Statutory guidance
National curriculum in England:
mathematics programmes of study
Upper "Year 6" Key stage 2  programme of study
Number  number and place value
Pupils should be taught to:

read, write, order and compare numbers up to 10,000,000 and determine the value of each digit

round any whole number to a required degree of accuracy

use negative numbers in context, and calculate intervals across 0

solve number and practical problems that involve all of the above
Notes and guidance (nonstatutory)
Pupils use the whole number system, including saying, reading and writing numbers accurately.
Number  addition, subtraction, multiplication and division
Pupils should be taught to:

multiply multidigit numbers up to 4 digits by a twodigit whole number using the formal written method of long multiplication

divide numbers up to 4 digits by a twodigit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context

divide numbers up to 4 digits by a twodigit number using the formal written method of short division where appropriate, interpreting remainders according to the context

perform mental calculations, including with mixed operations and large numbers

identify common factors, common multiples and prime numbers

use their knowledge of the order of operations to carry out calculations involving the 4 operations

solve addition and subtraction multistep problems in contexts, deciding which operations and methods to use and why

solve problems involving addition, subtraction, multiplication and division

use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy
Notes and guidance (nonstatutory)
Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division (see Mathematics appendix 1).They undertake mental calculations with increasingly large numbers and more complex calculations.Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50, etc, but not to a specified number of significant figures.Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.Common factors can be related to finding equivalent fractions.
Number  Fractions (including decimals and percentages)
Pupils should be taught to:

use common factors to simplify fractions; use common multiples to express fractions in the same denomination

compare and order fractions, including fractions >1

add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions

multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, × = ]

divide proper fractions by whole numbers [for example, ÷ 2 = ]

associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, ]

identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places

multiply onedigit numbers with up to 2 decimal places by whole numbers

use written division methods in cases where the answer has up to 2 decimal places

solve problems which require answers to be rounded to specified degrees of accuracy

recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
Notes and guidance (nonstatutory)
Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example, + = ] and progress to varied and increasingly complex problems.Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle.Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if quarter of a length is 36cm, then the whole length is 36 × 4 = 144cm).They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to 2 decimal places by onedigit and twodigit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.Pupils are introduced to the division of decimal numbers by onedigit whole numbers, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.
Ratio and proportion
Pupils should be taught to:

solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts

solve problems involving the calculation of percentages [for example, of measures and such as 15% of 360] and the use of percentages for comparison

solve problems involving similar shapes where the scale factor is known or can be found

solve problems involving unequal sharing and grouping using knowledge of fractions and multiples
Notes and guidance (nonstatutory)
Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).Pupils link percentages or 360° to calculating angles of pie charts.Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.Pupils solve problems involving unequal quantities, for example, ’for every egg you need 3 spoonfuls of flour’, ‘ of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.
Algebra
Pupils should be taught to:

use simple formulae

generate and describe linear number sequences

express missing number problems algebraically

find pairs of numbers that satisfy an equation with 2 unknowns

enumerate possibilities of combinations of 2 variables
Notes and guidance (nonstatutory)
Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:

missing numbers, lengths, coordinates and angles

formulae in mathematics and science

equivalent expressions (for example, a + b = b + a)

generalisations of number patterns

number puzzles (for example, what 2 numbers can add up to)
Measurement
Pupils should be taught to:

solve problems involving the calculation and conversion of units of measure, using decimal notation up to 3 decimal places where appropriate

use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal notation to up to 3 decimal places

convert between miles and kilometres

recognise that shapes with the same areas can have different perimeters and vice versa

recognise when it is possible to use formulae for area and volume of shapes

calculate the area of parallelograms and triangles

calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³]
Notes and guidance (nonstatutory)
Pupils connect conversion (for example, from kilometres to miles) to a graphical representation as preparation for understanding linear/proportional graphs.They know approximate conversions and are able to tell if an answer is sensible.Using the number line, pupils use, add and subtract positive and negative integers for measures such as temperature.They relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.Pupils could be introduced to compound units for speed, such as miles per hour, and apply their knowledge in science or other subjects as appropriate.
Geometry  properties of shapes
Pupils should be taught to:

draw 2D shapes using given dimensions and angles

recognise, describe and build simple 3D shapes, including making nets

compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons

illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius

recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles
Notes and guidance (nonstatutory)
Pupils draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles.Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements.These relationships might be expressed algebraically for example, d = 2 × r; a = 180 − (b + c).
Geometry  position and direction
Pupils should be taught to:

describe positions on the full coordinate grid (all 4 quadrants)

draw and translate simple shapes on the coordinate plane, and reflect them in the axes
Notes and guidance (nonstatutory)
Pupils draw and label a pair of axes in all 4 quadrants with equal scaling. This extends their knowledge of one quadrant to all 4 quadrants, including the use of negative numbers.Pupils draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. These might be expressed algebraically for example, translating vertex (a, b) to (a − 2, b + 3); (a, b) and (a + d, b + d) being opposite vertices of a square of side d.
Statistics
Pupils should be taught to:

interpret and construct pie charts and line graphs and use these to solve problems

calculate and interpret the mean as an average
Notes and guidance (nonstatutory)
Pupils connect their work on angles, fractions and percentages to the interpretation of pie charts.Pupils both encounter and draw graphs relating 2 variables, arising from their own enquiry and in other subjects.They should connect conversion from kilometres to miles in measurement to its graphical representation.Pupils know when it is appropriate to find the mean of a data set.
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