1. Developing subject knowledge and practice in fractions. To identify some of the difficulties and misconceptions which children have & implications for teaching To consider different models of fractions To consider implications in the new 2014 curriculum for the teaching of fractions

2. Throughout the session Reflect on development of teacher subject knowledge in your school. Planning sheet 2

3. What is a fraction? 3 3 4

4. What is a fraction?: multiple meanings A fraction as a division of a whole 4 into equal parts one ‘unit’a collection of ‘units’ / a set A fraction as an operator A fraction as a number (cardinal and ordinal) A fraction as a division A fraction as a ratio defining the relationship between two quantities

5. Potential misconceptions Misconception 1: Fractions are read as pieces rather than equal part/ whole relationships

6. Misconception 2: Fractional pieces have to be congruent (the same shape) to be the same fraction.

7. Misconception 3: Identical fractions of different ‘wholes’ are not the same

8. How do the following activities draw out the concept of equal parts within the context of fractions?

9. 9 With your partner take 3 pieces of A4 paper. Fold one in half and tear it into two equal parts. Take one each. Stick your small piece into the middle of an A4 piece so that it looks like a picture inside a frame. What fraction is the picture frame of the whole A4 piece of paper? Prove your conjecture. Picture Frame

10. 10 Picture Frame What fraction of the whole is the picture frame?

11. 11 Fair Feast http://nrich.maths.org/2361 http://nrich.maths.org/2361 Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have? What if three others join them for the picnic? What will each one have?

12. ●Consider how often do we vary the 'whole'? Is it always 'one'? nrich maths:Chocolate: the 'whole' is one, two or three bars of chocolate. Learners have to make a decision about the best table to stand at if the chocolate on it is shared between everyone at that table. Encouraging children to record their ideas themselves helps us 'see' their thinking and assess what they are doing.Chocolate: the 'whole' is one, two or three bars of chocolate. Learners have to make a decision about the best table to stand at if the chocolate on it is shared between everyone at that table. Encouraging children to record their ideas themselves helps us 'see' their thinking and assess what they are doing.

15. Partitioning

16. Where is a ? 0 1 0 2 1 4

17. Fractions as numbers: counting Pupils should count in fractions up to 10, starting from any number and using the and equivalence on the number line (e.g. 1, 1 (or 1 ), 1, 2). This reinforces the concept of fractions as numbers and that they can add up to more than one. 17 1 2 1 2 2 4 1 4 2 4 3 4

18. Counting in s 18 1 2

19. 19 Counting in s Not in the N.C. but … opportunity to generalise. 1 3

20. Making connections: the relationship between fractions and division IES Sharing 5 apples

21. Linking fractions with division Research: Students come to kindergarten with a rudimentary understanding of basic fraction concepts. They can share a set of objects equally among a group of people (i.e., equal sharing) 21 and identify equivalent proportions of common shapes (i.e., proportional reasoning). 22 The Institute of Education Sciences (IES) September 2010 Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. P12 Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.

22. Fractions: difficult but crucial in mathematics learning Can pupils who were starting to learn fractions in school use existing knowledge to help in their understanding? Year 4 and 5 pupils from 8 schools in London and Oxford- assess their awareness of two alternative ways of solving fractions problems. part–whole situations-the denominator indicates the number of equal parts into which a whole was cut and the numerator indicates the number of parts taken division situations, the numerator refers to the number of items being shared and the denominator refers to number of recipients

23. Quantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. But fractions cause difficulty to most people because they involve relations between quantities. What is ? One half of what? If Ali and Jazmine both spent of their pocket money on snacks, they may not have spent the same amount of money each. 23 1 2 1 2

24. Fractions and Division 24 Research on fractions has shown that many of the mistakes which pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One well- documented error that pupils make with fractions is to think that, for example, of a cake is smaller than because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. Because children do show good insight into some aspects of fractions when they are thinking about division, mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions. (Nunes - Fractions: difficult but crucial in mathematics learning) 1 3 1 5

25. Nunes - Fractions: difficult but crucial in mathematics learning Found that pupils were better at solving fractions problems about division than part–whole situations. The operation of division expresses a fraction in a context that makes sense to young children 25 Both the surveys and detailed analyses of pupils’ reasoning showed that primary school pupils have some insights about fractions that could be used in teaching when they solve division problems. They understand the relative nature of fractions: if one child gets half of a big cake and the other gets half of a small one, they do not receive the same amount. They also realise, for example, that you can share something by cutting it in different ways: this makes it ‘different fractions but not different amounts’. Finally, they understand the inverse relation between the denominator and the quantity: the more people there are sharing something, the less each one will get.

26. Exploring division and fractions ITP Fraction Strips 26

27. How does a fraction represent a division? 3/8 can also represent 3 divided by 8, thinking of division as ‘equal sharing between’. Consider… How can you share 3 chocolate bars equally between 8 people?

28. Addition of fractions with the same denominator 2 5 + 3 5 c.f.

29. Addition of fractions with different denominators 1 5 + 1 4 Why does this cause difficulties for pupils? What are the classic errors / misconceptions? 01:55:16 - end

30. Try these 1 4 + 1 3 1 4 + 1 5 1 4 + 2 3 1 6 + 2 3 1 7 + 2 3 2 5 + 2 3

31. Expectations for multiplication of fractions in Y6 multiply simple pairs of proper fractions, writing the answer in its simplest form (e.g. × = ) 1 4 1 2 1 8

32. Try these 1 4 × 1 3 1 4 × 1 4 1 2 × 1 3 1 4 × 2 3

33. Expectations for division of fractions divide proper fractions by whole numbers (e.g. ÷ 2 = ) 1 3 1 6

34. Try these ÷ 2 ÷ 3 ÷ 2 2 5 1 5 1 3 1 2

35. What do you notice? ÷ 2 ÷ 3 ÷ 2 2 5 1 5 1 3 1 2 1 3 1 2 × 2 5 1 5 1 3 1 2 1 2 1 2

36. Fractions in the new curriculum Use the programmes of study to identify key aspects of the fractions curriculum

37. ●Understand and recognise the concept of a fraction; ●Connect different types; ●Read, write and use the language of fractions; ●Round; ●Equivalence; ●Compare and order; ●Calculate; ●Connect to division. Fractions in the new curriculum

38. Throughout the session Reflect on development of teacher subject knowledge in your school. 38