# Maths

Mathematics

Aims

The national curriculum for mathematics aims to ensure that all pupils:

• Become fluent in the fundamentals of mathematics, including the varied and regular practice of increasingly complex problems over time.
• Reason mathematically by following a line of enquiry, understanding relationships and generalisations, and developing an argument, justification or proof using mathematical language.
• Can solve problems by applying their mathematics to a variety of problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

Mathematics is an interconnected subject in which pupils need to be able to move fluently between mathematical ideas. The programmes of study are, by necessity, organised into distinct areas, but pupils will make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They will also apply their mathematical knowledge to science and other subjects.

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress will always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly will be challenged through being offered rich and sophisticated problems before any acceleration through new content.

Those who are not sufficiently fluent with earlier material will consolidate their understanding, including additional practice, before moving on.

Year 1 programme of study

Number – 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 multiples of twos, fives and tens

·         When 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 one-digit and two-digit numbers to 20, including zero

• solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 =  ? – 9

Number Multiplication and division

Pupils should be taught to:

Solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher

Number 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

Measurement

Pupils should be taught to:

• compare, describe and solve practical problems for:
• lengths and heights [for example, long/short, longer/shorter, tall/short, double/half]
• mass/weight [for example, heavy/light, heavier than, lighter than]
• capacity and volume [for example, full/empty, more than, less than, half, half full, quarter]

·         time [for example, 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 [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

Geometry Properties of shape

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]

3D shapes [for example, cuboids (including cubes), pyramids and spheres

Geometry Position and direction

Pupils should be taught to:

describe position, direction and movement, including whole, half, quarter and three quarter turns

Year 2 programme of study

Number Number and place value

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 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

Pupils should be taught to:

• solve 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
•    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

Number 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

·         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

Number Fractions

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 ½

Measurement

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

Geometry Properties of shape

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

Geometry Position and direction

Pupils should be taught to:

·         order and arrange patterns 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 a  angles for quarter, half, three-quarter turns (clockwise and anti-clockwise)

Statistics

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.

Year 3 programme of study

Number Number and place value

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

o    a three-digit number and hundreds

o    add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction

o    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

Number 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 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

Number 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 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 [for example, 5/7 + 1/7 = 6/7]

·         compare and order unit fractions, and fractions with the same denominators

solve problems that involve all of the above

Measurement

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 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 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]

Geometry – Properties of shape

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 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 and vertical lines and pairs of perpendicular and parallel lines

Statistics

Pupils should be taught to:

·         interpret and present data using bar charts, pictograms and tables

solve one-step and two-step questions [for example, ‘How many more?’ and ‘How many    fewer?’] using information presented in scaled bar charts and pictograms and tables

Year 4 programme of study

Number – Number and place value

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 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 two-step problems in contexts, deciding which operations and methods to use and why

Number – 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 commutatively 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 to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects

Number – Fractions (and decimals)

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 non-unit 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 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 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

Measurement

Pupils should be taught to:

·         convert between different units of measure [for example, kilometre to metre; hour to minute]

·         estimate, compare and calculate different measures, including money in pounds and pence

·         measure and calculate the perimeter of a rectilinear figure (including squares) in centimetres and metres

·         find the area of rectilinear shapes by counting squares

·         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

Geometry – Properties of shape

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

Geometry – Position and direction

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

Statistics

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 programme of study

Number – Number and place value

Pupils should be taught to:

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

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

·         interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through 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 multi-step problems in contexts, deciding which operations and methods to use and why

Number – Multiplication and division

Pupils should be taught to:

·         Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers

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

·         Establish whether a number up to 100 is prime and recall prime numbers p to 19

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

·         multiply and divide numbers mentally drawing upon known facts

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

·         multiply and divide whole numbers and those involving decimals by 10, 100 and 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

Number – Fractions (including decimals and percentages)

Pupils should be taught to:

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

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

·         recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, 2/5 + 4/5 = 6/5 = 1 and 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 1/2, 1/4, 1/5, 2/5, 4/5 and those fractions with a denominator of a multiple of 10 or 25

Measurement

Pupils should be taught to:

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

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

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

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

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

·         solve problems involving converting between units of time

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

Geometry – Properties of shape

Pupils should be taught to:

·         Identify:

o    3D shapes, including cubes and other cuboids, from 2D representations

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

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

• Identify:
• angles at a point and one whole turn (total 360°)
• angles at a point on a straight line and 1/2 a turn (total 180°)

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

Geometry – Position and direction

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

Statistics

Pupils should be taught to:

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

complete, read and interpret information in tables, including timetables

Year 6 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 zero

solve number and practical problems that involve all of the above

Number – 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 formal written method of long multiplication

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

• identify common factors, common multiples and prime numbers
• perform mental calculations, including with mixed operations and large 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, an appropriate degree of accuracy
•

Number – 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

·         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, 1/4 × 1/2 = 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 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

recall and use equivalences between simple fractions, decimals and percentages, including in different contexts

Ratio and proportion

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

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

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 (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km

Geometry – properties of shape

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 angles³]

Geometry – position and direction

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

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

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