Questions: Fact Families: Multiplication and Division

Question 1

A student sees 56 ÷ 7 = ? and doesn't remember this division fact. Their teacher says: 'Think about what you know from multiplication.' Which multiplication fact should the student use?

• A. 7 + ? = 56
• B. 7 × ? = 56
• C. 56 × 7 = ?
• D. 56 − 7 = ?

Answer: (B)

Division and multiplication are inverse operations in the same fact family. '56 ÷ 7 = ?' asks the same question as '7 × ? = 56.' If you know that 7 × 8 = 56, then 56 ÷ 7 = 8 immediately. Using multiplication as a lookup table for division is exactly the strategy fact families teach — and why knowing multiplication facts makes division much easier.

Question 2

The numbers 6, 7, and 42 form a fact family. Which of the following is NOT a member of that fact family?

• A. 6 × 7 = 42
• B. 42 ÷ 6 = 7
• C. 7 × 6 = 42
• D. 6 ÷ 42 = 7

Answer: (D)

The four valid equations in the 6, 7, 42 fact family are: 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, and 42 ÷ 7 = 6. '6 ÷ 42 = 7' is false — 6 ÷ 42 is a tiny fraction, not 7. In a valid fact family, the product (the largest number, 42) is always the starting number in the division equations. You never divide a factor by the product.

Question 3

Knowing that 8 × 9 = 72 immediately tells you the answers to both 72 ÷ 8 and 72 ÷ 9.

• T. True
• F. False

Answer: True

The three numbers 8, 9, and 72 form a complete fact family. The single multiplication fact 8 × 9 = 72 gives you both division facts: 72 ÷ 8 = 9 and 72 ÷ 9 = 8. This is the payoff of fact families — one known multiplication fact unlocks two division facts simultaneously, without any separate memorization.

Question 4

In a fact family, you can create a valid division equation by dividing any one of the three numbers by either of the other two.

• T. True
• F. False

Answer: False

Only the product (the largest number) can serve as the dividend in a fact family's division equations. For the family 3, 4, 12: you can write 12 ÷ 3 = 4 and 12 ÷ 4 = 3, but 3 ÷ 12 and 4 ÷ 3 are not in the family — they would produce fractions, not whole-number answers. Division equations in a fact family always start with the product.

Question 5

Explain why a student who knows all their multiplication facts already knows most of their division facts. How does thinking about fact families make this work?

Model answer: Multiplication and division are inverse operations that undo each other. Every multiplication fact has two corresponding division facts in the same fact family: if 6 × 8 = 48, then 48 ÷ 6 = 8 and 48 ÷ 8 = 6. Fact families make this explicit by grouping all four equations together. So instead of memorizing division facts separately, you can use multiplication as a lookup table: for 48 ÷ 6, ask '6 × ? = 48' and your multiplication knowledge gives the answer.

This is the practical benefit of understanding fact families: division becomes a form of 'missing factor' multiplication rather than a separate set of facts to memorize. The key is understanding that the product (48) plays a special role — it is always the number being divided, and the two factors (6 and 8) are always the divisor and quotient, in either order.