Groups of the same size
Here are 12 squares in 3 rows, with 4 squares in each row. Each row is a group, and every group holds the same number of squares, 4. Groups that all hold the same number are called equal groups.
Equal groups can be counted a group at a time instead of one square at a time. Count the first row: 4. Add the second row: 4 + 4 = 8. Add the third row: 8 + 4 = 12.
One row is one group of 4, and the three rows together are 4 + 4 + 4 = 12.
Adding the same number again and again
Adding the same number again and again has a shorter way to write it. 4 + 4 + 4 is 3 groups of 4, and it is written 3 × 4. The sign × is read "times", so 3 × 4 is read "three times four".
The first number says how many groups there are, and the second says how many are in each group. So 4 + 4 + 4 = 3 × 4 = 12. This is what multiplication is: adding equal groups.
3 groups of 4 squares make 12 squares: 3 × 4 = 12.
Equal jumps
Equal groups can also be jumps of the same size along a number line. Start at 0 and jump 4 at a time: 4, 8, 12. Three jumps of 4 land on 12, which is 3 × 4 again. Keep jumping and the line gives the rest of the fours: 4 jumps land on 16, 5 jumps on 20, and 6 jumps on 24.
Each step along this line is a jump of 4. Three jumps from 0 land on 12: 3 × 4 = 12.
Groups, not just numbers
The usual mistake is to add the two numbers: 3 + 4 = 7. The 3 counts the groups and the 4 counts what is in each group, so the 4 is added three times, 4 + 4 + 4 = 12.
One more group adds a whole group, not 1. With a fourth row of 4, the total goes from 12 to 12 + 4 = 16, and 4 × 4 = 16.
Worked example: Cakes on Plates in Equal Groups
Question There are 5 plates on a table. Each plate has 3 cakes on it. (a) How many cakes are there altogether? (b) Mom brings 1 more plate with 3 cakes on it. How many cakes are there now?
1.Draw 5 plates. Draw 3 cakes on each plate. These are 5 equal groups of 3.
Each plate is one group of 3 cakes. There are 5 equal groups. 2.Count in threes, one plate at a time: 3, 6, 9, 12, 15.
Count in threes, one plate at a time: 3, 6, 9, 12, 15. 3.(a) 5 groups of 3 is 5 × 3 = 15. There are 15 cakes altogether.
(a) 5 groups of 3 is 5 × 3 = 15 cakes. 4.(b) One more plate brings 3 more cakes: 15 + 3 = 18. Now there are 6 groups of 3, and 6 × 3 = 18 cakes.
(b) One more plate is 3 more cakes: 15 + 3 = 18, and 6 × 3 = 18.
Answer: (a) 15 cakes; (b) 18 cakes
Common mistakes
- Adding the two numbers: 5 + 3 = 8. The 5 is the number of plates and the 3 is the number of cakes on each plate, so the cakes are 3 + 3 + 3 + 3 + 3, not 5 + 3.
- Adding 1 in part (b) to get 16. One more plate is one more group of 3, so 3 cakes are added, not 1.
More multiplication and division problems, worked step by step →
Worked example: Equal Jumps Along a Number Line
Question A number line is painted on the playground. Ravi starts at 0. Each jump takes him 3 numbers along the line. (a) Which number does he land on after 6 jumps? (b) How many jumps from 0 does he need to land on 30?
1.Draw the number line. Start at 0 and mark jumps of 3.
Start at 0. Every jump is 3 numbers long. 2.Six jumps is 3 + 3 + 3 + 3 + 3 + 3. The jumps land on 3, 6, 9, 12, 15, 18.
Six jumps is 3 + 3 + 3 + 3 + 3 + 3. The jumps land on 3, 6, 9, 12, 15, 18. 3.(a) 6 jumps of 3 is 6 × 3 = 18. He lands on 18.
(a) 6 jumps of 3 is 6 × 3 = 18. 4.Keep jumping in threes and count the jumps: 21, 24, 27, 30. These are jumps 7, 8, 9 and 10.
Keep jumping in threes: 21, 24, 27, 30. Count the jumps. 5.(b) He needs 10 jumps to land on 30, because 10 × 3 = 30.
(b) 10 jumps land on 30, because 10 × 3 = 30.
Answer: (a) 18; (b) 10 jumps
Common mistakes
- Adding the two numbers: 6 + 3 = 9. That is one jump of 3 starting from 6. Six jumps of 3 starting from 0 is 3 added six times.
- Counting the marks on the line and not the jumps. From 0 to 30 there are 11 marks but only 10 jumps, because the mark at 0 is where he starts.
More multiplication and division problems, worked step by step →