1. Fractions Louise Addison

2. Fraction starter One player is dots the other is crosses Number line from 0 to 6 Roll 2 dice and form a fraction, place this on number line (use materials if necessary) Aim is to get 3 marks uninterrupted by your opponent’s marks on the number line. If a player chooses a fraction that is equivalent to a mark that is already there they lose a turn.

3. Fractions in the new curriculum Level 4 Number strategies and knowledge NA4-2 Understand addition and subtraction of fractions, decimals, and integers. NA4-3 Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals. NA4-4 Apply simple linear proportions, including ordering fractions. NA4-5 Know the equivalent decimal and percentage forms for everyday fractions. NA4-5 Know the relative size and place value structure of positive and negative integers and decimals to three places. Probability S4-4Use simple fractions and percentages to describe probabilities.

4. Level 5 Number strategies and knowledge NA5-3 Understand operations on fractions, decimals, percentages, and integers. NA5-4 Use rates and ratios. NA5-5 Know commonly used fraction, decimal, and percentage conversions. Patterns and relationships NA5-8 Generalise the properties of operations with fractional numbers and integers. Probability S5-4Calculate probabilities, using fractions, percentages, and ratios.

5. Level 6 Number strategies and knowledge NA6-2 Extend powers to include integers and fractions. Patterns and relationships NA6-6 Generalise the properties of operations with rational numbers, including the properties of exponents.

6. A closer look… NA4-2 Understand addition and subtraction of fractions, decimals, and integers. NA4-3 Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals. NA5-3 Understand operations on fractions, decimals, percentages, and integers.

8. 5 views of fractions 3 over 73 : 7 3 out of 7 3 ÷ 7 3 sevenths

9. Implications for each… 3 pizzas are divided amongst 4 people. How much of a pizza does each person get? 3 out of 7

10. How would each of the views solve this problem? of 42 3 over 7 3 : 7 3 out of 7 3 ÷ 7 3 sevenths 3 out of 7 of 42 - ok if factor of 7 0.42857143 of 42 3 over 7 of 42? 1 seventh is 6 so 3 sevenths are 18 3:7 split 12.6 : 29.4

11. How would each of the views solve this problem? + 3 over 7 3 : 7 3 out of 7 3 ÷ 7 3 sevenths 0.42857143 + 0.21428571 6 out of 21 3:7 + 3: 14 = 6:21 3 over 7 + 3 over 14

12. How would each of the views solve this problem? 3 over 7 3 : 7 3 out of 7 3 ÷ 7 3 sevenths 1 - 0.42857143 1-3 out of 7 = -2/7 (or 2/7) 1 - 3:7? 1 - 3 over 7 = -2/7 (or 2/7) 1 - 3 sevenths = 4 sevenths

13. How would each of the views solve this problem? of 3 over 7 3 : 7 3 out of 7 3 ÷ 7 3 sevenths 0.42857143 of 0.333333333 3 out of 7 of 1 out of 3 3:7 of 1:3 3 over 7 of 1 over 3 3 sevenths of 1 third 1 third, split into 7 pieces gives ‘21ths’ So is three 21ths (3/21)

14. Key ideas of fractions

15. Fractional vocabulary One half One third One quarter Don’t know

16. Implications for teaching Use words (pattern and meaning needs to be taught) Always refer to the ‘whole’ Modelling with covered unifix cubes This needs to be understood before decimal fractions can be taught

18. NA5-5 Know commonly used fraction, decimal, and percentage conversions. 3  4 3  8

21. Use fraction strips to work out:

22. Two more questions…

23. Also…

26. Algebraic generalisation NA4-8 Generalise properties of multiplication and division with whole numbers. NA5-8 Generalise the properties of operations with fractional numbers and integers. NA6-6 Generalise the properties of operations with rational numbers, including the properties of exponents.

27. The EGG technique Explain the strategy Give other examples Generalise using Algebra

28. Example:

29. Task 1

30. Lightbulb Moments How useful is the algebra this generates? How could you use this in your classroom? Who could you use this with? What connections between ideas can you make? What thinking is involved? What issues could arise...

31. Task 2 15 + 16 = 15 + 15 +1 = 2  15 + 1 19 + 20 = 20 + 20 – 1 = 2  20 – 1 9 + 10 + 11 = 9 + (9+1) + (9+2) = 3  9 + 3 9 + 10 + 11 = (10 – 1) + 10 + (10+1) = 3  10 9 + 10 + 11 = (11 – 2) + (11 – 1) + 11 = 3  11 – 3

32. Task 3

33. Task 4

34. Task 5 32  42 = 30  40 + 2  40 + 30  2 + 2  2 32  48 = 30  50 + 2  50 + 30  -2 + 2  -2 39  42 = 40  40 + -1  40 + 40  2 + -1  2 39  49 = 40  50 + -1  50 + 40  -1 + -1  -1

35. Task 6 9  9  9  9  9  9  9 = 9 7 9 2  9 5 = (9  9)  (9  9  9  9  9) = 9 7 9 7 = 9  9  9  9  9  9  9 9 5 9  9  9  9  9 = 9 2 (9 4 ) 3 = 9 4  9 4  9 4 = 9 12

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