1. Teaching Fractions, Percentages and Proportions Presenter: Anna MacDougall annamac@waikato.ac.nz

2. Goals for today: Understand the importance of multiplicative thinking in the development of proportional reasoning. Understand the relationship between fractions and percentages Understand the difference between ratio and proportion Have strategies and ideas for teaching percentages and proportional reasoning and extending Stage 6+ students.

3. Fractional Language Denominator: bottom digit, number of parts in a whole, what is being counted (Page 11) Numerator: the top digit, shows the count, how many parts we have (Page 11) Unit Fraction: has a numerator of 1 eg ¼ (Page 11) Non-unit Fraction: has a numerator greater than 1 eg ¾ (Page 11) Improper Fraction: numerator greater than denominator eg 5/4 (Page 33) Mixed Fraction: whole number and fraction eg 1¾ (Page 33)

4. Stage 6-7  Ordering and rounding of decimals.  Ordering non-unit fractions for halves, quarters, thirds, fifths and tenths.  Use common factors to simplify fractions and find equivalent fractions  Fractions, decimals and percentages conversions for halves, quarters, thirds, fifths and tenths.  Use multiplication and addition facts to find answers to fraction problems eg. 7/8 of 64 = 56. 1/8 = 8 so 7/8 = 7x8 Use mental strategies to solve word problems  Including Place Value and Compensating from tidy numbers

5. Mystery Stars: Part to Whole ¼ of a number is 6. What is the number? If this is ½, what could the whole look like?

6. Mystery Stars: Whole to Part There are 24 sweets on the cake and you eat ¾ of the cake. How many sweets do you eat?

7. Equation to solve My Strategy/Explanation of solution Holly started with a whole basket of chocolates. She ate 5/8 of them. That left 21 chocolates for her friends to eat. How many chocolates were there to start with? My group’s most efficient strategy Equipment/diagram

8. Finding Common Denominators: 2/4 + 3/4 = 5/4 = 1 1/4 Use the fraction strips to solve this problem ⅝ + ¾ = ? ⅛⅛⅛⅛⅛ ¼¼¼¼¼¼ ⅛⅛⅛⅛⅛1

9. Multiplying Fractions This is ⅓ of one whole strip. If it is cut into quarters, four equivalent pieces, what will each new piece be called? ⅓ 1/4 x 1/3 = 1/12

10. Fractions as Operators Continued Scaffolding problems can help students work through to difficult number properties: ⅓ of 1 / 5 = ? So what is ⅔ of 1 / 5 = ? 1/5 1/15 1/ 5 1/15

11. Decimats for Decimals Decimals are special cases of equivalent fractions. The decimal system says that fractions must be expressed as tenths, hundredths, etc. Use your decimats to solve this problem: 7 ÷ 4 = ? How does your answer compare with what the calculator gives as the answer?

12. Stage 6-7 Use the Deci Pipes to first develop the concept of decimals at Stage 5. Use at Stage 6 for adding and subtracting of decimals. Double number line to develop links between fractions, decimals and percentages. 12/40 75%50%100%25% 1/43/4

13. Teaching Percentages Why do we have Percentages? It is a method of comparing fractions by giving both fractions a common denominator, namely hundredths. So it is useful to view percentages as hundredths.

14. In order to rename a common fraction as a percentage what do you need to know? = And in order to find equivalent fractions you need to be multiplicative thinking.

15. 1 / 10 10% 1 / 5 20% 1 / 2 50% 1 / 425% 1 / 812.5%

16. Estimate and find percentages of whole number amounts. E.g. Find 25% of $80 25% = 1 / 4 so 25% = 1 / 4 of 80 = $20 E.g. Find 35% of $80 (harder!) “Pondering Percentages” NS&AT 3-4.1(12-13)

17. Find 35% of 80 Let us make this into a word problem for our students to solve….. We can use: Square paper A double number line Find 10% strategy

18. Equation to solve My Strategy/Explanation of solution Wiremu wanted to buy a scooter, so he waited until it was on sale. It cost $80 and the discount was 35%. How much did Wiremu save? How much did he pay for the scooter? My group’s most efficient strategy Equipment/drawing

19. Double number line to develop links between fractions, decimals and percentages. 10% 80400 75%50%100%25% 20 60

20. Calculating Percentages Percentage strips help students to see that calculating percentages is like mapping a fraction onto a base of 100. Leonne got 18 out of her 24 shots in. What was her shooting percentage? 1020 24 0

21. Leonne’s Percentage So Leonne’s 18 out of 24 maps onto 75 out of 100 (75%). How does this relate to how you would calculate 18 / 24 as a percentage? 1020 24 0 10%20%30%40%50%60%70%80%90%100 % 0%

22. Using Percentage Strips Use your percentage strips to work out these percentage problems: Keith got 16 out of 25 for his spelling test. What percentage was that? Bubble Puzzles for quick learners. These are great as they have to create problems instead of just solving them. Renee paid $360 000 for her house.

23. Using What I Know to Calculate Percentages Mind map what I know about 240 What is 10% So what is 5% What else can you work out? Try this with 360 – so if an I pod cost $360.00 and there was a 36% discount offered, how much would the ipod cost now? Percentage games…..

24. Equation to solve My Strategy/Explanation of solution Room 13 decided to have a shared sushi lunch. There are 48 pieces of teryaki sushi and 16 pieces of avocado sushi. What percentage of the sushi was avocado sushi? My group’s most efficient strategy Equipment/diagram /table

25. Ratios and Rates What is the difference between a ratio and a rate? Both are multiplicative relationships. A ratio is a relationship between to things that are measured by the same unit, e.g. 4 shovels of sand to 1 shovel of cement (shovel vs shovel) A rate involves different measurement units, e.g. 60 kilometres in 1 hour (km vs hr) A rate involves different measurement units, )

26. Ratio and proportion RATIO compares part to part, ‘two to every three’ or ‘two for every three’ What is the ratio of red to white? What is the ratio of white to red?

27. Ratio and proportion RATIO compares part to part, ‘two to every three’ or ‘two for every three’ What is the ratio of red to white? What is the ratio of white to red?

28. Ratio and proportion RATIO compares part to part, ‘two to every three’ or ‘two for every three’ What is the ratio of red to white? one redto every two white What is the ratio of white to red? two white to every one red

29. Ratio and proportion RATIO compares part to part, ‘two to every three’ or ‘two for every three’ What is the ratio of red to white? 1 to 2 What is the ratio of white to red? 2 to 1

30. Ratios Make up this paint recipe using unifix cubes. 24 yellow cubes with 18 blue cubes (24:18). If this recipe makes a big pot of paint, what might go into smaller pots with the same colour? What is the key multiplication idea here?

31. Equivalent Ratios Equal subdivisions

32. Double Number Lines This could be shown on a double number line: 0 0 24 18 12 9 0 0 24 18 12 9 4 3

33. Paint Mixtures Make up these paint recipes. For each recipe show how it might be shown on the circular colour chart. 18 yellow 6 blue 14 yellow 21 blue

35. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? What proportion of this shape is white?

36. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? four reds out of twelve squares in total What proportion of this shape is white? eight whites out of twelve squares in total

37. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? four out of twelve What proportion of this shape is white? eight out of twelve

38. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? four / twelve What proportion of this shape is white? eight / twelve

39. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? What proportion of this shape is white? 4 12 8 12

40. Ratio and proportion PROPORTION compares a part to the whole, ‘two in every five’ (this is like a fraction) What proportion of this shape is red? What proportion of this shape is white? 4 12 8 12 1313 2323 = =

41. Contexts for proportional reasoning: Scaling, enlarging and shrinking Duplication, e.g. recipes Repeating, sequential patterns Calculating best deals Predicting probabilistic outcomes Fair sharing Measure, including unit conversion Speed Pulleys and gears Density, e.g. mass per volume or flavour of mixtures Consumption

42. How long is the short side of the grey rectangle? long is the short side of the grey rectangle? 18 cm 24 cm ? 16cm

43. Ratio tables × ⅔ Small Rectangle Large Rectangle 18 ? 24 16 12

44. Ratio tables × ¾ Small Rectangle Large Rectangle 18 ? 24 16 12 × ¾

45. Ratio and proportion Here is a recipe for raspberry ice cream. This recipe is for 8 people. Josie makes enough raspberry ice cream for 12 people. How much cream does she use? Fred makes raspberry ice cream in the same way. He uses 2½ kg of raspberries. How much sugar does he use?

46. Within measures M1M2 WeightCooking Time.510 2.5? x5 Cooking a chicken takes 10 minutes for every 500g. How long to cook a 2.5kg chook. Ratio and proportion

47. Double Number Lines It takes Cara 8 hours to groom 12 cars. Each car takes the same time. How long will it take her to groom 20 cars? 8 12 ? 20 0 0 Hours Cars Ratio and proportion

48. Equation to solve My Strategy/Explanation of solution Juice mix for camp consists of 2 cups of concentrate for every 3 cups of water. If there are 240 campers and each camper has ½ cup of juice, how much concentrate and water will be required? My group’s most efficient strategy Equipment/drawing

49. I have eight pizzas to share among three people. How much pizza does each person get?

50. Five girls share three pizzas equally and three boys share two pizzas equally. Each pizza is the same size. Who gets more pizza, a boy or a girl? How much more? Problem One

51. The elephants are in proportion. What is the height of the little elephant? Problem Two

52. Making bread! 40 loaves of bread can be made from 6 kg of flour. How many loaves can be made from 15 kg of flour? Ratio and proportion 14 bags of meal can feed 56 calves How many bags can feed 20 calves Try this!

53. Extra for Experts At Kiwi School 40% of the children are girls. At Tui School 70% of the children are girls. If the schools meet for sports day, what percentage of all the children are girls? Kiwi school has twice as many children as Tui School.

54. In Summary Children must be multiplicative to solve percentage and proportional problems They must be able to recognise common factors Give children a range of contexts in order to solve proportional problems

56. http://nrich.maths.org/public/