Primary National Strategy

1. PrimaryNational Strategy Mathematics3 plus 2 day course:Session 11

2. Objectives • To consider how the foundations of algebra are laid in Key Stage 2 • To further understanding of direct proportion • To consider ways of teaching direct proportion

3. The structure of a problem of direct proportion variable X variable Y There are two ‘variables’; three of the numbers (e.g. a, b and c) are known, and the fourth number d is to be found. Slide 11.2

4. The structure of a problem of direct proportion variable X variable Y The ratio of variable X to variable Y is always the same, so a is to b as c is to d. We say that the two variables are in ‘direct proportion’. Slide 11.3

5. The structure of a problem of direct proportion oranges cost Slide 11.4

6. The structure of a problem of direct proportion variable X variable Y It is also true that a is to c as b is to d. Slide 11.5

7. The structure of a problem of direct proportion oranges cost Slide 11.6

8. Solving direct proportion problems When you solve problems involving direct proportion in this way, you can work either left to right (across the variables) or top to bottom (within the variables). The most efficient way will depend on which numbers are known and the relationships between them.

9. × 2 16 Answer: 16 eggs A recipe for 6 people needs 12 eggs. Adapt it for 8 people. people eggs Slide 11.8

10. × 1.5 6 Answer: 6 eggs A recipe for 6 people needs 4 eggs. Adapt it for 9 people. people eggs Slide 11.9

11. × 20 140 Answer: 140g flour A recipe for 6 people needs 120g flour. Adapt it for 7 people. people flour (g) Slide 11.10

12. 3 4 × 375 Answer: 375g flour A recipe for 8 people needs 500g flour. Adapt it for 6 people. people flour (g) Slide 11.11

13. A recipe for 6 people needs 140g flour. Adapt it for 14 people. people flour (g) This time there is no obvious straightforward relationship. Slide 11.12

14. 326.666 Answer: approximately 327g A recipe for 6 people needs 140g flour. Adapt it for 14 people. people flour (g) 23.333 23.333×14 Slide 11.13

15. Informal working of the cost of 4.5kg potatoes Answer: £2.34 Slide 11.14

16.  4.5 234 Answer: £2.34 Efficient working of the cost of 4.5kg potatoes potatoes (kg) cost (p) Slide 11.15

17.  52 234 Answer: £2.34 Efficient working of the cost of 4.5kg potatoes potatoes (kg) cost (p) Slide 11.16

18. 2.4 Answer: 2.4kg Finding the weight of beans that £2.52 will buy cost (£) beans (kg) (4 ÷ 4.2) × 2.52 Slide 11.17

19. Solving direct proportion problems • To get started with a direct proportion problem, summarise the information in a four-cell diagram, making sure that the units are consistent for each variable. • It helps to arrange the diagram so that the unknown is in the bottom right corner but this is not essential. • Look for a relationship that attracts you. It can be across the variables (left to right) or within the variables (top to bottom). • Apply the same relationship to the other variable to find the unknown number.

20. Solving direct proportion problems • Pupils will move from informal methods using a four-cell diagram in Key Stage 2 to the generally applicable unitary method in Key Stage 3. • Some Key Stage 2 pupils may use the unitary method without being taught it, developing it for themselves.