Multiplication and Division

2

Key Messages – Mental Calculation

• Involves rapid recall• Use of rapid recall facts and strategies to derive

new facts• May involve pencil and paper jottings• Choice of strategy depends on the numbers

involved, known facts and personal preference

What else do you know?

6 x 7 =42

Lemonade

Mrs Jones bought 15 bottles of lemonade at 28p each.

15 x 28 = 420

So Mrs Jones paid £4.20

Lemonade

• Find the cost of 16 bottles of lemonade

15 x 28 = 420

Lemonade

• Find the cost of 8 bottles of lemonade

15 x 28 = 420

Lemonade

• Find the cost of 24 bottles of lemonade

15 x 28 = 420

Lemonade

• Find the cost of 15 bottles at 14p each

15 x 28 = 420

Lemonade

• Find the cost of 15 bottles at 7p each

15 x 28 = 420

Lemonade

• Find the cost of 15 bottles at 8p each

15 x 28 = 420

Lemonade

• Find the cost of 15 bottles at 29p each

15 x 28 = 420

Lemonade

• One tin of beans if 28 tins cost £4.20

15 x 28 = 420

Mental Calculation Strategies

General•Counting up and counting back•Partitioning

Particular•Reordering •Compensating•Factorising•Doubling and halving

14

Progression:

Mental with jottings

Non -standard written methods

Formal / compact / standard method

250 x 3 1 4 3 x 3 2 3 6 x 7

Mental Jottings Grid method

Write down What do I know? Partition into H, T and U the answer 143 x 2 = 286 200 + 30 + 6

750 What is the next step? Draw out a grid 286 + 143

What is the answer? Multiply each part by 7

280 + 140 = 420 420 + 9 = 429 Find the total 1400 + 210 + 42 = 1652

X 200 30 6

7

x 200 30 6

7 1400 210 42

Directed task

• Read Thompson, I. (1999) Written Calculation in Thompson, I. (ed) (1999) Issues in teaching numeracy in primary schools Buckingham: OUP

• Girling, M. (2001) ‘Towards a definition of basic numeracy’, Mathematics Teaching, 174, March 2001, pp.12-13, reflect on the key points made in reference to the seminar content.

• Possible points to consider:• What position does Girling take in the debate?• Consider the extent to which you agree or disagree

with his stance based on the reading you have undertaken so far and your personal experience.

Thompson

‘There are many studies which show that children's mental methods operate very differently from standard algorithms in several respects’

(Thompson, 1999, p. 170)

Multiplication

x 30 4

40 1200 160 1360

6 180 24 204

1564

3 4 3 4

x 4 6 4 6

2 4 (6 x 4) 2 0 4

1 8 0 (6 x 30) 1 3 6 0

1 6 0 (40 x 4) 1 5 6 4

1 2 0 0 (40 x 30)

1 5 6 4

Thompson

• Children are more likely to be able to understand these more ‘natural’ methods than the traditional standard algorithms which involve the automatic and unthinking manipulation of abstract symbols.’

(p.182)

Role of Calculators

‘At a time when a cheap calculator can be bought for the price of two good cabbages, we need to redefine our aims for numeracy. I suggest the following definition;

Basic numeracy is the ability to use a four function calculator sensibly.’

(Girling, 2001, p.12)

Position in the debate

‘I am not going to suggest that pencil and paper methods should not be taught , but that they should only be taught as part of the armoury of techniques that we have to help in an understanding of number and not because they are useful’