Number. Fractions. Teaching for mastery in primary maths

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1 Number Fractions Teaching for mastery in primary maths

2 Contents Introduction Introduction to halves and quarters Introduction to fractions Introduction to comparing, ordering and equivalent fractions Introduction to adding and subtracting fractions Introduction to finding fractions of an amount Common equivalent fractions Add and subtract fractions with the same denominator Fractions of an amount Compare, order and simplify fractions Introduction to adding and subtracting fractions with different denominators Multiply proper fractions and mixed numbers by whole numbers Adding and subtracting fractions with different denominators Multiply and divide fractions 9 2

3 Introduction As well as being useful in a real world context, the ability to manipulate and calculate with fractions is vital for pupils later success with algebra. In these chapters, pupils progress from working with unit fractions (where the numerator is 1) and fractions with small denominators, to comparing, ordering and simplifying any fraction. They also learn to add and subtract any fraction, multiply proper fractions and mixed numbers by integers, multiply pairs of proper fractions and divide proper fractions by integers. Pupils learn that fractions can be a number in themselves (ie, that it is possible to locate 3 5 at a particular point on a number line) and are also an operator (so, you can find 3 5 of something). When they begin to work with decimals and percentages, they realise that the dual nature of fractions (ie, that they are both a number and an operator) is what makes them distinctive. Decimals, for example, are generally used as numbers and rarely as operators ie, you can find 0.6 of something, but tend not to. Percentages, on the other hand, are more often used as operators, rather than as numbers ie, you could say that 60% is the number 0.6, but you usually wouldn t. Throughout this learning journey, it is vital that pupils build their understanding of fractions using varied concrete and pictorial representations. It is important that the use of these representations should not just be confined to the early chapters. 3

4 Chapter 1: Introduction to halves and quarters In this chapter, pupils are introduced to halves and quarters. Pupils know that a half is one of two equal parts of an object, shape or quantity and that a quarter is one of four equal parts of an object, shape or quantity. They can find halves and quarters of a shape or a number line marked from 0 to 1. In doing so, they learn that fractions can be a part of a whole. They also connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, by finding half or a quarter of a length, quantity or set of objects. Chapter 2: Introduction to fractions In this chapter, pupils begin to learn about how fractions can be numbers in their own right and how they can also be operators. Fractions as numbers Pupils recognise the equivalence of 2 4 and 1 2. They count in fractions up to 10, starting from any number (eg, 11 4, 12 4 (or 11 2), 13 4, 2) and, as such, can say where they are positioned on a number line. This reinforces the concept of fractions as numbers and that they can add up to more than 1. Fractions as operators Pupils use 1 3, 1 4, 2 4 and 3 4 to describe fractions of discrete and continuous quantities and write simple fractions (eg, 1 2 of 6 = 3). They connect unit fractions to equal sharing and grouping, to numbers when they can be calculated and to measures when finding fractions of lengths, quantities, sets of objects or shapes. They meet 3 4 as the first example of a non-unit fraction. Pupils may also become familiar with 2 3, as well as 2 2, 2 3 and 4 4, to support them in understanding the role played by the numerator and denominator. 4

5 Chapter 3: Introduction to comparing, ordering and equivalent fractions In this chapter, pupils recognise and use unit fractions and non-unit fractions with small denominators as numbers. They also compare and order unit fractions and fractions with the same denominators. Pupils begin to understand unit and non-unit fractions as numbers on the number line and deduce relationships of size and equivalence between them. They work with fractions greater than 1 and relate fractions to measure. It is vital that fractions are explored through concrete and pictorial means to secure understanding. They also begin to use diagrams to recognise and show equivalent fractions with small denominators. There is a particular focus on tenths in this chapter. Pupils count up and down in tenths. They connect tenths to division, recognising that they arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10. They also connect tenths to place value and decimal measures. Chapter 4: Introduction to adding and subtracting fractions In this chapter, pupils build on their understanding of simple fractions, and the knowledge that fractions can be a number in themselves, by starting to calculate with them. This chapter explores addition and subtraction of fractions with the same denominator within one whole (eg, = 6 7). This should be introduced using different concrete examples and visual representations so that pupils have a secure understanding of what happens when they add or subtract two fractions. 5

6 Chapter 5: Introduction to finding fractions of an amount In this chapter, pupils start to understand the relationship between unit fractions as operators (ie, as fractions of something) and division by integers. Pupils will recognise, find and write fractions of a discrete set of objects, in the form of unit fractions and non-unit fractions with a small denominator. They should use visual representations, such as the bar model, to help them understand the process they go through to find a simple fraction of a set of objects. This will help them in later chapters. Chapter 6: Common equivalent fractions In this chapter, pupils extend the use of the number line to connect fractions, numbers and measures. Building on introductory work on equivalence in chapter 3, pupils use diagrams to recognise and show families of common equivalent fractions. They use factors and multiples to recognise equivalent fractions and simplify where appropriate (eg, 6 9 = 2 3 or 1 4 = 2 8). Pupils also look in depth at tenths and hundredths. They count up and down in tenths and in hundredths, and recognise that hundredths arise when dividing an object by 100 and when dividing tenths by 10. Tenths and hundredths are recorded in fraction notation; decimal notation is introduced later, in chapter 1 of decimals. Chapter 7: Add and subtract fractions with the same denominator In this chapter, pupils build on their understanding of the addition and subtraction of fractions with the same denominator within one whole, by working with fractions that are greater than one whole. To further develop pupils understanding of what happens when adding and subtracting fractions from chapter 4, they should use different concrete examples and visual representations. It is essential that pupils are not just told to add the numerators and keep the denominators the same. They should be able to explain, using diagrams, how they have come to an answer when adding or subtracting two given fractions. 6

7 Chapter 8: Fractions of an amount In this chapter, pupils further their understanding of fractions as an operator, which builds on twork from chapter 5. Pupils find a fraction of an amount, including non-unit fractions of amounts, where the answer is a whole number. As in earlier chapters, they should use representations, such as the bar model, as a visual aid in helping them to understand the process of finding a fraction of an amount. Too often pupils are told to "divide by the numerator and multiply by the denominator, without any understanding of why this is the case. They should explore this in detail and be encouraged to solve more-involved problems, in order to check their understanding. Chapter 9: Compare, order and simplify fractions In this chapter, pupils compare and order fractions whose denominators are all multiples of the same number. As in previous chapters, pupils should use diagrams to explain their reasoning and understanding. This will help develop a deeper understanding of the content. Although they have already worked with fractions greater than one whole, pupils start to explore the connection between improper fractions and mixed numbers. They recognise mixed numbers and improper fractions, convert from one form to the other and write mathematical statements greater than 1 as a mixed number (eg, = 6 5 = 11 5). The rest of this chapter explores the links between decimals and fractions. Chapter 1 of decimals, which introduces decimal notation, is therefore a necessary prerequisite for this. Explicit links are made between tenths, hundredths and thousandths in their decimal and fraction notation. Pupils read and write decimal numbers as fractions, eg, 0.71 =

8 Chapter 10: Introduction to adding and subtracting fractions with different denominators In this chapter, pupils continue to practise counting forwards and backwards in simple fractions. Pupils should recognise and describe linear number sequences (eg, 3, 31 2, 4, ), including those involving fractions, and find the term-to-term rule in words (eg, add 1 2). They use their understanding of equivalent fractions to add and subtract fractions with the same denominator and denominators that are multiples of the same number. They extend their understanding of adding and subtracting fractions to include calculations that exceed 1 as a mixed number. Chapter 11: Multiply proper fractions and mixed numbers by whole numbers In this chapter, pupils extend their understanding of fractions of an amount, as well as their knowledge of multiplying fractions. Pupils extend their knowledge of fractions as an operator by finding fractions of numbers and quantities where the answer may not always be a whole number. They should explore this through the use of visual concepts, such as the bar model and other diagrams. This will help them solve more-involved problems. Supported by materials and diagrams, pupils multiply proper fractions and mixed numbers by whole numbers. They see the connection between multiplication and repeated addition and start to recognise the relationship between multiplying and finding a fraction of an amount. Pupils go on to solve problems involving multiplication and division, including scaling by simple fractions. 8

9 Chapter 12: Adding and subtracting fractions with different denominators In this chapter, pupils draw on all of their previous knowledge to confidently add and subtract fractions with different denominators. It is essential that pupils are secure in their understanding of what a fraction is and in the function of a fraction as division, as an operator, as a number and as a decimal. Pupils use their knowledge of equivalent fractions, and factors and multiples, to simplify fractions of any size. They then compare and order any number of fractions and build on their knowledge of fractions as division to be able to calculate decimal equivalents (eg, 3 8 = 0.375). They also add and subtract any fractions using the concept of equivalence. Concrete objects and visual representations should be used throughout to help explain key concepts and allow pupils to make the links between the calculations. Chapter 13: Multiply and divide fractions In this chapter, pupils build on their understanding of calculating with fractions. Pupils will be able to multiply a pair of proper fractions and use concrete and pictorial representations to explain and justify their answer. They may, for example, relate it to area. Once pupils are confident with multiplication, they start dividing proper fractions by integers. Again, they use diagrams to explain and justify their answer. Copyright TES Global Ltd

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