Two kinds of division

What you will learn

• Share a quantity equally and say how many each
• Make equal groups and say how many groups
• Write both situations with the division symbol
• Use a multiplication fact to check a division

Models used: counters, sharing, equal groups, number lines, number sentences, stories.

Before you start

• Equal groups become multiplication

What ÷ asks

Share 12 counters between 4 children. Nobody has any yet.

1. Each plate is one share. Counters outside the plates have not been shared yet.

   Share 12 counters between 4 children. Nobody has any yet.

   Equal sharing means every share has the same amount. Check both conditions: the shares match, and all the starting objects are accounted for.

   Friend 1: 0

   Friend 2: 0

   Friend 3: 0

   Friend 4: 0

2. Each plate is one share. Counters outside the plates have not been shared yet.

   Give each child one counter. Everyone must receive the same amount.

   Equal sharing means every share has the same amount. Check both conditions: the shares match, and all the starting objects are accounted for.

   Friend 1: 1

   Friend 2: 1

   Friend 3: 1

   Friend 4: 1

3. Each plate is one share. Counters outside the plates have not been shared yet.

   Keep going around. After three rounds, the counters are all shared.

   Equal sharing means every share has the same amount. Check both conditions: the shares match, and all the starting objects are accounted for.

   None left

   Friend 1: 3

   Friend 2: 3

   Friend 3: 3

   Friend 4: 3

4. There are 4 groups, with 3 objects in each group. Count a group before counting them all.

   Each child receives 3. Four equal shares of 3 make our original 12.

   12 ÷ 4 = 3

   3 + 3 + 3 + 3 = 4 × 3 = 12

Worked example

15 ÷ 5 = ?

1. What is the question asking?

   15 ÷ 5 can mean…

   15 shared into 5 equal parts

2. Make groups of 5 from 15 counters.

   None left

   Group 1: 5

   Group 2: 5

   Group 3: 5

3. Ask the multiplication question: 5 × ? = 15.

   5 × ? = 15

   What is 15 ÷ 5?

   3

4. So 15 ÷ 5 = 3.

   15 ÷ 5 = 3

5. Check with multiplication.

   3 × 5 = 15 ✓

Common mistakes to discuss

• Thinking the answer to a division is always the number of groups. Read what the question asks: how many each (sharing) or how many groups (grouping). The calculation is the same, but the answer means something different.
• Subtracting instead of dividing. 12 − 3 takes 3 away once. 12 ÷ 3 splits all 12 into equal parts.

Check your understanding

• share 12 between 3 and write 12 ÷ 3 = 4
• find how many groups of 4 fit into 12
• tell a sharing story from a grouping story
• check a division with its multiplication

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.