Year 4 Progression in mathematics

Using and applying mathematics

Children solve increasingly complex word problems. Where appropriate, they use a calculator to do so. They choose the calculations that they need to do and the best way to do them. They record their methods to explain them, and interpret their answers in the context of the problem.

When they solve problems and puzzles, children make and annotate tables, diagrams or text to identify which key bits of information to use. They begin to make notes to help them to keep track of their progress. They organise and interpret their data to compare their predictions with their recorded outcomes. They analyse the data that they collect through surveys, for example, to see if more children in the school have their birthday in the summer than in the winter. Children begin to pose their own ‘What if …?’ questions that they and others investigate. They recognise the need to have a go, to see what works. They become more systematic, choosing ways to organise their work, and testing and checking solutions to select an appropriate strategy. They refine their methods of recording as they become more confident about how to start to solve a problem and more familiar with ways of recording.

Children discuss their work, explaining patterns and rules using mathematical language and symbols. They explore the number sequences arising from a given rule such as ‘double the last number and subtract 1’. Children use examples to test a general statement such as: ‘The sum of a pair of numbers in the same column of a 100-square is always even’.

Counting and understanding number

Children develop their use of number vocabulary and notation. They meet negative numbers and position positive and negative numbers on a number line. They write inequalities using the signs < and >.

Children generate number sequences given the start number and the whole-number step size, using a grid or number line to continue the sequence on and back. They use their understanding of place value to add or subtract multiples of 1, 10, 100 or 1000 to or from any four-digit whole number and to partition the numbers in different ways. They understand the importance of zero as a place holder in numbers such as 2036, partitioning it as 2000 + 30 + 6 and 1900 + 100 + 20 + 16.

Children understand the meaning of the decimal point as a separator for whole numbers and parts of a whole. They recognise the representation of tenths and hundredths and partition numbers with up to two decimal places; for example, they partition 4.75 as 4 + 0.7 + 0.05 and recognise that this is equivalent to 4 + ⁷¤₁₀ + ⁵¤₁₀₀. They write the decimal number that is equivalent, say, to four tenths and six hundredths, and position decimals on a number line. They use decimals in context. For example, they calculate in metres the amount of ribbon that is left if 65cm is cut from a 2-metre length.

Children extend their mathematical vocabulary to enable them to read and write fractions such as ‘three fifths’ and mixed numbers such as ‘four and two thirds’. They count forwards and backwards in steps of one half, one quarter and one third, and position mixed numbers on a number line. They use diagrams or paper strips to identify equivalent fractions. They construct a ‘fraction wall’ to compare fractions.

Children use practical resources to establish that one fifth of 35kg is more than one seventh of 35kg. They recognise the decimal equivalents of common fractions, including halves, tenths and hundredths.

Children use the vocabulary of ratio and proportion to describe the relationship between two quantities. For example, they describe a tiling pattern as ‘one red tile for every three blue tiles’, or say that ‘one quarter of the tiles are red’

Knowing and using number facts

Children consolidate their knowledge of number pairs that total 100. They derive sums and differences of pairs of multiples of 10, 100 or 1000 e.g. ‘How many more is 1400 than 900?’. Children use their knowledge of halves of two-digit multiples of 10 and of even numbers to 20 to calculate half of any even two-digit number. They find doubles of two-digit numbers such as 48, adding 80 and 16 to get 96. They use this knowledge to derive the doubles of 480 and 4800 and the corresponding halves of 960 and 9600.

Children learn by heart multiplication tables to 10 × 10 and derive the associated division facts. They use the vocabulary ‘multiple’ and ‘factor’ when describing relationships between numbers, for example: ‘24 is a multiple of 6’ and ‘6 is a factor of 24’. They know the multiples of numbers to 10 up to the tenth multiple. They make use of doubling and halving, for example, to work out 45 × 4 by doubling and doubling again.

Children use their knowledge of rounding to estimate and check calculations. They round measurements to the nearest unit and amounts of money to the nearest pound. They recognise that the context can influence the need to round up or down.

Children use fractions to estimate and describe proportions, saying, for example, that: ‘This container holds about half as many cubes as that one’. They use diagrams and practical resources to identify pairs of fractions that sum to 1.

Calculating

Children add or subtract mentally pairs of two-digit whole numbers, such as 38 + 47 and 83 – 35. Some of them may need to make jottings to record the steps. They draw on their ability to partition numbers and count on or back. In the case of subtraction they count on, adding 5 to 35 and 43 to 40, then adding 5 and 43 to get the difference of 48. Alternatively, they subtract 30 from 83 to get 53, and a further 5 to get 48. They discuss their methods and look for methods that they can do most easily in their heads with little or no recording. Children use these mental methods to find the missing numbers in number sentences such as £ + 54 = 86.

Children apply their mental calculation skills to add and subtract multiples of 10 and 100. For example, they work out what to add to 370 to make 1000.They find the difference between two near numbers such as 7003 and 6988 by bridging across 7000 and adding 3 and 12 to get the answer 15. Where necessary, they continue to use jottings such as a number line to support mental calculations and to record methods that they explain to other children.

Children recognise the need for conventions and rules when carrying out calculations involving more than one addition or subtraction. For example, they recognise that the answer to the calculation 9 – 5 – 3 is 1 and that the calculation is carried out from left to right – otherwise a different answer is obtained (if 5 – 3 is carried out first the answer is 7). Children test the effect that changing the order in which they carry out the steps in the calculation has on the answer. They recognise that addition can be done in any order. They apply the rule to calculations such as 12 – 17 + 19 that they carry out mentally, rearranging this to 12 + 19 – 17 to avoid negative numbers.

Children build on their understanding of place value and partitioning to refine and use written methods of recording for the addition and subtraction of two- and three-digit numbers. They always check first to see if they can do the calculations in their heads. For example, they recognise that they can work out 50 + 76 and 60 – 28 in their heads, but that to answer 341 + 176 or 213 – 76 they need to record steps to help them. They begin to understand how the methods that they use relate to each other and, for particular calculations, why some methods are more efficient than others.

Children use their skills of partitioning to support the expanded method of calculation.

As children become more confident in using expanded methods, recording as much detail becomes less essential. Some children begin to use a more compact method of recording for addition and subtraction.

Children explain, for those numbers that give a whole-number answer, the effect of multiplying and dividing by 10 and 100. They recognise that multiplying by 10 and then by 10 again is equivalent to multiplying by 100. They use the language of scaling up and down, recognising, for example, that a centimetre is 100 times smaller than a metre. They derive answers to calculations such as 30 × 5, working this out as 3 × 5 × 10 = 15 × 10 = 150. They add fives to find 31 × 5, 32 × 5, and so on, and subtract fives to find 29 × 5, 28 × 5, and so on. They find £2400 ÷ 20 by dividing by 10 and then halving. They also recognise that dividing, say, 2400 by 10 gives 240 and dividing by 10 again gives 24, and that this is the same as dividing 2400 by 100.

Children develop written methods for short multiplication and division calculations such as 74 × 4 or 87 ÷ 6. They partition the two-digit number and use a grid method for multiplying.

Children divide by subtracting multiples of the divisor. They check to see if the divisor multiplied by a multiple of 10 can be used and look for the largest possible case. For example, when they divide 64 by 4, children recognise that the answer must lie between 40 ÷ 4 = 10 and 80 ÷ 4 = 20. They use this approximation to do a calculation by partitioning the two-digit number. Children recognise that a remainder represents what is left over after a division and that it is always smaller than the divisor. They give a remainder as a whole number and make sensible decisions about rounding up or down after division according to the context of the problem.

Children apply their knowledge of multiplication and division to one- and two-step calculations involving money, measures and time. For example, they calculate the change from £5 when they buy five oranges at 35p each. They work out the number of 250ml bowls of soup that can be filled from a 2-litre pan. They use calculators where appropriate, understanding and recording the steps involved, and checking and interpreting the number displayed in the context of the question. Children use their knowledge of multiplication and division to find fractions of numbers and quantities. For example, they find ¹¤₅ of £40 by dividing 40 by 5. They use practical resources or diagrams to find proper fractions. For example, they use squared paper or an interactive whiteboard to work out ⁵¤₈ of a 12-by-4 rectangle, first working out and colouring in the squares that represent one eighth of the rectangle and then finding and colouring four more such groups.

Understanding shape

Children describe and classify an increasing range of 2-D and 3-D shapes. They recognise and draw regular and irregular polygons using templates or grids. They identify lines of symmetry in the shapes and other properties such as equal sides and equal angles. They visualise familiar 3-D shapes from 2-D drawings and make and identify the nets of common solids such as the cube, cuboids and simple prisms.

Children know that angles are measured in degrees and that a quarter turn or one right angle is 90 degrees. They use this to work out that one whole turn is four right angles or 360 degrees.

They calculate that half a right angle is 45 degrees and one third of a right angle is 30 degrees. They compare and order angles less than 180 degrees and begin to estimate their size.

Children talk about horizontal and vertical lines and identify horizontal and vertical faces or edges of cuboids or prisms placed on a table. They use the eight compass points to describe direction and identify squares on a grid using a code such as A3. They interpret movement about a grid; for example: ‘If you move south east from square C6, which squares will you pass through?’

Measuring

Children choose and use standard units to measure. They record estimated and measured lengths, weights and capacities using decimal notation where appropriate, e.g. 1.35 metres. They understand and use the prefixes kilo-, centi- and milli- to write, say, 4125 grams as 4 kilograms and 125 grams. They relate standard measures to real-life contexts, as in: ‘My hand is about 6cm wide’

Children interpret partly numbered scales. They work out the value of each division and count between the numbered divisions to confirm their interpretation. They record their readings using whole numbers or decimals, for example as 5.6 grams or 1.45 litres.

Children read times to the nearest minute from analogue and digital clocks. They use 12-hour clock notation and am and pm, recognising that 2:45 pm is the same as a quarter to three in the afternoon. They choose the unit of time to estimate or measure time intervals. They interpret timetables and use them to calculate, say, how long a journey should take, using a time line as support.

Children understand ‘perimeter’ and ‘area’ and find the perimeter and area of a rectangle by measuring the sides and by counting squares. They create shapes on a square grid and find the area, giving the answer in square units..

Handling data

Children collect the information they need to answer a question. They organise the data and present and analyse it to look for patterns, trends or unusual outcomes. They extract data from tables, graphs and charts in order to highlight and illustrate points. For example, from temperature data collected in science or population data in geography, children describe the trend over time or look out for dips or peaks that signal some unusual event. They use a range of graphs and diagrams to present and explain their work to other children.

Children extend their skills by representing and interpreting data in pictograms where one symbol represents 2, 5, 10 or 20 units. They use bar charts with intervals labelled in 2s, 5s, 10s or 20s. They use ICT to compare the way that different scales or different diagrams can change the impact of the representations. They understand that the more items a symbol in their pictogram represents, the fewer symbols there will be. They recognise that as the size of the intervals on a bar chart changes so does the relative size of the bars, and that when an interval represents a larger number, differences in readings are less obvious.

Children continue to use two-way Venn and Carroll diagrams. They use these diagrams to display information about shapes and numbers, drawing on their knowledge and understanding of the properties.

Embedding key aspects of learning

Year 4 children are beginning to acquire the vocabulary and knowledge that enables them to talk in more depth about numbers and shapes and to explain their solutions, decisions and reasoning. This greater independence is supported by collaborative activity in which children work with others.

The introduction of written methods that build on earlier practical, mental and visual work, and consideration of why some calculation methods are more efficient than others, helps children to develop their evaluation skills.

Children extend their information processing skills. They learn more ways to organise and present information, drawing on ICT to help them. They identify patterns and pose and test conjectures. The analysis of information in written, pictorial and symbolic form takes on a more significant role, and challenges their thinking and reasoning.