Programmes of study:

Number and place value

Addition and subtraction

Multiplication and division

Fractions (inc decimals and percentages

Year 1

Pupils should be taught to:
 count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number
 count, read and write numbers to 100 in numerals; count in multiples of twos, fives and tens
 given a number, identify one more and one less
 identify and represent numbers using objects and pictorial representations including the number line, and use the language of: equal to, more than, less than (fewer), most, least
 read and write numbers from 1 to 20 in numerals and words.

Pupils should be taught to:
 read, write and interpret mathematical statements involving addition (+), subtraction (–) and equals (=) signs
 represent and use number bonds and related subtraction facts within 20
 add and subtract onedigit and twodigit numbers to 20, including zero
 solve onestep problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = – 9.

Pupils should be taught to:
 solve onestep problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

Pupils should be taught to:
 recognise, find and name a half as one of two equal parts of an object, shape or quantity
 recognise, find and name a quarter as one of four equal parts of an object, shape or quantity.

Year 2

Pupils should be taught to:
 count in steps of 2, 3, and 5 from 0, and in tens from any number, forward and backward
 recognise the place value of each digit in a twodigit number (tens, ones)
 identify, represent and estimate numbers using different representations, including the number line
 compare and order numbers from 0 up to 100; use <, > and = signs
 read and write numbers to at least 100 in numerals and in words
 use place value and number facts to solve problems.

Pupils should be taught to:
 solve problems with addition and subtraction:
o using concrete objects and pictorial representations, including those involving numbers, quantities and measures
o applying their increasing knowledge of mental and written methods
 recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100
 add and subtract numbers using concrete objects, pictorial representations, and mentally, including:
oa twodigit number and ones
oa twodigit number and tens
otwo twodigit numbers
oadding three onedigit numbers
 show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannot
 recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems.

Pupils should be taught to:
 recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
 calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs
 show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot
 solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts.

Pupils should be taught to:
 recognise, find, name and write fractions 1/3, 1/4, 2/4, 3/4 of a length, shape, set of objects or quantity
 write simple fractions for example, 1/2 of 6 = 3 and recognise the equivalence of 2/4 and 1/2.

Year 3

Pupils should be taught to:
 count from 0 in multiples of 4, 8, 50 and 100; find 10 or 100 more or less than a given number
 recognise the place value of each digit in a threedigit number (hundreds, tens, ones)
 compare and order numbers up to 1000
 identify, represent and estimate numbers using different representations
 read and write numbers up to 1000 in numerals and in words
 solve number problems and practical problems involving these ideas.

Pupils should be taught to:
 add and subtract numbers mentally, including:
o a threedigit number and ones
o a threedigit number and tens
o a threedigit number and hundreds
 add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction
 estimate the answer to a calculation and use inverse operations to check answers
 solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.

Pupils should be taught to:
 recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
 write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for twodigit numbers times onedigit numbers, using mental and progressing to formal written methods
 solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects.

Pupils should be taught to:
 count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing onedigit numbers or quantities by 10
 recognise, find and write fractions of a discrete set of objects: unit fractions and nonunit fractions with small denominators
 recognise and use fractions as numbers: unit fractions and nonunit fractions with small denominators
 recognise and show, using diagrams, equivalent fractions with small denominators
 add and subtract fractions with the same denominator within one whole [for example, 7/7 +1/7 = 6/7]
 compare and order unit fractions, and fractions with the same denominators
 solve problems that involve all of the above.

Year 4

Pupils should be taught to
 count in multiples of 6, 7, 9, 25 and 1000
 find 1000 more or less than a given number
 count backwards through zero to include negative numbers
 recognise the place value of each digit in a fourdigit number (thousands, hundreds, tens, and ones)
 order and compare numbers beyond 1000
 identify, represent and estimate numbers using different representations
 round any number to the nearest 10, 100 or 1000
 solve number and practical problems that involve all of the above and with increasingly large positive numbers
 read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value.

Pupils should be taught to:
 add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate
 estimate and use inverse operations to check answers to a calculation
 solve addition and subtraction twostep problems in contexts, deciding which operations and methods to use and why.

Pupils should be taught to:
 recall multiplication and division facts for multiplication tables up to 12 × 12
 use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers
 recognise and use factor pairs and commutativity in mental calculations
 multiply twodigit and threedigit numbers by a onedigit number using formal written layout
 solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects.

Pupils should be taught to:
 recognise and show, using diagrams, families of common equivalent fractions
 count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten.
 solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including nonunit fractions where the answer is a whole number
 add and subtract fractions with the same denominator
 recognise and write decimal equivalents of any number of tenths or hundredths
 recognise and write decimal equivalents to ¼, ½, ¾
 find the effect of dividing a one or twodigit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths
 round decimals with one decimal place to the nearest whole number
 compare numbers with the same number of decimal places up to two decimal places
 solve simple measure and money problems involving fractions and decimals to two decimal places.


Year 5

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 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.

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 multistep problems in contexts, deciding which operations and methods to use and why.

Pupils should be taught to:
 identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
 Know and use the vocabulary of prime numbers, prime factors and composite (nonprime) 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 twodigit number using a formal written method, including long multiplication for twodigit numbers
 multiply and divide numbers mentally drawing upon known facts
 divide numbers up to 4 digits by a onedigit 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 1000
 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.

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, 2/5 + 4/5 = 6/5 = 1 1/5 ]
 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 = 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
 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 100, and as a decimal
 solve problems which require knowing percentage and decimal equivalents of ½ , ¼, 1/5, 2/5, 4/5and those fractions with a denominator of a multiple of 10 or 25.

Year 6

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 zero
 solve number and practical problems that involve all of the above.

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

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, ¼ x ½ = 1/8
 divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6]
 associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8]
 identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places
 multiply onedigit numbers with up to two decimal places by whole numbers
 use written division methods in cases where the answer has up to two 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.

Ratio and proportion
(Y6 only)

Pupils should be taught to:
 solve problems involving the relative sizes of two 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

Algebra
(Y6 only)

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 two unknowns
 enumerate possibilities of combinations of two variables.


Measurement

Geometry  Properties of Shapes

Geometry  Position and Direction

Statistics

Year 1

Pupils should be taught to:
 compare, describe and solve practical problems for:
o lengths and heights [for example, long/short, longer/shorter, tall/short, double/half]
o mass/weight [for example, heavy/light, heavier than, lighter than]
o capacity and volume [for example, full/empty, more than, less than, half, half full, quarter]
o time [for example, quicker, slower, earlier, later]
 measure and begin to record the following:
o lengths and heights
o mass/weight
o capacity and volume
o time (hours, minutes, seconds)
 recognise and know the value of different denominations of coins and notes
 sequence events in chronological order using language [for example, before and after, next, first, today, yesterday, tomorrow, morning, afternoon and evening]
 recognise and use language relating to dates, including days of the week, weeks, months and years
 tell the time to the hour and half past the hour and draw the hands on a clock face to show these times.

Pupils should be taught to:
 recognise and name common 2D and 3D shapes, including:
o 2D shapes [for example, rectangles (including squares), circles and triangles]
o 3D shapes [for example, cuboids (including cubes), pyramids and spheres].

Pupils should be taught to:
 describe, position, direction and movement including whole, half, quarter and three quarter turns.


Year 2

Pupils should be taught to:
 choose and use appropriate standard units to estimate and measure length/height in any direction (m/cm); mass (kg/g); temperature (°C); capacity (litres/ml) to the nearest appropriate unit, using rulers, scales, thermometers and measuring vessels
 compare and order lengths, mass, volume/capacity and record the results using >, < and =
 recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value
 find different combinations of coins that equal the same amounts of money
 solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change
 compare and sequence intervals of time
 tell and write the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times
 know the number of minutes in an hour and the number of hours in a day.

Pupils should be taught to:
 identify and describe the properties of 2D shapes, including the number of sides and line symmetry in a vertical line
 identify and describe the properties of 3D shapes, including the number of edges, vertices and faces
 identify 2D shapes on the surface of 3D shapes, [for example, a circle on a cylinder and a triangle on a pyramid]
 compare and sort common 2D and 3D shapes and everyday objects.

Pupils should be taught to:
 order and arrange combinations of mathematical objects in patterns and sequences
 use mathematical vocabulary to describe position, direction and movement, including movement in a straight line and distinguishing between rotation as a turn and in terms of right angles for quarter, half and threequarter turns (clockwise and anticlockwise).

Pupils should be taught to:
 interpret and construct simple pictograms, tally charts, block diagrams and simple tables
 ask and answer simple questions by counting the number of objects in each category and sorting the categories by quantity
 ask and answer questions about totalling and comparing categorical data.

Year 3

Pupils should be taught to:
 measure, compare, add and subtract: lengths (m/cm/mm); mass (kg/g); volume/capacity (l/ml)
 measure the perimeter of simple 2D shapes
 add and subtract amounts of money to give change, using both £ and p in practical contexts
 tell and write the time from an analogue clock, including using Roman numerals from I to XII, and 12hour and 24hour clocks
 estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes and hours; use vocabulary such as o’clock, a.m./p.m., morning, afternoon, noon and midnight
 know the number of seconds in a minute and the number of days in each month, year and leap year
 compare durations of events [for example to calculate the time taken by particular events or tasks].

Pupils should be taught to:
 draw 2D shapes and make 3D shapes using modelling materials; recognise 3D shapes in different orientations and describe them
 recognise angles as a property of shape or a description of a turn
 identify right angles, recognise that two right angles make a halfturn, three make three quarters of a turn and four a complete turn; identify whether angles are greater than or less than a right angle
 identify horizontal and vertical lines and pairs of perpendicular and parallel lines.


Pupils should be taught to:
 interpret and present data using bar charts, pictograms and tables
 solve onestep and twostep questions [for example, ‘How many more?’ and ‘How many fewer?’] using information presented in scaled bar charts and pictograms and tables.

Year 4

Pupils should be taught to:
 Convert between different units of measure [for example, kilometre to metre; hour to minute]
 measure and calculate the perimeter of rectilinear figure (including squares) in centimetres and metres
 find the area of rectilinear shapes by counting squares
 estimate, compare and calculate different measures, including money in pounds and Pence
 read, write and convert time between analogue and digital 12 and 24hour clocks
 solve problems involving converting from hours to minutes; minutes to seconds; years to months; weeks to days.

Pupils should be taught to:
 compare and classify geometric shapes, including quadrilaterals and triangles, based on their properties and sizes
 identify acute and obtuse angles and compare and order angles up to two right angles by size
 identify lines of symmetry in 2D shapes presented in different orientations
 complete a simple symmetric figure with respect to a specific line of symmetry.

Pupils should be taught to:
 describe positions on a 2D grid as coordinates in the first quadrant
 describe movements between positions as translations of a given unit to the left/right
 and up/down
 plot specified points and draw sides to complete a given polygon.

Pupils should be taught to:
 interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs.
 solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs.

Year 5

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), and including using standard units, square centimetres (cm^{2}) and square metres (m^{2}) and estimate the area of irregular shapes
 estimate volume [for example, using 1 cm^{3}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.

Pupils should be taught to:
 identify 3D shapes, including cubes and other cuboids, from 2D representations
 know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles
 draw given angles, and measure them in degrees (^{o})
 identify:
o angles at a point and one whole turn (total 360^{o})
o angles at a point on a straight line and 1/2 a turn (total 180^{o})
o other multiples of 90^{o}
 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.

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.

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.

Year 6

Pupils should be taught to:
 solve problems involving the calculation and conversion of units of measure, using decimal notation up to three 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 three 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 (cm^{3}) and cubic metres (m^{3}), and extending to other units [for example, mm^{3} and km^{3}].

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.

Pupils should be taught to:
 describe positions on the full coordinate grid (all four quadrants)
 draw and translate simple shapes on the coordinate plane, and reflect them in the axes.

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.

