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# COURSE NM309

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1. COURSE NM309 FRACTIONS, DECIMALS & PERCENTAGES TEACHING FOR UNDERSTANDING From Vince

2. OBJECTIVES Developing understandings of fractions and decimals Discuss the difficulties and misperceptions Identify strategies at different stages Discuss learning processes Discuss teaching strategies Explore activities to use in the classroom

3. Which family has more girls? The Jones Family The King Family

4. Before, tree A was 8m tall and tree B was 10m tall. Now, tree A is 14m tall and tree B is 16m tall. Which tree grew more? A B A B Before Now

5. A fishy problem • Two-thirds of the goldfish are male • There are 24 male goldfish • How many goldfish are there altogether?

6. Share your strategy • How did you do it? • Discuss your method in groups • Who taught you how to do it this way?

7. You have a fish tank containing 200 fish and 99% of them are guppies. You will remove guppies until 98% of the remaining fish are guppies. How many will you remove? The Bill Gates question

8. Report on Numeracy Project 2003 • The performance of year 7 and 8 students on fractions and decimals is well below what would be wished. • Integration of fractions with proportional reasoning would aid understanding of those topics. • Decimals are of particular concern • Decimals need to be taught using the Numeracy principles: using materials and imaging before number properties

9. Why do students have difficulty with fractions? • Rational number ideas are sophisticated and different from natural number ideas • Natural numbers can be represented individually, rational numbers cannot. • Students’ whole number schemes can interfere with their efforts to learn fractions • Students have to learn new ways to represent, describe and interpret rational numbers • Rote procedures for manipulating fractions (eg making equivalent fractions) may not be enough

10. Initial Fraction Interview: Task 1 This is three-quarters of the lollies I started with. How many lollies did I start with? Why did you choose that many lollies?

11. Initial Fraction Interview: Task 2 221221 5 3 4 8 3 3 • Which of these pairs of fractions are equivalent (have the same value)? • How did you decide?

12. Initial Fraction Interview: Task 2 Typical responses: • One-quarter is equivalent to two-eighths ‘cos ‘1 goes into 4, four times, and 2 goes into 8, four times. • If you were to simplify it (2/8) it would go down to a quarter. You just halve it. • Double one-quarter to get two-eighths. All were successful except one student who said that one third and two thirds were equivalent because ‘the bottom is the same’

13. Initial Fraction Interview: Task 3 3 = 21 10 What number do you need to write in the box so that the fractions are equivalent? How did you decide?

14. Initial Fraction Interview: Task 4 0.5 0.25 0.1 0.4 1112 2 4 10 5 Match each fraction with the equivalent decimal. How did you decide?

15. Initial Fraction Interview: Task 4 • Most confidently matched fraction and decimal equivalents for one-half and one-tenth, were less confident with one-quarter and put two-fifths with 0.4 because it was ‘just the one left’ • Difficulties arose when students were asked to choose the larger of two fractions…

16. Probing Task 1 32 5 3 Which is larger, three-fifths or two-thirds? How did you decide?

17. Probing Task 2 35 5 8 Which is larger, three-fifths or five-eighths? How did you decide?

18. Probing Task 3 33 5 4 Which is larger, three-fifths or five-quarters? How did you decide?

19. Probing Task 4 • Pick one of the tasks where the student was incorrect. Hand the student one card and a number line marked 0 to 1. ‘Place this fraction on the number line.’ ‘How did you decide?’ • Hand the student the second card ‘Place this fraction on the number line.’ ‘What did you find when you placed your fractions on the number line?’

20. Misconceptions 1: ‘gap’ thinking Interviewer: ‘Which is larger: 3 or 5 ? 5 8 Student 1: ‘Three-fifths is larger because there is less of a gap between the three and the five than the five and the eight’.

21. Misconceptions 2: ‘comparing to a whole’ thinking Interviewer: ‘Which is larger: 3 or 5 ? 5 8 Student 2: ‘Three-fifths is larger because it is two numbers away from being a whole and five-eighths is three away from being a whole’.

22. Misconceptions 3: ‘larger is bigger’ thinking Interviewer: ‘Which is larger 2/3 or 3/5 ? Student 3: 261218 3 9 18 27 3 61218 5 10 20 30 18 is larger than 18 because 30 > 27 30 27

23. Probing with student A Chose 3 as larger than 3 4 5 Int: ‘Can you do it another way?’ A: ‘I automatically said it.’ He was given a sheet with empty number lines

24. Further probing with student A A placed 3 close to 1. 4 On the number line he put 3 twice as far away from 1 as 3 5 4 0 3/4 1 0 3/5 1

25. Probing with student B Student B said correctly that 2/3 was larger than 3/5. His reason was that three-fifths is ‘two numbers away from being a whole and two-thirds is one number away from being a whole’ He applied the same reasoning to 3/5 and 5/8 arguing that ‘three-fifths must therefore be bigger’

26. Probing with student B Int: ‘Think about 2/3 and 3/4.’ B: ‘I think they are equal. Not just because they are one away form being a whole. This (3/4) is 75% and 2/3 is about 75%.’ He didn’t have any idea of how he could check how close 2/3 was to 75%

27. Probing with student B He was given an empty number line. He marked the number line in fifths. On the second number line he marked one-half, one-quarter and three-quarters by eye. From his diagram he concluded that ¾ was bigger than 3/5 He reiterated that ¾ was 75% and used a calculator to show that 3/5 was 60%

28. Probing with student B 0 1 1/5 3/5 0 1 ¼ ½ ¾

29. Probing with student B To compare 3/5 and 5/8, B subdivided the second number line from quarters into eighths by eye He then said ‘5/8 is bigger- it is a bit ahead of 3/5. My old method doesn’t work.’ Int: ‘Consider one-half and four-eighths’ B: ‘ my old method would say that ½ is bigger but they are the same’

30. Probing with student C To compare 3/5 and 2/3, C said ‘Both go into 15’ and then wrote 2/3 as 10/15, and 3/5 as 9/15. To compare 3/5 and 5/8, C first said that ‘3/5 is bigger by one’. He then converted both fractions to the same denominator (24/40 and 25/40) and said that 5/8 was bigger.

31. Probing with student C He converted 3/5 and ¾ to 12/20 and 15/20 and correctly concluded that ¾ is bigger. Using number lines to compare ¾ and 3/5, he divided the first number line by eye into quarters and marked one-half and three-quarters. He placed one-half on the number line below in a corresponding position. He said that ‘three-fifths is smaller than three-quarters and marked three-fifths to the right of one-half and the left of three-quarters on the number line

32. Probing with student C 0 1 ½ ¾ 0 1 ½ 3/5

33. Probing with student C He placed 3/5 and ¾ approximately where we would expect.On a pencil and paper test his response would be OK… However it was not clear to the interviewer why student C had placed the fractions where he did. Further probing was required.

34. Further Probing with student C Int: ‘Can you place 3/5 on the number line? Int: ‘Where would 1/5 be?’ C: ‘one-fifth is more than one-half (I think)’ He then placed one-fifth to the right of one-half. Int: ‘where would one-third and one-quarter be on the number line?’ He placed these two fractions in between one-half and one-quarter.

35. Further Probing with student C 0 1 ½ 1/3 1/4 1/5

36. Findings Procedural competencecan disguise whole number thinking about fractions • eg scaling up to equivalent fractions is a rote technique and students may relate new numerators and denominators as discrete whole numbers Whole number thinking • Treats numerators and denominators as discrete whole numbers (gap thinking and larger is bigger) • Treats the ‘gap’ as a whole number not a fraction

37. Conclusions To overcome whole number thinking students need to: • Make multiple representations of fractions using discrete and continuous quantities • Use a number line to represent and compare fractions • Check results and estimate answers • Deal explicitly with whole number thinking

38. Models for fractions Discrete models Sets for counting counters, blocks, beans Continuous models Area for dividing and shading circles, triangles, rectangles Number lines rope and paper strips for folding double number lines

39. FRACTION NUMBER SENSE Developing an understanding of Fraction includes: Representing the fraction as an expression of a relationship between a part and a whole and relationships among parts and wholes. Regardless of the representation used for a fraction and regardless of the size, shape, colour, arrangement , orientation, and the number of equivalent parts, the student can focus on the relative amount Recognising that in the symbolic representation of a fraction the denominator indicates how many parts the whole has been divided into, and the numerator indicates how many parts of the whole have been chosen

40. FRACTION NUMBER SENSE Developing an understanding of Fraction Number Sense includes five different but interconnected subconstructs: (Kieran 1976,1980)

41. FRACTION NUMBER SENSE 1 • Part-Whole, • e.g. ‘3 parts out of every 4’ CLASSROOM EXAMPLES Fold a strip of paper into four equal parts (quarters). What are three of these called? Fold each quarter into three equal parts. What are the new parts called? What are three of these new parts called?

42. WHAT CAN YOU SEE ?

43. Can You See It? Can you see 3/5 of something? Can you see 5/3 of something? Can you see 2/3 of 3/5? Can you see 1 divided by 3/5? Can you see 3/5 divided by 2?

44. Big Stix Chocolate Bar Half the candy bar is how many sticks? 2 sticks is what part of the bar? If you have half and I have 1/3, who has more? How much more? How much is half and 1/3 together? What part remains for someone else? How much of the candy bar is half of a third? How many times will 1/3 fit into ½?

45. 2 • Operator, i.e. • ‘3/4 of something’ FRACTION NUMBER SENSE • MEANING 3/4 gives a rule that tells how to operate on a unit (or the result of a previous operation), that is find 3/4 of something. CLASSROOM EXAMPLES A photo measures 26cm x 15cm. You want a copy made which has each side three quarters of its original length. How big will the copy be? You have a collection of bubble gum cards. You divide the collection into 4 equal piles and give your friend three of the piles. How much of the whole collection do you give them?

46. Thinking Up And Down

47. 3 • Ratios and Rates, i.e. • ‘3 parts of one thing to 4 parts of another’ FRACTION NUMBER SENSE • MEANING 3:4 means 3 parts of A to 4 parts of B, where A and B are of like measure (ratio) or of different measure (rate) CLASSROOM EXAMPLES Sally mixes 12 tins of yellow paint with 9 tins of red paint. Tane mixes 8 tins of yellow paint with 6 tins of red paint. Each tin holds the same amount. Whose paint is the darkest shade of orange? How do you know?