Scottish Survey of Literacy & Numeracy Support Material Third Level – Fractions Classroom version Produced by Education Scotland Transforming lives through learning

Third Level Fractions Key Points: Pupils have difficulty with: Finding equivalent fractions, decimal fractions and percentages Working with fractions, decimal fractions and percentages in context We need to consider the reasons why these areas cause problems and look at some ways that these skills could be developed and improved upon. Pupils have difficulty with: Finding equivalent fractions, decimal fractions and percentages and Working with fractions, decimal fractions and percentages in context We need to consider the reasons why these areas cause problems and look at some ways that these skills could be developed and improved upon.

Equivalent Percentage Introduction Fractions Equivalent Decimal Equivalent Percentage

Problems with equivalent forms of S2 pupils experience problems Problems with equivalent forms Review & Reflect What concepts do learners find difficult? Support for understanding Look at your own practice Look at exemplars of effective practice Consider Language being used Decimal Fraction Decimal Allows pupils to make connections

Learning & Teaching Process Third Level Fractions How can we improve pupils’ understanding of fractions and help them develop strategies to solve problems involving fractions? Primary / Secondary Liaison Joint approach Did you know that recent surveys have shown that between 65 and 80% of pupils in S2 have difficulty correctly answering questions involving fractions? Fractions / Decimal Fractions / %

Effective strategies required for: Issues Fractions Decimals Decimal Fractions Percentages

Decimal fractions Fraction Effective Questioning Familiar Contexts To support understanding decimal notation % representation Significance of % sign 1% = 1 ÷ 100 A key skill when working with equivalences between fractions, decimal fractions and percentages is that pupils should be confident with the concept of 10ths and hundredths when talking about decimal fractions and should understand that ‘percent’ means ‘out of 100’ This will help them understand that turning a fraction into a decimal fraction (and vice-versa) will involve finding an equivalent fraction that is in tenths or hundredths and that turning a fraction into a percentage involves finding an equivalent fraction that has 100 as its denominator.

Decimal fraction Fraction To turn 0.65 into a fraction, consider reinforcing place value by showing 0.65 visually as: Pupils can then read this number as 65 hundredths. And so can then write 0.65 as To simplify this fraction, use strategies developed previously. In this case, we can divide both the numerator and denominator by 5, so 0.65 = = To turn 0.65 into a fraction, consider reinforcing place value by showing 0.65 visually Remind pupils that just as fifths mean ‘out of 5’ and tenths mean ‘out of 10’, hundredths mean ‘out of a hundred. So, when turning this decimal fraction into a fraction, we are looking at a fraction out of a hundred, so our denominator will be 100. Pupils can then read this number as 65 hundredths and so write 0.65 as 65/100 To simplify this fraction, use strategies developed previously. In this case, we can divide both the numerator and denominator by 5 Remind pupils that this is actually dividing by 5/5 and so we are dividing by 1, thus not actually changing the size of the fraction. We therefore get 65/100 = 13/20 as the simplest form of the fraction. So, 0.65 = 13 twentieths

Fraction Decimal fractions Change into a decimal fraction. What are the key things for pupils to consider when linking fractions with decimal fractions? Changing a fraction into a decimal fraction can be tackled in a range of ways. What are the key things pupils should consider when linking fractions and decimal fractions?

Fraction Decimal fractions Encourage pupils to first think of tenths and hundredths when linking fractions with decimal fractions. Reading this as two tenths and twenty hundredths should enable pupils to understand that this is written as 0.2 in decimal fraction form. Prior knowledge: Pupils need to be confident with creating equivalent fractions before attempting to change fractions into decimal fractions (non calculator) Encourage pupils to first think of tenths and hundredths when linking fractions with decimal fractions. How would pupils find a fraction in tenths that is equivalent to one fifth? Once they have established that 1 out of 5 is equal to 2 out of 10, they can read this as 2 tenths, which should lead them to write it in decimal form as 0.2 How would you encourage pupils to find decimal equivalents for fractions that cannot be easily changed into tenths? What about ¾ or 17/20 or 3/8 or 2/3 ? Might there be some fractions that pupils will have to recognise as exceptions and just learn their decimal equivalent? Would percentage equivalents help in these circumstances?

. Fraction Percentage 10 x 10 grid split into blocks of 6 and shade For percentages, think ‘out of 100’. ? 10 x 10 grid split into blocks of 6 and shade 1 out of every 6 How would you deal with changing a fraction like 12/40 into a percentage? Or 1/6 ? How could you represent this visually to reinforce the equivalence? Work with pupils to develop more than one possible strategy for doing this. Would estimation and approximation help? Starting points to consider before thinking about 12/40 and 1/6 What strategies would you work on with pupils to enable them to change a fraction such as 4/5 to a percentage? For percentages, encourage pupils to think ‘out of a hundred’ Pupils should then realise they are looking for an equivalent fraction that has 100 as its denominator. What questions would you pose to pupils at this point? this works for the first 16 blocks but it is not possible to create the 17th block ? .

Percentage Fraction Change 64% to a fraction in its simplest form Percent means out of 100, so 64% means 64 out of 100 What are the key concepts pupils need to understand when dealing with percentages? What contexts could be used to aid understanding? Percent means out of a hundred, so using this knowledge, we are looking for a fraction with 100 as its denominator. 64% means 64 out of one hundred and so can be written as a fraction with 64 as the numerator and 100 as the denominator.

Percentage Fraction

Percentage Fraction We can then simplify this fraction using previously developed strategies to get 32/50 and 16/25 How could you represent this type of problem visually? How could you use the strategies discussed to help pupils change a decimal fraction to a percentage, or vice-versa?

Percentage Fraction

Strategies Consider playing games, such as matching pairs to develop pupils’ understanding of equivalent fractions Consider playing games, such as matching pairs to develop pupils’ understanding of equivalent fractions

Percentage Fraction

Why/how we use questions in context Effective Questioning Familiar Contexts Developing Higher Order Skills Creating Giving pupils the opportunity to create their own matching card game. Evaluating Giving pupils the opportunity to justify their answers to given problems. Analysing Giving pupils the opportunity to make the connection between fractions, decimal fractions and percentages.

Questions in Context In all types of fraction problem, ensure that pupils are able to transfer the skills they develop in answering simply worded questions, to problems written in context. For example, what steps would you encourage a pupil to go through to answer questions such as: ‘Emma saves 10% of her pocket money each week. What fraction of her pocket money does she save?’ In all types of fraction problem, ensure that pupils are able to transfer the skills they develop in answering simply worded questions, to problems written in context. For example, what steps would you encourage a pupil to go through to answer questions such as these? What range of strategies could be used to answer this problem? Would using a specific numerical example help, then generalising from there? How would you help visual learners deal with this?

Reflective Questions What range of strategies could be used to answer this problem? Would using a specific numerical example help, then generalising from there? How would you help visual learners deal with this?

Questions in Context In all types of fraction problem, ensure that pupils are able to transfer the skills they develop in answering simply worded questions, to problems written in context. For example, what steps would you encourage a pupil to go through to answer questions such as: ‘In a Science test, Lewis got 70% of the test correct. In the same test, Amy answered of the questions correctly. Amy’s actual mark in the test was 30. What was Lewis’s actual mark?’ What’s the crucial thing that pupils have to find here? What numerical strategies could be used? Why might 52 be a common incorrect answer given by pupils? Would a visual representation help? What’s the crucial thing that pupils have to find here? What numerical strategies could be used? Why might 52 ½ be a common incorrect answer given by pupils? Would a visual representation help? How can pupils check their answer?

Reflective Questions What’s the crucial thing that pupils have to find here? What numerical strategies could be used? Why might 52 be a common incorrect answer given by pupils? Would a visual representation help?

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