●        Use the angle sums of irregular polygons;

●        Calculate and use the sums of the interior angles of polygons, use the sum of angles in a triangle to deduce and use the angle sum in any polygon and to derive the properties of regular polygons;

●        Use the sum of the exterior angles of any polygon is 360°;

●        Use the sum of the interior angles of an n-sided polygon;

●        Use the sum of the interior angle and the exterior angle is 180°;

●        Find the size of each interior angle, or the size of each exterior angle, or the number of sides of a regular polygon, and use the sum of angles of irregular polygons;

●        Calculate the angles of regular polygons and use these to solve problems;

●        Use the side/angle properties of compound shapes made up of triangles, lines and quadrilaterals, including solving angle and symmetry problems for shapes in the first quadrant, more complex problems and using algebra;

●        Use angle facts to demonstrate how shapes would ‘fit together’, and work out interior angles of shapes in a pattern.

·         Factorise quadratic expressions in the form ax2 + bx + c;

·         Solve quadratic equations by factorisation and completing the square;

·         Solve quadratic equations that need rearranging;

·         Set up and solve quadratic equations;

Solve quadratic equations by using the quadratic formula 

·         Recall and use the formula for volume of pyramid;

·         Find the surface area of a pyramid;

·         Use the formulae for volume and surface area of spheres and cones;

·         Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones;

·         Find the surface area and volumes of compound solids constructed from cubes, cuboids, cones, pyramids, spheres, hemispheres, cylinders;

·         Give answers in terms of π;

Form equations involving more complex shapes and solve these equations 

 Calculate the length of a line segment given the coordinates of the end points;

●        Find the coordinates of points identified by geometrical information.

●        Find the equation of the line through two given points.

 . 

·         Recognise a linear, quadratic, cubic, reciprocal and circle graph from its shape;

·         Generate points and plot graphs of simple quadratic functions, then more general quadratic functions;

·         Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function;

·         Interpret graphs of quadratic functions from real-life problems;

·         Draw graphs of simple cubic functions using tables of values;

Interpret graphs of simple cubic functions, including finding solutions to cubic equations 

·         Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and pair of compasses constructions;

·         Solve angle problems by first proving congruence;

·         Understand similarity of triangles and of other plane shapes, and use this to make geometric inferences;

·         Prove that two shapes are similar by showing that all corresponding angles are equal in size and/or lengths of sides are in the same ratio/one is an enlargement of the other, giving the scale factor;

·         Use formal geometric proof for the similarity of two given triangles;

·         Understand the effect of enlargement on angles, perimeter, area and volume of shapes and solids;

·         Identify the scale factor of an enlargement of a similar shape as the ratio of the lengths of two corresponding sides, using integer or fraction scale factors;

·         Write the lengths, areas and volumes of two shapes as ratios in their simplest form;

·         Find missing lengths, areas and volumes in similar 3D solids;

·         Know the relationships between linear, area and volume scale factors of mathematically similar shapes and solids;

·         Use the relationship between enlargement and areas and volumes of simple shapes and solids;

·         Solve problems involving frustums of cones where you have to find missing lengths first using similar triangles.

       Factorise quadratic expressions of the form ax2 + bx + c;

 Factorise quadratic expressions using the difference of two squares.

·         Write probabilities using fractions, percentages or decimals;

·         Understand and use experimental and theoretical measures of probability, including relative frequency to include outcomes using dice, spinners, coins, etc;

·         Estimate the number of times an event will occur, given the probability and the number of trials;

·         Find the probability of successive events, such as several throws of a single dice;

·         List all outcomes for single events, and combined events, systematically;

·         Draw sample space diagrams and use them for adding simple probabilities;

·         Know that the sum of the probabilities of all outcomes is 1;

·         Use 1 – p as the probability of an event not occurring where p is the probability of the event occurring;

·         Work out probabilities from Venn diagrams to represent real-life situations and also ‘abstract’ sets of numbers/values;

·         Use union and intersection notation;

·         Find a missing probability from a list or two-way table, including algebraic terms;

·         Understand conditional probabilities and decide if two events are independent;

·         Draw a probability tree diagram based on given information, and use this to find probability and expected number of outcome;

·         Understand selection with or without replacement;

·         Calculate the probability of independent and dependent combined events;

·         Use a two-way table to calculate conditional probability;

·         Use a tree diagram to calculate conditional probability;

·         Use a Venn diagram to calculate conditional probability;

·         Compare experimental data and theoretical probabilities;

·         Compare relative frequencies from samples of different sizes.

·         linear / quadratic;

·         linear / x2 + y2 = r2;

·         Set up and solve a pair of simultaneous equations in two variables for each of the above scenarios, including to represent a situation;

Interpret the solution in the context of the problem 

·         Set up simple equations from word problems and derive simple formulae;

·         Understand the ≠ symbol (not equal), e.g. 6x + 4 ≠ 3(x + 2), and introduce identity ≡ sign;

·         Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation;

·         Solve linear equations which contain brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution;

·         Solve linear equations in one unknown, with integer or fractional coefficients;

·         Set up and solve linear equations to solve to solve a problem;

·         Derive a formula and set up simple equations from word problems, then solve these equations, interpreting the solution in the context of the problem;

·         By writing the denominator in terms of its prime factors, decide whether fractions can be converted to recurring or terminating decimals;

·         Convert a fraction to a recurring decimal;

·         Convert a recurring decimal to a fraction;

·         Find the reciprocal of an integer, decimal or fraction.

·         Know and apply Area = ab sin C to calculate the area, sides or angles of any triangle.

·         Know the sine and cosine rules, and use to solve 2D problems (including involving bearings).

·         Use the sine and cosine rules to solve 3D problems.

·         Understand the language of planes, and recognise the diagonals of a cuboid.

·         Solve geometrical problems on coordinate axes.

·         Understand, recall and use trigonometric relationships and Pythagoras’ Theorem in right-angled triangles, and use these to solve problems in 3D configurations.

·         Calculate the length of a diagonal of a cuboid.

•         Find the angle between a line and a plane.

       Continue a quadratic sequence and use the nth term to generate terms;

·         Find the nth term of quadratic sequences;

·         Distinguish between arithmetic and geometric sequences;

·         Use finite/infinite and ascending/descending to describe sequences;

·         Recognise and use simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd);

·         Continue geometric progression and find term to term rule, including negative, fraction and decimal terms;

·         Solve problems involving sequences from real life situations.

·         Calculate the upper and lowers bounds of numbers given to varying degrees of accuracy;

·         Calculate the upper and lower bounds of an expression involving the four operations;

·         Find the upper and lower bounds in real-life situations using measurements given to appropriate degrees of accuracy;

·         Find the upper and lower bounds of calculations involving perimeters, areas and volumes of 2D and 3D shapes;

·         Calculate the upper and lower bounds of calculations, particularly when working with measurements;

·         Use inequality notation to specify an error bound.

·         Recognise and interpret graphs showing direct and indirect proportion;

·         Identify direct proportion from a table of values, by comparing ratios of values, for
x squared and x cubed relationships;

·         Write statements of proportionality for quantities proportional to the square, cube or other power of another quantity;

·         Set up and use equations to solve word and other problems involving direct proportion;

·         Use y = kx to solve direct proportion problems, including questions where students find k, and then use k to find another value;

·         Solve problems involving inverse proportion using graphs by plotting and reading values from graphs;

·         Solve problems involving inverse proportionality;

·         Set up and use equations to solve word and other problems involving direct proportion or inverse proportion.

·         Use index notation for integer powers of 10, including negative powers;

·         Recognise powers of 2, 3, 4, 5;

·         Use the square, cube and power keys on a calculator and estimate powers and roots of any given positive number, by considering the values it must lie between, e.g. the square root of 42 must be between 6 and 7;

·         Find the value of calculations using indices including positive, fractional and negative indices;

·         Recall that n0 = 1 and n–1 =  for positive integers n as well as,  = √n and  = 3√n for any positive number n;

·         Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation to the power ;

·         Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers, fractional and negative powers, and powers of a power;

·         Solve problems using index laws;

·         Use brackets and the hierarchy of operations up to and including with powers and roots inside the brackets, or raising brackets to powers or taking roots of brackets;

·         Use an extended range of calculator functions, including +, –, ×, ÷, x², √x, memory, x y, , brackets;

·         Use calculators for all calculations: positive and negative numbers, brackets, powers and roots, four operations.

·         Draw graphs of the reciprocal function  with x ≠ 0 using tables of values;

·         Draw circles, centre the origin, equation x2 + y2 = r2.

·         Recall the definition of a circle and identify (name) and draw parts of a circle, including sector, tangent, chord, segment;

·         Prove and use the facts that:

·         the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference;

·         the angle in a semicircle is a right angle;

·         the perpendicular from the centre of a circle to a chord bisects the chord;

·         angles in the same segment are equal;

·         alternate segment theorem;

·         opposite angles of a cyclic quadrilateral sum to 180°;

·         Understand and use the fact that the tangent at any point on a circle is perpendicular to the radius at that point;

·         Find and give reasons for missing angles on diagrams using:

·         circle theorems;

·         isosceles triangles (radius properties) in circles;

·         the fact that the angle between a tangent and radius is 90°;

·         the fact that tangents from an external point are equal in length.

·         Factorise quadratic expressions in the form ax2 + bx + c;

·         Solve quadratic equations by factorisation and completing the square;

·         Solve quadratic equations that need rearranging;

·         Set up and solve quadratic equations;

Solve quadratic equations by using the quadratic formula 

·         Recall the definition of a circle and name and draw parts of a circle;

·         Recall and use formulae for the circumference of a circle and the area enclosed by a circle (using circumference = 2πr = πd and area of a circle = πr2) using a variety of metric measures;

·         Use π ≈ 3.142 or use the π button on a calculator;

·         Calculate perimeters and areas of composite shapes made from circles and parts of circles (including semicircles, quarter-circles, combinations of these and also incorporating other polygons);

·         Calculate arc lengths, angles and areas of sectors of circles;

·         Find radius or diameter, given area or circumference of circles in a variety of metric measures;

·         Give answers in terms of π;

Form equations involving more complex shapes and solve these equations 

·         Find the original amount given the final amount after a percentage increase or decrease (reverse percentages), including VAT;

·         Use calculators for reverse percentage calculations by doing an appropriate division;

·         Use percentages in real-life situations, including percentages greater than 100%;

·         Describe percentage increase/decrease with fractions, e.g. 150% increase means  times as big;

·         Understand that fractions are more accurate in calculations than rounded percentage or decimal equivalents, and choose fractions, decimals or percentages appropriately for calculations.

By the end of the sub-unit, students should be able to:

·         Convert large and small numbers into standard form and vice versa;

·         Add and subtract numbers in standard form;

·         Multiply and divide numbers in standard form;

·         Interpret a calculator display using standard form and know how to enter numbers in standard form;

·         Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2;

·         Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).

·         Show inequalities on number lines;

·         Write down whole number values that satisfy an inequality;

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

·         Solve two linear inequalities in x, find the solution sets and compare them to see which value of x satisfies both solve linear inequalities in two variables algebraically;

·         Use the correct notation to show inclusive and exclusive inequalities.

·         Find the surface area of prisms using the formulae for triangles and rectangles, and other (simple) shapes with and without a diagram;

·         Draw sketches of 3D solids;

·         Identify planes of symmetry of 3D solids, and sketch planes of symmetry;

·         Recall and use the formula for the volume of a cuboid or prism made from composite 3D solids using a variety of metric measures;

·         Convert between metric volume measures;

·         Convert between metric measures of volume and capacity, e.g. 1 ml = 1 cm3;

·         Use volume to solve problems;

·         Estimating surface area, perimeter and volume by rounding measurements to 1 significant figure to check reasonableness of answers.

·         Use π ≈ 3.142 or use the π button on a calculator;

Find the volume and surface area of a cylinder 

·       Write a ratio as a linear function;

·       Identify direct proportion from a table of values, by comparing ratios of values;

·       Use a ratio to compare a scale model to real-life object;

·       Use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36;

·       Scale up recipes;

·       Convert between currencies.

·         Know the appropriate uses of histograms;

·         Construct and interpret histograms from class intervals with unequal width;

·         Use and understand frequency density;

·         From histograms:

·         complete a grouped frequency table;

·         understand and define frequency density;

·         Estimate the mean from a histogram;

·         Estimate the median from a histogram with unequal class widths or any other information from a histogram, such as the number of people in a given interval.