Numeracy Professional Development Fractions, decimals and maybe a bit on percentages Michael Drake Victoria University of Wellington College of Education Learning fractions Something to think about • If I say to you that I understand what two thirds means, what am I telling you that I know? Discuss your answer with someone next to you Representations • A representation is one way of showing a mathematical idea • For fractions, a shape that is drawn and shaded is one representation of the idea two thirds Issues of Representation • Research has found that students have difficulty translating between different representations of a mathematical object. For example the symbol 1/3 and the fraction drawing showing 1/3, or the fraction circle 1/3 and the number line showing 1/3. • If I say I understand what two thirds means, I should be able to recognise the key representations of two thirds and translate between them • If I say I understand what two thirds means, I should be able to recognise that the representations are equivalent • Johnny says he understands what two thirds means. It means two out of three pieces. • What do you want Johnny to be able to do in relation to this concept of fractions? By the way Did you know? • In 2001 42% of year 7 & 8 students who sat the initial NUMPA could not name these symbols 1 2 1 4 1 3 • More could not place them in size order… What we want Johnny to be able to do is… Recognise a range of drawings for two thirds Be able to read the symbol – and relate it to the drawings And be able to talk about it Written words Verbal/mental picture Materials Symbol Fraction sub-constructs There are a number of ways we think of fractions, all of which involve slightly different understandings • Part-whole • Measure construct • Quotient construct • Operator construct • Ratio construct • Probability construct • So what does Johnny really need to understand if he is to be successful with fractions? Something to think about and discuss in school • How do you get these various interpretations across so students understand fractions better? Decimals materials imaging Using number properties Generalising from number • Decimals can be introduced by generalising from whole numbers For example • Addition by place value: 34 + 25 Does it work for decimals? Investigate 3.4 and 2.5 [etc] But several problems with student understanding need to be checked before hand • 85% of students who haven’t been formally taught decimals have a firm idea of how they work. Its wrong, but they have the idea Bruce Moody What’s the problem? • Discuss what the student is doing, and why it is a problem • What teaching strategies do you currently have that could help sort out the problem? 1) John reads the decimal 4.35 as four point thirty five 2) Carol says 1.50 is bigger than 1.5 3) Tupu says 0.23 is the same as 0.023 4) Martika and Karl are having an argument. Martika says 1.5 10 = 1.50, while Karl says its 10.50 5) On a trip you spend 98.5c on a Litre of petrol, and $0.95 on a cheeseburger. How much have you spent? The moral of the story… • Decimals need explicit teaching – assuming students understand them is asking for trouble • Don’t introduce them until students understand how whole numbers work, and have the concept of a fraction • Spend time on decimal place value

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