Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

(Nuffield Maths 3 Teachers’ Handbook: Longman 1991) Fractions Fractions have often been considered as one of the least popular areas of maths. Many children consider the concept of fractions as ‘difficult’ and too often children have had difficulty understanding why they are carrying out a particular procedure to solve a calculation involving fractions. “It has been said that ‘fractions’ have been responsible for putting more people off mathematics than any other single topic.” (Nuffield Maths 3 Teachers’ Handbook: Longman 1991) This is probably due to confusion caused by introducing calculations involving fractions too early, when certain children still require more experience with the visual and practical aspect of creating simple fractions of shapes in order to gain a more secure understanding of what a fraction actually is. Children need to have a firm understanding of what the denominator represents and the numerator represents through the use of visual (and kinaesthetic) resources.

The mission must be to not go back to the experiences of generations before, even though the New National Curriculum risks the past problems being repeated (as it is now crammed full of calculations with fractions). Do be aware it doesn't take much work to find fractions "support" which reinforces misconceptions...

Number and Measure Can the participants find half of the counters on the table? And find half of the water on the table? One of the strengths of the new curriculum is that number and measure are developed in parallel – together a connected relationship which is better particularly when developing calculation Two parallel paths – counting where you are enumerating and measurement where you have a continuous quantity Discrete/ Continuous

Picture Frame What fraction of the whole is the picture frame? Fold one in half and tear it into two equal parts – Ask what fraction have you made This activity was done with a class of Year 2 children who could all say that the frame must be half because you need to halves to make a whole and we know the window part is half. The activity was also done with a class of Year 6 children who randomly guessed at fractions such as sixths, 12ths etc. They didn’t think it could be half because the two pieces didn’t look the same. They cut out the ‘frame’ and stuck it onto the ‘window’ and could then see that it was half and therefore saw that fractions need to be the same size but that they don’t necessarily look the same. Question to ask is.... What happens to children’s understanding in KS2 to go from the Year 2 responses to the Year 6 responses? For example, worksheets and visuals which show fractions appearing to be the same can feed such misconceptions.

How many different ways can you fold an A4 sheet to represent a half? Children often come from KS1 halving and halving again – they do not build an idea of equal parts. New National Curriculum puts thirds into KS1 for the first time so this should improve over time. In Shanghai, they always write the denominator first – the number of parts.

Build a dragon Use cubes to make a dragon that is ½ red, 1/5 blue and the rest green. What fraction is green? Make a dragon that is twice the size Describe the colours using fractions Half of a big thing is bigger than half of a little thing – children understand that. Stress the ½, not the 1/5

The Fractions Journey Counting in Fractions Finding a Fraction of an Amount Equivalence of Fractions Mixed Numbers and Improper Fractions Number operations To be covered next session Try to have consistent approaches across year groups.

Counting in Fractions Year 2 Count up in ½ and ¼ up to 10 (differentiation opportunity?) Year 3 Count up and down in tenths. Year 4 Count up and down in hundredths (huge differentiation opportunity) http://www.mathsisfun.com/numbers/number-line-zoom.html Limitations of modelling with a circle – can be laying misconceptions as children cannot create easily. Play table races, Bead bar? This is introducing the concept of a fraction as a number in itself – usefully modelled on a number line Can be shown on a number track or a number line – benefits of both Fractions needs to be understood as numbers not objects.

Finding a Fraction of an Amount Year 3 Recognise, find and write fractions of a discrete set of objects, unit fractions and non-unit fractions with small denominators. e.g. Find ½ of 16 Find ¼ of 12 Find of 15 Year 4 Solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities including non unit fractions where the answer is a whole number) - main focus tenths and hundredths. e.g. Find of 30 Find of 28 2 3 Integral to this idea is the scaling model of multiplication – three times larger, three times smaller. Children can double but not half – because they are looking it as repeated addition – it is scaling. Historically students only halved and quartered – this meant 1/3 was a disaster in KS2. Now 1/3 in KS1. 1 10 3 7

Reflection... Are circles the easiest shape to tackle fractions with?! What resources might help the pupils?

Equivalence of Fractions Year 3 Recognise and show, using diagrams, equivalent fractions with small denominators. Year 4 Recognise and show, using diagrams, families of common equivalent fractions. Visually seeing 3 6 6 8 2 5 = 4 10 http://www.mathsisfun.com/numbers/fraction-number-line.html

Don’t forget Cuisenaire! Perfect illustration of a bar model Caroline Ainsworth video – 12a Year 2 children identifying and naming fractions

Cuisenaire Rods If the orange rod has a value of one, what is the value of the yellow rod? Can you find a quarter family? How did you find it? Is there more than one way? Can you find the value of all the other rods in relation to the orange rod? I have two sweets, Sam has four times as many – scaling idea of multiplication. Represent it with cuisenaire: 1:4 so I have one quarter of the amount that Sam has. Make a fraction wall – equivalences. Start with 12 – half, how many more? Or from 12 – twelfths, sixths, thirds, etc make them notations – what is the pattern. Important because then they understand why “cancelling” works – ratio Equivalence is vital – see the ratio relationship

NCETM 2014 (National Centre for Excellence in Teaching Mathematics) It is important that learners think about relationships between fractions, rather than just trying to memorise methods for processing them. The learning of fractions should include a strong emphasis on developing reasoning skills, comparing fractional amounts in a variety of contexts and exploring equivalence. https://www.ncetm.org.uk/resources/28795 Challenge the able with reasoning

Equivalence of Fractions Year 5 Compare and order fractions whose denominators are all multiples of the same number. Which are bigger than: 2 3 7 12 5 6 7 9 Do pupils need concrete still or are they OK – bar model?

Year 6 Use common factors to simplify fractions: Use common multiples to express fractions in the same denominator. Compare and order fractions (once previous is mastered!). 8 12 12 15 4 7 2 3 7 9 5 6 4 7 2 7 4 7 2 5 Reflection - inequalities

Mixed Numbers & Improper Fractions Year 5 Convert between the two. No "tricks" to be taughts. 4 2 3 Nice Example 12 5 Poor Example If I could eat four sixths of a cake or two thirds, which would give me more? 12 fifths – 2 and two fifths – how is the teacher going to spot that they have the quotient and the remainder the right way round. Card game – equivalence and conversions

Tangram Puzzle http://nrich.maths.org/1 Equivalence – Can you work out what fraction of the square each colour is? So half is one out of two pieces, two quarters is the same because two pieces out of four is the same proportion - ratio

Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

Where now? By the next meeting, I am going to trial/action... * Where next? Progress in fractions, decimals and percentages