Maths

•  Recognises some numerals of personal significance 

• Recognises numerals 1 to 5 

• Select the correct numeral to represent 1- 5, then 1-10 objects 

• Counts up to three or four objects by saying a number name for each item 

• Counts actions or objects which cannot be moved 

• Counts objects to 10, and beginning to count beyond 10 

• Counts out up to six objects from a larger group 

• Counts an irregular arrangement of up to ten objects 

• ELG – count reliably with numbers from one to 20 

• One more, one less 

• Says the number that is one more than a given number 

• ELG – with numbers from one to 20, say which number is one more or less than a given number

• Length, weight, and capacity 

• Orders two or three items by length or height 

• Orders two items by weight or capacity 

• Time 

• Orders and sequences of familiar events 

• Measures short periods of time in simple ways 

• ELG – children use everyday language to talk about time

Pattern 

• Uses familiar objects and common shapes to create and recreate patterns 

• ELG – recognise, create, and describe patterns 

Shape 

• Beginning to use mathematical names for ‘solid’ 3D shapes and ‘flat’ 2D shapes, and mathematical terms to describe shapes 

• Uses familiar objects and common shapes to create and recreate patterns and build models 

• ELG – explore the characteristics of everyday objects and shapes and use mathematical language to describe them 

Year 1 

• 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 

• 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 

• Given a number, identify one more and one less 

Year 2

• Count in steps of 2, 3, and 5 from 0, and in tens from any number, forward or backward 

• Recognise the place value of each digit in a two-digit number (tens, ones) 

• Identity, represent and estimate numbers using 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

Year 1 

• 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

Year 2

• 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

Year 1

• 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

• Recognise, find, name, and write fractions 1/3, ¼, 2/4 and ¾ of a length, shape, set of objects or quantity 

• Write simple fractions e.g., ½ of 6 = 3 and recognise the equivalence of 2/4 and 1/2

Year 1

• Compare, describe, and solve practical problems for: 

• lengths and heights [for example, long/short, longer/shorter, tall/short, double/half] 

  • mass or weight [for example, heavy/light, heavier than, lighter than] 

  • capacity/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 a half past the hour and draw the hands on a clock face to show these times

Year 2

• 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 the symbols for pounds (£) and pence (p); combine amounts to make a particular value 

• Find different combinations of coins that equal the same amount 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.

Year 1

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

  • 2-D shapes (for example, rectangles (including squares), circles and triangles) 

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

• POSITION 

• Describe position, direction, and movement, including whole, half, quarter, and three-quarter turns

Year 2

• Identify and describe the properties of 2-D shapes, including the number of sides and line 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 

• POSITION 

• 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 three-quarter turns (clockwise and anti-clockwise)

Year 2

• 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 totaling and comparing categorical data 

Year 3

• 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) 

• Identify, represent, and estimate numbers using different representations 

• Compare and order numbers up to 1000 

• Read and write numbers up to 1000 in numerals and in words 

• Find 1, 10 or 100 more or less than a given number 

• Solve number problems and practical problems involving these ideas

Year 4

• Count in multiples of 6, 7, 9, 25 and 1000  

• 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)  

• Identify, represent, and estimate numbers using different representations 

• Order and compare numbers beyond 1000 

• Round any number to the nearest 10, 100 or 1000 

• Find 1000 more or less than a given number 

• 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

Year 5

• 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. 

Year 6

• 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

Year 3

• 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

Year 4

• Recall multiplication and division facts for multiplication tables up to 12x12 

• Multiply two-digit and three-digit numbers by a one-digit number using formal written layout 

• 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 

• 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

Year 3

• 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 [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

Year 4

• Count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten 

• Recognise and show, using diagrams, families of common equivalent fractions 

• Add and subtract fractions with the same denominator 

• Recognise and write decimal equivalents to ¼; ½; ¾ 

• Recognise and write decimal equivalents of any number of tenths or hundredths 

• 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 problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where an answer is a whole number  

• Solve simple measures and money problems involving fractions and decimals to two decimal places

Year 5

• 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  

• 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  

• 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 the numbers of 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 percentages and decimal equivalents of 2 1, 4 1, 5 1, 5 2, 5 4 and those fractions with a denominator of a multiple of 10 or 25.

Year 6

• 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 

• Divide proper fractions by whole numbers  

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

• 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

Year 6

• 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.

Year 6

• Use simple formula 

• Generate and describe linear number sequences 

• Express missing number problems algebraically 

• Find pairs of numbers that satisfy an equation with two unknowns 

• Enumerate the possibilities of combinations of two variables

Year 3

• 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, 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]

Year 4

• Convert between different units of measure [for example, 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 squares 

• Estimate, compare, and calculate different measures, including money in pounds and pence 

• Read, write, and convert the 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

Year 5

• 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 include 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.

Year 6

• 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 km3]. 

Year 3

• Draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D 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 

Year 4

• 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 

• POSITION 

• 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.

Year 5

• Identify 3-D shapes, including cubes and other cuboids, from 2-D 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: 

• angles at a point and one whole turn (total 360o)  

• angles at a point on a straight line and ½ a turn (total 180o)  

• other multiples of 90o  

• 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.

Year 6

• Draw 2-D shapes using given dimensions and angles 

• 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 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

Year 5

• 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.

Year 6

• 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.

Year 3

• 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

• 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

• Solve comparison, sum and difference problems using information presented in a line graph 

• Complete, read and interpret information in tables, including timetables

Year 6

• Interpret and construct pie charts and line graphs and use these to solve problems 

• Calculate and interpret the mean as an average. 

Year 5

• 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. 

• 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 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

Year 6

• 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  

• 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, an appropriate degree of accuracy

• Understand and use place value for decimals, measures, and integers of any size.  

• Order positive and negative integers, decimals, and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, ≤, ≥ 

• Round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures] 

• Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b 

• Interpret and compare numbers in standard form A x 10n 1≤A< 10, where n is a positive or negative integer or zero 

• Use standard units of mass, length, time, money, and other measures, including with decimal quantities 

• Use a calculator and other technologies to calculate results accurately and then interpret them appropriately.  

• Appreciate the infinite nature of the sets of integers, real and rational numbers.

• Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property  

• Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative  

• Use conventional notation for the priority of operations, including brackets, powers, roots, and reciprocals  

• Recognise and use relationships between operations including inverse operations  

• Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations 

• Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b 

• Order positive and negative integers, decimals, and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, ≤, ≥ 

• Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 2 7 or 0.375 and 8 3)  

• Define percentage as ‘number of parts per hundred,’ interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%  

• Interpret fractions and percentages as operators

• Change freely between related standard units [for example time, length, area, volume/capacity, mass] 

• Use scale factors, scale diagrams and maps 

• Express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1  

• Use ratio notation, including reduction to simplest form 

• Divide a given quantity into two parts in a given part: part or part: whole ratio; express the division of a quantity into two parts as a ratio 

• Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction 

• Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions 

• Solve problems involving percentage change, including percentage increase, decrease and original value problems and simple interest in financial mathematics 

• Solve problems involving direct and inverse proportion, including graphical and algebraic representations 

• Use compound units such as speed, unit pricing and density to solve problems.

• Use and interpret algebraic notations, including: 

1. ab in place of a × b  

2. 3y in place of y + y + y and 3 × y  

3. a2 in place of a × a, a3 in place of a × a × a; a2 b in place of a × a × b  

4. b a in place of a ÷ b  

5. Coefficients written as fractions rather than as decimals  

6. Brackets 

• Substitute numerical values into formulae and expressions, including scientific formulae  

• Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors 

• Simplify and manipulate algebraic expressions to maintain equivalence by:  

• collecting like terms  

• multiplying a single term over a bracket 

• taking out common factors  

• expanding products of two or more binomials 

• Understand and use standard mathematical formulae; rearrange formulae to change the subject  

• Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs 

• Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement 

• Work with coordinates in all four quadrants 

• Recognise, sketch, and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane  

• Interpret mathematical relationships both algebraically and graphically Mathematics – key stage 3 7  

• Reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically, and algebraically  

• Use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations 

• Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential, and reciprocal graphs 

• Generate terms of a sequence from either a term-to-term or a position-to-term rule  

• Recognise arithmetic sequences and find the nth term  

• Recognise geometric sequences and appreciate other sequences that arise.

• Derive and apply formulae to calculate and solve problems involving perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)  

• Calculate and solve problems involving perimeters of 2-D shapes (including circles), areas of circles and composite shapes 

• Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line 

• Interpret mathematical relationships both algebraically and geometrically

• Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres to solve problems in 3-D 

• Draw and measure line segments and angles in geometric figures, including interpreting scale drawings 

• Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric 

• Use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles  

• Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies 

• Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids  

• Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles 

• Understand and use the relationship between parallel lines and alternate and corresponding angles 

• Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons 

• Apply angle facts, triangle congruence, similarity, and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs  

• Use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

• Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures 

• Describe, interpret, and compare observed distributions of a single variable through appropriate graphical representation involving discrete, continuous, and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)  

• Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data 

• Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs.

• Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale 

• Understand that the probabilities of all possible outcomes sum to 1  

• Enumerate sets and unions/intersections of sets systematically, using tables, grids, and Venn diagrams 

• Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

Foundation 

• Apply systematic listing strategies 

• Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations 

• Apply and interpret limits of accuracy when rounding or truncating, 

• Calculate with roots, and with integer indices 

• Calculate with numbers in standard form A 10n, where 1 ≤ A < 10 and n is an integer 

Higher

• Apply systematic listing strategies, including use of the product rule for counting 

• Estimate powers and roots of any given positive number 

• Apply and interpret limits of accuracy when rounding or truncating, including upper and lower bounds. 

• Calculate with roots, and with integer and fractional indices 

Foundation

• Calculate with roots, and with integer indices 

• Calculate exactly with fractions, and multiples of π.

Higher

• Calculate with roots, and with integer and fractional indices

• Calculate exactly with fractions, surds, and multiples of π; simplify surd expressions involving squares and rationalise denominators 

Foundation

Higher

• Change recurring decimals into their corresponding fractions and vice versa 

Foundation

• Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)  

• Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts  

• Understand that X is inversely proportional to Y is equivalent to X is proportional to 1 Y interpret equations that describe direct and inverse proportion  

• Interpret the gradient of a straight-line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion  

• Set up, solve, and interpret the answers in growth and decay problems, including compound interest

Higher

• Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios) 

• Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts 

• Understand that X is inversely proportional to Y is equivalent to X is proportional to 1 Y; {construct and} interpret equations that describe direct and inverse proportion 

• Interpret the gradient of a straight-line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion 

• Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic, and graphical contexts 

• Set up, solve, and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes

Foundation 

• Simplify and manipulate algebraic expressions (including those involving surds) by:  

1. factorising quadratic expressions of the form 2 x bx c + + 2 ax bx c + +, including the difference of two squares.  

2. simplifying expressions involving sums, products, and powers, including the laws of indices 

• Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments  

• Where appropriate, interpret simple expressions as functions with input and output.  

• Use the form y mx c = + to identify parallel lines; find the equation of the line through two given points, or through one point with a given gradient  

• Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically  

• Recognise, sketch, and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function 1 y = x y x = cos with x ≠ 0 

• Plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed, and acceleration 

• Solve quadratic equations algebraically by factorising, find approximate solutions using a graph  

• Solve two simultaneous equations in two variables (linear/) algebraically; find approximate solutions using a graph  

• Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s), and interpret the solution  

• Solve linear inequalities in one variable; represent the solution set on a number line,  

• Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (r n where n is an integer, and r is a positive rational number)  

• Deduce expressions to calculate the nth term of sequences.

Higher

• Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:  

• factorising quadratic expressions of the form 2 x bx c + + 2 ax bx c + +, including the difference of two squares; factorising quadratic expressions of the form 

• Simplifying expressions involving sums, products, and powers, including the laws of indices  

• Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and  

• Use algebra to support and construct arguments and proof 

• Where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function;’ interpret the succession of two functions as a ‘composite function’ 

• Use the form y mx c = + to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient  

• Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square 

• Recognise, sketch, and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function 1 y = x y x = cos with x ≠ 0, the exponential function x y k = y x = sin for positive values of k, and the trigonometric functions (with arguments in degrees), and y x = tan for angles of any size, sketch translations and reflections of the graph of a given function 

• Plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, 

• To find approximate solutions to problems such as simple kinematic problems involving distance, speed, and acceleration  

• Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs, and graphs in financial contexts  

• Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point 

• Solve quadratic equations including those that require rearrangement algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph 

• Solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph  

• Find approximate solutions to equations numerically using iteration 

• Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s), and interpret the solution 

• Solve linear inequalities in one or two variables, and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph 

• Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (r n where n is an integer, and r is a positive rational number or a surd) and other sequences 

• Deduce expressions to calculate the nth term of linear {and quadratic} sequences.

Foundation

• Calculate arc lengths, angles, and areas of sectors of circles  

• Calculate surface areas and volumes of spheres, pyramids, cones, and composite solids  

• Apply the concepts of congruence and similarity, including the relationships between lengths, in similar figures

Higher

• Calculate arc lengths, angles, and areas of sectors of circles  

• Calculate surface areas and volumes of spheres, pyramids, cones, and composite solids 

• Apply the concepts of congruence and similarity, including the relationships between lengths, areas, and volumes in similar figures

Foundation

• Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector, and segment 

• Construct and interpret plans and elevations of 3D shapes 

• Interpret and use bearings 

• Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles} in two dimensional figures 

• Know the exact values of sin and cos θ for 0, 30, 45, 60 90 and know the exact value of tanθ at 0, 30, 45, 60

Higher

• Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector, and segment  

• Apply and prove the standard circle theorems concerning angles, radii, tangents, and chords, and use them to prove related results} 

• Construct and interpret plans and elevations of 3D shapes  

• Interpret and use bearings 

• Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles and, where possible, general triangles in two- and three-dimensional figures 

• Know the exact values of sin and cos θ for 0, 30, 45, 60 90 and know the exact value of tanθ at 0, 30, 45 60 

• Know and apply the sine rule and cosine rule to find unknown lengths and angles 

• Know and apply area = 0.5 ab Sin C to calculate the area, sides, or angles of any triangle

Foundation

• Interpret and use fractional scale factors for enlargements 

• Describe translations as 2D vectors 

• Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors

Higher

• Interpret and use fractional {and negative} scale factors for enlargements 

• Describe the changes and invariance achieved by combinations of rotations, reflections, and translations 

• Describe translations as 2D vectors 

• Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors 

• Use vectors to construct geometric arguments and proof

Foundation

• Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling 

• Interpret and construct tables and line graphs for time series data 

• Interpret, analyse, and compare the distributions of data sets from univariate empirical distributions through: 

• Appropriate graphical representation involving discrete, continuous, and grouped data,  

• Appropriate measures of central tendency (including modal class) and spread 

• Apply statistics to describe a population  

• Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

Higher

• Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling 

• Interpret and construct tables and line graphs for time series data 

• Construct and interpret diagrams for grouped discrete data and continuous data, i.e., histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use 

• Interpret, analyse, and compare the distributions of data sets from univariate empirical distributions through:  

1. appropriate graphical representation involving discrete, continuous, and grouped data, including box plots  

2. appropriate measures of central tendency (including modal class) and spread including quartiles and inter-quartile range 

• Apply statistics to describe a population  

• Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

Foundation

• Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one  

• Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size  

• Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions

Higher

• Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one  

• Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size  

• Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions  

• Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagram.