Teaching for mastery in primary maths

In this chapter, pupils are introduced to the numbers zero to 10. It is essential that time is invested in this content so that pupils’ knowledge is secure before moving on.

Pupils count forwards and backwards within 10 and up to 10 objects. They begin to compare quantities of objects by using language such as “equal to”, “more than”, “less than”, “fewer”, “most” and “least”. Given a number, they also identify “one more” or “one less”.

Throughout this chapter, pupils read and write numbers to 10 in numerals and words. They ought to experience many different representations of numbers and should represent numbers using objects. For example, they may use cubes or Dienes to feel and see the size of a number and make comparisons. Additionally, they may use ten frames to represent numbers, as well as pictorial representations, such as the number line.

Although many pupils will already seem able to perform much of the above, this apparent fluency should not be mistaken for the required depth of conceptual understanding.

In this chapter, pupils are introduced to the numbers 11 to 20. It takes a similar approach to the previous chapter in that pupils continue to practise counting, comparing and ordering numbers.

The key to this chapter, however, is that pupils start to partition numbers above 10 into tens and ones. They use part-whole diagrams to reinforce their understanding and, as before, work with many different concrete and pictorial representations of numbers.

Pupils continue to use language, including “equal to”, “more than”, “less than”, “fewer”, “most” and “least”. They count forwards or backwards to 20, beginning at 0 or 1, or from any given number. They can also identify “one more” and “one less” than a given number.

This chapter lays the essential foundations of place value, as pupils begin to recognise the place value of each digit in a two-digit number (ie, in tens and ones).

In this chapter, pupils extend their representational work to include numbers up to 40 or 50. This helps to reinforce the place value of each digit in a two-digit number (ie, in tens and ones).

Pupils have opportunities to identify and represent numbers using objects, such as counters on a ten frame, cubes, Dienes and pictorial representations, including the number line and part-whole diagrams.

They continue to use “equal to”, “more than”, “less than”, “fewer”, “most” and “least” when talking about number. They count forwards and backwards to 40, beginning with 0 or 1, or from any given number. They can also identify “one more” and “one less” than a given number.

This chapter builds on the essential foundations of place value, in preparation for working with numbers to 100.

In this chapter, pupils begin to recognise place value in numbers beyond 40 or 50 by reading, writing, counting and comparing numbers up to 100, supported by objects and pictorial representations.

Pupils count forwards and backwards to and across 100, beginning with any given number including 0 and 1. They also count up to 100 objects. This is not simply a case of chanting the numbers by rote. Rather, pupils gain an understanding of a number, such as 68, by representing it in many different ways. They identify and represent numbers using objects such as Dienes, cubes and pictorial representations, including the number line and part-whole diagrams. They may also use a place value grid to show the value of each digit in a number.

In line with expectations in literacy, pupils read and write numbers to at least 100 in numerals and in words and use mathematical language, such as “equal to”, “more than”, “less than”, “fewer”, “most” and “least”.

In this chapter, pupils recognise the place value of each digit in a two-digit number (ie, in tens and ones) and begin to understand zero as a place holder.

This occurs through frequent opportunities to identify, represent and estimate numbers to 100 using different representations, including the number line.

Pupils partition numbers in different ways (eg, 23 = 20 + 3 and 23 = 10 + 13). They show this using part-whole diagrams or using Dienes, counters, cubes or place value grids to support their understanding.

Much of the mathematical content of this chapter involves deepening pupils’ understanding of concepts encountered previously. This might be through the introduction of a greater variety of representations (either pupils moving interchangeably between or selecting their own representations), practising for greater fluency and independence with language and asking and answering increasingly sophisticated mathematical questions.

This chapter also sees the introduction of inequality notation, as pupils compare and order numbers from 0 up to 100 using <, > and = signs.

In this chapter, pupils identify, represent and estimate numbers to at least 1,000 (ie, hundreds, tens and ones) using a variety of representations, including those related to measure.

They compare and order numbers up to 1,000 and they apply partitioning related to place value (eg, 146 = 100 + 40 + 6, 146 = 130 + 16). As ever, it is important that concrete and pictorial representations are used throughout.

Many children have been learning to say which number is “one more” or “one less” than a given number since before starting school; first with numbers within 5, then within 10, then within 100 and now within 1,000. This is extended further in this chapter by asking pupils to find 10 and 100 more or less than a given number. This helps pupils deepen their understanding of place value, particularly when working with numbers that move into the next 10 or 100.

In this chapter, pupils’ recognition of place value to four-digit numbers (ie, thousands, hundreds, tens and ones) is extended.

As always with work on number and place value, it is important that pupils identify, represent and estimate numbers using different representations. Once again, it is important that concrete and pictorial representations are used throughout.

Pupils learn to round to the nearest 10, 100 or 1,000 and connect estimation and rounding numbers to the use of measuring instruments. In this chapter, pupils are also asked to find 1,000 more or less than a given number in addition to finding 1, 10 and 100 more or less.

In this chapter, pupils extend their understanding of the number system to include larger integers. The skills and concepts build on those from all previous number and place value chapters.

Pupils learn to read and write numbers to at least 1 million. They determine the place value of each digit, using this to order and compare numbers. They count forwards or backwards in steps of powers of 10 and round to the nearest 10, 100, 1,000, 10,000 and 100,000.

In this final chapter, pupils work with any sized numbers in a variety of contexts. This is a chance for them to explore real-life situations and solve practical problems within authentic contexts.

Pupils will have solved plenty of real-life problems in previous chapters, but as many very large numbers appear in science and nature, opportunity can now be sought to explore these larger numbers across the curriculum. 

All of the work in this chapter reinforces the concepts from previous ones. For example, pupils should still be able to determine the place value of each digit within a number, compare and order numbers, as well as finding 10, 100 and 1,000 more or less than a given number.