1. Mathematics SKE Course
Fractions
PTSA and Marjon
Ruth Pitt S.L.E

2. SKE Training Afternoon
Agenda
Welcome, introductions and contents of the afternoon Examine 5 “Big Ideas” for developing understanding of fractions Exploration of the progression for fractions, decimals and percentages in the new curriculum Models and images to support conceptual understanding in relation to fractions Solving problems and fraction questions Where to find more information NCETM videos and exemplification Evaluation Close

3. Timings for this session:
Examine 5 “Big Ideas” for developing understanding of fractions
Exploration of the progression for fractions in the new curriculum
Models and images to support conceptual understanding in relation to fractions
Identifying potential misconceptions
New NC has higher expectations for understanding of fractions – some things are now earlier (eg. equivalence in y2 PoS) adding and subtracting in LKS2 and multiplying and dividing with fractions in UKS2

9. Fractions represent… A proportion of a ‘whole’ or unit
A point on a line
A proportion of a set
Where do we use fractions in every day life?
Size of the whole can vary (also issue for some children between whole and hole
2/3 of 12

10. Fractions represent… 2:3 A model of a division problem
= 2 ÷ 3 = 2 3
A ratio
2:3
Ratio: for every 2 blue squares there are 3 red squares. Expresses a comparison between 2 sets. Another way to view it is to say the number of blue squares is 2/3 of the number of red squares.
Ratio and proportion are introduced in y6 explicitly. Allows comparison between two integers.

11. Big Ideas Recognising fractions Fractions as numbers
Comparing and ordering
Equivalence
Calculating
Go through each in turn considering progression, models, images, misconceptions, teaching ideas
(NCETM 2014)

12. Recognising Fractions
Y1 Recognise, find and name a half as one of two equal parts of an object, shape or quantity
Recognise, find and name a quarter as one of four equal parts of an object, shape or quantity
Y2 Recognise, find, name and write fractions 1/3, 1/2, 2/4, 3/4 of a length, shape, set of objects or quantity
Y3 Recognise, find and write fractions of a discrete set of objects: unit fractions and non- unit fractions with small denominators
Recognise that tenths arise from dividing an object into 10 equal parts and in dividing one–digit numbers or quantities by 10
Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators
Y4 Recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten
Y5 Recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
Recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number

13. Recognising Fractions
Developing concept of equal parts / sharing
Denominator dictating how many equal parts the “whole” has been divided into

14. Recognising Fractions
Recognising halving as the shaded parts to the whole. Can halve in different ways.

15. nrich task: “Halving”
How might you check that each was correct? Can you think of other ways to split the squares in half?
Half of the square is not a particular shape but an amount of space.

16. Relating to quantities
Halving amounts/quantities
½ of …
Halve …
Discrete and continuous quantities
¼ of …
Quarter
Link of halve and halve again - investigate

17. nrich task: “Fractional Triangles
Progress into KS2

18. Recognising Fractions6 10 60
100
25 = 1

19. Cuisenaire
Have a go at the activity the children were doing. Choose a rod and give it a value. Work out related fractions

20. Misconceptions Pupils identify this as quarters
Pupils identify the separate parts of a whole and turn that into a fraction:
3 parts green, 5 parts white = 3/5 rather than relating the part to the whole.

21. Misconceptions
Fractional pieces have to be congruent (the same shape) to be the same fraction.
Paper activity: two pieces of different coloured paper – fold one in half (A5), place on other to create frame.
Explain how you know that the different coloured parts exposed are halves.

22. KS1 SATs 2016 example

23. KS1 SATs 2016 example
Is a learned procedure but children have understanding of what is happening with the numbers

24. Fractions as numbers
Y2 Pupils should count in fractions up to 10, starting from any number and using the 1/2 and 2/4 equivalence on the number line (Non Statutory Guidance)
Y3 Count up and down in tenths
Y4 Count up and down in hundredths

25. Counting in s 1 2 Stop at intervals and ask: How many ½ now?
Can you say this a different way? (What is ½ × n ?)
How many 1/2s are there in ?? (What is m ÷ ½ ?)

26. Counting in s 1 3

27. Counting in fractions 1 1 5 2 5 3 5 4 5 5
1 1 5
1 2 5

28. Counting in fractions Building up a visual representation 12 4
One whole
Two wholes
Three wholes
1 4
1 4
How many are 7 quarters etc.
Beginning to build up idea of improper fractions and mixed fractions
12 4
1 4

29. Position on a number line
Misconception 1: half is half way between any two integers. Need to understand the number on a number line as being for example, 3 and a half (not half on its own). Also ½ is also 5/10, 3/6 etc. It’s name is dependent on context.
Misconception 2: fractions have to be less than one.
Ordinal concept of fraction

30. Comparing and Ordering
Y3 Compare and order unit fractions, and fractions with the same denominators
Y5 Compare and order fractions whose denominators are all multiples of the same number
Y6 Compare and order fractions, including fractions >1

31. Comparing and Ordering
Use < and > to write number sentences comparing 2 fractions
How to extend? Strategies used?
Use commercial resources

32. Would you rather share the pizza with 4 people or 5?
Misconceptions
The larger the denominator the larger the portion
Would you rather share the pizza with 4 people or 5?
1 4
1 5
What would happen to the size of each piece if I shared the pizza between 8 people?
1 8

33. Comparing and Ordering
Leads into questions like:
Which is smaller or ?
Would you rather have of £24 or of £42 ?
Need for converting fractions to have same denominator to be able to compare fairly – leads to equivalence

34. Equivalence
Y2 Write simple fractions for example ½ of 6 = 3 and recognise the equivalence of and 1 2
Y3 Recognise and show, using diagrams, equivalent fractions with small denominators
Y4 Recognise and show, using diagrams, families of common equivalent fractions
Y5 Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
Y6 Use common factors to simplify fractions; use common multiples to express fractions in the same denomination

35. Equivalence
Concept using whole numbers: = = = Explore how Cuisenaire can support this understanding
Starts in y2 ½ = 2/4
Build on understanding that numbers can be made in many different ways.

36. Equivalence Extend to fractions: Fraction cards Fraction wall
Commercial resources
Pizza
Parts of a circle
Parts of a square (arrays)
Look at fraction cards on tables

37. Cuisenaire to explore equivalence
Demo using representation of 12.
Explore using representation of 24, 10 etc. what equivalent fractions do you find?

38. Equivalence 2 5 38

39. Equivalence 2 5 4 10 = 39

40. Equivalence 2 5 6 15 = 40

41. Equivalence 2 5 = 8 20 41

42. Equivalence 2 5 10 25 = 42

43. Equivalence 2 5 = 12 30 43

44. Equivalence 2 5 = 14 35 44

45. Equivalence 2 5 = 16 40
Using acetate. 4 10 6 15 8 20 25 12 30 14 35 16 40
. . . 45

46. Misconceptions
Adding the same number to the numerator and denominator will give you an equivalent fraction
Conversely because the
numerators and denominators are different.
3 5
4 6 =
2 4
4 8 ≠

47. Calculating with Fractions
Y3 Add and subtract fractions with the same denominator within one whole [for example, ]
Y4 Add and subtract fractions with the same denominator
Y5 Add and subtract fractions with the same denominator and multiples of the same number, for example
Y6 Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
Multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, × = ]
Divide proper fractions by whole numbers [for example,
1 3 ÷ 2 = ]

48. Calculating
Add and subtract fractions with the same denominator
= 4 5
1 5
1 5
Bar model
Cuisenaire representation

49. Calculating
Add and subtract fractions with the same denominator
3 5 − = 2 5
1 5
1 5
Bar model
Cuisenaire representation

50. KS2 SATs 2016 example

51. Calculating
Add and subtract fractions with different denominators where one is a multiple of the other
1 5
1 10
Using the bar model to start =

52. Calculating
Add and subtract fractions with different denominators where one is a multiple of the other
1 5
1 10
Using the bar model to start
3 5 − =

53. Calculating Add and subtract fractions with different denominators
Cuisenaire – ½ + 1/3 example:
Link back to equivalence activity.
Activity on acetate

54. Calculating
= =

55. KS2 SATs 2016 example

56. Calculating
Multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, × = ]
Language for understanding:
Quarter times half makes more sense if say a quarter of half. (“of” = multiply: ½ of 12 = ½ x 12 = 12 ÷ 2) A change in the language helps children to see that the answer is going to be smaller (even though multiplying)

57. Calculating Divide proper fractions by whole numbers
1 3 ÷ 2 = ÷ 2 = 1 8
Contexts such as – I have ¼ of my pizza left and two friends come along. If I share it between them, how much will they get each? ¼ ÷ 2 = 1/8
Children build up patterns to see where the numbers come from.

59. Problems Solving
Aileen had 12 eggs. She used 9 of them to bake a cake. What fraction of the eggs was left?
12 eggs
Used 9 eggs ?
Bar representation of problem.
1 – 9/12 = 12/12 - 9/12 = 3/12 = ¼
(Singapore Maths Teacher Website)

60. Problem Solving
There are 50 books in the class library. Rachel has read of the books. How many books has she read?
50 books
1 5
Use the previous method to solve the problem

61. Problem Solving
Eddy spent of his allowance and had £32 left. How much was his allowance?
Alice bought some stickers. She gave 45 stickers to Bernice. Then she had of the stickers left. How many stickers did Alice buy?

62. Problem Solving
Max had 120 stamps in a box. He gave of them to Ted and put of the remainder in a new album. The rest he gave to his sister. What fraction of stamps did his sister receive? How many stamps did he put in his album?
120 stamps
Remainder

63. May has 180 beads of them are red and of the remainder are purple. The rest are blue. What fraction of May’s beads are purple? How many blue beads does May have?
180 beads

64. 2 3 of Alice’s money is equal to 1 2 of Ben’s money
2 3 of Alice’s money is equal to of Ben’s money. If they have a total of £14, how much money does Ben have?
Alice
£14
Ben
7 units = £14
£14 ÷ 7 = 2
1 unit = £2
Ben has £8

65. 1 4 of Carrie’s weight is equal to 3 8 of David’s weight
1 4 of Carrie’s weight is equal to of David’s weight. If their total weight is 80 pounds, how much does David weigh? 80
20 units
80 ÷ 20 = 4
1 unit = 4
4 x 8 = 32 (David’s weight)

70. Reflection and goals
Set yourself a realistic target that you could achieve before the next meeting
Share this target with someone on the table
Write this target down