Year 5 programme of study (statutory requirements)

Notes and guidance (non-statutory)

Number, place value, approximation and estimation

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 through zero
• round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000
• solve number problems and practical problems that involve all of the above
• read Roman numerals to 1000 (M) and recognise years written in Roman numerals.

Number, place value, approximation and estimation

Pupils should identify the place value in large whole numbers.  They should 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, including those involving fractions and decimals, and find the term-to-term rule.

Addition and subtraction

Pupils should be taught to:

• add and subtract whole numbers with more than 4 digits, including using efficient 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.

Addition and subtraction

Pupils should practise using the efficient written methods of columnar addition and subtraction with increasingly large numbers to aid fluency.

They should practise mental calculations with increasingly large numbers to aid fluency (e.g. 12 462 – 2 300 = 10 162).

Multiplication and division

Pupils should be taught to:

• identify multiples and factors, including finding all factor pairs
• solve problems involving multiplication and division where larger numbers are used by decomposing them into their factors
• 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 an efficient 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 efficient 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 1000

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

• 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.

Multiplication and division

Pupils should practise and extend their use of the efficient written methods of short multiplication and short division. They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.

They should use and understand the terms factor, multiple and prime, square and cube numbers.

Pupils should 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 (e.g. 98 ÷ 4 = 24 r 2 = 24¹/₂ = 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 1000 in converting between units such as kilometres and metres.

 Fractions

Pupils should be taught to:

• compare and order fractions whose denominators are all multiples of the same number
• recognise mixed numbers and improper fractions and convert from one form to the other
• add and subtract fractions with the same denominator and related fractions; write mathematical statements >1 as a mixed number (e.g. ²/₅ + ⁴/₅ = ⁶/₅ = 11/5)
• multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams.

Fractions

Pupils should connect equivalent fractions >1 that simplify to integers with division and fractions >1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions.

Pupils should connect multiplication by a fraction to using fractions as operators, and to division, building on work from previous years.

This relates to scaling by simple fractions.

They should extend their knowledge of fractions to thousandths and connect to decimals and measures. Pupils continue to develop their understanding of fractions as numbers, measures and

operators by finding fractions of numbers and quantities, writing remainders as a fraction.

Pupils should practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They should extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number.

Pupils should read and write proper fractions and mixed numbers accurately and continue to practise counting forwards and backwards with mixed fractions.

Decimals and fractions

Pupils should be taught to:

• read and write decimal numbers as fractions (e.g. 0.71 = ⁷¹/₁00)
• recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
• round decimals with two decimal places to the nearest whole number and to one decimal place
• read, write, order and compare numbers with up to three decimal places
• solve problems involving number up to three decimal places.

Decimals and fractions

Pupils extend counting from Year 4, using decimals and fractions including bridging zero, for example on a number line.

They should add and subtract decimals including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (e.g. 0.83 + 0.17 = 1).

They should mentally add and subtract tenths, and one-digit whole numbers and tenths.

Pupils should say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and be confident in checking the reasonableness of their answers to problems.

Pupils should go beyond the measurement and money models of decimals, for example by solving puzzles involving decimals.

Percentages, decimals and fractions Pupils should be taught to:

• recognise the per cent symbol (%) and understand that per cent relates to “number of parts per hundred”, and write percentages as a fraction with denominator hundred, and as a decimal fraction
• solve problems which require knowing percentage and decimal equivalents of ¹/₂, ¹/₄, ¹/₅, ²/₅, ⁴/₅ and those with a denominator of a multiple of 10 or 25.

Percentages, decimals and fractions

Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing numbers.

Pupils should make connections between percentages, fractions and decimals (e.g. 100% represents a whole quantity and 1% is 1/100, 50% is 50/100, 25% is 25/100) and relate this to finding ‘fractions of’. They recognise that percentages are proportions of quantities as well as operators on quantities.

Measures

Pupils should be taught to:

• convert between different units of measure (e.g. kilometre and metre; metre and centimetre; centimetre and millimetre; kilogram and gram; litre and millilitre)
• understand and use basic equivalences between metric and common imperial units and express them in approximate terms
• measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres
• calculate and compare the area of squares and rectangles including using standard units, square centimetres (cm²) and square metres (m²) and estimate the area of irregular shapes

• recognise and estimate volume (e.g. using 1 cm³ blocks to build cubes and cuboids) and capacity (e.g. using water)

• solve problems involving converting between units of time

• solve problems involving addition and subtraction of units of measure (e.g. length, mass, volume, money) using decimal notation.

Measures

Pupils should use their knowledge of place value and multiplication and division to convert between standard units.

Pupils should calculate the perimeter of rectangles and related composite shapes, including using the relations of perimeter or area to find unknown lengths. Missing number questions such as these are the beginning of algebraic understanding. They should also calculate the area of scale drawings using given measurements.

Pupils should use all four operations in problems involving time and money, including conversions (e.g. days to weeks, leaving the answer as weeks and days).

Geometry: properties of shapes

Pupils should be taught to:

• identify 3-D shapes, including cubes and cuboids, from 2-D representations
• know angles are measured in degrees; estimate and measure them and draw a given angle, writing its size in degrees (^o)

To identify:

• multiples of 90^o
• angles at a point on a straight line and ¹/₂ a turn (total 180^o)
• angles at a point and one whole turn (total 360^o)
• reflex angles, and
• compare different angles
• draw shapes using given dimensions and angles
• state and use the properties of a rectangle (including squares) to deduce related facts
• distinguish between regular and irregular polygons based on reasoning about equal sides and angles.

Geometry: properties of shapes

Pupils should 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 should use the term diagonal and make conjectures about the angles formed by diagonals and sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools.

Pupils should use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems.

Geometry: position, direction, motion Pupils should be taught to:

• identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed.

Geometry: position, direction, motion

Pupils should 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.

Data

Pupils should be taught to:

• solve comparison, sum and difference problems using information presented in line graphs
• complete, read and interpret information in tables, including timetables.

Data

Pupils should connect their work on coordinates and scales to their interpretation of time graphs using ICT tools, except where data are easily calculable.

They should begin to decide which representations of data are most appropriate and why.