Statutory guidance

# mathematics programmes of study

## Number - number and place value

Pupils should be taught to:

• read, write, order and compare numbers to at least 1,000,000 and determine the value of each digit

• count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000

• interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through 0

• round any number up to 1,000,000 to the nearest 10, 100, 1,000, 10,000 and 100,000

• solve number problems and practical problems that involve all of the above

• read Roman numerals to 1,000 (M) and recognise years written in Roman numerals

## Notes and guidance (non-statutory)

Pupils identify the place value in large whole numbers.They continue to use number in context, including measurement. Pupils extend and apply their understanding of the number system to the decimal numbers and fractions that they have met so far.They should recognise and describe linear number sequences (for example, 3, 3 , 4, 4 …), including those involving fractions and decimals, and find the term-to-term rule in words (for example, add ).

## Number - addition and subtraction

Pupils should be taught to:

• add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)

• add and subtract numbers mentally with increasingly large numbers

• use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy

• solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

## Notes and guidance (non-statutory)

Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics appendix 1).They practise mental calculations with increasingly large numbers to aid fluency (for example, 12,462 – 2,300 = 10,162).

## Number - multiplication and division

Pupils should be taught to:

• identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers

• know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

• establish whether a number up to 100 is prime and recall prime numbers up to 19

• multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

• multiply and divide numbers mentally, drawing upon known facts

• divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context

• multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000

• recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)

• solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

• solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign

• solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

## Notes and guidance (non-statutory)

Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics appendix 1). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.They use and understand the terms factor, multiple and prime, square and cube numbers.Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 = = 24 r 2 = 24 = 24.5 ≈ 25).Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres.Distributivity can be expressed as a(b + c) = ab + ac.They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10).Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example 13 + 24 = 12 + 25; 33 = 5 x ?).

## Number - fractions (including decimals and percentages)

Pupils should be taught to:

• compare and order fractions whose denominators are all multiples of the same number

• identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths

• recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, + = = 1 ]

• add and subtract fractions with the same denominator, and denominators that are multiples of the same number

• multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams

• read and write decimal numbers as fractions [for example, 0.71 = ]

• recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents

• round decimals with 2 decimal places to the nearest whole number and to 1 decimal place

• read, write, order and compare numbers with up to 3 decimal places

• solve problems involving number up to 3 decimal places

• recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per 100’, and write percentages as a fraction with denominator 100, and as a decimal fraction

• solve problems which require knowing percentage and decimal equivalents of , , , , and those fractions with a denominator of a multiple of 10 or 25

## Notes and guidance (non-statutory)

Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions.They extend their knowledge of fractions to thousandths and connect to decimals and measures.Pupils connect equivalent fractions > 1 that simplify to integers with division and other fractions > 1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions.Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1.Pupils practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number.Pupils continue to practise counting forwards and backwards in simple fractions.Pupils continue to develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities. Pupils extend counting from year 4, using decimals and fractions including bridging 0, for example on a number line.Pupils say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and are confident in checking the reasonableness of their answers to problems.They mentally add and subtract tenths, and one-digit whole numbers and tenths.They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1).Pupils should go beyond the measurement and money models of decimals, for example, by solving puzzles involving decimals.Pupils should make connections between percentages, fractions and decimals (for example, 100% represents a whole quantity and 1% is , 50% is , 25% is ) and relate this to finding ‘fractions of’.

## Measurement

Pupils should be taught to:

• convert between different units of metric measure [for example, kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre]

• understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints

• measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres

• calculate and compare the area of rectangles (including squares), including using standard units, square centimetres (cm²) and square metres (m²), and estimate the area of irregular shapes

• estimate volume [for example, using 1 cm³ blocks to build cuboids (including cubes)] and capacity [for example, using water]

• solve problems involving converting between units of time

• use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling

## Notes and guidance (non-statutory)

Pupils use their knowledge of place value and multiplication and division to convert between standard units.Pupils calculate the perimeter of rectangles and related composite shapes, including using the relations of perimeter or area to find unknown lengths. Missing measures questions such as these can be expressed algebraically, for example 4 + 2b = 20 for a rectangle of sides 2 cm and b cm and perimeter of 20cm.Pupils calculate the area from scale drawings using given measurements.Pupils use all 4 operations in problems involving time and money, including conversions (for example, days to weeks, expressing the answer as weeks and days).

## Geometry - properties of shapes

Pupils should be taught to:

• identify 3-D shapes, including cubes and other cuboids, from 2-D representations

• know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles

• draw given angles, and measure them in degrees (°)

• identify:

• angles at a point and 1 whole turn (total 360°)

• angles at a point on a straight line and half a turn (total 180°)

• other multiples of 90°

• use the properties of rectangles to deduce related facts and find missing lengths and angles

• distinguish between regular and irregular polygons based on reasoning about equal sides and angles

## Notes and guidance (non-statutory)

Pupils become accurate in drawing lines with a ruler to the nearest millimetre, and measuring with a protractor. They use conventional markings for parallel lines and right angles.Pupils use the term diagonal and make conjectures about the angles formed between sides, and between diagonals and parallel sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools.Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems.

## Geometry - position and direction

Pupils should be taught to:

## Notes and guidance (non-statutory)

Pupils recognise and use reflection and translation in a variety of diagrams, including continuing to use a 2-D grid and coordinates in the first quadrant. Reflection should be in lines that are parallel to the axes.

## Statistics

Pupils should be taught to:

• solve comparison, sum and difference problems using information presented in a line graph

• complete, read and interpret information in tables, including timetables

## Notes and guidance (non-statutory)

Pupils connect their work on coordinates and scales to their interpretation of time graphs.They begin to decide which representations of data are most appropriate and why.

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