Questions: Division as the Inverse of Multiplication

Question 1

A student needs to solve 28 ÷ 4. Which thinking strategy best uses the relationship between multiplication and division?

• A. Subtract 4 from 28 repeatedly and count the subtractions
• B. Think: what number times 4 equals 28? Since 7 × 4 = 28, the answer is 7
• C. Divide 28 by 2 to get 14, then divide by 2 again to get 7
• D. Count up from 4 until you reach 28, recording each step

Answer: (B)

The 'think multiplication' strategy reframes 28 ÷ 4 as the missing-factor problem '? × 4 = 28.' Using multiplication knowledge (7 × 4 = 28), the answer is immediately 7. Options A and C can give correct answers but don't use the inverse relationship. Option B happens to work for this problem but isn't a general division strategy — it's a coincidence of halving.

Question 2

From the single fact 5 × 6 = 30, how many division equations can be written?

• A. One: 30 ÷ 6 = 5
• B. Two: 30 ÷ 5 = 6 and 30 ÷ 6 = 5
• C. Three: 30 ÷ 5, 30 ÷ 6, and 30 ÷ 30
• D. None — you need to learn division facts separately

Answer: (B)

The three numbers 5, 6, and 30 form a fact family with exactly two division equations: 30 ÷ 5 = 6 and 30 ÷ 6 = 5. Both come directly from rearranging the multiplication fact. This is why a single multiplication fact is worth two division facts — the fact family always contains two multiplication equations and two division equations.

Question 3

The division problem 42 ÷ 6 can be solved by asking: 'What number times 6 equals 42?'

• T. True
• F. False

Answer: True

This is the inverse-relationship strategy in action. Division 'undoes' multiplication, so any division problem a ÷ b can be rephrased as '? × b = a.' Knowing 7 × 6 = 42 directly answers 42 ÷ 6 = 7. This reframing lets students use their multiplication fluency to answer division questions.

Question 4

Division facts is expected to be memorized largely separately from multiplication facts because the two operations are unrelated.

• T. True
• F. False

Answer: False

Multiplication and division are inverse operations — they are deeply related, not separate. Every division fact is a rearrangement of a multiplication fact. Because of this, students who know their multiplication tables already know all their division facts; they just need to learn to access them from the other direction. Treating them as unrelated means doing twice the memorization work unnecessarily.

Question 5

What does it mean for multiplication and division to be 'inverse operations'?

Model answer: It means each operation undoes the other. If you multiply a number by 4 and then divide by 4, you return to the original number. The same three numbers always relate the same way: multiply two factors to get the product, or divide the product by one factor to get the other.

Inverse operations cancel each other out. Multiplication builds a total from groups; division breaks the total back into groups. Recognizing this symmetry is what allows the 'think multiplication' strategy: a division problem is just a multiplication problem with one factor unknown. This pattern — paired inverse operations — repeats in all of mathematics, from arithmetic through algebra.