Teaching for mastery in primary maths

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.

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.

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.

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, ⁵⁄7 + ¹⁄7 = 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.

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.

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.

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.

In this chapter, pupils further their understanding of fractions as an operator, which builds on work 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.

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, ²⁄5 + ⁴⁄5 = 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 = ⁷¹⁄100.

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, 4¹⁄2...), 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.

In this chapter, pupils identify, name and write equivalent fractions, as well as compare and order them.

Pupils compare and order fractions with denominators that are multiples of the same number, such as ⁷⁄9 and ⁵⁄12. In doing so, they understand that they must find the lowest common denominator to compare fractions.

To support pupils’ developing understanding of the concept of equivalence in the context of fractions, they use a range of both concrete and pictorial representations.​

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.

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. 

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.