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NUMBER Primes, Factors and Multiples Grade G C B Objective Find the factors of a number Recognise prime numbers Find the reciprocal of a number Find the Lowest Common Multiple (LCM) of two simple numbers Find the Highest Common Factor (HCF) of two simple numbers Write a number as a produce of its primes Find the Lowest Common Multiple (LCM) of two or more numbers Find the Highest Common Factor (HCF) of two or more numbers Rounding Grade Objective Round to the nearest integer Write an integer correct to the nearest 10 or the nearest 100 Estimate answers to problems involving decimals Round numbers to given powers of 10 and to given numbers of decimal places Estimate square roots E Round a number to one significant figure D Estimate answers to calculations involving division Estimate answers to calculations such as C Estimate answers to calculations involving division by numbers less than one Find minimum and maximum values Estimate answers to calculations such as B Round to a given number of significant figures G F Fractions Grade G F E D C Objective Find equivalent fractions Simplify fractions such as Arrange fractions in order of size Work out fractions of quantities such as Find one number as a fraction of another Do calculations with simple fractions involving addition Do calculations with simple fractions involving multiplication Do calculations with simple fractions involving subtraction Do calculations with simple fractions involving division Do calculations with mixed numbers 24 36 5 8 of £20 Decimals Grade Objective Write down the place value of a digit, for example, what is the value of the 4 in 0.24? Order decimals, for example, which is bigger, 0.24 or 0.3? E Add and subtract decimals D Multiply two decimals such as 2.4 x 0.7 Convert decimals to fractions and fractions to decimals C Divide a number by a decimal such as 1 ÷ 0.2 and 2.8 ÷ 0.07 B Identify recurring and terminating decimals Convert recurring decimals to fractions and fractions to recurring decimals F Percentages Grade Objective D Understand that percentage means ‘out of 100’ Change a percentage to a fraction or a decimal and vice versa Compare percentages, fractions and decimals Work out a percentage of a given quantity Calculate simple interest Increase or decrease a quantity by a given percentage Express one quantity as a percentage of another C Work out percentage increase or decrease B Understand how to use successive percentages Work out compound interest Work out reverse percentage problems F E Ratio and Proportion Grade Objective B Solve simple ratio and proportion problems, such as finding the ratio of teachers to students in a school Solve more complex ratio and proportion problems, such as sharing out money between two groups in the ratio of their numbers Solve ratio and proportion problems using the unitary method Calculate proportional changes using a multiplier A Solve direct and inverse proportion problems Interpret graphs of direct and inverse proportion relationships D C Negative Numbers Grade G F E Objective Understand positive and negative integers Add and subtract negative integers Multiply and divide negative integers Indices and Standard Form Grade Objective F E D C Calculate squares and square roots (with and without the use of a calculator) Calculate cubes and cube roots (with and without the use of a calculator) Use function keys on a calculator for powers and roots Use the terms square, positive square root, negative square root, cube and cube root Recall integer squares from 2 x 2 to 15 x 15 and the corresponding square roots Recall the cubes of 2, 3, 4, 5 and 10 Use index notation and index laws for positive and negative powers such as w3 x w5 and Use index notation and index laws for positive and negative powers such as 3w3y x 2w5y2 and B A A* Convert between numbers in ordinary and standard index form Use standard index form with and without a calculator Use index notation and index laws for fractional powers such as 16½ Use index notation and index laws for fractional powers such as 16¾ Surds Grade Objective A Rationalise the denominator of a surd, such as A* Simplify surds, such as write (3 - √5)² in the form a + b√5 Measures Grade G Objective Decide which metric unit to use for everyday measurements F Convert between imperial and metric units Know rough metric equivalents of pounds, feet, miles, pints and gallons Make sensible estimates of a range of measures in everyday settings E Solve simple speed problems Solve more difficult speed problems Understand and use compound measures such as speed and density Recognise accuracy in measurements given to the nearest whole unit Find the upper and lower bounds of simple calculations involving quantities given to a particular degree of accuracy Find the upper and lower bounds of more difficult calculations with quantities given to a various degrees of accuracy C B A-A* ALGEBRA Expanding and Factorising Grade F Simplify expressions with more than one variable such as 2a + 5b + a – 2b Multiply out expressions with brackets such as 3(x + 2) or 5(x – 2) Factorise expressions such as 24x + 6 Expand (and simplify) harder expressions such as x(x² - 5) and 3(x + 2) – 5(2x – 1) Expand (and simplify) quadratic expressions such as (x + 4)(x – 2), (2x + y)(3x – 2y) and (x + 2)² Factorise quadratic expressions such as 4x² + 6xy and x² - 8x – 16 Simplify rational expressions such as Factorise harder quadratics such as a² - 16b² and 5x² + 13x - 6 Factorise harder quadratics expressions such as x² - 10x + a, writing them in the form (x + b)² + c Simplifying harder rational expressions such as E D C B A A* Objective Simplify expressions with one variable such as a + 2a + 3a Sequences Grade G F E D C Objective Continue a sequence of numbers or diagrams Write down terms of a simple sequence Find a particular term in a sequence involving positive numbers Write the term-to-term rule in a sequence involving positive numbers Find a particular term in a sequence involving negative or fractional numbers Write the term-to-term rule in a sequence involving negative or fractional numbers Write the terms of a sequence or a series of diagrams given the nth term Write the nth term of a sequence or a series of diagrams Equations Grade F E D C B Objective Solve equations such as 3x 12 or x 5 9 x 7 or 3x 1 9 2 Solve equations such as 3x – 4 = 5 + x or 2(5x + 1) = 28 Solve equations such as Form and solve equations such as x 3 x 12 using trial and improvement methods 2x x 73 2 or 5 Solve equations such as 3 4 x Solve equations such as Formulae Grade Objective G F Use a formula written in words, such as cost = 20 distance travelled Use a simple formula such as P 2l 2 w Substitute positive numbers into a simple formula Write an expression from a problem Substitute negative numbers into a simple formula Use formulae from mathematics and other subjects D Substitute numbers into more complicated formulae such as C C Find a solution to a problem by forming an equation and solving it Rearrange linear formulae such as p 3q 5 Rearrange formulae that include brackets, fractions and square roots Rearrange formulae where the variable appears twice E B A ( A 1) D 9 Graphs of Linear Functions Grade E D C B A Objective Plot the graphs of straight lines such as x = 3 and y = 4 Complete a table of values for equations such as y 2 x 3 and draw the graph Solve problems involving graphs, such as finding where the line y x 2 crosses the line y = 1 Recognise the equations of straight-line graphs such as y 4 x 2 Find the gradients of straight-line graphs Explore the gradients of parallel straight-line graphs Explore the gradients of perpendicular straight-line graphs Quadratic Graphs Grade D C Objective Draw graphs of simple quadratic functions such as y 3x 2 and y x 2 4 Draw graphs of harder quadratic functions such as y x 2 2x 1 Find the points of intersection of quadratic graphs with lines Use graphs to find the approximate solutions of quadratic equations Use the points of intersection of a quadratic graph such as y x 2 2 x 4 A with lines such as y 2 x 1 to solve equations like x 2 2 x 4 2 x 1 and simplify this to x 2 4 x 5 0 Quadratic Functions Grade B A A* Objective Solve quadratic equations such as x 2 8x 15 0 by factorisation Solve equations such as x 2 2 x 1 0 by using the quadratic formula Solve equations such as Write quadratic expressions in forms like ( x a) 2 b (that is, complete the square) Use completing the square to solve equations and find maximum and minimum values 4 3 2 x 2 2x 1 Inequalities and Simultaneous Equations Grade C B Objective and 3x 2 y 4 A A* Solve inequalities such as 3x < 9 and 12 ≤ 3n < 20 Solve linear inequalities such as 4x – 3 < 10 and 4x < 2x + 7 Represent sets of solutions on the number line Solve linear inequalities such as x + 13 > 5x – 3 Solve a set of linear inequalities in two variables and represent the solution as a region of a graph Solve a pair of simultaneous equations in two unknowns such as 2 x y 5 Know that each equation can be represented by a line on a graph and that the point of intersection of the lines is the solution Solve a pair of simultaneous equations where one is linear and one is nonlinear such as y 3x 2 and y x 2 Solve a pair of simultaneous equations where one is linear and one is nonlinear such as x 5 y 13 and x 2 y 2 13 Non-linear Functions Grade B A* Objective Complete tables for, and draw graphs of cubic and reciprocal functions Use the graphs to solve equations Solve cubic equations by drawing appropriate lines on graphs Plot and sketch graphs of exponential functions Recognise the shapes of graphs of functions Transforming Functions Grade Objective A* Transform the graphs of y f ( x) , such as linear, quadratic, cubic, sine and cosine functions, using the transformations y f ( x) a , y f ( x a) , y af ( x) and y f (ax) Algebraic Proofs Grade E D C B A A* Objective Decide with a reason whether a simple statement is true or false Decide with a reason whether a harder statement is true or false Identify a counter example Understand the difference between a demonstration and a proof Show step-by-step deductions in providing a basic algebraic explanation Show step-by-step deductions in providing a full mathematical explanation Derive simple algebraic proofs using reasoning Derive harder algebraic proofs using reasoning and logic SHAPE & SPACE Area & Volume Grade Objective G F E D C A A* Find the perimeter of a shape by counting sides of squares Find the area of a square by counting squares Estimate the area of an irregular shape by counting squares and part squares Name the parts of a circle Find the volume of a solid by counting cubes and stating units Recognise and name three-dimensional (3-D) solids Sketch three-dimensional (3-D) solids Work out the area and perimeter of a simple rectangle, such as 3 m by 8 m Work out the area and perimeter of a harder rectangle, such as 3.6 m by 7.2 m Find the area and perimeter of compound shapes Find the volume of a cube or cuboid Find the height of a cuboid, given volume, length and breadth Draw a cuboid on an isometric grid and mark its dimensions Find the area of a triangle, parallelogram, kite and trapezium Find the area and perimeter of compound shapes Convert between measures of area Calculate the circumference of a circle to an appropriate degree of accuracy Calculate the area of a circle to an appropriate degree of accuracy Change between area measures such as m2 to cm2 Draw plans and elevations of three-dimensional (3-D) solids Calculate the surface area of prisms and cylinders Calculate volumes of triangular prisms, parallelogram-based prisms and cylinders Calculate the perimeter and area of a semi-circle Calculate volumes of prisms and cylinders Change between volume measures such as m3 to cm3 or cm3 to litres Calculate surface areas of prisms and cylinders Find the volume and surface area of pyramids, cones and spheres Find the volume of the top cone of a truncated cone Find the length of a major arc of a circle Find the area of a major sector of a circle Find the area of a segment of a circle Find the volume the frustum of a truncated cone Angles Grade F D Objective Express fractions of full turns in degrees and vice versa Recognise acute, obtuse and reflex angles, estimate angles and measure them accurately Use properties of angles at a point and angles on a straight line, understand the terms ‘perpendicular lines’ and ‘parallel lines’ Recognise corresponding angles and alternate angles Understand and use three-figure bearings Properties of Polygons Grade G Objective E C Identify isosceles, equilateral and right-angled triangles Use the word ‘congruent’ when triangles are identical Recognise and name shapes, such as isosceles triangle, parallelogram, rhombus, trapezium and hexagon Show that the angles of a triangle add up to 180° and use this to find angles Show that an exterior angle of a triangle is equal to the sum of the interior opposite angles Use angle properties of isosceles, equilateral and right-angled triangles Calculate interior and exterior angles of a quadrilateral Investigate tessellations Classify a quadrilateral by geometric properties Calculate exterior and interior angles of a regular polygon Coordinates Grade Objective G F E Use coordinates in the first quadrant, such as plot the point (3, 2) Use coordinates in all four quadrants, such as plot the points (3, –2), (–2, 1) and (–4, –3) Draw lines such as x = 3 and y = x + 2 D Solve problems involving straight lines Draw lines such as y = 2x + 3 Find the midpoint of a line segment Use and understand coordinates in three dimensions C Properties of Circles Grade B A Objective Use the tangent / chord properties of a circle Prove the tangent / chord properties of a circle Use and prove the alternate segment theorem Transformations Grade G F E D C Objective B A Draw the reflection of a shape in a mirror line Draw a line of symmetry on a 2-D shape Draw all the lines of symmetry on a 2-D shape Give the order of rotations symmetry of a 2-D shape Name, draw or complete 2-D shapes from information about their symmetry Draw the line of reflection for two shapes Give a scale factor of an enlarged shape Reflect shapes in the axes of a graph Enlarge a shape by a positive scale factor Find the measurements of the dimensions of an enlarged shape Use map scales to find a distance Reflect shapes in lines such as x = 2 and y = -1 Rotate shapes about the origin Describe fully reflections and rotations about the origin Identify reflection symmetry in 3-D solids Translate a shape using a description such as 4 units right and 3 units down Enlarge a shape by a positive scale factor from a given centre Compare the area of an enlarged shape with the original shape Translate a shape using a description such as 4 units right and 3 units down Reflect shapes in lines such as y = x and y = -x Rotate shapes about any point Describe fully reflections and rotations about any point Find the centre of rotation and describe it fully Combine reflections and rotations Enlarge a shape by a fractional scale factor 4 Translate a shape by a vector such as 3 Transform shapes by a combination of translation, reflection and rotation Compare the area of an enlarged shape with the original shape Distinguish between formulae for perimeter, area and volume by considering dimensions Enlarge a shape by a negative scale factor Compare areas and volumes of enlarged shapes Construction & Loci Grade G F E Objective Measure a line accurately to the nearest millimetre Recognise the net of a simple solid such as a cuboid Measure or draw an angle accurately to the nearest degree Draw the net of a simple solid such as a cuboid Draw a triangle given three sides, or two angles and a side, or two sides and the included angle Draw a quadrilateral such as a kite or a parallelogram with given measurements Understand that giving the lengths of two sides and a non-included angle may not produce a unique triangle Construct and recognise the nets of 3-D solids such as pyramids and triangular prisms Draw plans and elevations of 3-D solids Understand the idea of a locus Construct the perpendicular bisector of a line Construct the bisector of an angle Construct the perpendicular from a point to a line Construct the perpendicular from a point on a line Construct angles of 60° and 90° Construct accurately loci, such as those of points equidistant from two fixed points Solve loci problems, such as identifying points less than 3 cm from a point P Construct the graphs of loci, including the circle x 2 y 2 r 2 Solve simultaneous equations graphically, such as y = x – 1 and x 2 y 2 16 Solve simultaneous equations graphically, such as y = 2x – 1 and x 2 y 2 27 D C A A* Vectors Grade Objective A Add, subtract and multiply vectors to solve vector geometry problems Understand the relationship between parallel and perpendicular vectors A* Solve more difficult vector geometry problems Similarity and Congruence Grade C B A Objective Match one side and one angle of congruent triangles, given some dimensions Match sides and angles of similar triangles, given some dimensions Prove that two triangles are congruent Prove the construction theorems Find the area of a 2-D shape, given the area of a similar shape and the ratio Find the volume of a 3-D solid, given the volume of a similar solid and the ratio Pythagoras’ Theorem Grade Objective C Use Pythagoras’ theorem to find the hypotenuse of a right-angled triangle Use Pythagoras’ theorem to find any side of a right-angled triangle Use Pythagoras’ theorem to find the height of an isosceles triangle Use Pythagoras’ theorem in practical problems B Find the distance between two points from their coordinates A Use Pythagoras’ theorem in 3-D problems Trigonometry Grade B A A* Objective Use sine, cosine and tangent to calculate a side in a right-angled triangle Use sine, cosine and tangent to calculate an angle in a right-angled triangle Sketch and draw trigonometric graphs Use the sine rule and cosine rule to find the missing sides and missing angles in any triangle Use the formula for the area of a non right-angled triangle Use trigonometry to find sides and angles in three dimensions Find the angle between a line and a plane Understand the graphs of trigonometric functions for angles of any size HANDLING DATA Averages Grade G F E D C Objective Find the mode for a set of numbers Find the median for an odd set of numbers Work out the range for a set of numbers Calculate the mean for a set of numbers Find the median for an even set of numbers Write down the mode from a graph Compare the mean and range of two distributions Calculate the ‘fx’ column for a frequency distribution Calculate the mean for a frequency distribution Find the modal class for grouped data Find the mean for grouped data Find the median class for grouped data Scatter Graphs Grade D C Objective Draw a scatter graph by plotting points on a graph Interpret the scatter graph Draw a line of best fit ion the scatter graph Interpret the line of best fit Identify the type and strength of the correlation Real-life Graphs Grade F E D C B Objective Plot points of a conversion graph and read off positive values Read from a conversion graph for negative values Interpret distance–time graphs Calculate simple average speeds from distance–time graphs Calculate complex average speeds from distance–time graphs Interpret velocity–time graphs Discuss and interpret graphs modelling real situations Collecting Data Grade G E Objective Design and use tally charts for discrete and grouped data Design and use two-way tables for discrete and grouped data Classify and know the difference between various types of data Use a variety of different sampling methods Design and use data collection sheets and questionnaires C Identify possible sources of bias in the design and use of data collection sheets and questionnaires A Use stratified sampling methods D Representing Data Grade G F E D B A Objective Construct and interpret a pictogram Construct and interpret a bar Construct and interpret a dual bar chart Interpret a pie chart Construct a pie chart Interpret a stem-and-leaf diagram Construct a stem-and-leaf diagram (ordered) Construct a frequency diagram Interpret a time series graph Construct and interpret a cumulative frequency diagram Use a cumulative frequency diagram to estimate the median and interquartile range Construct and interpret a box plot Compare two sets of data using box plots Construct a time series and plot the moving averages Use the trend line to estimate other values Construct and interpret a histogram with unequal class intervals Probability Grade G F E D C B A A* Objective Understand and use the vocabulary of probability Understand and use a probability scale Express a probability as a fraction Display outcomes systematically Understand the difference between experimental and theoretical probabilities Understand and use relative frequency Use a two-way table to find a probability Understand mutually exclusive events Use the fact that the probabilities of mutually exclusive events add up to 1 Use probability to estimate outcomes for a population Use probability to estimate outcomes for a population Complete a tree diagram Understand dependent and independent outcomes Use tree diagrams to find probabilities of successive independent events Draw tree diagrams and use them to find probabilities of successive dependent events