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Math · Multiplication & Mathematical Reasoning

Multiplication and Division Facts: Arrays, Fact Families, Break Apart

Use arrays and fact families to link multiplication and division, and break hard facts like 7 × 8 into easier ones you already know.

Grades 3–5 · 32 practice questions · Original material
Learning objectiveShow a multiplication fact as an array of equal rows, write the four facts in a fact family, use multiplication to find missing factors in division, and work out facts you do not know by breaking them apart or using a nearby fact, then check with the inverse operation.

The idea

In the last few lessons you added, subtracted and estimated with bigger numbers. Today you go back to the facts that make all of that faster: multiplication and division facts. Knowing them well saves time, and knowing how to rebuild a fact you forget saves more.

An array is objects in equal rows and equal columns. If there are 6 rows with 4 counters in each row, there are 6 × 4 = 24 counters. Turn the array a quarter turn and it becomes 4 rows of 6, so 4 × 6 = 24 too. Multiplication can be done in either order.

Multiplication and division undo each other. Three numbers that belong together make a fact family. For 6, 7 and 42 the family is 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7 and 42 ÷ 7 = 6. The product is always the biggest number, and it is the number you divide. This is why a division fact is a missing-factor question: 56 ÷ 8 asks “8 times what is 56?”

When you do not remember a fact, break one factor apart into two easy facts and add. For 7 × 8, split the 8 into 5 + 3. Then 7 × 5 = 35 and 7 × 3 = 21, and 35 + 21 = 56. In the array, you cut 8 columns into a group of 5 columns and a group of 3 columns.

Other helpers work the same way. Doubles: 6 × 8 is double 3 × 8, so 24 + 24 = 48. Neighbour facts: 7 × 8 is one group of 8 less than 8 × 8, so 64 − 8 = 56. Nines: 9 × 7 is 10 × 7 take away one group of 7, so 70 − 7 = 63. Notice that you always add or take away whole groups, never just 1.

Be careful with order. Turn-around works for multiplication (6 × 8 = 8 × 6) but not for division (48 ÷ 6 = 8, but 6 ÷ 48 is less than 1). Always check a division answer by multiplying back: if 8 × 6 = 48, then 48 ÷ 6 = 8 and 48 ÷ 8 = 6.

Worked examples

An array split into two easier facts, and a fact familyLeft panel: an array of 7 rows and 8 columns of dots. A dashed line splits it after 5 columns. The left part has 7 rows of 5, which is 7 times 5 equals 35. The right part has 7 rows of 3, which is 7 times 3 equals 21. Together 35 plus 21 equals 56, so 7 times 8 equals 56. Right panel: the fact family for 6, 7 and 42. The multiplication facts are 6 times 7 equals 42 and 7 times 6 equals 42. The division facts are 42 divided by 6 equals 7 and 42 divided by 7 equals 6. The product 42 is the biggest number and is the number being divided.Arrays show how multiplication facts fit togetherOriginal schematic, drawn to the labelled values.1 · Break apart: 7 × 8 = 7 × 5 + 7 × 37 × 5 = 357 × 3 = 2135 + 21 = 56, so 7 × 8 = 56Check: 8 × 8 = 64, and 64 − 8 = 56 ✓2 · Fact family: 6, 7, 424267product6 × 7 = 427 × 6 = 4242 ÷ 6 = 742 ÷ 7 = 6The product is the number you divide.Multiplication turns around;division does not: 42 ÷ 6 ≠ 6 ÷ 42
Original schematic, drawn to the labelled values. Cutting the 8 columns into 5 and 3 turns one hard fact into two easy ones, and the parts are added because every dot is in exactly one part. In a fact family the product is the number that is divided. Multiplication can be turned around; division cannot.

Example 1 · Read an array

An array has 6 rows with 4 counters in each row. How many counters is that?

  1. Rows are equal groups, and each row holds 4.
  2. 6 rows of 4 is 6 × 4.
  3. Count by 4s six times: 4, 8, 12, 16, 20, 24.
  4. Turn the array: 4 rows of 6 is 4 × 6 = 24 as well.

24

Example 2 · Break apart a fact

Find 7 × 8 using facts you know.

  1. Split 8 columns into 5 columns and 3 columns.
  2. 7 × 5 = 35 and 7 × 3 = 21.
  3. Add the two parts: 35 + 21 = 56.
  4. Check with the neighbour fact: 8 × 8 = 64, and 64 − 8 = 56.

56

Example 3 · Fact family

Write the fact family for 6, 7 and 42.

  1. The product is the biggest number: 42.
  2. Two multiplication facts: 6 × 7 = 42 and 7 × 6 = 42.
  3. Two division facts divide 42 by each factor: 42 ÷ 6 = 7 and 42 ÷ 7 = 6.
  4. Every fact uses only 6, 7 and 42.

6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6

Example 4 · Division as a missing factor

Find 56 ÷ 8.

  1. Ask: 8 times what number is 56?
  2. Skip count by 8s: 8, 16, 24, 32, 40, 48, 56. That is 7 jumps.
  3. So 8 × 7 = 56 and 56 ÷ 8 = 7.
  4. Multiply back to check: 7 × 8 = 56.

7

Example 5 · Catch the mistake

Mia says 7 × 6 = 48 because 7 × 7 = 49 and 49 − 1 = 48. Is she right?

  1. 6 groups of 7 is one whole group of 7 fewer than 7 groups of 7.
  2. She took away 1, not 7.
  3. 49 − 7 = 42.
  4. Check: 6 × 7 = 42 and 42 ÷ 7 = 6.

No. The answer is 42.

Common mistakes

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

Adding the numbers instead of multiplying.

Why this fails: 6 rows of 4 is 6 × 4 = 24, not 6 + 4 = 10. Multiplication counts equal groups.

Adding or taking away 1 instead of a whole group.

Why this fails: 7 × 6 is 7 × 7 − 7 = 42, not 49 − 1 = 48. When you change a factor by 1, the product changes by a whole group.

Dividing the small number by the big one.

Why this fails: 48 ÷ 6 = 8, but 6 ÷ 48 is less than 1. The product, or the total, goes first in a division fact.

Using a nearby fact without adjusting.

Why this fails: 8 × 6 = 48 does not tell you 56 ÷ 8 is 6. Multiply back to check: 8 × 7 = 56, so the answer is 7.

Thinking turn-around works for division.

Why this fails: 4 × 9 = 9 × 4, but 12 ÷ 3 is not 3 ÷ 12. Only multiplication can be done in either order.

Skipping the check.

Why this fails: Multiply back to check a division, and divide to check a multiplication. If 8 × 7 = 56, then 56 ÷ 8 must be 7.

Foundation

Foundation practice

Read arrays, link repeated addition to multiplication, and use fact families for division.

Read an array

1. An array has 6 rows with 4 counters in each row. How many counters are there in all?

  1. 10
  2. 24
  3. 18
  4. 28
Repeated addition

2. 4 + 4 + 4 + 4 + 4 + 4 + 4 has seven 4s. Which multiplication fact matches it?

  1. 4 × 4 = 16
  2. 7 + 4 = 11
  3. 7 × 7 = 49
  4. 7 × 4 = 28
Fact family

3. 3 × 8 = 24. Which division fact belongs to the same fact family?

  1. 24 ÷ 8 = 3
  2. 8 ÷ 24 = 3
  3. 24 − 8 = 16
  4. 3 ÷ 8 = 24
Turn-around facts

4. You know that 9 × 6 = 54. What is 6 × 9?

  1. 45
  2. 15
  3. 54
  4. 63
Missing factor

5. What number makes this true? ___ × 7 = 56

  1. 7
  2. 9
  3. 49
  4. 8
Division as sharing

6. 35 stickers are shared equally among 5 friends. How many stickers does each friend get?

  1. 30
  2. 7
  3. 40
  4. 175
Skip counting

7. Count by 8s: 8, 16, 24, 32, ___. What comes next?

  1. 36
  2. 42
  3. 40
  4. 48
Zero and one

8. Which statement is true?

  1. 9 × 0 = 0
  2. 9 × 0 = 9
  3. 9 ÷ 1 = 1
  4. 0 ÷ 9 = 9

Practice

Practice practice

Break hard facts into easy ones: doubles, neighbour facts, the nines trick and word problems.

Break apart a fact

9. Which pair of smaller facts can you add to find 7 × 8?

  1. 7 × 5 + 7 × 4
  2. 7 × 4 + 3
  3. 7 × 5 + 7 × 3
  4. 7 × 5 + 3
Double a fact

10. You know 3 × 8 = 24. Use it to find 6 × 8.

  1. 48
  2. 27
  3. 32
  4. 24
Use a neighbour fact

11. You know 8 × 8 = 64. Use it to find 7 × 8.

  1. 72
  2. 63
  3. 57
  4. 56
Nines trick

12. Use 10 × 7 = 70 to find 9 × 7.

  1. 77
  2. 63
  3. 70
  4. 56
Array word problem

13. Chairs are set out in 9 rows of 6 for a school concert (invented example). How many chairs are there?

  1. 15
  2. 45
  3. 54
  4. 63
Division word problem

14. 48 pencils are packed in boxes of 6 (invented example). How many boxes are filled?

  1. 42
  2. 288
  3. 7
  4. 8
Name the fact family

15. Which number sentence belongs to the fact family for 6, 9 and 54?

  1. 54 ÷ 6 = 9
  2. 54 ÷ 9 = 9
  3. 6 + 9 = 54
  4. 9 ÷ 54 = 6
Same array, new view

16. Which description shows the same number of counters as 4 rows of 6?

  1. 4 rows of 10
  2. 6 rows of 4
  3. 5 rows of 5
  4. 6 rows of 6

Think

Think practice

Find the error, choose the operation and decide which turn-around ideas really work.

Find the error

17. Mia wrote 7 × 6 = 48 because “7 × 7 is 49, and 49 − 1 is 48.” What went wrong?

  1. She should have added 7 to 49.
  2. 7 × 7 is not 49.
  3. Nothing; 48 is correct.
  4. She took away 1 instead of one whole group of 7.
Find the error

18. Jay wrote 56 ÷ 8 = 6 because “8 × 6 = 48.” Which statement is correct?

  1. He should have multiplied 56 × 8.
  2. He used the wrong fact: 8 × 6 is 48, not 56. The answer is 7.
  3. He should have added 56 + 8.
  4. Nothing; 6 is correct.
Which are equal?

19. Which pair of expressions always has equal values?

  1. 12 ÷ 3 and 3 ÷ 12
  2. 20 − 8 and 8 − 20
  3. 4 × 9 and 9 × 4
  4. 10 ÷ 5 and 5 ÷ 10
Choose the operation

20. Which situation needs 36 ÷ 4?

  1. 36 cards are dealt equally to 4 players. How many cards does each player get?
  2. 4 packs each hold 36 cards. How many cards are there in all?
  3. You have 36 cards and find 4 more. How many cards now?
  4. You have 36 cards and give away 4. How many are left?
Compare products

21. Which product is the greatest?

  1. 6 × 8
  2. 9 × 5
  3. 7 × 7
  4. 4 × 11
Find the missing side

22. A rectangle of 45 tiles has 5 rows. How many tiles are in each row?

  1. 40
  2. 9
  3. 50
  4. 8
Equal rows

23. Which number of counters can NOT be arranged in 3 equal rows with none left over?

  1. 27
  2. 24
  3. 30
  4. 26
Skip-count error

24. Sam counted by 6s: 6, 12, 18, 24, 32, 36. Which number in his count is wrong?

  1. 32
  2. 12
  3. 24
  4. 36

Challenge

Challenge practice

Use two clues, two steps and your own explanations to show how the facts connect.

Two clues

25. Two whole numbers have a product of 54 and a sum of 15. What are they?

  1. 7 and 8
  2. 6 and 9
  3. 5 and 10
  4. 3 and 12
Between two numbers

26. Which number between 50 and 60 is in the 8 times table?

  1. 54
  2. 58
  3. 56
  4. 60
Two steps

27. 7 rows of 4 desks are set up (invented example). Then 8 desks are removed and the rest are put into groups of 5. How many groups of 5 are there?

  1. 5
  2. 20
  3. 8
  4. 4
Count the facts

28. A fact family uses 8, 9 and 72. How many different multiplication and division facts can you write with these three numbers?

  1. 4
  2. 2
  3. 3
  4. 6
Not equal

29. Which expression does NOT have the same value as 6 × 12?

  1. 6 × 10 + 6 × 2
  2. 6 × 10 + 2
  3. 3 × 12 + 3 × 12
  4. 6 × 6 + 6 × 6
Explain

30. Explain why 7 × 8 = 7 × 5 + 7 × 3. Describe or sketch an array to support your answer.

Write the fact family

31. Write all four facts in the fact family for 6, 7 and 42.

Is she right?

32. Maya says: “8 × 6 = 6 × 8, so 48 ÷ 6 must equal 6 ÷ 48.” Is she right? Explain.

Answer key

  1. 1. 24
    6 rows of 4 is 6 × 4 = 24. 10 comes from adding 6 + 4 instead of making equal groups, 18 is 6 × 3 (one column short), and 28 is 7 × 4 (one row too many).
  2. 2. 7 × 4 = 28
    Seven equal groups of 4 is 7 × 4 = 28. 4 × 4 counts only four fours, 7 + 4 adds the group count to the group size instead of multiplying, and 7 × 7 uses the number of groups as the size too.
  3. 3. 24 ÷ 8 = 3
    A fact family uses the same three numbers, and the product is the number you divide. 24 ÷ 8 = 3 undoes 3 × 8. 8 ÷ 24 and 3 ÷ 8 divide by or into the wrong numbers (neither equals the stated result), and 24 − 8 is subtraction, not division.
  4. 4. 54
    Multiplying in either order gives the same product: 6 rows of 9 is the same array as 9 rows of 6, turned a quarter turn. 45 swaps the digits of 54, 15 adds 9 + 6, and 63 is 7 × 9.
  5. 5. 8
    Think 56 ÷ 7 = 8, because 8 × 7 = 56. 7 would give 49, 9 would give 63, and 49 is 56 − 7 (taking away one group’s worth instead of dividing).
  6. 6. 7
    Equal sharing is division: 35 ÷ 5 = 7, because 5 × 7 = 35. 30 is 35 − 5 and 40 is 35 + 5, which do not describe equal sharing, and 175 multiplies instead of dividing.
  7. 7. 40
    Each count adds 8: 32 + 8 = 40. 36 adds 4 (counting by 4s), 42 adds 10, and 48 skips a number (that is two jumps of 8 from 32).
  8. 8. 9 × 0 = 0
    Zero groups, or groups of zero, hold nothing: 9 × 0 = 0. 9 × 0 = 9 mixes up multiplying by zero with adding zero. 9 ÷ 1 = 9 (not 1) because one group holds all 9. 0 ÷ 9 = 0 because sharing nothing gives nothing.
  9. 9. 7 × 5 + 7 × 3
    8 columns split as 5 + 3, so 7 × 8 = 7 × 5 + 7 × 3 = 35 + 21 = 56. 7 × 5 + 7 × 4 splits 8 as 5 + 4 and gives 63, and the last two add a plain 3 or use only part of the array instead of 3 whole columns of 7.
  10. 10. 48
    6 is double 3, so 6 × 8 is double 3 × 8: 24 + 24 = 48. 24 forgets to double, 27 adds 3 instead of 3 more groups of 8, and 32 is 4 × 8.
  11. 11. 56
    7 × 8 is one group of 8 fewer than 8 × 8, so subtract 8: 64 − 8 = 56. 72 adds a group instead, 57 subtracts 7 (one, not eight), and 63 is 7 × 9.
  12. 12. 63
    9 × 7 is one group of 7 less than 10 × 7: 70 − 7 = 63. 77 adds a group, 70 is the 10 × 7 fact itself, and 56 is 8 × 7 (two groups too few).
  13. 13. 54
    9 rows of 6 is 9 × 6 = 54. 15 adds 9 + 6, 45 is 9 × 5 (a near-miss fact), and 63 is 9 × 7.
  14. 14. 8
    Find how many 6s fit in 48: 48 ÷ 6 = 8, because 6 × 8 = 48. 42 subtracts 6, 288 multiplies instead of dividing, and 7 comes from 6 × 7 = 42, a near-miss fact.
  15. 15. 54 ÷ 6 = 9
    The product 54 is the number being divided: 54 ÷ 6 = 9. 54 ÷ 9 is 6, not 9, 6 + 9 is only 15, and 9 ÷ 54 divides the small number by the big one.
  16. 16. 6 rows of 4
    4 rows of 6 and 6 rows of 4 are both 24 counters. 4 rows of 10 is 40, 5 rows of 5 is 25 (close but not equal), and 6 rows of 6 is 36.
  17. 17. She took away 1 instead of one whole group of 7.
    6 groups of 7 is one group of 7 fewer than 7 groups of 7: 49 − 7 = 42. Taking away only 1 leaves 48. Adding 7 would give 56, 7 × 7 really is 49, and 7 × 6 is 42, so 48 is not correct.
  18. 18. He used the wrong fact: 8 × 6 is 48, not 56. The answer is 7.
    To check a division answer, multiply back: 8 × 6 = 48, which does not rebuild 56. 8 × 7 = 56, so 56 ÷ 8 = 7. Multiplying or adding would not find an equal-share amount, and 6 fails the multiply-back check.
  19. 19. 4 × 9 and 9 × 4
    Multiplication can be done in either order (4 × 9 = 9 × 4 = 36). Division and subtraction cannot: 12 ÷ 3 = 4 but 3 ÷ 12 is only one quarter, 20 − 8 = 12 but 8 − 20 is below zero, and 10 ÷ 5 = 2 but 5 ÷ 10 is one half.
  20. 20. 36 cards are dealt equally to 4 players. How many cards does each player get?
    Dealing 36 cards equally among 4 players is division: 36 ÷ 4 = 9. Four packs of 36 is 4 × 36 = 144, finding 4 more is 36 + 4 = 40, and giving away 4 is 36 − 4 = 32.
  21. 21. 7 × 7
    7 × 7 = 49, 6 × 8 = 48, 9 × 5 = 45 and 4 × 11 = 44. Having the largest factor (9 or 11) does not make the largest product; calculate each one.
  22. 22. 9
    Rows × tiles per row = 45, so 45 ÷ 5 = 9. 40 is 45 − 5, 50 is 45 + 5, and 8 comes from the near-miss fact 5 × 8 = 40.
  23. 23. 26
    Equal rows of 3 need a multiple of 3. 27 = 3 × 9, 24 = 3 × 8 and 30 = 3 × 10, but 26 ÷ 3 is 8 with 2 left over.
  24. 24. 32
    24 + 6 = 30, so the fifth number should be 30, not 32. The 12, 24 and 36 are all in the 6 times table (2 × 6, 4 × 6 and 6 × 6).
  25. 25. 6 and 9
    6 × 9 = 54 and 6 + 9 = 15. 7 and 8 also add to 15 but multiply to 56, 5 and 10 add to 15 but multiply to 50, and 3 and 12 add to 15 but multiply to 36. Both clues must work.
  26. 26. 56
    8 × 7 = 56 is the only multiple of 8 from 51 to 59. 54 = 9 × 6 is a different table, 58 is not a multiple of 8 (58 ÷ 8 leaves 2), and 60 is outside the range (and 60 ÷ 8 leaves 4).
  27. 27. 4
    7 × 4 = 28 desks, 28 − 8 = 20 desks left, and 20 ÷ 5 = 4 groups. 5 forgets to remove desks (28 ÷ 5 is about 5), 20 stops before grouping, and 8 is the number removed.
  28. 28. 4
    The four facts are 8 × 9 = 72, 9 × 8 = 72, 72 ÷ 8 = 9 and 72 ÷ 9 = 8. Two counts only one multiplication and one division, three leaves one out, and six counts facts that are repeats or do not exist.
  29. 29. 6 × 10 + 2
    6 × 12 = 72. 6 × 10 + 2 = 62 because the 2 was not multiplied by 6. The others all make 72: 60 + 12, 36 + 36 and 36 + 36.
  30. 30. An array of 7 rows and 8 columns can be cut into 5 columns and 3 columns. The 5-column part has 7 × 5 = 35 counters, the 3-column part has 7 × 3 = 21, and 35 + 21 = 56 = 7 × 8.
    Self-review with this: did you split one factor into two smaller parts (8 = 5 + 3), multiply each part by the other factor (7), and add the two products? Splitting works because every counter is in exactly one part. Check: 7 × 8 = 56 and 35 + 21 = 56.
  31. 31. 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6.
    The two multiplication facts use the same factors in either order, and the two division facts divide the product by each factor. Self-review: did every fact use only 6, 7 and 42, with 42 as the product or the number being divided?
  32. 32. No. Multiplication can be done in either order, but division cannot. 48 ÷ 6 = 8, while 6 ÷ 48 is less than 1 (one eighth).
    Self-review: did you test both divisions? 48 ÷ 6 = 8 because 6 × 8 = 48. Dividing 6 by 48 asks how many 48s fit in 6, which is less than one whole. Turn-around only works for multiplication.

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