Use multiplication to solve addition and subtraction problems.

Number Framework Stage 7

Differetn coloured transparent counters

Give each student a hundreds board. Ask them to put counters of the same colour on

any pair of numbers that add to 11. Tell them to continue the process, using a new

colour to mark each pair. The students should notice that there are five pairs, 5 + 6,

4 + 7, 3 + 8, 2 + 9, and 1 + 10. Challenge the students to add all of the numbers they have covered, that is, 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. Discuss strategies they may have used, like making tens. Note that the answer, 55, is the answer to 5 x 11. Ask the students why that is. They should be able to link this to the five pairs adding to 11, which they found through the counter exercise.

Challenge the students to find pairs of numbers that add to 21 and mark these pairs

with coloured counters on their hundreds board.

Tell them to add all the whole numbers, 1 + 2 + 3 + ... + 18 + 19 + 20. These form 10 pairs totalling 21, so the sum is 10 _ 21 = 210. Challenge the students to then add all of the whole numbers 1 + 2 + 3 + ...+ 22 + 23 + 24. Some students may realise that they could simply add 21 + 22 + 23 + 24 = 90 onto the previous answer of 210. This is a good check on the multiplicative strategy of finding 12 pairs of 25 (1 + 24, 2 + 23, 3 + 22, ...).

**Using Imaging**

Encourage imaging of adding arithmetic series (those with common differences) by

posing the following problems. Allow the students access to the hundreds boards if

necessary but encourage them to work out the problems without putting on counters.

Add all the whole numbers: 1 + 2 + 3 + ... + 98 + 99 + 100 (50 _ 101 gives 5 050).

Add all the odd numbers: 1 + 3 + 5 + ... + 95 + 97 + 99 (25 _ 100 gives 2 500).

Add all the even numbers: 2 + 4 + 6 + ... + 96 + 98 + 100 (25 _ 102 gives 2 550).

Note that the answers to the odd and even series should total the answer for all the

whole numbers: 2 500 + 2 550 = 5 050.

Add all these numbers: 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 (four pairs of 20 plus 10 gives 90).

### Using Number Properties

Promote generalisation of the sum of a series that has constant differences though

examples like:

5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 (five pairs of 40 plus 20 gives 220)

1/4 + 1/2 + 3/4 + 1 1/4 + 11/2 + 13/4 + 2 1/4 + 2 1/2 + 2 3/4 + 2 3/4 (5 x 3 + 1 1/2= 16 1/2)

1.4 + 1.3 + 1.2 + 1.1 +...+ 0.3 + 0.2 + 0.1 (7 x 1.5 = 10.5)

2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 (5 x 31 = 155)

5.0 + 4.6 + 4.2 + 3.8 + 3.4 + 3.0 + 2.6 + 2.2 + 1.8 + 1.4 (5 x 6.4 = 32.0)