Equal groups and sharing

Each group has the same number

Put three counters on each of four plates. Count 3 + 3 + 3 + 3 = 12. There are four groups of three.

Write 4 × 3 = 12. In this example the first number counts groups and the second tells how many are in each group. Read it as “four groups of three make twelve”.

Each dot stands for one counter. The plate numbers label the groups; they are not extra counters.

If one plate has only two counters, the groups are not equal. Counting four groups of three would then give the wrong total.

Use the multiplication and equals signs

The sign × means multiply. The sign = joins two equal amounts. Both 4 × 3 = 12 and 12 = 4 × 3 are true: four groups of three have the same total as twelve.

In 12 = 4 × 3, the equals sign comes first. It still means “has the same value as”; it does not tell you to do a calculation next.

Check your understanding 1 A total on the left

Fill the two blanks in 15 square 5 square 3. Use one × sign and one = sign. Check the amount on each side.

Your completed multiplication sentence

Show answer

15 = 5 × 3.

The first blank is =; the second is ×. Five groups of three make fifteen. 15 × 5 = 3 would not give the same amount on both sides.

Count in groups

For five bags with four blocks each, count 4, 8, 12, 16, 20. Five equal jumps of four give 5 × 4 = 20.

For four boxes with eight beads each, count 8, 16, 24, 32. The product is 32. Do not add four and eight: four counts boxes, while eight counts beads in each box.

Check your understanding 2 Count the whole collection

There are six cups. Each cup holds five buttons. How many buttons are there altogether?

Buttons altogether

Check answer

Show answer

6 × 5 = 30 buttons. The six cups are not six extra buttons.

Eight bags with five beads each have 8 × 5 = 40 beads. Count eight groups: 5, 10, 15, 20, 25, 30, 35, 40.

If every bag is empty, there are zero beads. Five groups with zero in each have 5 × 0 = 0. Five counts bags, not beads.

Share equally

Share 12 counters among three plates. Give one to each plate in turn until all counters have been shared. Each plate gets four.

Check: three groups of four make 12. Sharing equally means the plates get the same number, with no counter left over in this example.

Now change the question: make groups of three using the same 12 counters. You can make four groups. The total is the same, but this time you were finding the number of groups.

Check your understanding 3 What are you finding?

You have 20 blocks. First share them equally between five boxes. Then put the same 20 blocks into groups of five. What do you find each time?

Your answer or drawing on paper

Show answer

Sharing between five boxes gives four blocks in each box. Grouping five at a time gives four groups. In the first question five counts boxes; in the second it counts blocks in each group. Both use all 20 blocks.

Read a division calculation

“20 divided by 4” uses 20 as the whole and 4 as the number of equal groups. Find how many go in each group: five. Check with 4 × 5 = 20.

You can also find the answer by making groups of four from the same 20 objects. You make five groups. This time four tells you the size of each group.

Keep the given order: 20 is the whole, not the number of groups. If a calculation table lists 20 first and 4 second, divide 20 by 4.

Check your understanding 4 Use every object

18 divided by 3: share 18 counters between three plates. How many go on each plate?

Counters on each plate

Check answer

Show answer

Six counters on each plate. 3 × 6 = 18. Making groups of three from 18 would give six groups instead.

If there are no counters to share between four plates, each plate gets zero. Zero divided by four is zero. We use a positive number of groups; dividing by zero is not allowed.

Try with objects

Use 18 counters. Make three equal groups and say how many are in each. Then make groups of six and say how many groups you have. Check by counting all the counters again.