Questions: Division: Fair Sharing Model

Question 1

You know that 5 × 6 = 30. Which division fact does this multiplication fact directly tell you?

• A. 30 ÷ 2 = 15
• B. 30 ÷ 5 = 6
• C. 5 ÷ 30 = 6
• D. 30 ÷ 10 = 3

Answer: (B)

Multiplication and division are inverses — they describe the same three-number relationship. Because 5 × 6 = 30, you immediately know both 30 ÷ 5 = 6 and 30 ÷ 6 = 5. Division 'undoes' the multiplication. This is why knowing multiplication facts makes division directly accessible — no separate calculation is needed.

Question 2

A student shares 20 crayons equally into 4 boxes and gets 5 per box. She says the quotient is 4 because there are 4 boxes. Is she correct?

• A. Yes — 4 boxes means the quotient is 4
• B. No — the quotient is what each group receives (5), not the number of groups (4)
• C. No — the quotient refers to the total being shared (20)
• D. Yes — because 20 ÷ 4 = 4

Answer: (B)

The quotient is the answer to a division problem — what each group receives. In 20 ÷ 4 = 5, the dividend (20) is the total, the divisor (4) is the number of groups, and the quotient (5) is how many each group gets. Confusing the divisor with the quotient is one of the most common errors. The 4 boxes are the divisor, not the answer.

Question 3

Because 4 × 7 = 28, you can immediately know that 28 ÷ 4 = 7 without any separate calculation.

• T. True
• F. False

Answer: True

Division is the inverse of multiplication. The three numbers 4, 7, and 28 are locked together: multiply the two factors to get the product, or divide the product by either factor to get the other. Knowing the multiplication fact gives you both division facts (28 ÷ 4 = 7 and 28 ÷ 7 = 4) for free.

Question 4

In the problem 24 ÷ 6 = 4, the number 6 represents how many items each group gets.

• T. True
• F. False

Answer: False

6 is the DIVISOR — the number of groups you are sharing into. 4 is the QUOTIENT — what each group receives. 24 is the DIVIDEND — the total being shared. A helpful memory check: division answers 'how many per group?' The quotient (4) is the answer to that question; the divisor (6) is the number of groups you made.

Question 5

Explain how the fair sharing model of division connects to multiplication. Use the example 15 ÷ 3 = 5.

Model answer: In the fair sharing model, 15 is the total being shared (dividend), 3 is the number of groups (divisor), and 5 is how much each group gets (quotient). This works because 3 groups of 5 equals 15 — the multiplication fact 3 × 5 = 15. Division reverses the multiplication: instead of building up equal groups, you start with the total and find the group size. Knowing 3 × 5 = 15 immediately gives you 15 ÷ 3 = 5.

The relationship is the same fact seen from different directions. Multiplication builds up to the total; division starts from the total and finds a missing factor. This is why 3, 5, and 15 belong to the same 'fact family' — you can write four equations (3 × 5, 5 × 3, 15 ÷ 3, 15 ÷ 5) from the same underlying relationship.