(a)

explore, interpret, develop flexible approaches to, and persist with problems involving number and measures in a variety of forms

(a)

use correct language, symbols, and vocabulary associated with number and measures

(b)

explain and record methods and results in spoken, pictorial, and written form

(a)

present results in an organised way; sort and classify numbers according to given criteria

(b)

understand that some statements are exact and can be clearly demonstrated

(a)

count reliably at first up to 20 objects, later extending counting to 100 and beyond (to 120 say), remaining secure across “tens boundaries” [e.g. from 19 to 20, or from 99 to 100]; recognise the invariance of quantity

(b)

estimate a number of objects that can be checked by counting; round two digit numbers to the nearest 10

(a)

understand the groupings into units and 10s (and later into 100s) that underpin place value; know what each digit represents (including 0 as a number, and as a placeholder), and how the “value” represented by each digit is determined by its position

(b)

read and write two-digit and three-digit numbers in figures and words

(c)

order two-digit numbers and position them on a number line; use =, <, and > and the associated language

(a)

create, describe, and explore basic number patterns and sequences—including odd and even numbers, multiples of 2, multiples of 5, and multiples of 10

(a)

understand addition and use related vocabulary and notation; understand subtraction (as “take away” and as “difference”) and use the related vocabulary and notation; recognise that subtraction is the inverse of addition

(b)

identify and use the calculations needed to solve simple word problems and inverse problems [e.g. oral “I’m thinking of a number” problems]

(c)

understand simple instances of multiplication as repeated addition, and division as “grouping”, and as “sharing”; use the vocabulary and notation associated with multiplication and division; find one half, or one quarter of a familiar shape, or of a small set of objects

(a)

develop instant recall of number facts; know addition and subtraction facts with totals less than 10, and use these to derive other facts; learn addition facts with totals up to 20

(b)

know ×2, ×5, and ×10 multiplication tables, and derive the corresponding division facts; know the doubles of numbers to 20 and the corresponding halves

(c)

use practical and informal written methods to add and subtract two-digit numbers

(d)

develop mental methods which flexibly use known facts to calculate the answer to less familiar “sums” [e.g. working out 4 × 6 by doubling 2 × 6, or by doubling 4 × 3]; add 10 to any single digit number, then add and subtract a multiple of 10 to or from a two-digit number

(e)

make sense of number sentences involving all four operations

(f)

lay out and complete simple two-digit additions and subtractions in standard column format

(g)

use practical and informal written methods and related vocabulary to support multiplication and division, including calculations with remainder

(a)

choose sensible calculation methods to solve simple word problems involving whole numbers—including problems involving money or measures, drawing on their understanding of arithmetical operations

(a)

solve suitable problems using simple lists, tables, and charts to sort, classify, and organise information; discuss the methods they use and explain what they find

(a)

follow instructions to construct simple 2D and 3D objects; represent 3D objects via 2D drawings

(a)

use correct language and vocabulary for shape, space, and measures

(b)

measure objects using ad hoc informal as well as standard measures; record measurements in ordered tables

(a)

recognise simple spatial patterns and relationships; sort and classify shapes according to given criteria

(a)

describe relationships using the language “larger – smaller”, “higher – lower”, “longer – shorter”, “above – below”, “left of – right of”

(b)

draw and describe properties of 2D and 3D shapes; recognise, name, and sort common 2D and 3D shapes—including triangles, rectangles (including squares), circles, cubes, cuboids, hexagons, pentagons, cylinders, pyramids, cones, and spheres

(c)

recognise right angles; understand whole turns, and quarter- and half-turns (clockwise and anticlockwise)

(a)

use direct comparison to order objects by size, using appropriate language; put familiar events in chronological order

(b)

measure and draw straight lines accurate to the nearest centimetre

(c)

estimate, compare, and measure lengths, weights, and capacities; choose and use standard units (m, cm, kg, litre); compare durations (using seconds, minutes, hours, days); read and interpret numbers on scales to the nearest labelled division, interpreting the divisions between them; identify time intervals, including those that cross the hour

(a)

extract numerical, geometrical, and logical information from simple problems expressed in words

(b)

make connections; use integers, decimals, and fractions (and arithmetic) when solving problems involving measures, and in other settings

(c)

solve multi-step, and simple inverse problems

(d)

solve problems involving tables, lists, and information presented pictorially;

(e)

use knowledge of exact arithmetic to make good mental estimates

(a)

use notation, terminology, symbols, and language correctly

(b)

present results and solutions to problems clearly; explain reasoning, methods, and conclusions

(c)

interpret tables, lists, and charts; construct and interpret frequency tables

(a)

present results in an organised way; sort and classify numbers and shapes according to given criteria

(b)

investigate apparent patterns; understand that some statements are exact and can be clearly explained

(a)

count reliably beyond 100, passing smoothly from any given set of “90s” onto the next hundred

(b)

count on and back in steps of constant size, starting from any integer, extending to negative integers

(a)

use place value in representing numbers first up to 1000, then up to 10 000 and beyond; extend to decimals with up to three decimal places

(a)

recognise two- and three-digit multiples of 2, 5 and 10; find the factors of a given integer, and the common factors of two given integers; find the HCF and the LCM of two given integers; recognise prime numbers to 50, and square numbers to 10 × 10; find factor pairs and all the factors of any two-digit integer; double or halve any two-digit integer

(a)

read, write (in figures and words), and order whole numbers to 10 000

(b)

multiply, and divide, any integer by 10 or 100, and then by 1000; round integers to the nearest 10 or 100, and then 1000

(c)

understand and use negative integers; order a given set of positive and negative integers

(a)

use decimal notation for tenths, hundredths, and thousandths; order a set of numbers or measurements

(b)

compare and order integers and decimals in different contexts; locate integers (positive and negative), fractions, and decimals on the number line; use correctly the symbols =, ≠, <, ⩽, >, ⩾

(c)

multiply and divide, any integer or decimal by 10 or 100; round integers and decimals to the nearest integer, to the nearest ten, and to the nearest tenth

(a)

understand fractions; locate fractions on a number line; find fractional parts of shapes or quantities

(b)

understand equivalent fractions; simplify by cancelling common factors

(c)

order simple fractions

(d)

understand percentage; use simple percentages for comparison; find fractions and percentages of whole number quantities, and express part of a given whole as a percentage; express one whole number quantity as a fraction of another

(e)

divide a given quantity into two parts in a given ratio (both part-to-part and part-to-whole); compare quantities in a given (external) ratio; solve simple problems involving ratios

(a)

develop their understanding of the four number operations—including inverses, and operations with zero

(b)

find remainders after division; express a quotient as a fraction or decimal; relate to p ÷ q

(c)

know and use the conventions for the order of operations; understand and use structural arithmetic to simplify calculations; write numerical expressions involving brackets; group related terms in a sum and related factors in a product to simplify, and hence evaluate, numerical expressions

(a)

achieve instant recall of all addition and subtraction facts for integers up to 20

(b)

add or subtract any pair of two-digit integers; handle suitable three-digit and four-digit additions and subtractions presented in written form

(c)

add and subtract positive and negative integers mentally

(d)

achieve instant recall of (i.e. know by heart) multiplication tables to 10 × 10 and use them to derive division facts

(e)

multiply and divide in the range 1 to 100, then for larger numbers

(f)

derive multiplication and division facts involving decimals

(g)

relate fractions to multiplication and division; work with simple quotients as fractions and as decimals; switch freely between equivalent fractions; add and subtract simple fractions by reducing to a common denominator

(a)

use the standard written method in column format to add and subtract three-digit positive integers, then four-digit positive integers; add and subtract numbers involving decimals

(b)

use the standard written method in column format for short multiplication (of two- and three-digit integers by a single digit multiplier), then long multiplication of two-digit and three-digit integers by two-digit multipliers; extend to simple decimal multiplication

(c)

use short division of two-digit and three-digit integers by numbers up to 12

(d)

use approximations and other strategies to check that answers are reasonable

(a)

calculate reliably with standard measures, money, and time; convert measures from one unit to a related unit

(b)

relate distance, time, and speed in uniform rectilinear motion; work with other simple rates and compound measures

(a)

use the four number operations to solve word problems involving numbers, or money, or measures of length and area, mass, capacity, or time

(b)

solve multi-step and inverse problems with confidence

(c)

check that their results are reasonable; explain why their answers are correct

(a)

solve suitable problems using simple lists, tables, and charts to sort, classify, and organise information, discuss the methods they use, interpret their results, and explain what they find

(b)

explore the notions of “centre” and “spread” for numerical data sets

(a)

recognise standard geometrical figures; use their properties to select and perform appropriate calculations; measure and draw accurately to construct 2D and 3D figures

(b)

use standard units of measurement and simple compound measures; convert reliably between related units

(a)

use geometrical notation, terminology, and symbols correctly; interpret solutions to problems involving geometrical figures and measures; organise work and record findings clearly

(a)

analyse standard 2D and 3D figures; calculate efficiently and make simple deductions with angles, lengths, areas, volumes, time, and other measures

(a)

recognise right angles, perpendicular and parallel lines; know that angles at a point total 360°, that angles at a point on a straight line total 180°, and that angles in a triangle total 180°

(b)

describe relationships using the language “larger – smaller”, “higher – lower”, “longer – shorter”, “above – below”, “left of – right of”, “top – bottom”, “in front of – behind”, “closer – further away”, “between”

(c)

talk clearly about common 2D and 3D shapes; visualise 3D shapes from 2D drawings

(d)

make and draw shapes with increasing accuracy, and analyse their geometrical properties

(a)

read and plot coordinates—eventually in all four quadrants; draw, or locate, shapes with given properties in the coordinate plane

(b)

visualise, predict, and represent the position of a shape following a rotation, reflection, translation, or glide reflection

(a)

draw and measure lines to the nearest millimetre; combine linear measurements to measure perimeters

(b)

draw and measure acute and obtuse angles of a given size to the nearest degree; estimate the size of given angles and order them; draw angles reliably as parts of compound shapes

(c)

read the time to the nearest minute; calculate time intervals from clocks, from timetables, and from calendars

(d)

use standard units of length, area, volume, mass, and capacity; measure and weigh items; convert between related units

(e)

find areas of rectangles and of shapes composed of rectangles

(f)

measure and compare capacities; understand conservation of volume; find volumes of cuboids and of simple shapes composed of cuboids

(g)

read scales with increasing accuracy; record measurements using decimal notation

(a)

use numerical, geometrical, and logical information in analysing data and in solving simple problems

(b)

make connections; use the arithmetic of integers, decimals, and fractions when solving problems

(c)

regularly solve multi-step problems and inverse problems

(d)

solve problems involving measures, rates and compound measures, ratio and proportion; make and justify estimates

(a)

use spoken and written language, notation, diagrams, terminology, and symbols correctly

(b)

recognise when information is presented in a misleading way

(c)

present results and solutions to problems clearly, declare unknowns explicitly, and lay out solutions logically line-by-line

(d)

interpret tables, lists, and information presented graphically; construct and interpret frequency tables; use precise measures of “centre” and “spread”

(a)

understand that some statements can be clearly proved, and that other statements can be shown to be false

(b)

use place value and structural arithmetic to simplify calculations and expressions; recognise and use the fact that mathematical operations often come in “direct-inverse” pairs

(c)

use basic results and step-by-step deduction to draw conclusions; investigate apparent patterns and test the validity of statements—proving or disproving these statements conclusively where possible

(a)

count reliably forwards and backwards across hundreds and thousands boundaries

(b)

solve problems involving counting [e.g. How many pages from page 171 to 263?—inclusive and exclusive; How many dots are in a 5 by 7 rectangular array? How many chords are there joining 10 points on a circle?]

(c)

use place value in representing integers to 1 000 000, and decimals with up to four decimal digits; express position as a “power of 10”; choose the power of 10 to transform a given decimal to an integer (by multiplying)

(a)

recognise multiples of 2, 4, 5, 10, 20, 25, 50, 100; factorise instantly any output from multiplication tables to 10 × 10; recognise (or test quickly) prime numbers to 100 and test possible primes up to 500; recognise square numbers to 20 × 20; find all the factors of a given integer

(b)

recognise powers of 2, powers of 3, and powers of 5; recognise square and cube roots of familiar squares and cubes; understand and find, or estimate, the square root of any positive number; use index notation for small positive integer powers

(c)

find specified terms of a sequence given a term-to-term or a position-to-term rule; guess the simplest position-to-term rule for the nth term given the first few terms of a sequence

(a)

read, write (in figures and words), and order whole numbers and decimals with up to six digits; understand, use, and calculate freely with (positive and negative) integers; use correctly the symbols =, ≠, <, ⩽, >, ⩾ and the associated language; order a set of positive and negative integers and decimals, or measurements

(b)

use correctly the terms factor, multiple, common factor, common multiple; find and use the HCF and LCM of two given integers; test for divisibility by 2, by 3, by 4, by 5, by 9, by 10

(c)

multiply, and divide, any integer or decimal by 10, 100, 1000, or 10 000; know the multiplicative complements for 10 (2 × 5), for 100, and for 1000, and the corresponding decimals [e.g. = 0.5, = 0.2, = 0.125]; recognise as alternative representations the decimal and fraction forms of simple fractions

(d)

express any given large number as a number less than 10 times a power of 10, and a small number as a number greater than or equal to 1 times a power of 10

(e)

compare measurements (in various contexts); round integers and decimals

(a)

understand general fractions in terms of unit fractions; switch freely between mixed numbers (with fractional part < 1) and standard fractional form

(b)

find fractional parts of shapes and quantities, and recognise the fractional part represented; solve simple ratio problems

(c)

understand equivalent fractions; express two given fractions with a common denominator; simplify a given fraction; order a list of integers and fractions

(d)

understand “percentage” as a fractional operator with denominator 100; find fractions and percentages of given quantities; express one quantity as a fraction of another; use the multiplicative character of percentage as an operator in calculations involving percentage increase and decrease; distinguish between absolute and relative increase and decrease

(e)

reduce a ratio to its simplest form, and establish the connection with “fractional parts”; divide a given quantity into two parts in a given ratio; solve problems involving ratio and proportion

(a)

extend existing mental calculation to include negative numbers, decimals and fractions

(b)

calculate effectively in solving problems

(a)

use multiplication tables freely to simplify fractional expressions; convert fractions to decimals and terminating decimals to their simplified fraction equivalents

(b)

obtain the prime-power factorisation of a given integer by successive division

(c)

understand and use place value, inverse operations [e.g. cancellation], and structural arithmetic to simplify calculations; represent numbers and roots as powers, including fractional powers; work with powers of 10

(d)

understand why (−1) × (−1) = 1 and why a −(−b) = a + b; use these to simplify and to evaluate numerical expressions

(e)

use the idea of choosing a suitable (common) denominator to add, subtract, multiply and divide fractions

(f)

solve word problems involving rates and ratios, including the unitary method

(g)

give both roots of simple quadratic equations; simplify numerical expressions involving simple surds [e.g. because both are positive and have the same square]

(a)

substitute numerical values into formulae and expressions and evaluate; multiply out brackets, collect like terms, identify and take out common factors to simplify expressions; recognise that different-looking expressions may be identical; prove simple algebraic identities, and explain why two given expressions are not identical

(a)

relate decimal arithmetic to integer arithmetic; use standard written methods in column format for addition and subtraction, short and long multiplication, short (and long) division of integers and decimals

(a)

solve simple linear inequalities in one variable and represent solutions on a number line

(a)

calculate and work with perimeters, areas, volumes, durations, capacities, and simple compound measures; use standard units of length, area, volume, mass, and capacity; read scales with appropriate rounding; record and order measurements using decimal notation; convert between related units

(b)

estimate the size of any given angle; draw and measure angles reliably to the nearest degree

(c)

calculate reliably with measures; extract and use information from tables and charts; solve word problems involving money, time, length, and compound measures (speed, rates)

(a)

set up and solve linear equations in complete generality [e.g.

reduce a linear equation in two variables to standard form (ax + by = c, or y = mx + c); eliminate a variable from two simultaneous linear equations in two unknowns; solve linear inequalities in one unknown

(b)

change the subject of a formula; draw the graph of a linear function, identifying its gradient, and interpreting its position; construct linear functions arising from real problems, sketch and interpret their graphs; establish the link to ratio and proportion

(c)

use letters in general expressions; use index notation for small positive integer powers; simplify given expressions

(d)

use algebra to find the exact solution of two simultaneous linear equations in two unknowns by eliminating a variable

(e)

sketch the graphs of simple quadratic functions; solve simple quadratic equations

(a)

solve arithmetical problems, word problems, and geometry problems involving numbers and measures; check that their results are reasonable

(b)

solve multi-step and inverse problems with confidence

(c)

use the unitary method to solve proportion problems and problems involving ratios and rates

(d)

use algebraic formulae; set up and solve equations

(a)

solve problems involving lists, tables, charts, and graphs; sort, classify, and organise information; discuss the methods they use and explain what they find

(b)

find the average (i.e. mean) and other measures of “centre”, and measures of spread for small datasets; identify the modal class for grouped data; interpret frequency diagrams and histograms; use cumulative frequency

(c)

use counting where each outcome is “equally likely” to calculate probabilities

(a)

solve geometrical problems involving standard geometrical figures in 2D and 3D, and angles, length, area, and volume

(b)

measure and calculate accurately to construct and analyse 2D and 3D figures; use standard units in geometry

(a)

use geometrical language, notation, terminology, and symbols correctly

(b)

work in all four quadrants of the coordinate plane

(c)

lay out calculations, constructions, and proofs line-by-line

(a)

use basic geometrical principles to justify each step in a calculation or deduction

(b)

analyse 2D and 3D configurations in terms of triangles

(a)

recognise right angles, perpendicular and parallel lines, and use the associated language precisely; know that angles at a point total 360°, and that angles at a point on a straight line total 180°

(b)

know that two lines are parallel precisely when alternate angles are equal (or, equivalently, when corresponding angles are equal); prove and use the usual consequences (including the angle-sum in any triangle)

(c)

use known angles and angle properties to find unknown angles in given configurations (i.e. angle-chasing)

(d)

motivate the formula for the circumference of the circle and estimate π; solve related problems

(e)

talk about and work with common 2D and 3D shapes (including triangles [e.g. right angled, isosceles, and equilateral], quadrilaterals [e.g. parallelograms, rhombuses, rectangles, squares, and trapezia], cuboids, and prisms); correctly copy drawings from the board; make and draw shapes with increasing accuracy, and analyse their geometrical properties

(a)

use ruler and protractor to draw triangles with given data; extract and apply the basic congruence criteria (SAS, SSS, ASA; RHS) to prove standard results (including that the base angles in any isosceles triangle are equal, and the converse)

(b)

draw specified figures using “ruler” (i.e. straightedge) and compasses only; use the basic ruler and compass constructions to complete other constructions

(a)

find the area of rectangles and shapes made from rectangles; find the area of right angled triangles and of general triangles; find the area of a general parallelogram

(b)

relate the formula for the area of a circle to the formula for the circumference; use the formula to solve related problems

(c)

state, prove, and use Pythagoras’ Theorem

(a)

understand and use the terms centre, radius, chord, diameter, circumference, tangent, arc, sector, segment

(b)

prove the basic properties of a circle [e.g. centre and any chord form an isosceles triangle; angle in a semicircle is a right angle; tangent is perpendicular to radius; tangents form an external point are equal]; apply these results to solve problems

(a)

calculate volumes of cuboids and shapes made of cuboids; calculate volumes of a “wedge” (half a cuboid), polygonal right prisms, and cylinders

(b)

find lengths and angles in simple 3D figures by considering 2D cross-sections

(a)

draw figures to scale; interpret distances, angles, and areas on maps and other scale drawings

(a)

interpret a circle as a locus; interpret the perpendicular bisector of a given line segment as a locus

(a)

read and plot coordinates in all four quadrants

(b)

use Pythagoras’ Theorem to calculate the distance between two given points (simple cases); find the coordinates of the midpoint of a line segment (simple cases)

(c)

establish the link between straight lines in the coordinate plane and linear equations in x and y; understand that parallel lines have the same gradient; find the intersection of two given straight lines

(d)

sketch the graphs of simple quadratic functions

(e)

explore and use coordinates in 3D

(a)

use the terms factor, multiple, common factor, common multiple; find the HCF and LCM of given integers

(b)

recognise (or test quickly) prime numbers to 120; use the “square root test” to identify primes to 1000

(c)

obtain the prime power factorisation of a given integer; list all factors of a given integer

(d)

move freely between “mixed” fractions and fractions in standard fractional form ; reduce a given fraction to lowest terms; rewrite two given fractions with a common denominator; order a list of fractions

(e)

use factorisation to simplify surd expressions [e.g.

(a)

understand the unit fraction as “that part, of which q identical copies make 1”; understand a general fraction as a multiple of a unit fraction; move freely from a given fraction to a suitable equivalent fraction

(b)

add and subtract fractions; multiply and divide fractions; simplify, and hence evaluate, compound expressions involving fractions

(a)

move freely between terminating decimals and decimal fractions

(b)

know the equivalence of the exact (unevaluated) fraction notation and the result of evaluating p ÷ q; find the decimal of any given fraction; understand why the decimal form of must terminate, or recur

(c)

change any terminating decimal into a fraction in its lowest terms; change any recurring decimal into a fraction

(a)

recognise (for k > 0) as the exact positive real number whose square is equal to k; given k > 0 find the exact or approximate value of ; use the algebra of surds—including rationalising denominators [e.g.

use surds (and π) to calculate exactly in geometric contexts; give lengths arising from applications of Pythagoras’ Theorem and solutions to quadratic equations in exact (mixed surd) form

(b)

use the standard notation for, and calculate with, cube roots;

(a)

factorise instantly any output from multiplication tables to 10 × 10; recognise square numbers to 25 × 25; recognise cubes to 6³.

(b)

recognise powers of 2, 3, 4, 5; recognise square and cube roots of familiar squares and cubes; extend powers and roots to simple fractions and decimals; find, or estimate, the square root or cube root of any positive number

(c)

know, understand, and use the index laws; use index notation to present expressions in simplified power form; calculate freely with numerical and algebraic expressions involving powers

(d)

write any given number in standard form and translate a given standard form into the (approximate) number it represents; calculate with numbers given in standard form “as though they are exact”

(a)

find and recognise fractional parts of shapes and quantities; express one quantity as a fraction of another

(b)

understand percentage as a fractional operator with denominator 100; know and use the percentage equivalents of familiar fractional parts; work freely with percentages; use the multiplicative character of percentage increase and decrease; solve problems involving percentage change (including inverse problems and compound interest)

(a)

work with standard integer sequences; generate terms given a term-to-term rule, or a position-to-term rule; guess the simplest position-to-term rule for the nth term given the first few terms of a sequence

(b)

use a given term-to-term rule to find a closed formula for the position-to-term rule

(c)

find the term-to-term rule and the position-to-term rule for sequences defined intrinsically

(d)

understand that when x < 1 (or |x| < 1) the sequence of powers (xⁿ) tends rapidly to 0, and when x > 1 (or |x| > 1) the sequence of powers (xⁿ) grows rapidly without bound; link to compound interest, to population growth, to doubling times and to radioactive half-life

(a)

divide a given quantity into two parts in a given part-to-part, or part-to-whole ratio; express the division of a quantity into two parts as a ratio; work with separate quantities in a given (external) ratio; reduce a ratio to its simplest form

(b)

calculate the result of a change of units; draw and use scale diagrams and maps; understand the effect of scaling and enlargement on different quantities (including angles, lengths, areas, and volumes)

(c)

solve proportion problems (where three of the four variables are given, determine the fourth); use the unitary method, and then the general method, to solve proportion problems

(d)

understand and use “X is inversely proportional to Y” as meaning “X is proportional to ”

(a)

substitute numerical values into formulae and expressions

(b)

multiply out brackets, collect like terms, and take out common factors to simplify linear, quadratic and higher order expressions; simplify general expressions (possibly involving powers and roots) by using additive simplification, the distributive law, and cancellation—giving answers in factorised form; work with algebraic fractions having linear and quadratic denominators

(c)

rearrange formulae; solve problems using standard formulae

(d)

set up linear equations; solve the general linear equation in one unknown

(e)

set up linear equations in two unknowns; interpret a linear equation in two unknowns in the coordinate plane as representing a straight line; draw the graph of a linear function, identifying its gradient, and interpreting its position; find the gradient from an equation given in any form; transform a given equation into the form y = mx + c (or x = a); construct linear functions arising from real problems, sketch and interpret their graphs; establish the link to ratio and proportion

(f)

solve any pair of simultaneous linear equations by eliminating a variable; interpret the analytic solution as “finding the point of intersection” (if any) of the two lines

(g)

factorise quadratic expressions in one variable; solve quadratic equations by factorising; interpret solutions as those points where the graph crosses the x-axis; solve fractional equations that reduce to quadratics [e.g.

(h)

factorise and use the difference of two squares; conclude that, if k > 0, the equation x² = k has two solutions (±); interpret this as a statement about the graph of y = x² − k

(i)

know and use the expansion of (x + a)²; extend to (x + a)³; use this to “complete the square” for any given quadratic; obtain the formula for the solutions of the general quadratic

y = ax² + bx + c;

use this formula to solve quadratic equations; deduce the symmetry of the graphs of quadratic functions

(j)

solve two simultaneous equations where one is linear and the other quadratic; use completing the square to find the centre and radius, given the equation of a circle; find the points where two circles intersect

(k)

understand the difference between an equation and an identity; decide whether two given expressions are identical or not—then prove they are, or show that they are not

(l)

solve linear inequalities in one and two variables; interpret the solution graphically

(a)

read and plot coordinates in all four quadrants; move freely between straight lines in the coordinate plane and linear equations in x and y; derive the equation of a line through two given points, and the equation of a line through a given point with a given gradient

(b)

find the coordinates of the midpoint of a line segment; calculate the distance between two points in 2D or 3D

(c)

interpret straight line graphs arising in real situations

(d)

know and use the general form y = mx + c (or x = a) for a straight line; use gradient and intercept; find the point of intersection of two given straight lines

(e)

know that parallel lines have the same gradient; prove and use the fact that two lines with gradients m and m′ are perpendicular precisely when m · m′ = −1

(f)

for particular values of m and c interpret the standard form y = mx + c as ‘Y = mX’ relative to an origin at (0, c) or at (− ,0)

(g)

sketch the graph of any given quadratic function by completing the square

(h)

sketch other graphs—including simple cubic functions, the reciprocal function y = , the exponential function y = k^x for easy (positive) values of k, the circular functions y = sin x, y = cos x, and y = tan x

(i)

use coordinates to solve simple problems in 3D

(a)

engage in practical and theoretical work to construct and interpret tables, lists, and information presented graphically; use precise measures of “centre” and spread; sort, classify, and organise information

(b)

discuss variability; distinguish between data representing a single idealised measure and informal “random variables” sampled from a population or distribution

(c)

calculate the mean of a set of numbers or measures; use mode or median as appropriate to summarise the “centre”; identify the modal class for grouped data; refine measures of spread and “central tendency”

(d)

introduce ideas of probability via standard examples of discrete sample spaces in which each outcome is equally likely; explore the general notion of an “event”

(e)

understand why

prob(A ∪ B) = prob(A) + prob(B)

for disjoint events A, B; use simple counting to calculate probabilities in discrete sample spaces; understand and use the inclusion/exclusion formula

prob(A ∪ B) = prob(A) + prob(B) − prob(A ∩ B)

for events which are not necessarily disjoint

(a)

know and understand basic ruler and compass constructions; use these to devise simple constructions

(b)

use standard units in geometry; solve geometrical problems in 2D and 3D involving calculation, construction, and deduction

(c)

measure and calculate accurately to construct and analyse 2D and 3D figures in terms of triangles; use known results to construct simple proofs

(a)

use geometrical language, notation, terminology, and symbols correctly

(b)

work in all four quadrants of the coordinate plane; interpret a given equation as the graph of a function or a circle

(c)

lay out calculations, constructions, and proofs line-by-line

(a)

use the basic principles of Euclidean geometry and results derived from them to justify each step in a calculation, construction, or deduction

(b)

analyse 2D and 3D configurations [e.g. by singling out, and using known properties of triangles]

(a)

know that two points A, B determine a line AB, a line segment , and a circle with centre A and radius ; relate this to ideal ruler and compass constructions

(b)

know and use the conventional notation for labelling the angles and sides of ΔABC

(c)

accept and use the SAS, SSS, ASA (and later RHS) congruence criteria; prove the basic properties of isosceles triangles; justify the basic ruler and compass constructions and use them to devise other constructions

(d)

prove that the perpendicular bisector of a given line segment is the locus of points X equidistant from B and C; construct the circumcentre of any triangle

(e)

recognise the “perpendicular distance” from a point X to a line as the (shortest) distance to the line; prove the angle bisector of ∠BAC is the (part-) locus of points equidistant from the lines AB and AC; construct the incentre of ΔABC

(f)

prove that the three altitudes of a triangle are concurrent

(a)

know that angles at a point total 360°, and that angles at a point on a straight line total 180°; conclude that “vertically opposite angles are equal”

(b)

recognise that “two lines are parallel precisely when alternate angles (or equivalently, when corresponding angles) created by any transversal are equal”; derive the basic properties of a parallelogram and of a rhombus; where possible prove the converse results; prove and use the Midpoint Theorem

(c)

prove that the angles in any triangle add to 180° and that the exterior angle at any vertex is equal to the sum of the two interior opposite angles; deduce that the angles in any quadrilateral add to 360°; calculate the angle-sum in an n-gon, and the angle size in a regular n-gon

(d)

combine known results about angles to find unknown angles, and to show that certain pairs of lines are parallel

(e)

know and use the fact that the tangent and radius at a point on a circle are perpendicular; conclude that tangents from an external point are equal; prove that the angle subtended by a chord on the major arc is half the angle subtended at the centre O; conclude that angles subtended in the same segment are equal, and that opposite angles of a cyclic quadrilateral add to 180°; prove and use the Alternate Segment Theorem

(f)

prove that the area of a parallelogram is equal to that of a rectangle on the same base and between the same parallels and deduce the formula for the area of a triangle; use this to prove Pythagoras’ Theorem

(a)

establish and use the AAA similarity criterion and the SAS similarity criterion for general triangles; prove basic results using similarity

(b)

extend the Midpoint Theorem to divide a given segment into any number of equal parts; prove and use the Intercept Theorem

(a)

show that the standard trig ratios for acute angles θ depend only on the angle θ; understand that sin θ, cos θ take values between 0 and 1

(b)

find the exact values for θ = 0°, 30°, 45°, 60°; plot graphs of y = sin θ, y = cos θ, and y = tan θ for 0° ⩽ θ < 90°; understand why cos θ = sin(90° − θ)

(c)

calculate missing lengths and angles in a given triangle ABC

(d)

given triangle ABC, derive and use the formula

deduce the Sine Rule and use it to “solve triangles”; prove that

where R is the circumradius of triangle ABC

(e)

show that on the unit circle with centre at the origin O, the point P for which the radius OP makes an angle θ with the positive x-axis has coordinates (cos θ, sin θ); apply Pythagoras’ Theorem to derive the identity

sin² θ + cos² θ = 1;

use this identity to find values of cos θ given the value of sin θ (and vice versa), and the value of tan θ given the value of cos θ

(f)

prove the Cosine Rule, and use it to find unknown lengths and angles in triangles and other 2D and 3D figures

(g)

extend the definition of sin θ and cos θ to θ > 90°; extend the graphs of y = sin θ, y = cos θ to 180° < θ < 0°, and to −180° < θ < 0°

(h)

show that in the “ambiguous (ASS) case”, the data may determine two possible triangles

(a)

work freely with standard 2D figures

(b)

draw figures to scale; interpret maps and other scale drawings; apply similarity in analysing problems; understand how enlargement and scaling (or similarity) affects angles, lengths, areas, and volumes

(c)

find lengths and angles in 3D figures by considering 2D cross-sections; calculate the angle between two planes

(d)

calculate surface areas and volumes of standard figures

(a)

understand and use the terms centre, radius, chord, diameter, circumference, tangent, arc, sector, segment

(b)

understand and use the formula for the circumference of a circle; calculate the length of circular arcs

(c)

relate the formula for the area of a circle to the formula for the circumference; calculate the area of a sector

(d)

calculate the circumradius and inradius of a triangle

(e)

use Pythagoras’ Theorem to find the equation of a circle of radius r centred at the origin and at the point (c, d); complete the square to identify easy quadratic equations as circles and find their centre and radius

(f)

find the equation of the tangent to a given circle at a specified point