Use percentages in real-life situations
Solve percentage problems, including increase and decrease, and reverse percentage
Represent repeated proportional change using a multiplier raised to a power
Use calculators for reverse percentages calculations by doing an appropriate division
Use calculators to explore exponential growth and decay, using a multiplier and power key

Four operations of number
Awareness that percentages are used in everyday life
Percentages (Unit 2)

By the end of the module the student should be able to:
• Find a percentage increase/decrease of an amount (3.1, 3.2)
• Find a reverse percentage, eg find the original cost of an item given the cost after a 10% deduction (3.4)
• Use a multiplier to increase by a given percent, eg 1.1 ´ 64 increases 64 by 10% (3.2, 3.3)
• Calculate simple and compound interest for two, or more, periods of time (3.3)
• Solve a whole host of Functional Elements involving percentages (Throughout
Chapter 3)

Combine multipliers to simplify a series of percentage changes
Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100%
Problems which lead to the necessity of rounding to the nearest penny, eg real-life contexts
Comparisons between simple and compound interest calculations
Formulae in simple interest/compound interest methods
Increase and decreases leading to a combined multiplier to use, eg 10% decrease then 5% increase

Amounts of money should always be rounded to the nearest penny where necessary, except where such rounding is premature eg in successive calculations like in compound interest
In preparation for this unit, students should be reminded of basic percentages and be able to recognise their fraction and decimal equivalents from Unit 2
Use percentages in real-life situations:
– VAT
– Simple interest
– Income tax calculations
– Annual rate of inflation
– Compound interest
– Depreciation
– Find prices after a percentage increase or decrease
– Percentage profit and loss

3-2

Upper & lower bounds

By using calculators, or written methods, calculate the upper and lower bounds, particularly when working with measurements
Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction
Find the upper and lower bounds of calculations involving perimeter, areas and volumes of 2-D and 3-D shapes

Some experience of rounding to a specified number of decimal places or significant figures
Measurement and units
Perimeter and area

By the end of the module the student should be able to:
• Understand that measurements can not be precise, and write down the maximum and minimum possible values (2.1, 2.2)
• Work out the maximum/minimum possible error in a calculation involving measures (2. 2)
• Find when numbers are given to a specific degree of accuracy, the upper and lower bounds of perimeters and areas (2.2)
• Apply upper and lower bounds to compound units, eg speed (2.2)
• Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation (2.2)

Calculating areas and volumes upper and lower bounds using formulae
Use examples that demonstrate finding the maximum/minimum values for a – b and a ÷ b

An effective starter is to bring in a towel and ask the class to measure it. Since it stretches, this is a good application of something not having a defined length
Mention that bridges have expansion joints

3-3

Using fractions

Calculate a given fraction of a quantity, expressing the answer as a fraction
Understand ‘reciprocal’ as multiplicative inverse
Multiply and divide a given fraction by an integer, by a unit fraction or by a general fraction
Recognise that each terminating decimal is a fraction
Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals
Convert any recurring decimal to a fraction (proof)
Use a calculator to solve real-life problems involving Fractions

Fractions (Units 1 & 2)

By the end of the module the student should be able to:
• Convert a fraction to a decimal, or a decimal to a fraction (1.1 – 1.4)
• Find the reciprocal of whole numbers, fractions, and decimals (1.1)
• Multiply and divide a fraction by an integer, by a unit fraction and by a general fraction (expressing the answer in its simplest form) (1.2 – 1.3)
• Convert a fraction to a recurring decimal (and visa versa) (1.4)
• Use fractions in contextualised problems (1.3)

Use a calculator to find fractions of given quantities
Revise addition of fractions (Unit 1 and 2)
Use combinations of the four operations with fractions and in real-life problems, eg to find areas using fractional values
Revise algebraic fractions with very able students

Constant revision of this aspect is needed
All work needs to be presented clearly with the relevant stages of working shown, even if a calculator is used
Use Functional Elements problems as a source of questions involving fractions in a real-life context

3-4

Standard form

Use standard form, expressed in standard notation and on a calculator display
Calculate with standard form
Convert between ordinary and standard form representations
Convert to standard form to make sensible estimates for calculations involving multiplication and/or division
Use standard form display and know how to enter numbers in standard form

Rounding decimals to a given number of decimal places or significant figure
Multiplying decimal numbers with, and without, a calculator
Some experience with powers of 10, eg know that 102 = 100, 103 = 1000, 10–1 = 0.1
Negative indices and laws of indices
Standard form (Unit 2)

By the end of the module the student should be able to:
• Convert numbers to, and from, standard form (4.1)
• Calculate with numbers given in standard form with, and without, a calculator (4.1)
• Interpret a calculator display using standard form (4.1)
• Round numbers given in standard form to a given number of significant figures (4.1)
• Use standard form in real-life situations

eg stellar distances, sizes of populations and atomic sizes for small numbers

Make sure students know how to enter numbers in standard form on their particular calculator model, how to read the calculator display
Ensure that students never to write down the answer as displayed by the calculator,
eg 1.3 -03
This work can be enriched by using examples drawn from the sciences, eg Avogadro’s Constant 6.02 × 10^ -23

3-5

Factorising & algebraic fractions

Simplify terms, products and sums
Multiply a single term over a bracket
Take out common factors
Expand the product of two linear expressions
Factorise quadratic expressions
Manipulate Algebraic Fractions

By the end of this Unit the student should be able to:
• Simplify expressions with like terms, eg ; (7.2 Unit 2)
• Expand and factorise expressions with one pair of brackets,
eg expand x(2x +3y); factorise 3xy2 - 6x2y (8.1, 8.2, Unit 2)
• Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (8.3 Unit 2)
• Factorise quadratic expressions (including the difference of two squares) (8.4 Unit 2)
• Further examples in factorising quadratic expression with non-unitary values of a (including fractional values) (8.4 Unit 2)
• Simplify algebraic fractions, eg (11.1 Unit 2)
• Simplify algebraic fractions, including the addition of fractions and to solve problems (11.2 Unit 2)

Expand algebraic expressions involving three pairs of brackets
Link difference of two squares with surds and the rationalisation of denominators

Emphasise correct use of symbolic notation, eg 3x2 rather than 3 ´ x2
Present all work neatly, writing out the questions with the answers to aid revision at a later stage
Practise some basic fraction additions, with numerical values, then compare this with the algebraic version
Encourage students to look out for difference of two squares when working with algebraic fractions
Remind student to always look to take out a common factor first

3-6

Solving linear equations & inequalities

Solve equations by using inverse operations or by transforming both sides in the same way
Solve linear equations with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation
Solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Solve simple linear inequalities in one variable, and represent the solution set on a number line

Experience of finding missing numbers in calculations
The idea that some operations are ‘opposite’ to each other
Experience of using letters to represent quantities
Some experience of ‘balancing’ equations in order to solve them
Be able to draw a number line

By the end of the module the student should be able to:
• Solve linear equations with one, or more, operations (including fractional coefficients) (Throughout
Chapter 6)
• Solve linear equations involving brackets and/or variables on both sides (6.2, 6.3)
• Solve linear inequalities in one variable and present the solution set on a number line (7.1)
• Form linear equations from worded problems in a variety of contexts and relating the answer back to the original problem (Throughout
Chapter 6)

Use of inverse operations and rounding to 1 significant figure could be applied to more complex calculations
Derive equations from practical situations (such as finding unknown angles in polygons or perimeter problems)
Solve equations where manipulation of fractions (including the negative fractions) is required
Solve linear inequalities where manipulation of fractions is required

Students need to realise that not all linear equations can easily be solved by either observation or trial and improvement, and hence the use of a formal method is vital
Students can leave their answers in fractional form where appropriate
Interpreting the direction of an inequality is a problem for many
Students should use the correct notation when showing inequalities on a number line, eg a filled in circle to show inclusion of a point, an empty circle to show exclusion of a point
Inequalities in two variables will be covered again after simultaneous equations

3-7

Substitution & changing the subject

Substitute numbers into formulae
Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Change the subject of a formula including where the subject occurs once or more than once
Generate a formula

Understanding of the mathematical meaning of the words expression, simplifying, formulae and equation
Experience of using letters to represent quantities
Substituting into simple expressions using words
Using brackets in numerical calculations and removing brackets in simple algebraic expressions
Solving linear equations

By the end of the module the student should be able to:
• Use letters or words to state the relationship between different quantities (Revision of
prior knowledge)
• Substitute positive and negative numbers into simple algebraic formulae (Revision of
prior knowledge)
• Substitute positive and negative numbers into algebraic formulae involving powers (Revision of
prior knowledge)
• Find the solution to a problem by writing an equation and solving it (Revision of
prior knowledge)
• Simple change of subject of a formula, eg convert the formula for converting Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade (7.5)
• Change the subject of the formula when the variable appears more than once (questions could involve powers, roots, fractions or reciprocals) (7.6)
• Generate a formula from given information, eg find the formula for the perimeter of a rectangle given its area A and the length of one side (Revision of
prior knowledge)

Use negative numbers in formulae involving indices
Various investigations leading to generalisations
Further problems in generating formulae form given information
Use equation of a straight line, eg what is the gradient of the line 4x + 2y = 12?
Apply changing the subject to physics formulae, ie pendulum, equations of motion, lens formula

Emphasise good use of notation, eg 3ab means 3 ´ a ´ b
Students need to be clear on the meanings of the words expression, equation, formula and identity
Simple changing the subject is covered in Unit 2
Show a linear equation first and follow the same steps for the similarly structured formula to be rearranged
Functional Elements material is a rich source of formulae given in everyday contexts
Link with formulae for area, volume, surface area, etc

3-8

Straight line graphs

Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients.
Find the gradient of lines given equations of the form y = mx + c values are given for m and c)
Analyse problems and use gradients to see how one variable changes in relation to another
Calculate the length of a line segment between two coordinates

Experience at plotting points in all quadrants
Linear sequences and straight line graphs

By the end of the module the student should be able to:
• Draw linear graphs from tabulated data, including real-world examples (Revision of
prior knowledge)
• Interpret linear graphs, including conversion graphs and distance-time graphs (Revision of
prior knowledge)
• Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (Revision of
prior knowledge)
• Understand that lines are parallel when they have the same value of m (Revision of
prior knowledge)
• Find the gradient and intercept of a straight line graph (Revision of
prior knowledge)
• Find the distance between any two coordinates (link with Pythagoras’ theorem) (18.2)

Plot graphs of the form y = mx + c where pupil has to generate their own table and set out their own axes
Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines
Use a graphical calculator or graphical ICT package to draw straight-line graphs
Link with length of line to Pythagoras’ theorem and contrast with midpoint formula (Unit 2)

Clear presentation with axes labelled correctly is vital
Recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, ie science, geography etc
Interpret straight line graphs for Functional situations and link gradient to cost per unit or rate of change and intercept to initial cost etc
– Ready reckoner graphs
– Conversion graphs
– Fuel bills
– Fixed charge (standing charge) and cost per unit

3-9

Simultaneous equations and inequalities

Find the exact solution of two (linear) simultaneous equations in two unknowns by eliminating a variable and interpreting the equation as lines and their common solution as the point of intersection
Solve several linear inequalities in two variables and finding the solution set

Algebraic manipulation
Ability to solve simple linear equations
Some experience with solving inequalities
Straight line graphs

By the end of the module the student should be able to:
• Solve algebraically two simultaneous equations (9.2, 9.3)
• Interpret the solution of two simultaneous equations as the point of intersection the corresponding lines (9.4)
• Model worded problems as a pair of linear simultaneous equations and interpret the answer (9.3)
• Draw the graphs of linear inequalities in two variables and interpret the solution sets given by regions in the coordinate plane, or to identify all the integer coordinates with crosses (7.4)

Solve two simultaneous equations with fractional coefficients
Solve two simultaneous equations with second order terms, eg equations in x and y2 (link to further simultaneous equations)

Build up the algebraic techniques slowly
Link the graphical solutions with straight line graphs (Units 1 and 2) and changing the subject (above)
Inaccurate graphs could lead to incorrect solutions, encourage substitution of answers to check they are correct
Clear presentation of working out is essential
Students should use the correct notation when giving graphical solutions to inequalities,
eg a dotted boundary line for < or >

3-10

Direct and inverse proportion

Use repeated proportional change
Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relating algebraic solutions to graphical representations of the equations
Calculate an unknown quantity from quantities that vary in direct or inverse proportion

Substitute numbers into algebraic formulae
Rearrange the subject of a formula

By the end of the module the student should be able to:
• Solve simple direct & inverse proportion problems using the Unitary method or by proportional change (from a table of values (Chapter 5)
• Interpret direct and inverse proportions as algebraic functions,
eg y µ x2 as y = kx2 (Chapter 10)
• Use given information to find the value of the constant of proportionality (Chapter 10)
• Use algebraic functions for direct and inverse proportionality, with their value of k, to find unknown values (Chapter 10)
• Recognise and sketch the graphs for direct and inverse proportions
(y µ x, y µ x2, y µ x3, y µ , y µ ) (Chapter 10)

Link Unitary method with Ratio topic (Unit 1 and 2)
Problems involving other types of proportionality, eg surface area to volume of a sphere
Link to graphs to show direct and inverse proportion, eg etc

Students should be encouraged to show all steps in their working
Students often forget the “square” in inverse square proportionality

3-11

Trial and improvement

Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them

Substituting numbers into algebraic expressions
Dealing with decimals on a calculator
Ordering decimals and decimal place approximations.

By the end of the module the student should be able to:
• Solve quadratic and cubic functions by successive substitution of values of x

Solve functions of the form (link with ‘changing the subject’)

Students should be encouraged to use their calculators efficiently, by using the "replay" or ANS/EXE functions
The square/cube function on a calculator may not be the same for different makes
Take care when entering negative values to be squared (always use brackets)
Students should write down all the digits on their calculator display and only round the final answer declared to the degree of accuracy

3-12

Quadratic functions

Generate points and plot graphs of quadratic functions
Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function
Factorise quadratic expressions
Solve simple quadratic equations by factorising, completing the square and using the quadratic formula

Graphs
Factorising Quadratics

By the end of the module the student should be able to:
• Plot the graphs of quadratic functions for positive and negative values of x (8.1)
• Find graphically the solutions of quadratic equations by considering the intercept on the x-axis (9.10)
• Solve quadratic equations by factorising (including values of a not equal to 1) (9.5)
• Use the quadratic formula to solve quadratic equations giving the answers to a specified degree of accuracy (9.7)
• Use the quadratic formula to solve quadratic equations leaving the answer in surd form or decimal form (9.7)
• Complete the square of a quadratic function (using this to write down the maximum/minimum of the function) (9.1, 9.6)

Solve equations involving algebraic fractions which lead to quadratic equations
Solve quadratic equations by completing the square
Derive the quadratic equation by completing the square
Use graphical calculators or ICT graph package where appropriate to enable students to get through examples more rapidly
Show how the value of ‘b2 – 4ac’ can be useful in determining if the quadratic factorises or not (ie square number)
Extend to discriminant’s properties and roots (for those going on to C1)

There may be a need to remove the HCF (numerical) of a trinomial before factorising it to make the factorisation easier to do
Students should be reminded that factorisation should be tried before the formula is used
In problem-solving, one of the solutions to a quadratic may not be appropriate, eg negative length

3-13

Further simultaneous equations

Find the intersection points of the graphs of linear and quadratic functions, knowing that these are the approximate solutions of the corresponding simultaneous equation representing the linear and quadratic functions
Construct the graph of x2 + y2 = r2 for a circle of radius r centred at the origin of coordinates
Find graphically the intersection points of a given straight line and a circle and knowing that this corresponds to solving the simultaneous equations representing the line and the circle
Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, of where the second is of the form x2 + y2 = r2

Quadratic functions
Straight line graphs
Algebraic manipulation and solving linear and quadratic equations

By the end of the module the student should be able to:
• Find graphically the approximate solutions of linear and quadratic simultaneous equations (9.10)
• Find the exact solutions of linear and quadratic simultaneous equations (9.11)
• Draw a circle of radius r centred at the origin (9.10)
• Find the approximate solutions of linear and circular simultaneous equations graphically (9.11)
• Find the exact solutions of linear and circular simultaneous equations (9.10)

Find the approximate solutions of quadratic and circular simultaneous equations graphically
Find the exact solutions of quadratic and circular simultaneous equations using algebraic methods
Look at circles whose centre is not the origin (x – 2)2 + (y – 3)2 = 4 (link with transforming graphs)

Clear presentation of workings is essential
Stress which variable it is easiest to work with when assessing the linear equation
ICT graph drawing packages make this topic more dynamic and easier to picture
Further examples and questions can be obtained from A-Level (C1) texts

3-14

Curved graphs

Generate points and plot graphs of simple quadratic functions
Plot graphs of simple cubic functions, the reciprocal function y = with x 0, the exponential function, y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x
Recognise the characteristic shapes of all these functions

Straight line graphs
BIDMAS

By the end of the module the student should be able to:
• Plot and recognise quadratic, cubic, reciprocal, exponential and circular (trig) functions (see above) within the range –360º to +360º (8.1 – 8.4)
• Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa) (8.1 – 8.4)
• Match equations with their graphs (8.1 – 8.4)
• Sketch graphs of given functions (8.1 – 8.4)

Explore the function y = ex (perhaps relate this to y = ln x)
Explore the function y = tan x
Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise)
Start to investigate transformations, eg y = sin(2x) or y = x2 + 5

This work should be enhanced by drawing graphs on graphical calculators and appropriate software
Group work with each group assigned a different type of graph is an effective way to share the graphs’ properties. Each group reports to the whole class and creates a display
There are plenty of old exam papers with matching tables testing knowledge of the ‘Shapes of Graphs’

3-15

Transformations

Transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations
Understand that translations are specified by a distance and direction (or a column vector), and enlarging by a centre and a scale factor
Rotate a shape about the origin, or any other point
Measure the angle of rotation using right angles, simple fractions of a turn or degrees
Understand that rotations are specified by a centre and an (anticlockwise) angle
Understand that reflections are specified by a mirror line, at first using a line parallel to any axis, then a mirror line such as y = x or y = –x
Recognise, visualise and construct enlargements of objects using positive and negative scale factors greater and less than one
Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations

Recognition of basic shapes
Line and rotational symmetry (Unit 2)
An understanding of the concept of rotation and enlargement
Coordinates in four quadrants
Linear equations parallel to the coordinate axes

By the end of the module the student should be able to:
• Understand translation as a combination of a horizontal and vertical shift including signs for directions (17.2)
• Understand rotation as a (clockwise) turn about a given origin (17.4)
• Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = –x (17.3)
• Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater and less than one (and understand the effects that negative and fractional scale factors have on the image) (17.5, 17.6)
• Understand that shapes produced by translation, rotation and reflection are congruent to its image (17.1 – 17.4)

The tasks set should be extended to include combinations of transformations

Emphasise that students describe the given transformation fully
Diagrams should be drawn carefully
The use of tracing paper is allowed in the examination (although students should not have to rely on the use of tracing paper to solve problems)

3-16

Transforming graphs

Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a),
y = af(x) for linear, quadratic, sine and cosine functions f(x)
Draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both the x and y directions

Transformations
Curved graphs (Unit 1 and 3)

By the end of the module the student should be able to:
• Understanding of the notation y = f(x) (11.1)
• Represent translations in the x and y direction, reflections in the x-axis
and the y axis, and stretches parallel to the x-axis and the y-axis (11.2 – 11.4)
• Apply to general graphs or specific curves such as trigonometric functions (ie curved graphs) (11.2 – 11.4)
• Sketch the graph of y = 3sin(2x), given the graph of y = sinx (11.2 – 11.4)
• Sketch the graph of y = f(x + 2), y = f(x) + 2, y=2f(x), y = f(2x)
given the shape of the graph y = f(x) (11.2 – 11.4)
• Find the coordinates of the minimum of y = f(x + 3), y = f(x) + 3 given
the coordinates of the minimum of y = x2 – 2x (11.2 – 11.4)

Complete the square of quadratic functions and relate this to transformations of the
curve y = x2
Use a graphical calculator/software to investigate transformations
Investigate curves which are unaffected by particular transformations
Investigations of the simple relationships such as sin(180 – x) = sin x, and sin(90 – x) = cos x

Make sure the students understand the notation y = f(x). Perhaps start with comparing y = x2 with y = x2 + 2 before mentioning y = f(x) + 2
Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit
Link with trigonometry and curved graphs

3-17

Constructions, loci, similarity and congruency

Draw triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and angles
Understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA are not
Use straight edge and compasses to do standard constructions
Construct loci
Understand similarity of triangles and of other plane figures and use this to make geometric inferences
Recognise that enlargements preserve angle but not length
Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles by using formal arguments, and also to verify standard ruler and compass constructions

An ability to use a pair of compasses
The special names of triangles (and angles)
Understanding of the terms perpendicular, parallel and arc
Transformations (particularly enlargements)

By the end of the module the student should be able to do a range of standard constructions including:
• An equilateral triangle with a given side (16.2 – 16.5)
• The mid point and perpendicular bisector of a line segment (16.1)
• The perpendicular from a point on a line (16.1)
• The bisector of an angle (16.2)
• The angles 60°, 30° and 45° (16.2)
• A regular hexagon inside a circle, etc (16.2)
• A region bounded by a circle and an intersecting line (16.4)
• A path equidistant from two points or two line segments (16.3)
• Prove geometric properties of triangles formally, eg that the base angles of an isosceles triangle are equal (16.5)
• Prove that two triangles are congruent formally (14.1 – 16.5)
• Use integer and non-integer scale factors to find the length of a missing side in each of two similar shapes, given the lengths of a pair of corresponding sides (14.2 – 14.3)

Solve loci problems that require a combination of loci
Construct combinations of 2-D shapes to make nets
Link with tessellations (Unit 2) and enlargements (Unit 3)
Link with similar areas and volumes
Use harder problems in congruence
Relate this unit to circle theorems

All working should be presented accurately and clearly
A sturdy pair of compasses are essential
Construction lines should not be erased as they carry method marks
+H54

3-18

Pythagoras’ theorem & Trigonometry in 2-D

Understand, recall and use Pythagoras’ theorem in 2-D
Understand, recall and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings
Find angles of elevation and angles of depression

Names of triangles & quadrilaterals
Knowledge of the properties of rectangles, parallelograms and triangles
Indices, equations and changing the subject
Similarity of triangles and other plane figures

By the end of the module the student should be able to:
• Find missing sides of right-angle triangles by using Pythagoras (18.1 – 18.2)
• Find the distance between two coordinates using Pythagoras (18.2)
• Giving answers as decimals or surds for Pythagoras problems (18.2)
• Use trigonometric ratios (sin, cos and tan) to calculate angles in
right-angled triangles (18.3)
• Use the trigonometric ratios to calculate unknown lengths in
right-angled triangles (2-D) (18.4)
• Understand how bearings work and solve problems involving bearings using Pythagoras/trigonometry (16.6, 18.4)
• Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras and/or trigonometry (18.2, 18.4)

Introduce 3-D trigonometry and show that the trigonometric ratios will only work for
right-angled triangles

Students should be encouraged to become familiar with one make of calculator
Calculators should be set to “deg” mode
Emphasise that scale drawings will score no marks for this type of question
A useful mnemonic for remember trig ratios is “Sir Oliver’s Horse, Came Ambling Home, To Oliver’s Aunt” or ‘SOH/CAH/TOA’
Calculated angles should be given to at least 1 dp and sides are determined by the units used or accuracy asked for in the question
Students should not forget to state the units in their answers
Organise a practical surveying lesson to find the heights of buildings/trees around your school grounds. All that is needed is a set of tape measures (or trundle wheels) and clinometers

3-19

Applications of Pythagoras’ theorem & Trigonometry in 3-D

Understand and recall and use trigonometric relationships in right angled triangles and use these to solve problems in 3-D (include the use of Pythagoras’ Theorem)
Use the trigonometric ratios and the sine and cosine rules to solve 3-D problems
Find the angles between a line and a plane

Knowledge of Pythagoras’ theorem and trigonometry
3-D solids (Unit 2)

By the end of the module the student should be able to:
• Calculate the length of a diagonal of a rectangle given the lengths of the sides of the rectangle (18.2)
• Calculate the diagonal through a cuboid, or across the face of a cuboid (19.1, 19.2)
• Find the angle between the diagonal through a cuboid and the base of the cuboid (19.1, 19.2)
• Find the angle between a sloping edge of a pyramid and the base of the pyramid (19.1, 19.2)

Use harder problems involving multi-stage calculations

The angle between two planes or two skew lines is not required

3-20

Trigonometry for non-right-angled triangles

Use the sine and cosine rules to solve 2-D problems
Calculate the area of a triangle using absinC

Use the sine and cosine rules to solve 3-D problems

Trigonometry
Formulae

By the end of the module the student should be able to:
• Find the unknown lengths, or angles, in non right-angle triangles
(in 2-D and 3-D) using the sine and cosine rules (19.3, 19.5 – 19.9)
• Find the area of triangles given two lengths and an included angle (19.4)

• Use trigonometrical ratios to solve problems in 3-D and decide if it is easier to extract right-angle triangles to use simple trigonometry
• Establish strategies for solving problems in 3-D. Decide when it is more appropriate to use cosine rule rather than sine rule

Students find the cosine rule more difficult for obtuse angles. Demonstrate an example of the ambiguous case
Reminders of simple geometrical facts will always be helpful, eg angle sum of a triangle, the shortest side is opposite the smallest angle
Show the form of the cosine rule on the formula page and re-arrange it to show the form which finds missing angles

3-21

Circle theorems

Explain why the perpendicular from the centre to a chord bisect the chord
Prove and use the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference
Prove and use the fact that the angle subtended at the circumference by a semicircle is a right angle
Prove and use the fact that angles in the same segments are equal
Prove and use the fact that opposite angles of a cyclic quadrilateral sum to 180°
Prove and use the alternate segment theorem

Recall the words centre, radius, diameter and circumference
Have practical experience of drawing circles with compasses
Circle theorem covered in Unit 2 (tangent/radius property)
Recall that the tangent at any point on a circle is perpendicular to the radius at that point
Recall and use the fact that tangents from an external point are equal in length

By the end of the module the student should be able to:
• Understand, prove and use circle theorems (see above) (Chapter 15)
• Use circle theorems to find unknown angles and explain their method -
quoting the appropriate theorem(s) (Chapter 15)

Any proof required will be in relation to a diagram, not purely by reference to a named theorem
Reasoning needs to be carefully constructed as ‘Quality of Written Communication’ marks are allotted

3-22

Circles, cones, pyramids and spheres

Find circumferences of circles and areas enclosed by circles, recalling relevant formulae
Solve problems involving surface areas and volumes of prisms and cylinders
Solve problems involving volumes of prisms and cylinders
Solve problems involving surface areas and volumes of cones and pyramids
Solve problems involving more complex shapes, including segments of circles, length of a chord and frustums of cones
Use surds and π in exact calculations, without a calculator
Find the surface areas and volumes of compound solids constructed from; spheres, hemispheres and cylinders (include examples of solids in everyday use)

Perimeter and area (Unit 2)
Surface area and volume (Unit 2)
Formulae, substitution and changing the subject
Surds

By the end of the module the student should be able to:
• Learn that the ratio between the circumference and diameter is always
constant for a circle (ie π) (12.1)
• Learn the formula for circumference and area of a circle (12.1)
• Solve problems involving the circumference and area of a circle (and simple fractional parts of a circle) (12.1 – 12.4)
• Solve problems involving the volume of a cylinder (13.1)
• Find the surface area and the volume of more complex shapes, eg find the volume of an equilateral triangular prism (13.2 – 13.6)
• Solve more complex problems, eg given the surface area of a sphere find the volume (13.2 – 13.6)

Find the volume of a cylinder given its surface area, leaving the answer in terms of l
Find the volume of a right hexagonal cone of side x and height h (researching the method for finding the volume of any cone)

‘Now! I Know Pi’ is a good way to learn the approx value (the number of letters of each word and the ! is the
decimal point)
Also ‘Cherry Pie Delicious’ is C = πD and ‘Apple Pies are too’ is A = πr2
Answers in terms of π may be required or final answers rounded to the required degree of accuracy
Need to constantly revise the expressions for area/volume of shapes
Students should be aware of which formulae are on the relevant page on the exam paper and which they need to learn

3-23

Similar Shapes

Identify the scale factor of an enlargement as a ratio of the lengths of any two corresponding line segments
Understand the implications of enlargement for perimeter
Understand and use the effect of enlargement on areas and volumes of shapes and solids

Use ruler and compasses to construct triangles with given dimensions
Some concept of enlargement (magnification)
Similar triangles

By the end of the module the student should be able to:
• Use integer and non-integer scale factors to find the length of a missing
side in each of two similar shapes, given the lengths of a pair of
corresponding sides (14.2 – 14.3)
• Know the relationship between linear, area and volume scale factors of
similar shapes (14.4 – 14.6)
• Prove formally geometric properties of triangles, eg that the base angles of an isosceles triangle are equal (14.5)

Find algebraic formulae for the areas and volumes of similar shapes
Extend to questions which give the ratio for area and the need to square root the ratio to get back to length first etc

Students will need to be reminded of this work on a regular basis, link it to ratios
1 : L (Length)
1: L2 (Area)
1: L3 (Volume)
A good starter is to bring in a small bottle of water and a larger bottle (preferably for the larger bottle to be twice the length). Show by pouring that eight small bottles will fill the larger bottle. [This can also be done with one small cube and a larger box with lengths twice as long] Initially the class will say two bottles will fill the larger double size bottle

3-24

Vectors

Understand and use vector notation (revise column vectors)
Calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector
Calculate the resultant of two vectors
Understand and use the commutative and associative properties of vector addition
Solve simple geometrical problems in 2-D using vector methods

Vectors to describe translations.
Algebraic manipulation

By the end of the module the student should be able to:
• Understand that 2a is parallel to a and twice its length (20.1 – 20.3, 20.5)
• Understand that a is parallel to -a and in the opposite direction (20.1 – 20.3, 20.5)
• Use and interpret vectors as displacements in the plane (with an
associated direction) (20.1)
• Use standard vector notation to combine vectors by addition (20.4)
• Represent vectors, and combinations of vectors, in the plane (20.1 – 20.6)
• Solve geometrical problems in 2-D, eg show that joining the mid points
of the sides of any quadrilateral forms a parallelogram (20.6)

Harder geometric proof, eg show that the medians of a triangle intersect at a single point
Illustrate use of vectors by showing ‘Crossing the flowing River’ example or navigation examples
Vector problems in 3-D (for the most able)
Use i and j (and k) notation (preparation for A-Level but not in GCSE specification)
Lengths are sometimes given in the ratio of 1:3. It is important to explain this concept carefully

Students often find the pictorial representation of vectors more difficult than the manipulation of column vectors
The geometry of a hexagon provides a rich source of parallel, reverse and vectors with double the magnitude
Stress that parallel vectors are equal to one another
Link with ‘like terms’ and ‘brackets’ when simplifying
Show there is more than one route round a geometric shape, but that the answer simplifies to the same vector
Remind students to underline vectors or they will be regarded as just lengths with no direction
Extension material can be found in M1 texts

3-25

Compound Units

Understand and use compound measures, including density
Convert measurements from one unit to another (area and volume)

Knowledge of metric units of length, area, volume, and weight, eg 1 m = 100 cm, etc
Speed calculations
Experience of multiply by powers of 10, eg 100 ´ 100 = 10 000
Substitution and changing the subject

By the end of the module the student should be able to:
• Use the relationship between distance, speed and time to solve problems (revision of Unit 2) (Revision of
prior knowledge)
• Know that density is found by mass ÷ volume (13.6)
• Use the relationship between density, mass and volume to solve problems (13.6)
• Convert between metric units of density, eg kg/m to g/cm (13.7)
• Convert between area measures, eg 1 m2 = 100 00 cm2 (12.4)
• Convert between volume measures (using metric units) (13.5)
• Convert between volume and capacity measures,
eg 1 ml = 1 cm³ (1cc) (13.5)

Perform calculations on a calculator by using standard form
Convert imperial units to metric units, eg mph into km/h
Mention other units (not on course) like hectares

Use a formula triangle to help students see the relationship between the variables for density
Borrow a set of electronic scales and a Eureka Can from Physics for a practical density lesson
Look up densities of different elements from the internet
Link converting area and volume units to similar shapes
Draw a large grid made up of 100 by 100 cm squares to show what 1 square metre looks like
Use of Calculators (Spec Ref:- Nv)
Students are well advised to regularly bring and use their particular calculator to lessons and get used to its functions throughout this Unit 3 course.
They should be confident in entering a range of calculations including those involving time and money.
They should realise that 2.33333333…. hrs is 2 hours 20 minutes etc

## Unit 3 Higher Maths

Welcome to the Unit 3 Higher page for Edexcel GCSE Maths. You will find all sorts of useful resources to help with your revisionUnit 3 Higher Maths | Past Papers | Revision Resources | Higher Revision Booklets | Topic Checklist | Unit 3 Exam Question Solutions | Specification VIDEO SOLUTIONS | Detailed Specification

## Past Papers

## Revision Resources

Try these links for revision relating to Unit 3:Revision Booklet

## Higher Revision Booklets

Find a set of exam questions collected by topic including the answers for most:(solving equations, expandind and factorising, expressions)

(plotting, using and applying)

(factorising, formula, completing the square)

(2 linear, 1 linear 1 quadratic)

(reflection, rotation, translation, enlargement)

(SOH CAH TOA, Sine rule, Cosine rule, Area of triangle)

## Topic Checklist

A list of Unit 3 topics:Percentage of an amount

Percentage of an amount 2

One thing as a % of another

Reverse percentages

Reverese percentages 2

Compound interest

Adding Mixed Fractions

Multiplying Mixed Fractions

Dividing Fractions

Expand & Factorise Quadratics

Plotting Inequalities

MM Shading Inequalities

Using formulae

Solving Simultaneous Equations

MM Simultaneous 1

MM Simultaneous 2

MM Simultaneous 3

Expand & Factorise Quadratics

Notes and Videos

Factorising Quadratics

Find the function

Reflection

Scale Drawing

Constructing an Equilateral Triangle

Constructing an Isosceles Triangle

Constructing a Perpendicular Bisector

Constructing a Perpendicular line to a point on a line

Constructing a Perpendicular line to a point NOT on line

Pythagoras - Finding a short side

Pythagoras' theorem and Trigonometry

SohCahToa or Pythagoras?

Pythagoras' Theorem

Trig - Missing angles

Trig - Missing sides

Trig - Angles of elevation

3D Trigonometry

Using Cosine rule to find missing side

Using Sine rule to find missing angle

Using Sine rule to find missing side

All Theorems reminder

Area & Circumference of circles

Length Area Volume scale factors

MM Volume scale factors

Unit 3 Higher Maths | Past Papers | Revision Resources | Higher Revision Booklets | Topic Checklist | Unit 3 Exam Question Solutions | Specification VIDEO SOLUTIONS | Detailed Specification

## Unit 3 Exam Question Solutions

Try these questions then use the video solutions to check your understanding.Unit 3 Higher Maths | Past Papers | Revision Resources | Higher Revision Booklets | Topic Checklist | Unit 3 Exam Question Solutions | Specification VIDEO SOLUTIONS | Detailed Specification

## Specification VIDEO SOLUTIONS

## Detailed Specification

A detailed breakdown:Solve percentage problems, including increase and decrease, and reverse percentage

Represent repeated proportional change using a multiplier raised to a power

Use calculators for reverse percentages calculations by doing an appropriate division

Use calculators to explore exponential growth and decay, using a multiplier and power key

Awareness that percentages are used in everyday life

Percentages (Unit 2)

• Find a percentage increase/decrease of an amount (3.1, 3.2)

• Find a reverse percentage, eg find the original cost of an item given the cost after a 10% deduction (3.4)

• Use a multiplier to increase by a given percent, eg 1.1 ´ 64 increases 64 by 10% (3.2, 3.3)

• Calculate simple and compound interest for two, or more, periods of time (3.3)

• Solve a whole host of Functional Elements involving percentages (Throughout

Chapter 3)

Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100%

Problems which lead to the necessity of rounding to the nearest penny, eg real-life contexts

Comparisons between simple and compound interest calculations

Formulae in simple interest/compound interest methods

Increase and decreases leading to a combined multiplier to use, eg 10% decrease then 5% increase

In preparation for this unit, students should be reminded of basic percentages and be able to recognise their fraction and decimal equivalents from Unit 2

Use percentages in real-life situations:

– VAT

– Simple interest

– Income tax calculations

– Annual rate of inflation

– Compound interest

– Depreciation

– Find prices after a percentage increase or decrease

– Percentage profit and loss

Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction

Find the upper and lower bounds of calculations involving perimeter, areas and volumes of 2-D and 3-D shapes

Measurement and units

Perimeter and area

• Understand that measurements can not be precise, and write down the maximum and minimum possible values (2.1, 2.2)

• Work out the maximum/minimum possible error in a calculation involving measures (2. 2)

• Find when numbers are given to a specific degree of accuracy, the upper and lower bounds of perimeters and areas (2.2)

• Apply upper and lower bounds to compound units, eg speed (2.2)

• Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation (2.2)

Use examples that demonstrate finding the maximum/minimum values for a – b and a ÷ b

Mention that bridges have expansion joints

Understand ‘reciprocal’ as multiplicative inverse

Multiply and divide a given fraction by an integer, by a unit fraction or by a general fraction

Recognise that each terminating decimal is a fraction

Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals

Convert any recurring decimal to a fraction (proof)

Use a calculator to solve real-life problems involving Fractions

• Convert a fraction to a decimal, or a decimal to a fraction (1.1 – 1.4)

• Find the reciprocal of whole numbers, fractions, and decimals (1.1)

• Multiply and divide a fraction by an integer, by a unit fraction and by a general fraction (expressing the answer in its simplest form) (1.2 – 1.3)

• Convert a fraction to a recurring decimal (and visa versa) (1.4)

• Use fractions in contextualised problems (1.3)

Revise addition of fractions (Unit 1 and 2)

Use combinations of the four operations with fractions and in real-life problems, eg to find areas using fractional values

Revise algebraic fractions with very able students

All work needs to be presented clearly with the relevant stages of working shown, even if a calculator is used

Use Functional Elements problems as a source of questions involving fractions in a real-life context

Calculate with standard form

Convert between ordinary and standard form representations

Convert to standard form to make sensible estimates for calculations involving multiplication and/or division

Use standard form display and know how to enter numbers in standard form

Multiplying decimal numbers with, and without, a calculator

Some experience with powers of 10, eg know that 102 = 100, 103 = 1000, 10–1 = 0.1

Negative indices and laws of indices

Standard form (Unit 2)

• Convert numbers to, and from, standard form (4.1)

• Calculate with numbers given in standard form with, and without, a calculator (4.1)

• Interpret a calculator display using standard form (4.1)

• Round numbers given in standard form to a given number of significant figures (4.1)

• Use standard form in real-life situations

Ensure that students never to write down the answer as displayed by the calculator,

eg 1.3 -03

This work can be enriched by using examples drawn from the sciences, eg Avogadro’s Constant 6.02 × 10^ -23

Multiply a single term over a bracket

Take out common factors

Expand the product of two linear expressions

Factorise quadratic expressions

Manipulate Algebraic Fractions

• Simplify expressions with like terms, eg ; (7.2 Unit 2)

• Expand and factorise expressions with one pair of brackets,

eg expand x(2x +3y); factorise 3xy2 - 6x2y (8.1, 8.2, Unit 2)

• Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (8.3 Unit 2)

• Factorise quadratic expressions (including the difference of two squares) (8.4 Unit 2)

• Further examples in factorising quadratic expression with non-unitary values of a (including fractional values) (8.4 Unit 2)

• Simplify algebraic fractions, eg (11.1 Unit 2)

• Simplify algebraic fractions, including the addition of fractions and to solve problems (11.2 Unit 2)

Link difference of two squares with surds and the rationalisation of denominators

Present all work neatly, writing out the questions with the answers to aid revision at a later stage

Practise some basic fraction additions, with numerical values, then compare this with the algebraic version

Encourage students to look out for difference of two squares when working with algebraic fractions

Remind student to always look to take out a common factor first

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

Solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution

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

The idea that some operations are ‘opposite’ to each other

Experience of using letters to represent quantities

Some experience of ‘balancing’ equations in order to solve them

Be able to draw a number line

• Solve linear equations with one, or more, operations (including fractional coefficients) (Throughout

Chapter 6)

• Solve linear equations involving brackets and/or variables on both sides (6.2, 6.3)

• Solve linear inequalities in one variable and present the solution set on a number line (7.1)

• Form linear equations from worded problems in a variety of contexts and relating the answer back to the original problem (Throughout

Chapter 6)

Derive equations from practical situations (such as finding unknown angles in polygons or perimeter problems)

Solve equations where manipulation of fractions (including the negative fractions) is required

Solve linear inequalities where manipulation of fractions is required

Students can leave their answers in fractional form where appropriate

Interpreting the direction of an inequality is a problem for many

Students should use the correct notation when showing inequalities on a number line, eg a filled in circle to show inclusion of a point, an empty circle to show exclusion of a point

Inequalities in two variables will be covered again after simultaneous equations

Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution

Change the subject of a formula including where the subject occurs once or more than once

Generate a formula

Experience of using letters to represent quantities

Substituting into simple expressions using words

Using brackets in numerical calculations and removing brackets in simple algebraic expressions

Solving linear equations

• Use letters or words to state the relationship between different quantities (Revision of

prior knowledge)

• Substitute positive and negative numbers into simple algebraic formulae (Revision of

prior knowledge)

• Substitute positive and negative numbers into algebraic formulae involving powers (Revision of

prior knowledge)

• Find the solution to a problem by writing an equation and solving it (Revision of

prior knowledge)

• Simple change of subject of a formula, eg convert the formula for converting Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade (7.5)

• Change the subject of the formula when the variable appears more than once (questions could involve powers, roots, fractions or reciprocals) (7.6)

• Generate a formula from given information, eg find the formula for the perimeter of a rectangle given its area A and the length of one side (Revision of

prior knowledge)

Various investigations leading to generalisations

Further problems in generating formulae form given information

Use equation of a straight line, eg what is the gradient of the line 4x + 2y = 12?

Apply changing the subject to physics formulae, ie pendulum, equations of motion, lens formula

Students need to be clear on the meanings of the words expression, equation, formula and identity

Simple changing the subject is covered in Unit 2

Show a linear equation first and follow the same steps for the similarly structured formula to be rearranged

Functional Elements material is a rich source of formulae given in everyday contexts

Link with formulae for area, volume, surface area, etc

Find the gradient of lines given equations of the form y = mx + c values are given for m and c)

Analyse problems and use gradients to see how one variable changes in relation to another

Calculate the length of a line segment between two coordinates

Linear sequences and straight line graphs

• Draw linear graphs from tabulated data, including real-world examples (Revision of

prior knowledge)

• Interpret linear graphs, including conversion graphs and distance-time graphs (Revision of

prior knowledge)

• Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (Revision of

prior knowledge)

• Understand that lines are parallel when they have the same value of m (Revision of

prior knowledge)

• Find the gradient and intercept of a straight line graph (Revision of

prior knowledge)

• Find the distance between any two coordinates (link with Pythagoras’ theorem) (18.2)

Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines

Use a graphical calculator or graphical ICT package to draw straight-line graphs

Link with length of line to Pythagoras’ theorem and contrast with midpoint formula (Unit 2)

Recognise linear graphs and hence when data may be incorrect

Link to graphs and relationships in other subject areas, ie science, geography etc

Interpret straight line graphs for Functional situations and link gradient to cost per unit or rate of change and intercept to initial cost etc

– Ready reckoner graphs

– Conversion graphs

– Fuel bills

– Fixed charge (standing charge) and cost per unit

Solve several linear inequalities in two variables and finding the solution set

Ability to solve simple linear equations

Some experience with solving inequalities

Straight line graphs

• Solve algebraically two simultaneous equations (9.2, 9.3)

• Interpret the solution of two simultaneous equations as the point of intersection the corresponding lines (9.4)

• Model worded problems as a pair of linear simultaneous equations and interpret the answer (9.3)

• Draw the graphs of linear inequalities in two variables and interpret the solution sets given by regions in the coordinate plane, or to identify all the integer coordinates with crosses (7.4)

Solve two simultaneous equations with second order terms, eg equations in x and y2 (link to further simultaneous equations)

Link the graphical solutions with straight line graphs (Units 1 and 2) and changing the subject (above)

Inaccurate graphs could lead to incorrect solutions, encourage substitution of answers to check they are correct

Clear presentation of working out is essential

Students should use the correct notation when giving graphical solutions to inequalities,

eg a dotted boundary line for < or >

Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relating algebraic solutions to graphical representations of the equations

Calculate an unknown quantity from quantities that vary in direct or inverse proportion

Rearrange the subject of a formula

• Solve simple direct & inverse proportion problems using the Unitary method or by proportional change (from a table of values (Chapter 5)

• Interpret direct and inverse proportions as algebraic functions,

eg y µ x2 as y = kx2 (Chapter 10)

• Use given information to find the value of the constant of proportionality (Chapter 10)

• Use algebraic functions for direct and inverse proportionality, with their value of k, to find unknown values (Chapter 10)

• Recognise and sketch the graphs for direct and inverse proportions

(y µ x, y µ x2, y µ x3, y µ , y µ ) (Chapter 10)

Problems involving other types of proportionality, eg surface area to volume of a sphere

Link to graphs to show direct and inverse proportion, eg etc

Students often forget the “square” in inverse square proportionality

Dealing with decimals on a calculator

Ordering decimals and decimal place approximations.

• Solve quadratic and cubic functions by successive substitution of values of x

The square/cube function on a calculator may not be the same for different makes

Take care when entering negative values to be squared (always use brackets)

Students should write down all the digits on their calculator display and only round the final answer declared to the degree of accuracy

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

Factorise quadratic expressions

Solve simple quadratic equations by factorising, completing the square and using the quadratic formula

Factorising Quadratics

• Plot the graphs of quadratic functions for positive and negative values of x (8.1)

• Find graphically the solutions of quadratic equations by considering the intercept on the x-axis (9.10)

• Solve quadratic equations by factorising (including values of a not equal to 1) (9.5)

• Use the quadratic formula to solve quadratic equations giving the answers to a specified degree of accuracy (9.7)

• Use the quadratic formula to solve quadratic equations leaving the answer in surd form or decimal form (9.7)

• Complete the square of a quadratic function (using this to write down the maximum/minimum of the function) (9.1, 9.6)

Solve quadratic equations by completing the square

Derive the quadratic equation by completing the square

Use graphical calculators or ICT graph package where appropriate to enable students to get through examples more rapidly

Show how the value of ‘b2 – 4ac’ can be useful in determining if the quadratic factorises or not (ie square number)

Extend to discriminant’s properties and roots (for those going on to C1)

Students should be reminded that factorisation should be tried before the formula is used

In problem-solving, one of the solutions to a quadratic may not be appropriate, eg negative length

Construct the graph of x2 + y2 = r2 for a circle of radius r centred at the origin of coordinates

Find graphically the intersection points of a given straight line and a circle and knowing that this corresponds to solving the simultaneous equations representing the line and the circle

Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, of where the second is of the form x2 + y2 = r2

Straight line graphs

Algebraic manipulation and solving linear and quadratic equations

• Find graphically the approximate solutions of linear and quadratic simultaneous equations (9.10)

• Find the exact solutions of linear and quadratic simultaneous equations (9.11)

• Draw a circle of radius r centred at the origin (9.10)

• Find the approximate solutions of linear and circular simultaneous equations graphically (9.11)

• Find the exact solutions of linear and circular simultaneous equations (9.10)

Find the exact solutions of quadratic and circular simultaneous equations using algebraic methods

Look at circles whose centre is not the origin (x – 2)2 + (y – 3)2 = 4 (link with transforming graphs)

Stress which variable it is easiest to work with when assessing the linear equation

ICT graph drawing packages make this topic more dynamic and easier to picture

Further examples and questions can be obtained from A-Level (C1) texts

Plot graphs of simple cubic functions, the reciprocal function y = with x 0, the exponential function, y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x

Recognise the characteristic shapes of all these functions

BIDMAS

• Plot and recognise quadratic, cubic, reciprocal, exponential and circular (trig) functions (see above) within the range –360º to +360º (8.1 – 8.4)

• Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa) (8.1 – 8.4)

• Match equations with their graphs (8.1 – 8.4)

• Sketch graphs of given functions (8.1 – 8.4)

Explore the function y = tan x

Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise)

Start to investigate transformations, eg y = sin(2x) or y = x2 + 5

Group work with each group assigned a different type of graph is an effective way to share the graphs’ properties. Each group reports to the whole class and creates a display

There are plenty of old exam papers with matching tables testing knowledge of the ‘Shapes of Graphs’

Understand that translations are specified by a distance and direction (or a column vector), and enlarging by a centre and a scale factor

Rotate a shape about the origin, or any other point

Measure the angle of rotation using right angles, simple fractions of a turn or degrees

Understand that rotations are specified by a centre and an (anticlockwise) angle

Understand that reflections are specified by a mirror line, at first using a line parallel to any axis, then a mirror line such as y = x or y = –x

Recognise, visualise and construct enlargements of objects using positive and negative scale factors greater and less than one

Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations

Line and rotational symmetry (Unit 2)

An understanding of the concept of rotation and enlargement

Coordinates in four quadrants

Linear equations parallel to the coordinate axes

• Understand translation as a combination of a horizontal and vertical shift including signs for directions (17.2)

• Understand rotation as a (clockwise) turn about a given origin (17.4)

• Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = –x (17.3)

• Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater and less than one (and understand the effects that negative and fractional scale factors have on the image) (17.5, 17.6)

• Understand that shapes produced by translation, rotation and reflection are congruent to its image (17.1 – 17.4)

Diagrams should be drawn carefully

The use of tracing paper is allowed in the examination (although students should not have to rely on the use of tracing paper to solve problems)

y = af(x) for linear, quadratic, sine and cosine functions f(x)

Draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both the x and y directions

Curved graphs (Unit 1 and 3)

• Understanding of the notation y = f(x) (11.1)

• Represent translations in the x and y direction, reflections in the x-axis

and the y axis, and stretches parallel to the x-axis and the y-axis (11.2 – 11.4)

• Apply to general graphs or specific curves such as trigonometric functions (ie curved graphs) (11.2 – 11.4)

• Sketch the graph of y = 3sin(2x), given the graph of y = sinx (11.2 – 11.4)

• Sketch the graph of y = f(x + 2), y = f(x) + 2, y=2f(x), y = f(2x)

given the shape of the graph y = f(x) (11.2 – 11.4)

• Find the coordinates of the minimum of y = f(x + 3), y = f(x) + 3 given

the coordinates of the minimum of y = x2 – 2x (11.2 – 11.4)

curve y = x2

Use a graphical calculator/software to investigate transformations

Investigate curves which are unaffected by particular transformations

Investigations of the simple relationships such as sin(180 – x) = sin x, and sin(90 – x) = cos x

Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit

Link with trigonometry and curved graphs

Understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA are not

Use straight edge and compasses to do standard constructions

Construct loci

Understand similarity of triangles and of other plane figures and use this to make geometric inferences

Recognise that enlargements preserve angle but not length

Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles by using formal arguments, and also to verify standard ruler and compass constructions

The special names of triangles (and angles)

Understanding of the terms perpendicular, parallel and arc

Transformations (particularly enlargements)

• An equilateral triangle with a given side (16.2 – 16.5)

• The mid point and perpendicular bisector of a line segment (16.1)

• The perpendicular from a point on a line (16.1)

• The bisector of an angle (16.2)

• The angles 60°, 30° and 45° (16.2)

• A regular hexagon inside a circle, etc (16.2)

• A region bounded by a circle and an intersecting line (16.4)

• A path equidistant from two points or two line segments (16.3)

• Prove geometric properties of triangles formally, eg that the base angles of an isosceles triangle are equal (16.5)

• Prove that two triangles are congruent formally (14.1 – 16.5)

• Use integer and non-integer scale factors to find the length of a missing side in each of two similar shapes, given the lengths of a pair of corresponding sides (14.2 – 14.3)

Construct combinations of 2-D shapes to make nets

Link with tessellations (Unit 2) and enlargements (Unit 3)

Link with similar areas and volumes

Use harder problems in congruence

Relate this unit to circle theorems

A sturdy pair of compasses are essential

Construction lines should not be erased as they carry method marks

+H54

Understand, recall and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings

Find angles of elevation and angles of depression

Knowledge of the properties of rectangles, parallelograms and triangles

Indices, equations and changing the subject

Similarity of triangles and other plane figures

• Find missing sides of right-angle triangles by using Pythagoras (18.1 – 18.2)

• Find the distance between two coordinates using Pythagoras (18.2)

• Giving answers as decimals or surds for Pythagoras problems (18.2)

• Use trigonometric ratios (sin, cos and tan) to calculate angles in

right-angled triangles (18.3)

• Use the trigonometric ratios to calculate unknown lengths in

right-angled triangles (2-D) (18.4)

• Understand how bearings work and solve problems involving bearings using Pythagoras/trigonometry (16.6, 18.4)

• Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras and/or trigonometry (18.2, 18.4)

right-angled triangles

Calculators should be set to “deg” mode

Emphasise that scale drawings will score no marks for this type of question

A useful mnemonic for remember trig ratios is “Sir Oliver’s Horse, Came Ambling Home, To Oliver’s Aunt” or ‘SOH/CAH/TOA’

Calculated angles should be given to at least 1 dp and sides are determined by the units used or accuracy asked for in the question

Students should not forget to state the units in their answers

Organise a practical surveying lesson to find the heights of buildings/trees around your school grounds. All that is needed is a set of tape measures (or trundle wheels) and clinometers

Use the trigonometric ratios and the sine and cosine rules to solve 3-D problems

Find the angles between a line and a plane

3-D solids (Unit 2)

• Calculate the length of a diagonal of a rectangle given the lengths of the sides of the rectangle (18.2)

• Calculate the diagonal through a cuboid, or across the face of a cuboid (19.1, 19.2)

• Find the angle between the diagonal through a cuboid and the base of the cuboid (19.1, 19.2)

• Find the angle between a sloping edge of a pyramid and the base of the pyramid (19.1, 19.2)

Calculate the area of a triangle using absinC

Use the sine and cosine rules to solve 3-D problems

Formulae

• Find the unknown lengths, or angles, in non right-angle triangles

(in 2-D and 3-D) using the sine and cosine rules (19.3, 19.5 – 19.9)

• Find the area of triangles given two lengths and an included angle (19.4)

• Establish strategies for solving problems in 3-D. Decide when it is more appropriate to use cosine rule rather than sine rule

Reminders of simple geometrical facts will always be helpful, eg angle sum of a triangle, the shortest side is opposite the smallest angle

Show the form of the cosine rule on the formula page and re-arrange it to show the form which finds missing angles

Prove and use the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference

Prove and use the fact that the angle subtended at the circumference by a semicircle is a right angle

Prove and use the fact that angles in the same segments are equal

Prove and use the fact that opposite angles of a cyclic quadrilateral sum to 180°

Prove and use the alternate segment theorem

Have practical experience of drawing circles with compasses

Circle theorem covered in Unit 2 (tangent/radius property)

Recall that the tangent at any point on a circle is perpendicular to the radius at that point

Recall and use the fact that tangents from an external point are equal in length

• Understand, prove and use circle theorems (see above) (Chapter 15)

• Use circle theorems to find unknown angles and explain their method -

quoting the appropriate theorem(s) (Chapter 15)

Reasoning needs to be carefully constructed as ‘Quality of Written Communication’ marks are allotted

Solve problems involving surface areas and volumes of prisms and cylinders

Solve problems involving volumes of prisms and cylinders

Solve problems involving surface areas and volumes of cones and pyramids

Solve problems involving more complex shapes, including segments of circles, length of a chord and frustums of cones

Use surds and π in exact calculations, without a calculator

Find the surface areas and volumes of compound solids constructed from; spheres, hemispheres and cylinders (include examples of solids in everyday use)

Surface area and volume (Unit 2)

Formulae, substitution and changing the subject

Surds

• Learn that the ratio between the circumference and diameter is always

constant for a circle (ie π) (12.1)

• Learn the formula for circumference and area of a circle (12.1)

• Solve problems involving the circumference and area of a circle (and simple fractional parts of a circle) (12.1 – 12.4)

• Solve problems involving the volume of a cylinder (13.1)

• Find the surface area and the volume of more complex shapes, eg find the volume of an equilateral triangular prism (13.2 – 13.6)

• Solve more complex problems, eg given the surface area of a sphere find the volume (13.2 – 13.6)

Find the volume of a right hexagonal cone of side x and height h (researching the method for finding the volume of any cone)

decimal point)

Also ‘Cherry Pie Delicious’ is C = πD and ‘Apple Pies are too’ is A = πr2

Answers in terms of π may be required or final answers rounded to the required degree of accuracy

Need to constantly revise the expressions for area/volume of shapes

Students should be aware of which formulae are on the relevant page on the exam paper and which they need to learn

Understand the implications of enlargement for perimeter

Understand and use the effect of enlargement on areas and volumes of shapes and solids

Some concept of enlargement (magnification)

Similar triangles

• Use integer and non-integer scale factors to find the length of a missing

side in each of two similar shapes, given the lengths of a pair of

corresponding sides (14.2 – 14.3)

• Know the relationship between linear, area and volume scale factors of

similar shapes (14.4 – 14.6)

• Prove formally geometric properties of triangles, eg that the base angles of an isosceles triangle are equal (14.5)

Extend to questions which give the ratio for area and the need to square root the ratio to get back to length first etc

1 : L (Length)

1: L2 (Area)

1: L3 (Volume)

A good starter is to bring in a small bottle of water and a larger bottle (preferably for the larger bottle to be twice the length). Show by pouring that eight small bottles will fill the larger bottle. [This can also be done with one small cube and a larger box with lengths twice as long] Initially the class will say two bottles will fill the larger double size bottle

Calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector

Calculate the resultant of two vectors

Understand and use the commutative and associative properties of vector addition

Solve simple geometrical problems in 2-D using vector methods

Algebraic manipulation

• Understand that 2a is parallel to a and twice its length (20.1 – 20.3, 20.5)

• Understand that a is parallel to -a and in the opposite direction (20.1 – 20.3, 20.5)

• Use and interpret vectors as displacements in the plane (with an

associated direction) (20.1)

• Use standard vector notation to combine vectors by addition (20.4)

• Represent vectors, and combinations of vectors, in the plane (20.1 – 20.6)

• Solve geometrical problems in 2-D, eg show that joining the mid points

of the sides of any quadrilateral forms a parallelogram (20.6)

Illustrate use of vectors by showing ‘Crossing the flowing River’ example or navigation examples

Vector problems in 3-D (for the most able)

Use i and j (and k) notation (preparation for A-Level but not in GCSE specification)

Lengths are sometimes given in the ratio of 1:3. It is important to explain this concept carefully

The geometry of a hexagon provides a rich source of parallel, reverse and vectors with double the magnitude

Stress that parallel vectors are equal to one another

Link with ‘like terms’ and ‘brackets’ when simplifying

Show there is more than one route round a geometric shape, but that the answer simplifies to the same vector

Remind students to underline vectors or they will be regarded as just lengths with no direction

Extension material can be found in M1 texts

Convert measurements from one unit to another (area and volume)

Speed calculations

Experience of multiply by powers of 10, eg 100 ´ 100 = 10 000

Substitution and changing the subject

• Use the relationship between distance, speed and time to solve problems (revision of Unit 2) (Revision of

prior knowledge)

• Know that density is found by mass ÷ volume (13.6)

• Use the relationship between density, mass and volume to solve problems (13.6)

• Convert between metric units of density, eg kg/m to g/cm (13.7)

• Convert between area measures, eg 1 m2 = 100 00 cm2 (12.4)

• Convert between volume measures (using metric units) (13.5)

• Convert between volume and capacity measures,

eg 1 ml = 1 cm³ (1cc) (13.5)

Convert imperial units to metric units, eg mph into km/h

Mention other units (not on course) like hectares

Borrow a set of electronic scales and a Eureka Can from Physics for a practical density lesson

Look up densities of different elements from the internet

Link converting area and volume units to similar shapes

Draw a large grid made up of 100 by 100 cm squares to show what 1 square metre looks like

Use of Calculators (Spec Ref:- Nv)

Students are well advised to regularly bring and use their particular calculator to lessons and get used to its functions throughout this Unit 3 course.

They should be confident in entering a range of calculations including those involving time and money.

They should realise that 2.33333333…. hrs is 2 hours 20 minutes etc