
  2. 20 × 4 = 80 and 3 × 10 = 30
  3. Nothing; the work is correct
  4. Only 3 × 4 = 12
Estimate

23. Which is closest to 38 × 21?

  1. 600
  2. 80
  3. 800
  4. 1,200
Compare

24. Which is greater: 26 × 14 or 24 × 16?

  1. 26 × 14 is greater
  2. They are equal
  3. You cannot tell without a calculator
  4. 24 × 16 is greater
Word problem

25. A store packs 48 pens in each case. It fills 12 cases. How many pens is that?

  1. 576
  2. 60
  3. 536
  4. 96
Equal expressions

26. Which expression equals 17 × 12?

  1. (17 × 10) + 2
  2. (17 × 10) + (17 × 2)
  3. (10 × 12) + (7 × 2)
  4. 17 + 12
Word problem

27. A teacher puts 16 marbles in each of 25 bags. How many marbles are there?

  1. 230
  2. 280
  3. 400
  4. 350
Two-step

28. A class buys 24 packs of 12 pencils and then gives away 18 pencils. How many pencils are left?

  1. 306
  2. 288
  3. 194
  4. 270

Challenge

Challenge practice

Missing digits, greatest products and multi-step reasoning.

Missing digit

29. In 3■ × 12 = 432, what digit goes in the box?

  1. 6
  2. 2
  3. 4
  4. 5
Greatest product

30. Use each digit 2, 3, 4 and 5 once to make two 2-digit numbers. Which pair has the greatest product?

  1. 54 × 32
  2. 52 × 43
  3. 53 × 42
  4. 54 × 23
Add the boxes

31. An area model of 46 × 23 has boxes 800, 120, 120 and 18. What is 46 × 23?

  1. 938
  2. 920
  3. 1,058
  4. 1,178
Work backward

32. 15 × ? = 240. What number goes in the box?

  1. 24
  2. 15
  3. 26
  4. 16
Error analysis

33. A student writes 35 × 12 = (35 × 10) + 2 = 352. What is the correct product?

  1. 420
  2. 352
  3. 370
  4. 350
Halve and double

34. Which product equals 25 × 16?

  1. 25 × 8
  2. 50 × 8
  3. 50 × 16
  4. 12 × 32
Word problem

35. A school orders 32 boxes of crayons at $14 a box. What is the total cost?

  1. $428
  2. $328
  3. $448
  4. $46
Area

36. A square garden is 35 m on each side. What is its area?

  1. 1,075 m²
  2. 925 m²
  3. 140 m²
  4. 1,225 m²

Answer key

  1. 1. 20 × 10, 20 × 4, 3 × 10, 3 × 4
    Two numbers with two parts each make a 2 by 2 area model, so there are 2 × 2 = 4 boxes: 200, 80, 30 and 12 (total 322). 20 × 10 and 3 × 4: skips the two middle boxes. 20 + 10, 20 + 4, 3 + 10, 3 + 4: adds instead of multiplying. 23 × 10 and 14 × 4: does not break either number into tens and ones.
  2. 2. 200
    2 tens × 1 ten = 2 hundreds, so 20 × 10 = 200. 20: forgets the zero in 10. 2,000: adds one zero too many. 30: adds instead of multiplying.
  3. 3. 1,200
    3 tens × 4 tens = 12 hundreds, so 30 × 40 = 1,200 (3 × 4 = 12, then two zeros). 120: drops one zero: 3 × 4 = 12 is the number of hundreds, not the answer. 12,000: uses three zeros instead of two. 70: adds instead of multiplying.
  4. 4. 80
    20 × 4 = 80 because 2 tens × 4 = 8 tens. 8: treats the 2 as 2 ones instead of 2 tens. 800: treats the 4 as 4 tens. 24: adds the digits instead of multiplying.
  5. 5. 156
    100 + 30 + 20 + 6 = 156. Every box is part of the rectangle, so all four are added. 126: leaves out the 30 box. 136: leaves out the 20 box. 150: leaves out the 6 box.
  6. 6. 900
    Round to the nearest ten: 28 is about 30 and 31 is about 30, and 30 × 30 = 900 (the exact answer is 868). 600: rounds 31 down to 20 instead of 30. 90: loses a zero; 30 × 30 is 900, not 90. 1,200: rounds 28 up to 40 instead of 30.
  7. 7. 273
    20 × 10 = 200, 20 × 3 = 60, 1 × 10 = 10 and 1 × 3 = 3, and 200 + 60 + 10 + 3 = 273. 203: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 219: misreads 20 × 3 as 6 instead of 60. 213: leaves out the 60 box (20 × 3).
  8. 8. True
    12 = 10 + 2, so 14 groups of 12 are 14 × 10 = 140 plus 14 × 2 = 28, and 140 + 28 = 168 = 14 × 12. Choosing False would say you cannot split 12 into 10 + 2, but you can; 12 is 10 + 2.
  9. 9. 300
    15 × 2 = 30, so 15 × 20 = 30 tens = 300. 30: forgets the zero in 20. 3,000: adds one zero too many. 35: adds instead of multiplying.
  10. 10. 312
    200 + 60 + 40 + 12 = 312, so 24 × 13 = 312. 212: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 252: leaves out the 60 box (20 × 3). 272: leaves out the 40 box (4 × 10).
  11. 11. 420
    300 + 60 + 50 + 10 = 420, so 35 × 12 = 420. 310: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 360: leaves out the 60 box (30 × 2). 370: leaves out the 50 box (5 × 10).
  12. 12. 742
    500 + 200 + 30 + 12 = 742, so 53 × 14 = 742. 512: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 542: leaves out the 200 box (50 × 4). 712: leaves out the 30 box (3 × 10).
  13. 13. 270
    100 + 50 + 80 + 40 = 270, so 18 × 15 = 270. 140: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 220: leaves out the 50 box (10 × 5). 190: leaves out the 80 box (8 × 10).
  14. 14. 60
    6 × 10 = 60. The other boxes are 40 × 10 = 400, 40 × 2 = 80 and 6 × 2 = 12, for 552 in all. 6: ignores the 10. 600: adds one zero too many. 16: adds instead of multiplying.
  15. 15. 1,000
    5 tens × 2 tens = 10 hundreds, so 50 × 20 = 1,000. 100: drops a zero: 5 × 2 = 10 is the number of hundreds. 10,000: uses three zeros instead of two. 70: adds instead of multiplying.
  16. 16. 20
    The missing box is 2 × 10 = 20. The total is 300 + 120 + 20 + 8 = 448. 2: forgets the 10 in 14. 200: uses 2 × 100 instead of 2 × 10. 12: adds 2 + 10 instead of multiplying.
  17. 17. 682
    600 + 60 + 20 + 2 = 682, so 62 × 11 = 682. 602: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 622: leaves out the 60 box (60 × 1). 662: leaves out the 20 box (2 × 10).
  18. 18. 810
    27 × 3 = 81, so 27 × 30 = 81 tens = 810. 81: forgets the zero in 30. 8,100: adds one zero too many. 57: adds instead of multiplying.
  19. 19. 360 cm²
    Area is length × width: 200 + 100 + 40 + 20 = 360 cm². Area is measured in square centimetres. 220 cm²: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 260 cm²: leaves out the 100 box (20 × 5). 320 cm²: leaves out the 40 box (4 × 10).
  20. 20. 432
    24 × 18: 200 + 160 + 40 + 32 = 432 seats. 42: adds the numbers; equal rows of seats call for multiplication. 332: is 100 too small, as if 20 × 10 were only 100. 392: leaves out the 40 box (4 × 10).
  21. 21. $330
    22 × 15: 20 × 15 = 300 and 2 × 15 = 30, so the total is $330. $37: adds the price and the number of tickets. $300: uses 20 × 15 and ignores the extra $2 on each ticket. $320: adds only 2 × 10 = 20 for the extra $2 instead of 2 × 15 = 30.
  22. 22. 20 × 4 = 80 and 3 × 10 = 30
    The student added only the corner boxes (20 × 10 and 3 × 4). The two middle boxes, 80 and 30, make 200 + 80 + 30 + 12 = 322. 20 × 3 = 60 and 4 × 10 = 40: are not the missing boxes; 20 × 3 and 4 × 10 do not appear in 23 × 14. Nothing; the work is correct: is wrong because 23 × 14 is 322, not 212. Only 3 × 4 = 12: is wrong because the 12 was already included.
  23. 23. 800
    38 is about 40 and 21 is about 20, so 40 × 20 = 800. The exact product is 798. 600: rounds 21 up to 30. 80: loses a zero; 40 × 20 = 800. 1,200: rounds 38 up to 60.
  24. 24. 24 × 16 is greater
    26 × 14 = 260 + 104 = 364 and 24 × 16 = 240 + 144 = 384, so 24 × 16 is greater. 26 × 14 is greater: is wrong because 26 × 14 = 364, which is less than 24 × 16 = 384. They are equal: is wrong because 364 and 384 are different. You cannot tell without a calculator: is wrong because two quick area models are enough to compare them.
  25. 25. 576
    48 × 12 = 480 + 96 = 576 pens (40 × 10 = 400, 40 × 2 = 80, 8 × 10 = 80, 8 × 2 = 16). 60: adds 48 + 12. 536: is 40 too small, as if 8 × 10 were only 40. 96: counts only the 8 × 12 part.
  26. 26. (17 × 10) + (17 × 2)
    12 = 10 + 2, so 17 × 12 = (17 × 10) + (17 × 2) = 170 + 34 = 204. (17 × 10) + 2: multiplies 17 by 10 but only adds 2 for the other part. (10 × 12) + (7 × 2): uses only the tens × tens and ones × ones boxes. 17 + 12: adds instead of multiplying.
  27. 27. 400
    25 × 16: 200 + 120 + 50 + 30 = 400 marbles. 230: adds only the tens × tens and ones × ones boxes and skips both middle boxes. 280: leaves out the 120 box (20 × 6). 350: leaves out the 50 box (5 × 10).
  28. 28. 270
    First 24 × 12 = 200 + 40 + 40 + 8 = 288 pencils. Then 288 − 18 = 270. 306: adds 18 instead of subtracting it. 288: forgets the second step of giving pencils away. 194: gets 24 × 12 wrong as 212 (skipping the middle boxes) before subtracting.
  29. 29. 6
    36 × 12 = 360 + 72 = 432. The other digits give 384, 408 and 420. 2: gives 32 × 12 = 384, which is too small. 4: gives 34 × 12 = 408, which is too small. 5: gives 35 × 12 = 420, which is too small.
  30. 30. 52 × 43
    52 × 43 = 2,236, the greatest. Put the largest digits in the tens places, then give the larger tens digit the smaller ones digit. 54 × 32: gives 1,728, because putting the two largest digits together in one number leaves the other number small. 53 × 42: gives 2,226, which is close but 10 less than 52 × 43. 54 × 23: gives 1,242, because it pairs the biggest tens digit with the smallest ones digits.
  31. 31. 1,058
    800 + 120 + 120 + 18 = 1,058, so 46 × 23 = 1,058. 938: leaves out one 120 box. 920: adds only 800 + 120. 1,178: counts the 120 box three times.
  32. 32. 16
    15 × 16 = 150 + 90 = 240, so the missing factor is 16 (also 240 ÷ 15 = 16). 24: comes from 240 ÷ 10 and ignores the 15. 15: repeats the known factor; 15 × 15 = 225. 26: is a guess; 15 × 26 = 390.
  33. 33. 420
    12 = 10 + 2, so both parts multiply 35: 350 + 70 = 420. 352: repeats the mistake: the 2 should be 35 × 2 = 70, not just 2. 370: adds 350 + 20, using 10 × 2 instead of 35 × 2. 350: ignores the 2 in 12 completely.
  34. 34. 50 × 8
    Halving 16 gives 8 and doubling 25 gives 50, so 25 × 16 = 50 × 8 = 400. 25 × 8: halves only one factor. 50 × 16: doubles only one factor. 12 × 32: halves 25 to 12, which is not exactly half.
  35. 35. $448
    30 × 10 = 300, 30 × 4 = 120, 2 × 10 = 20 and 2 × 4 = 8, so the total is $448. $428: leaves out the 2 × 10 = 20 box. $328: leaves out the 30 × 4 = 120 box. $46: adds 32 + 14.
  36. 36. 1,225 m²
    35 × 35 = 900 + 150 + 150 + 25 = 1,225 m². The perimeter 140 m is a different measure. 1,075 m²: leaves out one 150 box. 925 m²: adds only the corner boxes 900 + 25. 140 m²: finds the perimeter, 4 × 35, instead of the area.

Related practice

Multiply a 2-digit number by 1 digit: area modelDivide a 2-digit number by 1 digit: area modelPlace value to 100,000: expanded formEstimate sums and differences