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Scottish Survey of Literacy & Numeracy Support Material Third Level Fractions, decimal fractions and percentages Produced by Education Scotland Transforming Transforming lives lives through through learning learning Introduction Fractions, decimal fractions & percentages Knowing which one to use ? Demonstrating a deep understanding 25% of £48 1 of £48 4 8.6 x 1.5 8.6 half of 8.6 48 ÷ 4 8.6 4.3 £12 12.9 Nearly 80% of S2 pupils are unable calculate 9.5 x 0.5 If asked in a different way What is half of 9.5 Would 4 out of 5 pupils still get it wrong? Reflective Questions • How are problems in context presented? • How can we help pupils translate the text into understandable numeracy operations? • Can they make the connections between fractions, decimal fractions and percentages? 1 Taking into account that 0.5 = 2 Reflect on other examples where pupils need to decide on the most appropriate form to use. 12.5% of £160 1 of £160 8 160 ÷ 8 £20 Strategy Changing 0.5 1 to 2 might help answer 9.5 x 0.5 This changes the questions to 9.5 x 1 2 Do pupils recognise that 9.5 is the same as 9.50? Pupils then need to realise that this is the same as asking 1 ‘How much is 2 of 9.5’ So, all that is required is dividing 9.50 by 2 Points to reflect on Does this simplify the problem for pupils? Is there a visual representation that might help? How would you suggest pupils tackle questions such as 4.8 x 0.25 ? Could a similar approach be used for percentage problems, such as ‘Find 75% of 10.8’? Equivalent fractions, decimal fractions and percentages Katie is changing some money from pounds into dollars The exchange rate is £1 = $1.58 How much will Katie get in dollars if she exchanges £4.50? Did you know… Recent surveys have shown that more than 80% of pupils cannot find the correct answer to problems like the above. Why might this be difficult for pupils? Strategy Strategy Pupils may realise they need to multiply £4.50 by 1.58 Is trying to do this the best strategy? Without a calculator, long multiplication can prove tricky calculation for pupils. Using direct proportion might help. A key step might be for pupils to think of £4.50 as 4 This need not involve multiplying by 4 1 2 1 pounds 2 though. Pupils can apply direct proportion to solve the problem, as well as some basic number strategies: Strategies Double it 1 2 1 2 0.79 Double it So £4.50 is worth $6.32 + $0.79 = $7.11 Direct proportion can also be a really useful approach for percentage calculations. it Step by step approach Pupils should be encouraged to look for the simplest calculations. Finding 10% and 1% is generally something which pupils find straightforward. 10% 1% . This combination allow us to calculate any percentage. ÷2 5% 10% ÷ 10 1% In a local election, 17% of voters voted for the Green Party. 28000 people voted in the election. How many voted for the Green Party? ½ it So 4760 people voted for the Green Party. Double it Investigate 3 75% 4 10% 1 50% 2 1% 5% How can you apply your knowledge of the above to calculate the following percentages? 43% 74% 62% 81% 56% Misconceptions of the link between fractions & ratio 2:5 2 5 3: 2 3 2 This needs to be investigated. or 2 3 Link between ratio & fractions 1 Shade 4 of the circles Shaded : Not shaded 1: 3 4=1+3 Link between ratio & fractions The ratio is Carol : James 3 : 2 Think of this as 3 for Carol and 2 for James, and shade the rectangle accordingly until it is completely coloured in. Let’s make Carol’s share red and James’s share yellow: So, Carol has 9 squared coloured and James has 6. So, Carol gets 9 crayons and James gets 6 crayons. Link between ratio & fractions What numerical strategies developed for working with fractions could we use to solve this ratio problem? The ratio is 3:2, meaning Carol gets 3 shares and James gets 2 shares, giving 5 in total. This means that Carol gets 3 of the crayons and James gets 2 of them. 5 5 Use skills developed in finding a fraction of an amount to find that 1 5 of the 15 crayons is 3 crayons Carol gets 3 , so she gets 3 lots of 3, which is 9 5 James gets 2 , so he gets 2 lots of 3, which is 6. 5 How could pupils check their answer? www.educationscotland.gov.uk