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Primary National Strategy Mathematics 3 plus 2 day course: Session 11

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© Crown copyright 2003 Primary National Strategy Slide 11.1 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

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© Crown copyright 2003 Primary National Strategy Slide 11.2 The structure of a problem of direct proportion Slide 11.2 ab cd variable Xvariable 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.

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© Crown copyright 2003 Primary National Strategy Slide 11.3 The structure of a problem of direct proportion Slide 11.3 ab cd variable Xvariable 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.

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© Crown copyright 2003 Primary National Strategy Slide 11.4 The structure of a problem of direct proportion Slide 11.4 5£1 10£2 orangescost

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© Crown copyright 2003 Primary National Strategy Slide 11.5 The structure of a problem of direct proportion Slide 11.5 ab cd variable Xvariable Y It is also true that a is to c as b is to d.

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© Crown copyright 2003 Primary National Strategy Slide 11.6 The structure of a problem of direct proportion Slide 11.6 5£1 10£2 orangescost

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© Crown copyright 2003 Primary National Strategy Slide 11.7 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.

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© Crown copyright 2003 Primary National Strategy Slide 11.8 A recipe for 6 people needs 12 eggs. Adapt it for 8 people. Slide 11.8 612 8? peopleeggs 16 Answer: 16 eggs × 2

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© Crown copyright 2003 Primary National Strategy Slide 11.9 A recipe for 6 people needs 4 eggs. Adapt it for 9 people. Slide 11.9 64 9? peopleeggs 6 Answer: 6 eggs × 1.5

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© Crown copyright 2003 Primary National Strategy Slide 11.10 A recipe for 6 people needs 120 g flour. Adapt it for 7 people. Slide 11.10 6120 7? peopleflour (g) 140 Answer: 140 g flour × 20

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© Crown copyright 2003 Primary National Strategy Slide 11.11 A recipe for 8 people needs 500 g flour. Adapt it for 6 people. Slide 11.11 8500 6? peopleflour (g) 375 Answer: 375 g flour × 3 4

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© Crown copyright 2003 Primary National Strategy Slide 11.12 A recipe for 6 people needs 140 g flour. Adapt it for 14 people. Slide 11.12 6140 14? peopleflour (g) This time there is no obvious straightforward relationship.

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© Crown copyright 2003 Primary National Strategy Slide 11.13 A recipe for 6 people needs 140 g flour. Adapt it for 14 people. Slide 11.13 6140 1140 ÷ 6 14? peopleflour (g) 23.333 23.333 × 14 326.666 Answer: approximately 327 g

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© Crown copyright 2003 Primary National Strategy Slide 11.14 Informal working of the cost of 4.5 kg potatoes Slide 11.14 weight (kg)cost (£) 10.52 21.04 42.08 0.50.26 4.52.34 Answer: £2.34

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© Crown copyright 2003 Primary National Strategy Slide 11.15 Efficient working of the cost of 4.5 kg potatoes Slide 11.15 152 4.552 × 4.5 potatoes (kg)cost (p) 234 Answer: £2.34 4.5

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© Crown copyright 2003 Primary National Strategy Slide 11.16 Efficient working of the cost of 4.5 kg potatoes Slide 11.16 152 4.5 4.5 × 52 potatoes (kg)cost (p) 234 Answer: £2.34 52

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© Crown copyright 2003 Primary National Strategy Slide 11.17 4.204 14 ÷ 4.2 2.52? cost (£)beans (kg) (4 ÷ 4.2) × 2.52 2.4 Answer: 2.4 kg Finding the weight of beans that £2.52 will buy

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© Crown copyright 2003 Primary National Strategy Slide 11.18 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.

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© Crown copyright 2003 Primary National Strategy Slide 11.19 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.

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