Maths

Number and place value

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 different multiples including ones, twos, fives and tens
• given a number, identify one more and one less
• identify and represent numbers using concrete 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 digits and words.

Number and place value

Pupils should practise counting (1, 2, 3), ordering (e.g. first, second, third), or to indicate a quantity (e.g. 3 apples, 2 centimetres), including solving simple concrete problems, until they are fluent.

They should practise counting as reciting numbers and counting as enumerating objects, and counting in ones, twos, fives and tens from different multiples to develop their recognition of patterns in the number system (e.g. odd and even numbers). They connect these patterns with objects and with shapes, including through varied and frequent practice of increasingly complex questions.

Pupils begin to recognise place value in numbers beyond 20 by reading, writing, counting and comparing numbers up to 100, supported by concrete objects and pictorial representations.

Addition and subtraction

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 one-digit and two-digit numbers to 20 (9 + 9, 18 - 9), including zero
• solve simple one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems.

Addition and subtraction

Pupils should memorise and reason with number bonds to 10 and 20 in several forms (e.g. 9 + 7 = 16; 16 – 7 = 9; 7 = 16 - 9). They should realise the effect of adding or subtracting zero.

Pupils should combine and increase numbers, counting forwards and backwards.

They should discuss and solve problems in familiar practical contexts, including using quantities. Problems should include the terms put together, add, altogether, total, take away, difference between, more than and less than so that pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.

Multiplication and division

Pupils should be taught to:

• solve simple one-step problems involving multiplication and division, calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

Multiplication and division

Through grouping and sharing small quantities, pupils should begin to understand multiplication and division; doubling numbers and quantities, and finding simple fractions of objects, numbers and quantities.

They should make connections between arrays, number patterns, and counting in twos, fives and tens.

Fractions

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.

Fractions

Pupils should be taught ¹/₂ and ¹/₄ as operators on discrete and continuous quantities by solving problems using shapes, objects and quantities. For example, they could recognise and find half a length, quantity, set of objects or shape. Pupils connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole.

Measures

Pupils should be taught to:

• compare, describe and solve practical problems for:

• lengths and heights (e.g. long/short, longer/shorter, tall/short, double/half)
• mass or weight (e.g. heavy/light, heavier than, lighter than)
• capacity/volume (full/empty, more than, less than, quarter)
• time (quicker, slower, earlier, later)

•  measure and begin to record the following:

• lengths and heights
• mass/weight
• capacity and volume
• time (hours, minutes, seconds)

• recognise and know the value of different denominations of coins and notes
• sequence events in chronological order using language such as: 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.

Measures

The terms mass and weight, volume and capacity are used interchangeably at this stage

Pupils should move from using and comparing different types of quantities and measures using non-standard units, including discrete (e.g. counting) and continuous (e.g. liquid) measures, to using manageable common standard units. They should understand the difference between non-standard and standard units.

In order to become familiar with standard measures, pupils begin to use measuring tools such as a ruler, weighing scales and containers.

Pupils should use the language of time, including telling the time throughout the day, first using o’clock and then half past.

Geometry: properties of shapes

Pupils should be taught to:

• recognise and name common 2-D and 3-D shapes, including:

• 2-D shapes (e.g. rectangles (including squares), circles and triangles)
• 3-D shapes (e.g. cuboids (including cubes), pyramids and spheres).

Geometry: properties of shapes

Pupils should handle common 2-D and 3-D shapes, naming these and related everyday objects fluently. They should recognise these shapes in different orientations and sizes, and know that rectangles, triangles, cuboids and pyramids can be different shapes.

Geometry: position, direction, motion

Pupils should be taught to:

• order and arrange combinations of objects and shapes in patterns
• describe position, directions and movements, including half, quarter and three-quarter turns.

Geometry: position, direction, motion

Pupils should create, copy, describe and reorganise patterns.

They should use the language of position, direction and motion, including: left and right, top, middle and bottom, on top of, in front of, above, between, around, near, close and far, up and down, forwards and backwards, inside and outside.

Pupils should make turns to show they understand half, quarter and three-quarter turns and routinely make these turns in a clockwise direction.

Number and place value

Pupils should be taught to:

• count in steps of 2, 3, and 5 from 0, and count in tens from any number, forward or backward
• recognise the place value of each digit in a two-digit 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.

Number and place value

Using materials and a range of representations, pupils should practise counting, reading, writing and comparing numbers to at least 100 and solving a variety of related problems to develop fluency. They should count in multiples of three to support their later understanding of a third.

As they become more confident with numbers up to 100, pupils should be introduced to larger numbers to develop further their recognition of patterns within the number system and represent them in different ways, including spatial representations.

Pupils should partition numbers in different ways (e.g. 23 = 20 + 3 and 23 = 10 + 13) to support subtraction. They become fluent and apply their knowledge of numbers to reason with, discuss and solve problems that emphasise the value of each digit in two-digit numbers. They begin to understand zero as a place holder.

Addition and subtraction

Pupils should be taught to:

• solve simple one-step problems with addition and subtraction: 

• using concrete objects and pictorial representations, including those involving numbers, quantities and measures
• 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:

• a two-digit number and ones
• a two-digit number and tens
• two two-digit numbers
• adding three one-digit 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 missing number problems.

Addition and subtraction

Pupils should extend their understanding of the language of addition and subtraction to include sum and difference.

Pupils should practise addition and subtraction to 20 to become increasingly fluent in deriving facts such as using 3 + 7 = 10, 10 - 7 = 3 and 7 = 10 - 3 to calculate 30 + 70 = 100, 100 - 70 = 30 and 70 = 100 - 30. They should check their calculations, including by adding to check subtraction and adding numbers in a different order to check addition (e.g. 5 + 2 + 1 = 1 + 5 + 2 = 1 + 2 + 5).

Recording addition and subtraction in columns supports place value and prepares for efficient written methods with larger numbers.

Multiplication and division

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
• recognise and use the inverse relationship between multiplication and division in calculations
•  show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot
• solve one-step problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts.

Multiplication and division

Pupils should use a variety of language to describe multiplication and division. They are taught multiplication and division with larger numbers through equal grouping and sharing out quantities, relating multiplication tables to arrays and repeated addition and finding more complex fractions of objects, numbers and quantities.

Pupils should be introduced to the multiplication tables. They should practise to become fluent in the 2, 5 and 10 multiplication tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They begin to use other multiplication tables and recall multiplication facts, including using related division facts to perform written and mental calculations.

Pupils should work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete and continuous quantities, relating these to fractions and measures (e.g. 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (e.g. 4 × 5 = 20 and 20 ÷ 5 = 4).

Fractions

Pupils should be taught to:

• recognise, find, name and write fractions ¹/₃, ¹/₄, ²/₄ and ³/₄ of a length, shape, set of objects or quantity
• write simple fractions e.g. ¹/₂ of 6 = 3 and recognise the equivalence of two quarters and one half.

Fractions

Pupils should use additional fractions as operators on discrete and continuous quantities by solving problems using shapes, objects and quantities. They connect unit fractions to equal sharing and grouping, to numbers when they can be calculated, and to measures, finding fractions of lengths, quantity, a set of objects or shapes. They meet ³/₄ as the first example of a non-unit fraction.

Pupils should count in fractions up to 10, starting from any number and using the ¹/₂ and ²/₄ equivalence on the number line (e.g. 1¹/₄, 1²/₄ , (or 1¹/₂), 1³/₄, 2). This reinforces the concept of fractions as numbers and that they can add up to more than one.

Measures

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 =
• read relevant scales to the nearest numbered unit
• recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value and match different combinations of coins to equal the same amounts of money; add and subtract money of the same unit, including giving change
• solve simple problems in a practical context involving addition and subtraction of money
• 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.

Measures

Pupils should use standard units of measurement with increasing accuracy, using their knowledge of the number system. They should use the appropriate language and record using standard abbreviations.

They should become fluent in telling the time on analogue clocks and recording it.

Pupils should also become fluent in counting and recognising coins. They should use the symbols £ and p accurately and say the amounts of money confidently.

Geometry: properties of shapes

Pupils should be taught to:

• identify and describe the properties of 2-D shapes, including the number of sides and symmetry in a vertical line
• identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces
• identify 2-D shapes on the surface of 3-D shapes, for example a circle on a cylinder and a triangle on a pyramid
• compare and sort common 2-D and 3-D shapes and everyday objects.

Geometry: properties of shapes

Pupils should handle and name a wider variety of common 2-D and 3-D shapes including: quadrilaterals and cuboids, prisms, cones and polygons, and identify the properties of each shape (e.g. number of sides, number of faces). Pupils identify, compare and sort shapes on the basis of their properties and use vocabulary precisely, such as sides, edges, vertices and faces.

Pupils should read and write names for shapes that are appropriate for their word reading and spelling.

Pupils should draw lines and shapes using a straight edge.

Geometry: position, direction, motion

Pupils should be taught to:

• order and arrange combinations of mathematical objects in patterns
• use mathematical vocabulary to describe position, direction and movement, including distinguishing between rotation as a turn and in terms of right angles for quarter, half and three-quarter turns (clockwise and anti-clockwise), and movement in a straight line.

Geometry: position, direction, motion

Pupils should work with patterns of shapes, including those in different orientations.

Pupils should use the concept and language of angles to describe ‘turn’ by applying rotations, including in practical contexts (e.g. pupils themselves moving in turns, giving instructions to other pupils to do so, and programming robots using instructions given in right angles).

Data

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 compare categorical data.

Data

At this stage, pupils’ recording and interpretation become more sophisticated as they collate, organise and compare information (e.g. using many-to-one correspondence in pictograms and using simple ratios 2, 5, 10).

Number, place value and rounding

Pupils should be taught to:

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

Number, place value and rounding

Pupils should work with larger numbers, applying partitioning related to place value using varied and increasingly complex problems, building on work in Year 2 (e.g. 46 = 40 and 6, 46 = 30 and 16).

Using a variety of representations, including those related to measure, pupils should continue to count in ones, tens and hundreds, so that they become fluent in the order and place value of numbers to 1000.

Addition and subtraction

Pupils should be taught to:

•  add and subtract numbers mentally, including: 

• a three-digit number and ones
• a three-digit number and tens
• a three-digit number and hundreds

• add and subtract numbers with up to three digits, using the efficient 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.

Addition and subtraction

Pupils should practise solving varied addition and subtraction questions. For mental calculations with two-digit numbers, the answers could exceed 100.

Pupils should use their understanding of place value and partitioning, and practise using columnar addition and subtraction with increasingly large numbers up to three digits to become fluent.

Multiplication and division

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 two-digit numbers times one-digit numbers, using mental and progressing to efficient written methods
• solve problems, including missing number problems, involving multiplication and division, including integer scaling problems and correspondence problems in which n objects are connected to m objects.

Multiplication and division

Pupils should continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables.

Pupils should develop efficient mental methods, for example, using commutativity (e.g. 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and multiplication and division facts (e.g. using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).

Pupils should develop reliable written methods for multiplication and division, starting with calculations of two-digit numbers by one-digit numbers and progressing to the efficient written methods of short multiplication and division.

Pupils should solve simple problems in contexts, deciding which of the four operations to use and why, including measuring and scaling contexts, and correspondence problems in which m objects are connected to n objects (e.g. 3 hats and 4 coats, how many different outfits; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children).

Fractions

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 one-digit numbers or quantities by 10
• recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
• recognise and use fractions as numbers: unit fractions and non-unit 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 (e.g. ⁵/₇ + ¹/₇ = ⁶/₇)
• compare and order unit fractions with the same denominator
• solve problems that involve all of the above.

Fractions

Pupils should connect tenths to place value and decimal measures, not restricted to decimals between 0 and 1 inclusive and to division by 10.

They should begin to understand unit and non-unit fractions as numbers on the number line, and deduce relations between them, such as size and equivalence. They should go beyond the [0, 1] interval, and ¹/₄ + ³/₄ = 1 for example, relating this to measure.

Pupils should understand the relation between unit fractions as operators and division by integers.

They should continue to recognise fractions in the context of parts of a whole, numbers, measurements, a shape, or unit fractions as a division of a quantity.

Pupils should practise adding and subtracting fractions with the same denominator through a variety of increasingly complex problems to improve fluency.

Measures

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 2-D 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 12-hour and 24-hour clocks
• estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes, hours and o’clock; use vocabulary such as 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.

Measures

Pupils should continue to measure using the appropriate tools and units, progressing to using a wider range of measures, including comparing and using mixed units (e.g. 1 kg and 200g) and simple equivalents of mixed units (e.g. 5m = 500cm).

The comparison of measures should also include simple scaling (e.g. a given quantity or measure is twice as long or five times as high) and connect this to multiplication.

Pupils should continue to become fluent in recognising the value of coins, by adding and subtracting amounts, including mixed units, and giving change using manageable amounts. They should record £ and p separately. The decimal recording of money is introduced formally in Year 4.

Pupils should use both analogue and digital 12-hour clocks and record their times. In this way they become fluent in and prepared for using digital 24-hour clocks in Year 4.

Geometry: properties of shapes

Pupils should be taught to:

• draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations; and describe them with increasing accuracy
• recognise angles as a property of shape and associate angles with turning
• identify right angles, recognise that two right angles make a half-turn, 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, vertical, perpendicular and parallel lines in relation to other lines.

Geometry: properties of shapes

Pupils’ knowledge of the properties of shapes is extended at this stage to symmetrical and non-symmetrical polygons and polyhedra. Pupils extend their use of the properties of shapes. They should be able to describe the properties of 2-D and 3-D shapes using accurate language, including lengths of lines and acute and obtuse for angles greater or lesser than a right angle.

Pupils should draw and measure straight lines in centimetres.

Data

Pupils should be taught to:

• interpret and present data using bar charts, pictograms and tables
• solve one-step and two-step questions such as ‘How many more?’ and ‘How many fewer?’ using information presented in scaled bar charts and pictograms and tables.

Data

Pupils should understand and use simple scales (e.g. 2, 5, 10 units per cm) in pictograms and bar charts with increasing accuracy.

They should continue to interpret data presented in many contexts.

Number, place value and rounding

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 four-digit 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 understand how, over time, the numeral system changed to include the concept of zero and place value.

Number, place value and rounding

Using a variety of representations, including measures, pupils should become fluent in the order and place value of numbers beyond 1000, including counting in tens and hundreds, and maintaining fluency in other multiples through varied and frequent practice.

They begin to extend their knowledge of the number system to include the decimal numbers and fractions that they have met so far.

Roman numerals should be put in their historical context so pupils understand that there have been different ways to write whole numbers and that the important concepts of zero and place value were introduced over a period of time.

Addition and subtraction

Pupils should be taught to:

• add and subtract numbers with up to 4 digits using the efficient written methods of columnar addition and subtraction where appropriate
• estimate and use inverse operations to check answers to a calculation
• solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why.

Addition and subtraction

Pupils should continue to practise both mental methods and columnar addition and subtraction with increasingly large numbers to aid fluency.

Multiplication and division

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 two-digit and three-digit numbers by a one-digit number using formal written layout
• solve problems involving multiplying and adding, including using the distributive law and harder multiplication problems such as which n objects are connected to m objects.

Multiplication and division

Pupils should continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Pupils should practise mental methods and extend this to three-digit numbers to derive facts, for example 200 × 3 = 600 into 600 ÷ 3 = 200, to become fluent.

Pupils should practise to become fluent in the efficient written method of short multiplication for multiplying using multi-digit numbers, and short division with exact answers when dividing by a one-digit number.

Pupils should write statements about the equality of expressions (e.g. use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)).

Pupils should solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as three cakes shared equally between 10 children.

Fractions

Pupils should be taught to:

• count up and down in hundredths; recognise that hundredths arise when dividing an object by a hundred and dividing tenths by ten
• solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number
• identify, name and write equivalent fractions of a given fraction, including tenths and hundredths
• add and subtract fractions with the same denominator.

Fractions

Pupils should connect hundredths to tenths and place value and decimal measure.

They should extend the use of the number line to connect fractions, numbers and measures.

Pupils should understand the relation between non-unit fractions and multiplication and division of quantities, with particular emphasis on tenths and hundredths.

Pupils should associate fractions of a length, of a shape and as a representation of one whole or set of quantities. Pupils should use factors and multiples to recognise equivalent fractions and simplify where appropriate (e.g. ⁶/₉ = ²/₃ or ¹/₄ = ²/₈).

Pupils should continue practice in adding and subtracting fractions with the same denominator, to become fluent through a variety of increasingly complex problems beyond one whole.

They should practise counting using simple fractions and decimal fractions, both forwards and backwards.

Decimals and fractions

Pupils should be taught to:

• recognise and write decimal equivalents of any number of tenths or hundredths
• recognise and write decimal equivalents to ¹/₄; ¹/₂; ³/₄
• find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as units, 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.

Decimals and fractions

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

Pupils’ understanding of the number system and decimal place value is extended at this stage to tenths and then hundredths. This includes relating the decimal notation to division of whole numbers by 10 and later 100.

Pupils should learn decimal notation and the language associated with it, including in the context of measurements. They make comparisons and order decimal amounts and quantities that are expressed to the same number of decimal places. They should be able to represent numbers with one or two decimal places in multiple ways, such as on number lines.

Measures

Pupils should be taught to:

• convert between different units of measure (e.g. kilometre to metre; hour to minute)
• measure and calculate the perimeter of a rectilinear figure (including squares) in centimetres and metres
• find the area of rectilinear shapes by counting
• estimate, compare and calculate different measures, including money in pounds and pence
• read, write and convert time between analogue and digital 12 and 24-hour clocks
• solve problems involving converting from hours to minutes; minutes to seconds; years to months; weeks to days.

Measures

Pupils should use multiplication and their knowledge of place value to convert from larger to smaller units.

They should relate area to arrays and multiplication.

Pupils should build on their understanding of decimal notation to record measures.

Geometry: properties of shapes

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 2-D shapes presented in different orientations
• complete a simple symmetric figure with respect to a specific line of symmetry.

Geometry: properties of shapes

Pupils should continue to classify shapes using geometrical properties, extending to classifying different triangles (e.g. isosceles, equilateral, scalene) and quadrilaterals (e.g. parallelogram, rhombus, trapezium).

Pupils should compare and order angles in preparation for using a protractor and compare lengths and angles to decide if a polygon is regular or irregular.

Pupils should draw symmetric patterns using a variety of media to become familiar with different orientations of lines of symmetry; and recognise line symmetry in a variety of diagrams.

Geometry: position, direction, motion

Pupils should be taught to:

• describe positions on a 2-D 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.

Geometry: position, direction, motion

Pupils should draw a pair of axes in one quadrant, with equal scales and integer labels. They should read, write and use pairs of coordinates (2, 5), including using coordinate-plotting ICT tools.

Data

Pupils should be taught to:

• interpret and present discrete data using bar charts and continuous data using line graphs
• solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and simple line graphs.

Data

Pupils should understand and use a greater range of scales in their representations. Pupils should begin to relate the graphical representation of data to recording change over time.

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)
•  identify:

• multiples of 90^o
• angles at a point on a straight line and ½ 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.

Number, place value and rounding

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 problems and practical problems that involve all of the above.

Number, place value and rounding

Pupils should use the whole number system, including saying, reading and writing numbers accurately.

Addition, subtraction, multiplication and division

Pupils should be taught to:

• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the efficient written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the efficient written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for 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 multi-step 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, levels of accuracy.

Addition, subtraction, multiplication and division

Pupils should practise addition, subtraction, multiplication and division for larger numbers, using the efficient written methods of columnar addition and subtraction, short and long multiplication, and short and long division.

They should undertake mental calculations with increasingly large numbers and more complex calculations.

Pupils should continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

Pupils should round answers to a specified degree of accuracy.

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.

Fractions

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
• associate a fraction with division to calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. ³/₈)
• 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 (e.g. ¹/₄ × ¹/₂ = ¹/₈)
• divide proper fractions by whole numbers (e.g. ¹/₃ ÷ 2 = ¹/₆ ).

Fractions

Pupils should 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 (e.g. if ¼ of a length is 36cm, then the whole length is 36 × 4 = 144cm).

They should practise with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators. Denominators of given fractions should not exceed 12, with the exception of 100 and 1000.

Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (e.g. 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils should learn about rounding the decimal to three decimal places.

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 (e.g. ¹/₂ + ¹/₈ = ⁵/₈) 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, as numbers, and as equal parts of objects, for example as parts of a rectangle.

Decimals and fractions

Pupils should be taught to:

• identify the value of each digit to three decimal places and multiply and divide numbers by 10, 100 and 1000 where the answers are up to three decimal places
• multiply one-digit 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.

Decimals and fractions

Pupils should begin to multiply and divide numbers with up to two decimal places by one-digit and two-digit 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 should also be introduced to the division of decimal numbers by one-digit whole numbers and, initially, in practical contexts involving measures and money. They should recognise division calculations as the inverse of multiplication.

Pupils should 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.

Percentages, decimals and fractions

Pupils should be taught to:

• solve problems involving the calculation of percentages of whole numbers or measures such as 15% of 360 and the use of percentages for comparison
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts.

Percentages, decimals and fractions

Pupils should understand that calculating a percentage of a quantity is the same as calculating a fraction of a quantity.

Ratio and proportion

Pupils should be taught to:

• solve problems involving the relative sizes of two quantities, including similarity
• solve problems involving unequal sharing and grouping.

Ratio and proportion

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They may use the notation a:b to record their work.

Pupils should recognise proportionality in contexts when the relations between quantities are in the same ratio (e.g. similar shapes, recipes).

Algebra

Pupils should be taught to:

• express missing number problems algebraically
• use simple formulae expressed in words
• generate and describe linear number sequences
• find pairs of numbers that satisfy number sentences involving two unknowns.

Algebra

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
• arithmetical rules (e.g. a + b = b + a)
• generalisations of number patterns
• number puzzles (e.g. what two numbers can add up to).

Measures

Pupils should be taught to:

• solve problems involving the calculation and conversion of units of measure, using decimal notation 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 three decimal places
• convert between miles and kilometres
• recognise that shapes with the same areas can have different perimeters and vice versa
• calculate the area of parallelograms and triangles
• recognise when it is necessary to use the formulae for area and volume of shapes
• calculate, estimate and compare volume of cubes and cuboids using standard units, including centimetre cubed (cm³) and cubic metres (m³) and extending to other units, such as mm³ and km³.

Measures

Using the number line, pupils should use, add and subtract positive and negative integers for measures such as temperature.

They should know approximate conversions and be able to tell if an answer is sensible.

They should relate the area of rectangles to parallelograms and triangles, and be able to calculate their areas, understanding and using the formula to do this.

Pupils could be introduced to other 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:

• recognise, describe and build simple 3-D 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
• find unknown angles where they meet at a point, are on a straight line, and are vertically opposite.

Geometry: properties of shapes

Pupils should draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles.

Pupils should describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements.

Geometry: position, direction, motion

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.

Geometry: position, direction, motion

Pupils should draw and label a pair of axes in all four quadrants with equal scaling. This extends their knowledge of one quadrant to all four quadrants, including the use of negative numbers.

Pupils should draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes.

Data

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.

Data

Pupils should connect their work on angles, fractions and percentages to the interpretation of pie charts.

Pupils should both encounter and draw graphs relating two variables, arising from their own enquiry and in other subjects. They should connect conversion from kilometres to miles in measure to its graphical representation.

Pupils should know when it is appropriate to find the mean of a data set.