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Math · Number Sense

Divide a 2-Digit Number by 1 Digit: Area Model, Partial Quotients

Break a two-digit number into friendly parts, divide by one digit with an area model and partial quotients, check by multiplying, and solve word problems.

Grades 4–5 · 28 practice questions · Original material
Learning objectiveDivide a two-digit whole number by a one-digit number using an area model and partial quotients, check the quotient by multiplying, handle a small remainder, and solve one- and two-step division word problems.

The idea

Dividing 78 ÷ 6 means finding how many groups of 6 fit into 78. You do not have to find them all at once. Take out a big, friendly chunk first: 10 groups of 6 is 60. That leaves 78 − 60 = 18, and 18 ÷ 6 = 3 more groups. The two partial quotients are added: 10 + 3 = 13.

An area model shows this as a rectangle. The height is the divisor (6) and the area is the number being divided (78). The width is the unknown quotient. The rectangle is cut into a 60 piece with width 10 and an 18 piece with width 3, so the whole width is 10 + 3 = 13.

Division undoes multiplication, so you can always check: quotient × divisor should equal the number you started with. 13 × 6 = 60 + 18 = 78, so 13 is correct.

Estimate first. 78 is close to 80, and 80 ÷ 6 is a little more than 13, so an answer near 13 is reasonable; an answer such as 103 is not. Partial quotients are added, never written side by side.

Sometimes the last piece does not divide evenly. For 83 ÷ 4, take 20 groups (80) and 3 are left, so 83 ÷ 4 = 20 R3. A remainder must always be smaller than the divisor. What the remainder means depends on the story: you may share it, ignore it, or round the number of groups up.

Worked examples

An area model and partial quotients for 78 divided by 6Left panel: a rectangle with height 6 and total area 78 is cut into two pieces. The first piece has area 60, so its width is 60 divided by 6 equals 10. The second piece has area 18, so its width is 18 divided by 6 equals 3. The whole width is 10 plus 3 equals 13. Right panel: the same division as partial quotients. 78 minus 60 leaves 18, which is 10 groups of 6. 18 minus 18 leaves 0, which is 3 groups of 6. Adding 10 and 3 gives the quotient 13. Check: 13 times 6 equals 78.Take friendly chunks out, then add the groupsOriginal schematic, drawn to the labelled values (not to scale).1 · Area model: 78 ÷ 6width = 60 ÷ 6 = 1018 ÷ 6 = 36area 60area 1810 groups of 63 groupsWhole width: 10 + 3 = 13So 78 ÷ 6 = 13Check: 13 × 6 = 60 + 18 = 782 · Partial quotients78start− 6010 groups of 618left− 183 groups of 60left: done10 + 3 = 13 groupsAdd the groups; do not write them side by side.
Original schematic, drawn to the labelled values. The divisor is the height and the number being divided is the area, so the width is the quotient. Take out friendly chunks (60 is 10 groups of 6), see what is left, and keep going until what is left is smaller than the divisor. Add the partial quotients: 10 + 3 = 13. Check by multiplying: 13 × 6 = 78.

Example 1 · Area model

Find 84 ÷ 4 with an area model.

  1. Break 84 into 80 + 4, because both parts divide evenly by 4.
  2. 80 ÷ 4 = 20 and 4 ÷ 4 = 1.
  3. Add the partial quotients: 20 + 1 = 21.

21

Example 2 · Partial quotients

Find 78 ÷ 6 with partial quotients.

  1. Take 10 groups of 6: 10 × 6 = 60. Left: 78 − 60 = 18.
  2. Take 3 groups of 6: 3 × 6 = 18. Left: 0.
  3. Add the groups: 10 + 3 = 13.

13

Example 3 · Check and remainder

Find 83 ÷ 4 and check it.

  1. Take 20 groups of 4: 20 × 4 = 80. Left: 83 − 80 = 3.
  2. 3 is less than 4, so no more groups fit. 83 ÷ 4 = 20 R3.
  3. Check: 20 × 4 = 80, and 80 + 3 = 83.

20 R3

Example 4 · Word problem

58 students go on a trip. Each van holds 6 students. How many vans are needed?

  1. 58 ÷ 6 = 9 R4: nine full vans carry 54 students.
  2. 4 students are left over and still need a ride.
  3. Round up: 9 + 1 = 10 vans.

10 vans

Common mistakes

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

Writing the partial quotients side by side (78 ÷ 6 = 103).

Why this fails: The partial quotients 10 and 3 are counts of groups, so they are added: 10 + 3 = 13. An estimate of 80 ÷ 6 shows that 103 is far too big.

Dividing the tens digit as if it were ones (80 ÷ 4 = 2).

Why this fails: The 8 in 84 means 8 tens. 8 tens ÷ 4 = 2 tens = 20, so the zero must come back.

Stopping after the first partial quotient (85 ÷ 5 = 10).

Why this fails: After taking 50, 35 is still left. 35 ÷ 5 = 7 more groups, so 85 ÷ 5 = 10 + 7 = 17. Keep going until what is left is smaller than the divisor.

Leaving a remainder that is too big (83 ÷ 4 = 19 R7).

Why this fails: A remainder must be less than the divisor. If 7 are left and groups of 4 are being made, one more group fits: 19 R7 becomes 20 R3.

Choosing the wrong operation or ignoring the remainder in a word problem.

Why this fails: Equal groups mean divide. Then read the story: 9 R4 vans means 10 vans are needed, but 9 R4 sticker packs may only be 9 full packs.

Foundation

Foundation practice

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

Break apart

1. Which expression shows 84 ÷ 4 broken into tens and ones that divide evenly by 4?

  1. (8 ÷ 4) + (4 ÷ 4)
  2. (80 − 4) ÷ 4
  3. (80 ÷ 4) + (4 ÷ 4)
  4. (84 ÷ 8) + (84 ÷ 4)
Tens divided

2. What is 80 ÷ 4?

  1. 20
  2. 2
  3. 200
  4. 76
Area model

3. An area model has height 3 and total area 63. It is cut into a 60 part and a 3 part. What are the widths of the two pieces?

  1. 2 and 1
  2. 20 and 3
  3. 21 and 3
  4. 20 and 1
Division fact

4. What is 48 ÷ 6?

  1. 9
  2. 8
  3. 7
  4. 42
Estimate

5. About how much is 83 ÷ 4? Use the nearby multiple 80 ÷ 4.

  1. 2
  2. 20
  3. 200
  4. 40
Leftover

6. To find 72 ÷ 5, start by taking 10 groups of 5, which is 50. How much is left to divide?

  1. 20
  2. 67
  3. 12
  4. 22
Check

7. Which multiplication checks the answer 91 ÷ 7 = 13?

  1. 13 × 7 = 91
  2. 13 + 7 = 91
  3. 91 × 7 = 13
  4. 91 × 13 = 7

Practice

Practice practice

Divide two-digit numbers by one digit with partial quotients.

Quotient

8. What is 96 ÷ 4?

  1. 20
  2. 204
  3. 24
  4. 23
Partial quotients

9. In 78 ÷ 6, you take 10 groups of 6 first, which is 60. How much is left to share?

  1. 72
  2. 12
  3. 68
  4. 18
Quotient

10. What is 85 ÷ 5?

  1. 107
  2. 17
  3. 15
  4. 10
Area model

11. A rectangle has area 92 and height 4. What is its width?

  1. 23
  2. 203
  3. 20
  4. 32
Quotient

12. What is 68 ÷ 4?

  1. 107
  2. 16
  3. 17
  4. 7
Area model

13. An area model for 54 ÷ 3 is cut into areas 30 and 24. What are the two widths?

  1. 10 and 24
  2. 1 and 8
  3. 10 and 8
  4. 30 and 24
Quotient

14. What is 84 ÷ 7?

  1. 12
  2. 102
  3. 11
  4. 10
Quotient

15. What is 75 ÷ 3?

  1. 205
  2. 25
  3. 20
  4. 15

Think

Think practice

Word problems, errors, estimates and comparisons.

Word problem

16. A bakery packs 72 muffins into boxes of 6. How many boxes does it fill?

  1. 66
  2. 13
  3. 78
  4. 12
Word problem

17. 84 stickers are shared equally among 6 friends. How many stickers does each friend get?

  1. 14
  2. 504
  3. 13
  4. 78
Error analysis

18. Sam wrote 91 ÷ 7 = 103. What did Sam most likely do?

  1. He added 91 and 7
  2. He divided only the 9 tens
  3. He multiplied 91 by 7
  4. He wrote the partial quotients 10 and 3 side by side instead of adding them
Reasonable?

19. Mia says 84 ÷ 4 = 201. Use 80 ÷ 4 to decide if that is reasonable.

  1. Yes, because 201 is bigger than 84
  2. No. 80 ÷ 4 = 20, so the answer should be close to 20, not 201
  3. Yes, because 4 × 201 ends in 4
  4. No, because the answer must be less than 4
Compare

20. Which is greater, 72 ÷ 4 or 72 ÷ 6?

  1. 72 ÷ 6
  2. They are equal
  3. 72 ÷ 4
  4. Not enough information
Equal expressions

21. Which expression has the same value as 96 ÷ 3?

  1. (9 ÷ 3) + (6 ÷ 3)
  2. (90 ÷ 3) + (6 ÷ 3)
  3. (96 ÷ 6) + (96 ÷ 3)
  4. (90 − 3) ÷ 6

Challenge

Challenge practice

Missing digits, remainders and multi-step reasoning.

Missing digit

22. In 8■ ÷ 4 = 21, what is the missing digit?

  1. 0
  2. 2
  3. 4
  4. 8
Remainder

23. What is 83 ÷ 4?

  1. 19 R7
  2. 2 R3
  3. 23
  4. 20 R3
Round up

24. 58 students go on a trip. Each van holds 6 students. How many vans are needed?

  1. 10
  2. 9
  3. 9 R4
  4. 52
Two-step

25. Team A has 48 cards and Team B has 36 cards. They put all the cards into packs of 6. How many packs is that?

  1. 54
  2. 44
  3. 84
  4. 14
Work backward

26. A number divided by 6 is 15. What is the number?

  1. 90
  2. 21
  3. 9
  4. 60
Greatest quotient

27. Which division has the greatest quotient?

  1. 90 ÷ 6
  2. 96 ÷ 4
  3. 72 ÷ 2
  4. 84 ÷ 3
Reasoning

28. Sara splits 96 ÷ 6 into 60 ÷ 6 and 36 ÷ 6. Is that a good split, and what is the answer?

  1. Yes. 10 + 6 = 106
  2. Yes. 10 + 6 = 16
  3. No. 96 can only be split as 90 + 6
  4. No. Division cannot be split into parts

Answer key

  1. 1. (80 ÷ 4) + (4 ÷ 4)
    84 is 80 + 4, and both 80 and 4 are multiples of 4, so 84 ÷ 4 = (80 ÷ 4) + (4 ÷ 4) = 20 + 1 = 21. Using 8 ÷ 4 loses the place value because the 8 means 8 tens. Subtracting 4 changes the number. Dividing 84 by 8 as well as by 4 changes the divisor.
  2. 2. 20
    8 tens ÷ 4 = 2 tens = 20. The answer 2 forgets that the 8 is tens. 200 adds a zero too many. 76 subtracts instead of dividing.
  3. 3. 20 and 1
    Width = area ÷ height. 60 ÷ 3 = 20 and 3 ÷ 3 = 1, so the whole width is 21 and 63 ÷ 3 = 21. A width of 2 forgets the zero. Using 3 as a width repeats the area, and 21 is the total width, not a piece.
  4. 4. 8
    6 × 8 = 48, so 48 ÷ 6 = 8. 6 × 9 = 54 and 6 × 7 = 42, so 9 and 7 are neighbouring facts. 42 comes from subtracting 6.
  5. 5. 20
    83 is close to 80, and 80 ÷ 4 = 20, so the answer is about 20. 2 forgets the place value. 200 and 40 are far too big: 4 groups of 40 would be 160.
  6. 6. 22
    72 − 50 = 22. The 22 is then divided into more groups of 5. 20 ignores the ones digit and subtracts 70 − 50. 67 subtracts 5 instead of 50. 12 is a subtraction slip.
  7. 7. 13 × 7 = 91
    Division and multiplication are inverse operations: quotient × divisor = the number you divided. 13 × 7 = 91. Adding 13 and 7 gives 20. The other two switch the roles of the numbers.
  8. 8. 24
    80 ÷ 4 = 20 and 16 ÷ 4 = 4, so 20 + 4 = 24. The answer 20 stops after the first partial quotient. 204 writes 20 and 4 side by side instead of adding them. 23 treats 16 ÷ 4 as 3, but 4 × 3 = 12.
  9. 9. 18
    78 − 60 = 18. Then 18 ÷ 6 = 3 more groups, so 78 ÷ 6 = 10 + 3 = 13. 72 is 12 groups of 6, not the amount left. 12 is the number of groups in 72, and 68 subtracts only 10.
  10. 10. 17
    50 ÷ 5 = 10 and 35 ÷ 5 = 7, so 10 + 7 = 17. 107 writes the partial quotients side by side. 15 stops at 75 and forgets the last 10. 10 is only the first partial quotient.
  11. 11. 23
    The width is 92 ÷ 4. 80 ÷ 4 = 20 and 12 ÷ 4 = 3, so the width is 20 + 3 = 23. 203 puts the two widths side by side. 20 is only the first piece. 32 swaps the digits.
  12. 12. 17
    40 ÷ 4 = 10 and 28 ÷ 4 = 7, so 10 + 7 = 17. 107 writes 10 and 7 side by side. 16 is one fact short because 4 × 16 = 64, not 68. 7 forgets the tens.
  13. 13. 10 and 8
    Width = area ÷ height. 30 ÷ 3 = 10 and 24 ÷ 3 = 8, so 54 ÷ 3 = 18. The numbers 24 and 30 are areas, not widths. A width of 1 loses a zero.
  14. 14. 12
    70 ÷ 7 = 10 and 14 ÷ 7 = 2, so 10 + 2 = 12. 102 writes the parts side by side. 11 is wrong because 7 × 11 = 77. 10 is only the first partial quotient.
  15. 15. 25
    60 ÷ 3 = 20 and 15 ÷ 3 = 5, so 20 + 5 = 25. 205 writes 20 and 5 side by side. 20 forgets the 15 left over. 15 is the second partial dividend, not the quotient.
  16. 16. 12
    Equal groups of 6 mean divide: 72 ÷ 6 = 12 (60 ÷ 6 = 10 and 12 ÷ 6 = 2). 66 subtracts 6 and 78 adds 6, which are not equal-group operations. 13 is one group too many, since 6 × 13 = 78.
  17. 17. 14
    84 ÷ 6 = 14 because 60 ÷ 6 = 10 and 24 ÷ 6 = 4. 504 multiplies instead of dividing. 13 is off by one: 6 × 13 = 78. 78 subtracts 6.
  18. 18. He wrote the partial quotients 10 and 3 side by side instead of adding them
    70 ÷ 7 = 10 and 21 ÷ 7 = 3, so the quotient is 10 + 3 = 13. Sam joined 10 and 3 to make 103. Adding 91 and 7 would give 98, multiplying would give 637, and dividing only the tens would give about 1 ten, not 103.
  19. 19. No. 80 ÷ 4 = 20, so the answer should be close to 20, not 201
    80 ÷ 4 = 20, so 84 ÷ 4 should be a little more than 20: it is 21. A quotient is smaller than the number being divided, so 201 cannot be right, and the quotient does not have to be less than the divisor.
  20. 20. 72 ÷ 4
    72 ÷ 4 = 18 and 72 ÷ 6 = 12. Sharing the same amount among fewer groups gives more in each group. The two are not equal, and there is enough information to decide.
  21. 21. (90 ÷ 3) + (6 ÷ 3)
    96 = 90 + 6 and both parts divide evenly by 3, so 96 ÷ 3 = 30 + 2 = 32. Using 9 ÷ 3 drops the place value. The other choices change the divisor or the operation.
  22. 22. 4
    4 × 21 = 84, so the missing digit is 4. 80 ÷ 4 is only 20. 82 ÷ 4 and 88 ÷ 4 do not give 21 (20 R2 and 22).
  23. 23. 20 R3
    4 × 20 = 80 and 83 − 80 = 3, so 83 ÷ 4 = 20 R3. A remainder must be smaller than the divisor, so 19 R7 is wrong: 7 is more than 4, so another group fits. 2 R3 loses a zero. 23 is not a quotient of 83 ÷ 4.
  24. 24. 10
    58 ÷ 6 = 9 R4. Nine vans carry 54 students, and 4 students still need a ride, so a tenth van is needed. 9 leaves 4 students behind. 9 R4 is not a number of vans. 52 subtracts 6.
  25. 25. 14
    First add: 48 + 36 = 84. Then divide: 84 ÷ 6 = 14. The answer 54 divides only 36 and adds 48. 44 divides only 48 and adds 36. 84 forgets to divide.
  26. 26. 90
    Quotient × divisor = number, so 15 × 6 = 90. Check: 90 ÷ 6 = 15. 21 adds the two numbers. 9 divides 15 by 6 or guesses. 60 is 10 × 6 and misses the 5 groups of 6.
  27. 27. 72 ÷ 2
    72 ÷ 2 = 36, 84 ÷ 3 = 28, 96 ÷ 4 = 24 and 90 ÷ 6 = 15. The biggest number being divided does not make the biggest quotient; a small divisor does.
  28. 28. Yes. 10 + 6 = 16
    Any split into parts that each divide evenly by 6 works. 60 ÷ 6 = 10 and 36 ÷ 6 = 6, so 96 ÷ 6 = 16 (check: 16 × 6 = 96). Writing 10 and 6 side by side gives 106, which is wrong. Other splits work too (90 + 6 gives 15 + 1 = 16), so 96 is not limited to one split.

Related practice

Multiply a 2-digit number by 1 digit: area modelMultiplication and division facts: arrays and fact familiesDivision remainders: share, pack, or round upFactors and multiples