Mathematics SKE Course Fractions PTSA and Marjon 2017-18 Ruth Pitt S.L.E Email: ruth.pitt@princerock.co.uk Twitter: @ruth_pitt

SKE Training Afternoon Agenda 1.30-1.40 Welcome, introductions and contents of the afternoon. 1.40-2.00 Examine 5 “Big Ideas” for developing understanding of fractions 2.00-2.30 Exploration of the progression for fractions, decimals and percentages in the new curriculum. 2.30-3.00 Models and images to support conceptual understanding in relation to fractions. 3.00-3.10 Solving problems and fraction questions. 3.10-3.20 Where to find more information NCETM videos and exemplification https://www.ncetm.org.uk/resources/43609 15.20-15.30 Evaluation 15.30 Close

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

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

Fractions represent… 2:3 A model of a division problem 2 3 = 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.

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)

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

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

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

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.

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

nrich task: “Fractional Triangles Progress into KS2

Recognising Fractions 6 10 60 100 25 = 1 100 4

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

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.

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.

KS1 SATs 2016 example

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

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

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 ?)

Counting in s 1 3

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

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

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

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

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

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

Comparing and Ordering Leads into questions like: Which is smaller 5 9 or 3 5 ? Would you rather have 5 8 of £24 or 1 3 of £42 ? Need for converting fractions to have same denominator to be able to compare fairly – leads to equivalence

Equivalence Y2 Write simple fractions for example ½ of 6 = 3 and recognise the equivalence of 2 4 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

Equivalence Concept using whole numbers: 4 + 5 = 7 + 2 = 1 + 8 = 6 + 3 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.

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

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

Equivalence 2 5 38

Equivalence 2 5 4 10 = 39

Equivalence 2 5 6 15 = 40

Equivalence 2 5 = 8 20 41

Equivalence 2 5 10 25 = 42

Equivalence 2 5 = 12 30 43

Equivalence 2 5 = 14 35 44

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

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 ≠

Calculating with Fractions Y3 Add and subtract fractions with the same denominator within one whole [for example, 5 7 + 1 7 ] 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 1 5 + 1 10 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, 1 4 × 1 2 = 1 8 ] Divide proper fractions by whole numbers [for example, 1 3 ÷ 2 = 1 6 ]

Calculating Add and subtract fractions with the same denominator 3 5 + 1 5 = 4 5 1 5 1 5 Bar model Cuisenaire representation

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

KS2 SATs 2016 example

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 + 3 10 = 9 10

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 − 3 10 = 3 10

Calculating Add and subtract fractions with different denominators 3 4 + 2 5 Cuisenaire – ½ + 1/3 example: http://www.youtube.com/watch?v=1_E_SrpyPvU Link back to equivalence activity. Activity on acetate

Calculating 3 4 + 2 5 = 23 20 = 1 3 20

KS2 SATs 2016 example

Calculating Multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1 4 × 1 2 = 1 8 ] 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)

Calculating Divide proper fractions by whole numbers 1 3 ÷ 2 = 1 6 1 4 ÷ 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.

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)

Problem Solving There are 50 books in the class library. Rachel has read 3 5 of the books. How many books has she read? 50 books 1 5 Use the previous method to solve the problem

Problem Solving Eddy spent 3 5 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 2 7 of the stickers left. How many stickers did Alice buy?

Problem Solving Max had 120 stamps in a box. He gave 1 4 of them to Ted and put 2 3 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

May has 180 beads. 2 3 of them are red and 2 5 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

2 3 of Alice’s money is equal to 1 2 of Ben’s money 2 3 of Alice’s money is equal to 1 2 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

1 4 of Carrie’s weight is equal to 3 8 of David’s weight 1 4 of Carrie’s weight is equal to 3 8 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)

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