GCSE Module 5 Higher
- what you need to know

This is a list of all the topics that you need to know for Module 5 Higher level. An idea of the grade level is given in brackets

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Area & volume | Properties of polygons | Sequences | Co-ordinates | Equations | Reflections and rotations | Properties of circles | Trial and improvement | Translation and enlargement | Measures | Real-life graphs | Formulae | Construction | Vectors | Graphs of linear functions | Similarity and congruence | Pythagoras' theorem | Quadratic functions | Inequalities and simultaneous equations | Trigonometry | Other functions | Loci | Transforming functions | Algebraic proofs

Area & Volume

• Find the area of a triangle, parallelogram, kite and trapezium (D)
• Find the area and perimeter of compound shapes (D)
• Calculate the circumference of a circle to an appropriate degree of accuracy (D)
• Find the perimeter of a semicircle (C)
• Calculate the area of a circle to an appropriate degree of accuracy (D)
• Find the area of a semicircle (C)
• Calculate volumes of triangular prisms, parallelogram-based prisms and cylinders (C)
• Convert between measures of volume (C)
• Solve problems involving surface areas of prisms and cylinders (C)
• Convert between measures of area (C)
• Calculate volumes of spheres and Cones (A)
• Calculate areas of sectors and segments (A)
• Calculate the length of arcs (A)

Properties of polygons

• Classify a quadrilateral by geometric properties (C)
• Calculate exterior and interior angles of a regular polygon (C)

Sequences

• Write the terms of a sequence or a series of diagrams given the nth term (D)
• Write the nth term of a sequence or a series of diagrams (C)

Co-ordinates

• Draw lines such as y = 2x + 3 (D)
• Solve problems involving straight lines (D)
• Find the midpoint of a line segment (C)
• Use and understand coordinates in three dimensions (C)

Equations

• Solve equations such as 3x – 4 = 5 + x (D)
• Solve equations such as 2(5x + 1) = 28 (D)
• Solve equations such as 3x – 12 = 2(x – 5) (C)
• Solve equations such as 7 - x /3 or 2x/3 - x/4 = 5 (C)
• Solve equations such as 2x-1/6 + x+3/3 = 5/2 (B)

Reflections and rotations

• Reflect shapes in lines such as x = 2 or y = –1 (D)
• Rotate shapes about the origin (D)
• Reflect shapes in the lines y = x and y = –x (C)
• Rotate shapes about any point (C)
• Describe fully reflections and rotations about the origin (D)
• Describe fully reflections and rotations about any point (C)
• Find the centre of a rotation and describe it fully (C)
• Combine reflections and rotations (C)
• Identify reflection symmetry in 3-D solids (D)

Properties of circles

• Use the angle properties of a circle (B)
• Prove the angle properties of a circle (A)
• Use the tangent / chord properties of a circle (B)
• Prove the tangent / chord properties of a circle (A)
• Use and prove the alternate segment theorem (A)

Trial and improvement

• Form and solve equations such as x³ + x = 12 using trial and improvement methods

Translation and enlargement

• Translate a shape using a description such as 4 units right and 3 units down (D)
• Translate a shape by a vector such as (4/-3) (C)
• Transform shapes by a combination of translation, reflection and rotation (C)
• Compare the area of an enlarged shape with the original shape (C)
• Enlarge a shape by a positive scale factor from a given centre (D)
• Enlarge a shape by a fractional scale factor (C)
• Enlarge a shape by a negative scale factor (A)
• Compare areas and volumes of enlarged shapes (A)
• Distinguish between formulae for perimeter, area and volume by considering dimensions (B)

Measures

• Solve more difficult speed problems (C)
• Understand and use compound measures such as speed and density (C)
• Recognise accuracy in measurements given to the nearest whole unit (C)
• Find the upper and lower bounds of simple calculations involving quantities given to a particular degree of accuracy (B)
• Find the upper and lower bounds of more difficult calculations with quantities given to a various degrees of accuracy (A-A*)

Real-life graphs

• Calculate simple average speeds from distance–time graphs (D)
• Calculate complex average speeds from distance–time graphs (C)
• Interpret velocity–time graphs (B)
• Discuss and interpret graphs modeling real situations (B)

Formulae

• Substitute numbers into more complicated formulae such as C = (A+1)D / 9 (D)
• Find a solution to a problem by forming an equation and solving it (C)
• Rearrange linear formulae such as P=3q+5 (C)
• Rearrange formulae that include brackets, fractions and square roots (B)
• Rearrange formulae where the variable appears twice (A)

Construction

• Draw a quadrilateral such as a kite or a parallelogram with given measurements (D)
• Understand that giving the lengths of two sides and a non-included angle may not produce a unique triangle (D)
• Construct the perpendicular bisector of a line (C)
• Construct the bisector of an angle (C)
• Construct the perpendicular from a point to a line (C)
• Construct angles of 60° and 90° (C)
• Construct and recognise the nets of 3-D solids such as pyramids and triangular prisms (D)
• Draw plans and elevations of 3-D solids (D)

Vectors

• Add, subtract and multiply vectors to solve vector geometry problems (A)
• Understand the relationship between parallel and perpendicular vectors (A)
• Solve more difficult vector geometry problems (A*)

Graphs of linear functions

• Solve problems involving graphs, such as finding where the line y = x+2 crosses the line y = 1 (D)
• Recognise the equations of straight-line graphs such as y = -4x+2 (C)
• Find the gradients of straight-line graphs (C)
• Explore the gradients of parallel straight-line graphs (B)
• Explore the gradients of perpendicular straight-line graphs (A)

Similarity and congruence

• Match sides and angles of similar triangles, given some dimensions (B)
• Find the area of a 2-D shape, given the area of a similar shape and the ratio (A)
• Find the volume of a 3-D solid, given the volume of a similar solid and the ratio (A)
• Match one side and one angle of congruent triangles, given some dimensions (C)
• Prove that two triangles are congruent (A)
• Prove the construction theorems (A)

Pythagoras' theorem

• Use Pythagoras' theorem to find the hypotenuse of a right-angled triangle (C)
• Use Pythagoras' theorem to find any side of a right-angled triangle (C)
• Use Pythagoras' theorem to find the height of an isosceles triangle (C)
• Use Pythagoras' theorem in practical problems (C)
• Find the distance between two points from their coordinates (B)
• Use Pythagoras' theorem in 3-D problems (A)

Quadratic functions

• Solve quadratic equations such as x² - 8x + 15 = 0 by factorisation (B)
• Solve equations such as (4/x+2) + (3/2x-1) = 2 (A*)
• Solve equations such as x² - 2x - 1 = 0 by using the quadratic formula (A)
• Write quadratic expressions in forms like (x+a)²+b (that is, complete the square) (A*)
• Use completing the square to solve equations and find maximum and minimum values (A*)

Inequalities and simultaneous equations

• Solve inequalities such as 3x , 9 and 12 = 3n < 20 (C)
• Solve linear inequalities such as 4x – 3 < 10 and 4x < 2x + 7 (C)
• Represent sets of solutions on the number line (C)
• Solve linear inequalities such as x + 13 > 5x – 3 (B)
• Solve a set of linear inequalities in two variables and represent the solution as a region of a graph (B)
• Solve a pair of simultaneous equations in two unknowns such as 2x+y=5 and 3x-2y=4 (B)
• 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 (B)
• Solve a pair of simultaneous equations where one is linear and one is non-linear such as y=3x-2 and y=x² (A)
• Solve a pair of simultaneous equations where one is linear and one is non-linear such as x+5y=13 and x²+y²=13 (A*)

Trigonometry

• Use sine, cosine and tangent to calculate a side in a right-angled triangle (B)
• Use sine, cosine and tangent to calculate an angle in a right-angled triangle (B)
• Use trigonometry to find sides and angles in three dimensions (A*)
• Find the angle between a line and a plane (A*)
• Sketch and draw trigonometric graphs (A)
• Understand the graphs of trigonometric functions for angles of any size (A*)
• Use the sine rule to find the missing sides and missing angles in any triangle (A)
• Use the cosine rule to find the missing sides and missing angles in any triangle (A)
• Use the formula for the area of a non right-angled triangle (A)

Other functions

• Complete tables for, and draw graphs of cubic functions (B)
• Use cubic graphs to solve equations (B)
• Solve cubic equations by drawing appropriate lines on graphs (A*)
• Complete tables for, and draw graphs of reciprocal functions (B)
• Use reciprocal graphs to solve equations (B)
• Plot and sketch graphs of exponential functions (A*)
• Recognise the shapes of graphs of functions (A*)

Loci

• Understand the idea of a locus (D)
• Construct accurately loci, such as those of points equidistant from two fixed points (C)
• Solve loci problems, such as identifying points less than 3 cm from a point P (C)
• Construct the graphs of loci, including the circle x² + y² = r²
• Solve simultaneous equations graphically, such as y = x – 1 and x² + y² = 16 (A)
• Solve simultaneous equations graphically, such as y = 2x – 1 and x² + y² = 2 (A*)

Transforming functions

• Transform the graphs of y=f(x), such as linear, quadratic, cubic, sine and cosine functions, using the transformations y=af(x), y=f(x)+a, y=f(x+a), y=f(ax) (A*)

Algebraic proofs

• Decide with a reason whether a harder statement is true or false (D)
• Identify a counter example (D)
• Understand the difference between a demonstration and a proof (C)
• Show step-by-step deductions in providing a basic algebraic explanation (C)
• Show step-by-step deductions in providing a full mathematical explanation (B)