Download Year 10 and 11 - Higher GCSE - Hertfordshire Grid for Learning

Year 9
SUMMER TERM B
Topic:
TOPIC 1
Target Grade: C/B/A
Exploring Numbers
Edexcel Content:
NA2b: Recognise triangular, square and cube numbers
NA3g: Recalling integer squares and corresponding
square roots to 15 x 15
NA3g: Recalling the cubes 2, 3, 4, 5 and 10
NA2a: Finding multiples, factors, primes and prime
factors
NA2a: Finding prime factor composition of positive
integers
NA2a: Using prime factors to find HCFs and LCMs
NA2b: Using index notation
NA5d: Using indices in expressions
NA2b: Using index laws for multiplication and division
(integer powers)
NA3a: Fractional and negative powers
NA3g: Recognising the relationships between fractional
powers and roots
NA3g: Recalling the zero power
Prior Knowledge:
Basic number bonds and multiplication/division facts.
Experience of using powers of numbers.
Learning Objectives:
Recognise the different types of numbers, and find multiples and factors
Write numbers in terms of their prime factors and use prime factors to find the HCF, and
LCM
Calculate squares and cubes, and recall the relevant facts
Find square and cube roots of numbers
Know and use the rules of indices (adding, subtracting and multiplying indices)
Evaluate fractional and negative indices
Evaluate integer, fractional negative and zero powers with or without a calculator
Understand the term reciprocal, and use this in calculations involving powers
Understand the concept of a root being an irrational number, and leave answers to
problems in surd form
Differentiation & Extension:
Trial and improvement for roots of integers.
Estimating answers of complex calculations involving surds, or powers.
Simplify expressions involving complex indices.
Notes:
All of the work in this chapter can be reinforced by starter and end activities.
London Reference:
Chapter 1 p.1-17
Other references:
Ray (H) p.179
Discussion opportunities:
Why do the rules for indices work? What is the best way to find the HCF and LCM?
Pair / Group Work:
Factors/multiples board game could be used to introduce the topic.
ICT Links:
Powerpoint presentations on laptop
Investigation:
Rules for indices can be approached investigatively.
Spiritual, Moral and Citizenship Links:
Numbers have properties – what properties do you have?
Time:
5 lessons
SUMMER TERM B
Topic:
Brackets in Algebra
Edexcel Content:
NA5b: Collecting like terms
NA5b: Removing a single pair of brackets
NA5b: Factorising with a single pair of brackets
NA5e: Using inverse operations to solve equations
NA5f: Linear equations with integer or fractional
coefficients
NA5e: Equations combining operations
TOPIC 2
Target Grade: C/B/A
NA5f: Solving equations with the unknown on both
sides
NA5f: Solving equations using brackets and negative
solutions
NA5e: Using algebraic equations to solve problems
NA6a: Use inverse operations to rearrange formulae
NA5j: Solving linear inequalities in one variable
NA5b: Expanding brackets – the product of two linear
expressions
Prior Knowledge:
Topic 1. Experience of solving simple linear equations.
Learning Objectives:
Simplify algebra by collecting like terms – answers may involve negative coefficients
Remove and factorise a single pair of brackets – including cases where variables are removed as
part of the factor
Solve linear equations including those with an unknown on both sides, those that require prior
simplification (e.g. brackets), fractional equations, and those where the answers are either negative or a
fraction
Derive algebraic expressions from information given and extend this to derive equations, solving
problems
Solve linear inequalities through both algebraic methods and listing possible integer values
Rearrange formulae, including those where the potential subject occurs more than once
Solve by substitution a pair of simultaneous equations
Expand and simplify two pairs of linear brackets, e.g. (x + 2)(x – 4), (3x + 2y)(4x + y), (x - p)(a
+ g) etc
Expand the square of a linear expression
Make efficient use of techniques covering signs, products and sums
Differentiation & Extension:
Derive equations from practical situations (such as angle calculations)
Expanding triple brackets. The binomial expansion.
Notes:
Do both chapters together. Chapter 2 is very straightforward; you could start on chapter 10.
Answers can be left in fractional form where appropriate.
Students need to realise that not all linear equations can easily be solved by observation or trial
and improvement, so use of a formal method is vital.
London Reference:
Other references:
Chapter 2 p.18-32
Ray (H) p.58, 78-87
Chapter 10 p.200-214
Discussion opportunities:
Which is the best method for expanding double brackets (FOIL, smiley face etc)? Show various
methods and let the students decide.
Pair / Group Work:
Equation loop cards could be played as a class.
Domino/Bingo/Snap games – match expressions with their factorised equivalent
ICT Links:
Algebra Foundations/ Algebra Tutor (Outware Education) on department laptop.
Autograph or graphics calculators could be used to show that, for example, x2 – 5x + 6 is
equivalent to (x – 6)(x + 1) or www.waldomaths.com.
Investigation:
Beyond Pythagoras involves a lot of rearranging equations, as does Opposite Corners and T-Total
Spiritual, Moral and Citizenship Links:
Why do we try to make expressions as simple as possible? Inequality in the world.
Time:
8 - 9 lessons
SUMMER TERM B
Topic:
TOPIC 3
Target Grade: C/B
Shapes
Edexcel Content:
SSM2d:
SSM2k:
SSM2e:
SSM2e:
SSM3d:
Angles in polygons
Nets of simple solids
Understand similarity and congruence
Prove the congruence of triangles using formal arguments
Enlargement calculations
Prior Knowledge:
It will help if the students already know the names and properties of 2-D and 3-D shapes and
simple angle facts.
Learning Objectives:
Solve problems involving interior/exterior angles of polygons (regular and irregular), and
understand the concept and limitations of tessellation
Construct nets of simple solids
Understand the names and properties of 3-D shapes
Calculate the missing sides of triangles and other shapes (e.g. regular polygons) using
similarity
Prove the congruence of triangles using formal arguments*
Use scale factors to solve problems involving similar shapes
Differentiation & Extension:
Attempt to draw up to 3 shapes each which have exactly 1, 2, 3, …8 lines of symmetry.
Find all possible nets of a cube.
Notes:
*19b – add more formal argument for congruency.
Most of this chapter is revision. Concentrate mainly on congruence and similarity.
London Reference:
Chapter 3 p.33-62
Other references:
Ray (H) p.119
Discussion opportunities:
“What is the largest angle you can have in a triangle?” – this should set off a lively debate.
Discuss reasons for congruency.
Pair / Group Work:
Mental imaging – in pairs the students must describe a shape for their partner to draw.
ICT Links:
Cabri Geometry 2 Plus
Investigation:
Polygons could be approached investigatively.
Spiritual, Moral and Citizenship Links:
People can be described as 2-Dimensional or 3-Dimensional.
Time:
5 lessons
AUTUMN TERM A
TOPIC 4
Topic: Using Basic Number Skills
Target Grade: C/B/A/A*
Edexcel Content:
NA3a: Multiplying by a number between 0 and 1
NA3e: Understanding the multiplicative nature of
percentages as operators
NA3e: Finding 100% when another amount is known
NA3e: Solving percentage problems
NA3e: Solving reverse percentage problems
NA3k: Solving problems involving compound interest
NA2f: Simplifying ratios
NA2f: Relating ratio form to fractions
NA3f: Dividing in a given ratio
NA3n: Unitary method
NA2b: Using standard index form
NA3h: Converting between ordinary and standard index
form representations
NA3h: Using standard index form to make estimates
NA3m: Calculating with standard index form
NA3r: Using a calculator for standard index form
NA3e: Interchanging between percentages, fractions and
decimals
NA3j: Finding percentages and percentage changes
NA3j: Finding VAT, a percentage profit or loss
NA3j: Using simple interest
Prior Knowledge:
Multiplying and dividing by powers of 10. Understanding of indices and percentages.
Learning Objectives:
Recognise that some numbers are too large or too small to be represented normally on a calculator
Represent standard form as a number between 1 and 10 multiplied by a positive or negative power of ten
Convert between standard form and ‘normal’ numbers
Solve problems involving standard form, using the correct calculator method and making estimates
Interpret a calculator display showing a number in standard form
Change between percentages, fractions and decimals
Find percentages of quantities using both mental mathematics and calculator method, and solve percentage
problems
Increase and decrease quantities by a percentage, including within contexts of VAT, profit and loss
Find one quantity as a percentage of another, and calculate the percentage when an actual profit or loss is given
Calculate simple and compound interest
Solve problems using percentages e.g. taxation, bills
Recognise that an increase of e.g. 15% leads to 115% and decrease of e.g. 15% leads to 85%
Find the original amount e.g. price before a sale, price before VAT
Use multipliers to solve reverse percentage and compound interest problems
Recognise a ratio as a way of showing the relationship between two numbers
Relate a ratio to a fraction
Simplify a ratio by dividing both its numbers by a common factor and recognise when it is in its lowest terms
Use the unitary method as a way of solving ratio and proportion problems (e.g. recipes)
Divide a quantity into a given ratio (in two or three parts)
Differentiation & Extension:
Calculate the original price before compound interest. Combine multipliers to simplify a series of percentage changes.
Comparisons between simple and compound interest calculations (e.g. real-life contexts).
Notes:
Amounts of money should be rounded to the nearest penny (but only at the end of the question).
Working out should always be shown.
London Reference:
Chapter 5 p.95-123
Discussion opportunities:
Other references:
Ray (H) p.7, 15,
Plenty of opportunity with percentages, especially when dealing with real-life situations. Why use standard form?
Pair / Group Work:
Domino/Bingo/Snap games – match equivalent fractions, decimals and percentages.
ICT Links:
Use of Excel for budgeting.
How do scientific/graphics calculators represent standard form?
Distance of planets from sun can be looked up on the Internet.
Investigation:
VAT, credit, interest, and mortgages – the students can research all these applications of percentages.
Spiritual, Moral and Citizenship Links:
How much should we be taxed? Facts relating to the third world often involve percentages.
Explain how the economy functions, including the role of business and financial services.
What percentage of 18 year olds voted in the last general election? Discuss.
Time:
10 lessons
AUTUMN TERM A
TOPIC 5
Topic: Transformations and Loci
Target Grade: C/B/A
Edexcel Content:
SSM2a: Reflecting 2D shapes
SSM2a: Rotating shapes though various angles and about various centres of rotation
SSM2a: Using translations that are specified by a vector
SSM3c: Enlarging assorted shapes using various centres of enlargement and integer scale factors
SSM3c: Enlarging assorted shapes using non-integer scale factors
SSM3b: Transforming 2-D shapes using a combination of transformations
SSM3b: Recognising properties which are preserved under transformations
SSM4c: Constructing triangles
SSM4c: Constructing a perpendicular bisector and finding the mid-point of a line segment
SSM4c: Construct perpendiculars to a line
SSM4c: Bisecting an angle
SSM4e: Finding Loci
NA6h: Constructing graphs of simple loci
Prior Knowledge:
Plotting co-ordinates, equations of simple straight lines, symmetry.
An ability to use a pair of compasses. Understanding the terms perpendicular, bisecting, parallel.
Learning Objectives:
Reflect shapes and identify equations of mirror-lines, including diagonal lines (y = x, y = -x)
Rotate 2-D shapes following instruction and describe a rotation in full
Recognise translations as sliding movements, and translate simple 2-D shapes within a plane using words or
vector notation
Understand which are the invariant properties of enlargements
Enlarge shapes using a variety of positive, negative, integer and non-integer scale factors
Work on tasks involving these transformations
Use scale to interpret maths and complete scale drawings
Recognise properties that are preserved, and those that are changed under transformations
Recognise combinations of transformations and describe them in full
Construct shapes from given information using only compasses and a ruler
Construct perpendicular bisectors, and angle bisectors using only compasses and a ruler
Construct LOCI in terms of distance from a point, equidistance from two points, distance from a line,
equidistance from two lines and line of sight
Shade regions using LOCI to solve problems e.g. vicinity to lighthouse/port
Differentiation & Extension:
Scale drawing of the classroom. Solve loci problems that require a combination of loci.
Notes:
Emphasis needs to be placed on ensuring that students describe the given transformation fully.
All working should be presented clearly and accurately.
For loci, try to use past exam papers to cover objectives.
London Reference:
Chapter 6 p.124-155
Discussion opportunities:
Other references:
Ray (H) p.96, 232
Mental imaging could be used with loci – discuss what the students imagined.
Pair / Group Work:
Practical work on scale drawing or loci could be done in pairs or groups.
ICT Links:
Geometry Transformations (Outware Education) on department laptop. LOGO
Powerpoint presentation on laptop. Cabri Geometry (see ICT folder).
Translation (see Edexcel Teachers’ guide – ICT)
Investigation:
Investigation into different ways of transforming an object into a particular image.
Loci work can be investigative e.g. find where on a map buried treasure is located given some clues.
Spiritual, Moral and Citizenship Links:
When shapes are transformed, do they still have same properties? We can also transform our personalities – are we
still the same?
Time:
7 lessons
COURSEWORK 1: Choose a task from the Edexcel coursework folder (in the maths staff room)
All groups will do this piece at the same time.
Time: 2 weeks + half term.
AUTUMN TERM B
TOPIC 6
Topic: Lines, Simultaneous Equations
and Regions
Target Grade: C/B/A
Edexcel Content:
NA6b:
NA6c:
NA6c:
NA6c:
NA6b:
NA6b:
NA5j:
Plotting graphs of functions where y is expressed in terms of x, leading to a straight line
Calculating gradients of straight lines, and exploring gradients of parallel lines
Recognising the y-intercept of a straight line
Exploring graphs of the form y = mx + c
Solving simultaneous equations using elimination
Solving simultaneous equations using a graphical method
Solving linear inequalities in two variables and finding the solution set
Prior Knowledge:
The ability to plot points that follow a simple rule (in four quadrants).
The ability to substitute values into algebraic formulae.
Learning Objectives:
Plot a straight-line graph from a given set of values
Realise that an equation of the type y = mx + c represents a straight line graph, and plot
this graph
Understand the relevance of m and c in the above equation
From a given graph, find the gradient and y-intercept and hence the equation of the graph
Draw a straight-line graph without plotting points
Solve linear simultaneous equations by eliminating a variable and by graphical methods,
using them to solve problems
Use regions on a graph to solve inequality problems in two variables
Differentiation & Extension:
Use gradient and intercept to draw lines.
Simultaneous equations that need rearranging before one of the methods can be used.
The most able students could try to solve simultaneous equations in 3 variables.
Notes:
Try to use ICT in this topic, especially graphics calculators and Autograph.
Many pupils find locating regions difficult – it is often useful to choose a particular point on one side of
the line to check if it fits the inequality.
London Reference:
Chapter 7 p.156-174
Other references:
Ray (H) p.156-174
Discussion opportunities:
Real-life situations give opportunity for this, as does the interpretation of y = mx + c.
Discuss which is the best way to solve a simultaneous equation.
Pair / Group Work:
ICT work could be done in pairs (one person investigates a change in m, the other investigates a change in c).
ICT Links:
Graphs and inequalities (see Edexcel Teachers’ guide – ICT)
Using graphical calculators or Autograph to draw lines and solve simultaneous equations (see ICT folder).
www.mathslessons.co.uk - for simultaneous equations (see ICT folder).
Investigation:
Car hire, Mobile Phones (coursework tasks)
y = mx + c can be approached investigatively – see ICT work.
Links with the Science department could yield many experiments that would give rise to straight-line
relationships.
Spiritual, Moral and Citizenship Links:
How can simultaneous equations be used to solve business problems?
Time:
6 – 7 lessons
AUTUMN TERM B
Topic:
Pythagoras’ Theorem
TOPIC 7
Target Grade: C/B/A/A*
Edexcel Content:
SSM2f:
SSM2f:
SSM2f:
SSM2f:
NA6h:
Using Pythagoras’ Theorem to find the hypotenuse
Using Pythagoras’ Theorem to find the shorter sides
Using Pythagoras’ Theorem to solve problems
Calculating lengths of lines on a grid
Constructing the graphs of simple loci, including the circle
Prior Knowledge:
Knowledge of different types of triangle. Using a calculator to find squares and square roots.
Knowledge of simple bearings and rounding off answers.
Learning Objectives:
Identify the hypotenuse of a right-angles triangle
Recall Pythagoras’ theorem
Pick out right-angled triangles from diagrams (e.g. circles, isosceles triangles)
Use Pythagoras’ theorem to find the length of any side of a right angled triangle
Use Pythagoras’ theorem to solve problems such as bearings, areas of triangles, diagonals
of rectangles etc
Express a circle of radius r, and centre (0,0) in algebraic form
Differentiation & Extension:
Further work on applying Pythagoras in three-dimensional problems.
Find Pythagorean triples, looking for a general case.
Proof of why the theorem works (e.g. Perigal’s dissection is straightforward and practical)
Does the theorem still work if you have equilateral triangles or regular pentagons on the sides
(rather than squares).
Find a formula for the area of a right-angled isosceles triangle with hypotenuse x cm.
Investigate how to draw a line of exactly 5 cm
Notes:
Consult GCSE papers for types of questions, depending on the orientation of the triangle and
whether the hypotenuse or shorter side is required.
London Reference:
Other references:
Chapter 8 p.175-187
Ray (H) p.100
Discussion opportunities:
What degree of accuracy is required in answers?
Pair / Group Work:
Investigative work could be in pairs.
ICT Links:
Pythagorean Shoe Laces (see ICT folder, Excel section)
The history of Pythagoras – research on the Internet
Perigal’s dissection is on www.mathsnet.net
Autograph – Drawing Circles (see ICT folder)
Powerpoint presentations on the laptop.
Investigation:
Beyond Pythagoras coursework task (investigating Pythagorean triples)
Spiritual, Moral and Citizenship Links:
Pythagoras’ theorem can be proved from simple axioms – so much in life requires faith rather
than proof (religion, love etc).
What was life like for Pythagoras?
Time:
6 lessons
AUTUMN TERM B
TOPIC 8
Topic: Estimation and Approximation Target Grade: C/B/A
Edexcel Content:
NA2f:
NA2f:
NA2f:
NA2f:
Rounding to the nearest integer, to decimal places and to significant figures
Selecting and justifying appropriate degrees of accuracy
Recognising the limitations on the accuracy of measurement
Checking and estimating answers to problems
Prior Knowledge:
Understanding place value in whole numbers and decimals.
Learning Objectives:
Round any number to a specified accuracy, or justify their own choice of accuracy e.g.
nearest integer, significant figures or decimal places
Calculate upper and lower bounds of measurements or rounded numbers
Use rounding methods to estimate answers to complex expressions
Differentiation & Extension:
Discuss appropriateness of types of rounding in particular contexts.
Real-life estimations.
Notes:
Concentrate on upper and lower bounds.
Only round off at the end of a calculation.
It is essential to ensure that students understand the difference between significant figures and
decimal places and take note of the required degree of accuracy for questions.
London Reference:
Chapter 12 p.255-266
Other references:
Ray (H) p.373-384
Discussion opportunities:
Why do we need to round numbers? Why use significant figures?
Pair / Group Work:
Students can show each other their methods or check each other’s answers; compare estimations.
ICT Links:
Number Foundations (Outware Education on laptop).
Demonstration of Excel facilities for rounding off.
Investigation:
Investigate the difference it makes to an answer if you round off too early.
Spiritual, Moral and Citizenship Links:
Human beings can never be rounded off – every part of us is unique.
Time:
3 - 4 lessons
AUTUMN TERM B
Topic:
Basic Trigonometry
TOPIC 9
Target Grade: C/B/A/A*
Edexcel Content:
SSM2g: Tangent, sine and cosine ratios
SSM2g: Uses of the three ratios
SSM2g: Angles of elevation and depression
SSM2g: Bearings and trigonometry
SSM2g: Drawing, sketching and describing the graphs of trigonometric functions
NA6g: Applying transformations to the graphs of trigonometric functions
Prior Knowledge:
Rounding to decimal places and significant figures.
Ability to change fractions to decimals using a calculator.
Pythagoras’ theorem. Basic concepts of ratio.
Learning Objectives:
Identify appropriately the various sides of a right-angled triangle as the Hypotenuse,
Opposite and Adjacent
Recall the ratios for sine, cosine, and tangent and identify which are required to solve a
problem
Use information given to find angles using the appropriate ratio
Use the appropriate ratio to find the lengths of sides in a right-angles triangle
Find angles of elevation and depression using the appropriate ratio
Apply trigonometric ratios and Pythagoras Theorem to solve assorted problems, including
those involving bearings
Sketch the curves y = sin x, y = cos x and y = tan x
Draw such graphs as y = a + b sin x
Use graphs to aid the solution of equations such as a + b cos x = 1, for angles between 0
and 360 degrees
Differentiation & Extension:
Given two properties of a right-angled triangle find the others.
The study of curves such as y = a + b sin (cx + d)
Notes:
Emphasise the importance that a calculator is in ‘Degree mode’, and that scale drawings will
score 0 marks for this type of question.
The graph work may be done investigatively using graphics calculators or Autograph.
London Reference:
Other references:
Chapter 13 p.267-290
Ray (H) p.122-137
Discussion opportunities:
Plenty of opportunity here, especially when solving problems and initial investigations. You can
also get some interesting mnemonics for SOHCAHTOA.
Where is trigonometry used in real life?
Pair / Group Work:
Investigations could be done in pairs. Practical work (e.g. using a clinometer) could be done in
groups. You could get the students to make their own Powerpoint presentations in groups.
ICT Links:
Autograph or graphical calculators for trigonometric graphs.
There are many Powerpoint presentations on trigonometry (laptop). You could get students to
make their own.
Investigation:
Investigation of simple relationships such as sin (180 – x) = sin x, and sin (90 – x) = cos x can
help. The Fencing Problem requires trigonometry to reach higher levels.
Spiritual, Moral and Citizenship Links:
Awe and wonder
Time:
4 lessons
AUTUMN TERM B
Topic:
Sequences and Formulae
TOPIC 10
Target Grade: C/B/A
Edexcel Content:
NA4a:
NA3a:
NA5g:
NA5g:
NA6a:
NA6a:
NA6a:
NA5a:
NA6a:
Using the order of operations, and the commutative, associative and distributive laws
Four rules with negative numbers
Substituting into algebraic formulae
Generating a formula
Generating common number sequences
Generating number sequences using term-to-term and position-to-term definitions
Finding the nth term (linear expressions)
Use function notation
Use inverse operations to rearrange formulae
Prior Knowledge:
BIDMAS. Negative numbers.
Learning Objectives:
Confidently use the order of operations, and the commutative, associative and distributive
laws for positive and negative numbers
Substitute numbers into any expression or formula
Derive a formula from given information
Find a designated term of a sequence given a pattern or a formula
Find the nth term of a linear expression*
Rearrange formulae, including those where the potential subject occurs more than once
Differentiation & Extension:
Extend the nth term to quadratic sequences. You could bring in suffix notation tn = …and its use
in Fibonacci-type sequences (especially if you are planning to do the Flagging coursework).
See the investigations below for more ideas.
Estimate answers before attempting complex substitutions.
Notes:
*Giving the nth term of a quadratic sequence as an algebraic expression is not in the syllabus but
is useful in many investigations.
London Reference:
Other references:
Chapter 14 p.291-304
Ray (H) p.48
Discussion opportunities:
Why are the second differences the same in quadratic sequences?
Pair / Group Work:
Some pattern spotting could be done in pairs. Students could make up sequences for a partner to
find the rule.
ICT Links:
Graphics calculators and Excel can be used to generate quadratic sequences (see ICT folder and
Edexcel Teachers’ guide – ICT).
Powerpoint – students put each number in their own sequence on a different slide (see ICT
folder).
Investigation:
You could bring in some short investigations which lead to simple number sequences – see
Rayner p.48-51. Lots of investigations give rise to quadratic sequences. Such as Bad Tomatoes,
Borders, Lines, Regions and Crossovers.
Flagging and The Pay Phone Problem involve recurrence sequences.
Spiritual, Moral and Citizenship Links:
Fibonacci’s “Golden Ratio”.
The numbers in sequences all follow the same rule – this is an excellent example of teamwork
Can population growth be (simplistically) modelled using quadratic expressions?.
Time:
5 lessons
SPRING TERM A
TOPIC 8
Topic: The Collection of Data
Target Grade: C/B/A
Edexcel Statistics Content:
1(c) use stratified sampling
1(c) understand the concept of bias
1(c) understand and use systematic, quota and cluster
sampling
1(d) collect data by observation, surveys, experiments,
convenience sampling, questionnaires and measurements
1(d) understand the effects of accuracy on
measurements
1(d) understand the rationale behind pilots and pre-tests
1(d) the uses of open and closed questions
1(d) understand the meaning of explanatory and
response variables
1(a) specify a hypothesis to be tested
1(a) select an appropriate method of obtaining data and
justify the choice
1(b) recognise the difference between quantitative and
qualitative variables
1(b) recognise the difference between discrete and
continuous data
1(b) use class boundaries
1(b) understand the meaning of bivariate data
1(b) understand grouped and ungrouped data
1(c) understand the reasons for sampling
1(c) understand the terms random, randomness and
random sample
1(c) generate and use random numbers
1(c) design and use a sampling frame
Prior Knowledge:
An understanding of why data needs to be collected.
Learning Objectives:
Recognise the difference between quantitative and qualitative variables
Recognise the difference between discrete and continuous data
Understand the meaning of bivariate data and primary/secondary data
Use experiments and surveys to collect data
Understand the advantages and disadvantages of using a sampling technique and a census
Understand the different methods of sampling, including stratified random sampling
Use and analyse questionnaires (open/closed questions, pilot surveys and pre-tests)
Understand the effects of accuracy on measurements
Differentiation & Extension:
The last section on accuracy has been taught in the previous topic on Estimation and Accuracy
and so can be left out or revised briefly.
Notes:
Pre-tests and stratified random sampling are particularly important in the Data Handling
coursework. Simulations such as the rolling of a die can be obtained using the RAN# key.
Edexcel Statistics Reference:
Other references:
Chapter 2 p.3-28
Discussion opportunities:
What makes a good questionnaire? Discuss bias in questionnaires.
Pair / Group Work:
Questionnaires can be done in pairs and then presented to a group or the whole class for
constructive criticism.
ICT Links:
Excel – collect data in tables and draw different types of graphs.
www.censusatschool.co.uk - provides interesting raw data to take samples from and analyse
Investigation:
Carry out a statistical investigation of their own including; designing an appropriate means of
gathering the data.
Spiritual, Moral and Citizenship Links:
How do opinion polls use sampling to predict e.g. election results?
Understand the problems of ambiguity and bias, for example when emotions, finance, politics or
criminal activity are involved.
Time:
5 lessons
SPRING TERM A
Topic: Representing and Processing
Discrete Data
TOPIC 9
Target Grade: C/B/A
Edexcel Statistics Content:
2(a)
2(a)
2(a)
2(a)
2(a)
2(b)
2(b)
2(b)
2(b)
construct frequency tables by tallying raw data
use class intervals, including open ended classes and classes of varying width
use various forms of grouping data
read and interpret data presented in tabular or graphical form
design suitable tables, including summary tables and two-way tables
use and interpret pictograms, bar charts and pie charts for qualitative, quantitative and discrete data
vertical line graphs for discrete data
stem and leaf diagrams
comparative pie charts with area proportional to frequency
Prior Knowledge:
Measuring and drawing angles, fractions of quantities, co-ordinates, ratio.
Learning Objectives: For discrete data…
Construct frequency tables by tallying raw data
Group data into classes, including classes of varying width and open-ended classes
Organise data into a two-way table
Use spreadsheet formulae to construct two-way tables
Use databases and summary tables
Draw and interpret pictograms and understand their advantages and disadvantages
Draw and interpret bar charts and vertical line graphs
Use multiple and compound bar charts to make comparisons
Use ICT to draw bar charts
Draw and interpret pie charts
Use ICT to draw pie charts
Use comparative pie charts with area proportional to frequency to compare two sets of
data of different sizes
Draw and interpret stem and leaf diagrams
Differentiation & Extension:
Comparative pie charts are quite tricky and will stretch the more able.
Notes:
The reasons for choosing one form of representation will be expected. An awareness of problems
associated with creating categories that are too broad, too narrow or redundant. Poorly presented
data can be misleading e.g. 3-D angled pie charts and 3-D pie charts with slices pulled out.
Edexcel Statistics Reference:
Other references:
Chapter 3 p.29-77
Discussion opportunities:
What are the advantages and disadvantages of the different forms of representation?
Pair / Group Work:
Collect real-life data and process it.
ICT Links:
www.censusatschool.co.uk - provides interesting raw data to take samples from and draw graphs.
Excel (see ICT folder).
Investigation:
e.g. do students perform better at GCSE French or Italian? See Ex 3M Q1
Spiritual, Moral and Citizenship Links:
Consider the uses of statistics in politics. How the same data can be interpreted differently,
depending on politician’s agenda.
Time:
4 - 5 lessons
SPRING TERM A
TOPIC 10
Topic: Representing and Processing
Continuous Data
Target Grade: C/B/A/A*
Edexcel Statistics Content:
2(a)
2(b)
2(b)
2(b)
2(b)
2(b)
2(b)
construct frequency tables by tallying raw data
grouped frequency diagrams (histograms) with equal and unequal class intervals
frequency polygons
stem and leaf diagrams
population pyramids
using line graphs to predict trends
chloropleth maps (shading)
Prior Knowledge:
Inequalities, rounding numbers.
Learning Objectives: For continuous data…
Sort data into a frequency table using inequalities to define the class boundaries
Sort rounded data into a frequency table
Draw histograms with equal class intervals
Draw histograms with unequal class intervals
Analyse histograms with unequal class intervals
Draw and interpret stem and leaf diagrams
Display information on a population pyramid
Use line graphs to predict trends
Use chloropleth maps to classify regions
Differentiation & Extension:
Draw and shade a chloropleth map of the school population at lunchtime.
Notes:
Emphasise the continuous scale is used on the horizontal axis. Also emphasise that the areas of
the rectangles in a histogram are equal to the frequencies.
Edexcel Statistics Reference:
Other references:
Chapter 4 p.79-108
Discussion opportunities:
All of this chapter lends itself well to discussion work, particularly the uses of population
pyramids and chloropleth maps.
What are the differences between a bar chart and a histogram?
Pair / Group Work:
Practical work e.g. draw and shade a chloropleth map of the school population at lunchtime.
See investigations below.
ICT Links:
Autograph – especially useful for histograms.
www.censusatschool.co.uk - provides interesting raw data to take samples from and draw graphs.
Microsoft Excel (see ICT folder).
Investigation:
Investigate the uses of population pyramids and chloropleth maps in Geography.
“Gary’s car sales” coursework
Spiritual, Moral and Citizenship Links:
Population pyramids can be used to discuss the third world, see Ex 4G Q2.
Time:
6 - 7 lessons
SPRING TERM A
TOPIC 11
Topic: Summarising Data: Measures Target Grade: C/B/A/A*
of Central Tendency and Dispersion
Edexcel Statistics Content:
2(c) work out the mean, mode and median from a list and a frequency distribution
2(c) identify the modal class interval for grouped frequency distributions
2(c) work out and use estimates for the mean and median of grouped frequency distributions
2(c) make a reasoned choice of a measure of central tendency
2(d) work out the range, quartiles, percentiles, deciles and interquartile range. Identify outliers.
2(d) construct, interpret and use box plots
2(d) calculate and use variance and standard deviation
2(d) understand the advantages and disadvantages of each of the measures of dispersion
2(d) & 3 compare distributions of data using an appropriate measure of both central tendency and dispersion
2(d) calculate, interpret and use standardised scores
2(e) simple index numbers, chain base numbers, weighted index numbers, Retail Price Index
Prior Knowledge:
An understanding of the concept of an average. Experience of reading from graphs.
Learning Objectives:
Find the mode, median, and mean of a set of data
Find the mode, median, and mean from a frequency table
Find the modal class, and estimate the median and mean from a grouped frequency table or cumulative
frequency diagram
Understand the effects of transformations of the data on the mean, mode and median
Understand the advantages and disadvantages of the mode, median and mean
Calculate the weighted mean and the geometric mean
Use simple index numbers, chain base index numbers, weighted index numbers and the retail price index
Work out the range, quartiles, percentiles and interquartile range from frequency tables and cumulative
frequency diagrams
Construct, interpret and use box plots
Understand how to discover errors in data and formally identify outliers
Use a cumulative frequency polygon
Calculate and understand variance and standard deviation
Compare values from different data sets using standardised scores
Differentiation & Extension:
Investigate the effect, if any, of transformations of the data on measures of dispersion.
Notes:
Students must be aware that a full comparison needs at least both a measure (or measures) of central tendency and of
dispersion (important for coursework). Transformations will be restricted to those of the type x  ax + b.
 notation will be expected. Cumulative frequency – emphasise the use of the upper boundary and that the median
and quartiles are read of as values from the non-cumulative frequency axis.
Edexcel Statistics Reference:
Chapter 5 p.109-144
Discussion opportunities:
Other references:
Why are there 3 types of average? Discuss occasions when one average is more appropriate, and the limitations of
each average.
Pair / Group Work:
Collect data from class – children per family etc
ICT Links:
Autograph calculates averages and range very quickly. Also easy to draw and compare cumulative frequency curves.
Excel also calculates averages.
www.censusatschool.co.uk - many good worksheets on averages and has relevant raw data.
Investigation:
Investigate what effects, if any, to (i) the median, (ii) the interquartile range if you (a) add 10, (b) subtract 10, (c)
multiply by 10, (d) divide by 10, to all the data.
Hypothesis testing e.g. Girls get higher effort grades than boys
Spiritual, Moral and Citizenship Links:
Is there an average person in Year 10?
No one is average – everyone is unique!
Time:
7 - 8 lessons
SPRING TERM B
Topic: Scatter Diagrams and
Correlation
TOPIC 12
Target Grade: C/B/A/A*
Edexcel Statistics Content:
2(f)
2(f)
2(f)
2(f)
2(f)
2(f)
2(f)
2(f)
2(f)
3
plot data as points on a scatter diagram
recognise positive, negative and zero correlation by eye
understand the distinction between correlation, causality and a non-linear relationship
fit a line of best fit through the mean point to the points on a scatter diagram, by eye may be required
find the equation of a line of best fit in the form y = ax + b and interpret a and b
understand the pitfalls of interpolation and extrapolation
interpret data presented in the form of a scatter diagram
fit non-linear models of the form y = axn + b and y = kan
calculate and interpret Spearman’s rank correlation coefficient
interpret correlation as a measure of the strength of the association between two variables
Prior Knowledge:
Co-ordinates. An understanding of the concept of a variable.
Learning Objectives:
Use scatter diagrams to show whether two sets of data are related
Recognise positive, negative and zero correlation by eye
Understand the distinction between correlation, causality and a non-linear relationship
Draw a line of best fit on a scatter diagram using the mean point
Use a line of best fit for interpolation and extrapolation (and understand the pitfalls)
Find the equation of a line of best fit in the form y = ax + b and interpret a and b
Use ICT to plot scatter diagrams and lines of best fit
Fit a line of best fit to a non-linear model of the form y = axn + b and y = kan
Calculate and interpret Spearman’s rank correlation coefficient
Differentiation & Extension:
Questions on tied ranks, see p.179 (they will not be in the exam, but may occur in coursework)
Notes:
The labelling and scaling of axes may be required. Terms such as strong or weak will be
expected. The points lying on the circumference of a circle are related but show zero correlation.
Beware the use of correlation in small samples.
Edexcel Statistics Reference:
Other references:
Chapter 5 p.147-182
Discussion opportunities:
Discuss correlation. How do we know where to put the line of best fit?
Pair / Group Work:
“Compatibility Test” – list the 8 things you look for in a partner (in order) and see how well
correlated you are to others in the class by working out Spearman’s rank correlation coefficient.
Practical work e.g. do tall people also have large hand spans?
ICT Links:
Autograph – especially useful for scatter graphs (many good worksheets)
www.censusatschool.co.uk - provides interesting raw data to take samples from and draw graphs.
StatsAid (Outware Education) on department laptop. Excel (see p.167).
Investigation:
See Pair / Group work above
Spiritual, Moral and Citizenship Links:
Is there any correlation between wealth and happiness?
The “Compatibility Test” throws up a few moral dilemmas.
Time:
6 – 7 lessons
SPRING TERM B
Topic: Time Series
TOPIC 13
Target Grade: C/B/A/A*
Edexcel Statistics Content:
2(g)
2(g)
2(g)
2(g)
2(g)
2(h)
plot points as a time series; draw a trend line by eye and use it to make a prediction
calculate and use moving averages
identify and discuss the significance of seasonal variation by inspection of time series graphs
establish a trend line, with its equation, based on moving averages
recognise seasonal effect at a given data point and average seasonal effect
plot sample means, median and ranges over time
Prior Knowledge:
Averages, co-ordinates, equation of a straight line.
Learning Objectives:
Draw and use line graphs
Plot points as a time series
Draw a trend line by eye and use it to make a prediction
Identify and discuss the significance of seasonal variation by inspection of time series
graphs
Use ICT to draw line graphs
Calculate and use appropriate moving averages* and draw a trend line through them
Use ICT to calculate moving averages
Recognise seasonal effect at a given data point and average seasonal effect
Make predictions using a trend line and the estimated mean seasonal variations
Calculate the equation of a trend line
Understand quality control by plotting sample means, median and ranges over time**
Differentiation & Extension:
Calculate moving averages and trend lines for real life time series such as world records for the
mile.
Notes:
*up to and including a seven-point moving average.
**to view consistency and accuracy against a target value in cases where a process is off-target.
Edexcel Statistics Reference:
Other references:
Chapter 7 p.184-216
Discussion opportunities:
Discuss interpretations of seasonal effect and average seasonal effect.
Pair / Group Work:
Perform quality control experiments for real life products e.g. weights of packets of crisps,
amount of coke in a can.
ICT Links:
Excel for calculating moving averages.
Autograph for plotting time series and trend lines.
Investigation:
Students could design their own quality control experiments.
“The World Record for the Mile” investigates how time series can be used to predict future
records.
Spiritual, Moral and Citizenship Links:
Why do we need quality assurance?
Time:
6 - 7 lessons
COURSEWORK 2 (Data Handling): Newspapers
Use Chapter 10 in the textbook to introduce the data handling project
Time: 2 weeks (Hand in before Easter).
SUMMER TERM A
Topic: Probability
TOPIC 14
Target Grade: C/B/A
Edexcel Statistics Content:
4
4
4
4
4
4
4
4
4
4
4
4
understand the words event and outcome
put probabilities in order on a probability scale
understand the terms ‘random’ and ‘’equally likely’
the relationship between ‘odds’ and probability
use limiting frequency to measure probability
use simulation to estimate more complex probabilities
use probability to assess risk
produce, understand and use a sample space
understand and use Venn diagrams
understand terms mutually exclusive and exhaustive and use the addition law
use the rules p = 1 and P(not A) = 1 – P(A) for mutually exclusive, exhaustive events
form and use tree diagrams for both independent events and conditional cases
Prior Knowledge:
Writing probabilities as fractions, decimals or percentages. Manipulating fractions and decimals.
Learning Objectives:
Understand the probability scale and find the properties of simple events
Understand the meaning of the words event and outcome
Use experiments and simulation to estimate probabilities
Use ICT to simulate experiments
Use probability to assess risk
Produce, understand and use a sample space
Understand and use Venn diagrams
Understand terms mutually exclusive and exhaustive and use the addition law
Use the rules p = 1 and P(not A) = 1 – P(A) for mutually exclusive, exhaustive events
Form and use tree diagrams for both independent events and conditional cases
Differentiation & Extension:
Some students will find it hard to draw tree diagrams – initial questions could have the diagrams
already drawn.
Calculate the probability of winning the National Lottery.
Notes:
Students can often lose marks at probability due to inability to manipulate fractions.
The relationship between p = 1 and ‘odds’ may be tested.
Tree diagrams can include with and without replacement for up to three outcomes and three sets
of branches.
Edexcel Statistics Reference:
Other references:
Chapter 8 p.256-271
Discussion opportunities:
Deciding if events are mutually exclusive or independent.
Pair / Group Work:
Make predictions of outcomes for probability games and then test the predictions.
ICT Links:
See the Probability section in the ICT folder.
Powerpoint presentation on laptop.
Investigation:
Investigate the use of risk assessment in insurance.
Spiritual, Moral and Citizenship Links:
National Lottery – is it moral to gamble if the money raised goes to charity?
Time:
6 lessons
SUMMER TERM A
TOPIC 15
Topic: Probability Distributions
Target Grade: C/B/A/A*
Edexcel Statistics Content:
4 use simple cases of the binomial and discrete uniform distribution
4 the shape and simple properties of the normal distribution
Prior Knowledge:
Expanding brackets, percentages.
Learning Objectives:
Understand probability distributions
Use simple cases of the discrete uniform distribution and the binomial distribution
Know and use the shape and simple properties of the normal distribution
Work out the standard deviation and variance of a normal distribution
Differentiation & Extension:
Investigate how to use the binomial expansion to expand (p + q)n for n > 2.
Use probability distribution tables. Positive and negative skew.
Notes:
The expansion of (p + q)2 will be expected. In all other cases the expansion of (p + q)n will be
given. Use of the normal distributions to model some populations. Use of Normal distribution
tables will not be required.
Edexcel Statistics Reference:
Other references:
Chapter 9 p.256-271
Discussion opportunities:
Discuss which real live data can be modelled by a normal distribution, see Ex 9D Q1.
Pair / Group Work:
Find out if the heights of students in the class can be modelled by a normal distribution.
ICT Links:
www.censusatschool.co.uk - provides interesting raw data to take samples from.
Investigation:
Investigate how Pascal’s triangle relates to binomial expansions.
Spiritual, Moral and Citizenship Links:
The fact Pascal’s triangle helps use calculate probabilities is quite amazing.
Time:
5 - 6 lessons
REVISION FOR GCSE STATISTICS EXAM
Use past papers and specimen papers to revise for the exam.
The exam practice papers in the textbook are very good.
YEAR 10 EXAMS
There is no mathematics exam for the Higher sets. The students will be on study leave for about a week.
SUMMER TERM B
TOPIC 19
Topic: Measure and Mensuration
Target Grade: C/B/A/A*
Edexcel Content:
SSM4d: Finding volume of prisms
SSM2h: Recalling terms relating to a circle
SSM4d: Calculating circumferences
SSM4d: Calculating areas of circles
NA3n: Using pi in exact calculations
SSM4d: Finding areas of plane shapes using formulae
SSM4d: Finding surface area of solids with triangular
and rectangular faces
SSM4d: Developing, knowing and using the formula
for the volume of a cuboid
SSM4d: Finding volume of solids made from cuboids
Prior Knowledge:
Knowledge of basic circle vocabulary. Perimeter and area of triangles and rectangles.
Learning Objectives:
Find the perimeter and area of simple shapes, such as rectangles, squares, triangles,
parallelograms, trapezia, kites and composites of rectangles and triangles
Know the formulae for area and volume of the shapes mentioned
Work confidently with 3-D shapes and be able to calculate the volume of cuboids, prisms,
pyramids, cones, spheres, and solids made from cuboids
Calculate the surface area of solids with triangular and rectangular faces
Find how many boxes of a given size fit into a larger box
Use the vocabulary of a circle (circumference, radius, diameter, sector, segment, chord
and tangent)
Recall and apply the formulae for the area and circumference of a circle given either the
radius or diameter, using various approximations of  or leaving  as part of an irrational
answer
Solve problems involving surface areas of prisms, cylinders, pyramids, and spheres
Solve problems involving volumes of prisms, cylinders, pyramids, cones and spheres
Recognise whether a formula represents a length, area or volume by considering its
dimensions
Differentiation & Extension:
Proofs of area formulae e.g. trapezium.
Notes:
Make sure the students know which formulae are given in the examination formulae sheet.
Circles could be on non-calculator paper – leave answers in terms of .
 can be 3 or 3.14 or 22/7 depending on accuracy or style of answer required.
London Reference:
Other references:
Chapter 16 p.335-348
Ray (H) p.104
Discussion opportunities:
Dimensional analysis – debate whether a formula represents a length, area, volume or none of
these.
Pair / Group Work:
There is a practical “proof” for the area of a circle, which could be done in pairs.
Domino/Bingo/Snap games – match formulae for length, area, volume or none of these.
ICT Links:
Solving problems involving volumes of cylinders (see ICT folder, Excel section).
Research the history of  on the Internet. Powerpoint presentations on laptop.
Investigation:
The Fencing Problem. The Carpet.
Spiritual, Moral and Citizenship Links:
 is a special number and was even mentioned in the bible.
Time:
6 lessons
SUMMER TERM B
Topic:
TOPIC 20
Target Grade: B/A
Proportion
Edexcel Content:
NA3l & NA5h: Using direct proportion
NA3l & NA5h: Using inverse proportion
NA5h: Setting up equations involving proportion
NA5h: Graphical representations of equations involving proportion
Prior Knowledge:
Topics 1 and 2, fractions, calculator skills.
Learning Objectives:
Recognise when two variables are in direct proportion or inverse proportion
Use the unitary method as a way of solving ratio and proportion problems
Define inverse or direct proportion in terms of a formula, finding the constant using given
information
Differentiation & Extension:
Use of three variables in proportion questions e.g. y is directly proportional to the square of x and
x is directly proportional to the cube of z; write a statement to show the proportionality between y
and z.
Notes:
Emphasise the importance of finding the constant k.
London Reference:
Chapter 17 p.349-367
Other references:
Ray (H) p.150
Discussion opportunities:
Deciding whether two quantities are in proportion.
Pair / Group Work:
See computer work and investigation below
ICT Links:
Autograph or graphics calculators - Graphs could be used to check if two quantities are
proportional.
Investigation:
Investigation based on the cylinder e.g. A closed cylinder, half filled with water, is placed on a
flat surface. Mark the position of the water level. Repeat when the cylinder is a quarter full.
Generalise for any fraction.
Spiritual, Moral and Citizenship Links:
Is a human’s health inversely proportional to his/her age? Is intelligence directly proportional to
age?
Proportional representation in government.
Time:
6 – 7 lessons
Year 11
AUTUMN TERM A
Topic:
Graphs and Higher Order
Equations
TOPIC 21
Target Grade: C/B
Edexcel Content:
NA5m:
NA6d:
NA6d:
NA6e:
NA6f:
NA6f:
NA6e:
NA6f:
Using trial and improvement to solve non-linear equations
Plotting linear graphs from real-life problems
Interpret graphs representing real-life problems
Plotting the graph of a quadratic function
Plotting graphs of cubic, reciprocal and exponential functions
Recognising characteristics of graphs
Finding approximate solutions to quadratics using graphs
Finding approximate solutions to problems using graphs of complex functions
Prior Knowledge:
The ability to substitute positive and negative values into a non-linear formula.
Learning Objectives:
Find square and cube roots of numbers including decimals, and solve non-linear equations using trial and
improvement
Interpret travel graphs, and calculate with speed, distance and time (including decimal divisions of an hour)
Understand compound measures such as density or rate of flow, and interpret this from a graph
Evaluate exponential functions
Plot curves from given quadratic, cubic, reciprocal and exponential functions
Interpret and plot real-life graphs such as conversion graphs and distance/time graphs
Recognise graphs e.g. filling different shaped containers
Solve quadratic and cubic equations using a given graph or where one has to be drawn
Use terms like ‘minimum point’, ‘maximum point’, ‘quadratic function’
Use graphical methods to find the maximum or minimum of a quadratic function
Solve problems using cubic or exponential graphs
Differentiation & Extension:
Investigate how the graph of a cubic equation can be used to solve it.
Notes:
This is an excellent opportunity to use ICT (see below).
London Reference:
Other references:
Chapter 18 p.368-383
Ray (H) p.162
Discussion opportunities:
Real-life situations give opportunity for this.
How can you tell which graph will match which equation?
Pair / Group Work:
Travel graphs can be introduced as a group activity (Dicey moves)
Dominoes/Pairs – match graphs to their equations.
ICT Links:
Graphics calculators can be used to draw graphs.
Autograph (see ICT folder)
Excel for trial and improvement (see ICT folder)
The Open Box Problem (see Edexcel Teachers’ guide – ICT)
Investigation:
Drawing graphs may help to analyse the Open Box problem.
Spiritual, Moral and Citizenship Links:
Quadratics can be “happy” or “sad”.
Time:
6 lessons
AUTUMN TERM A
Topic:
TOPIC 22
Advanced Mensuration
Target Grade: B/A/A*
Edexcel Content:
SSM4d:
SSM4d:
SSM2i:
SSM2i:
SSM2i:
SSM3d:
SSM4d:
Calculating lengths of arcs
Calculating areas of sectors
Solving problems involving surface areas of more complex solids
Solving problems involving volumes of more complex solids
Solving problems involving more complex shapes and solids
Understanding the effect of enlargement on areas and volumes
Converting between area and volume measures
Prior Knowledge:
Topics 3 and 19. The ability to substitute positive and negative numbers into formulae.
Learning Objectives:
Find the length of an arc, the area and perimeter of a sector, and the area of a segment
Find the area of compound shapes or shaded areas that include part of a circle
Solve problems involving surface areas of prisms, cylinders, pyramids and spheres
Find the surface area of a cone from a given net
Solve problems involving volumes of prisms , cylinders, pyramids, cones and spheres
Understand the effect an enlargement has on the area and volume of a shape, by
considering scale factors
Convert between units of area and volume e.g. square metres to square centimetres*
Differentiation & Extension:
Find the volume of a can of soup given only it’s label.
Find the area of an annulus, or volume of a prism with a cross-section that is part of a circle.
Use trigonometry calculations to solve problems of volume and surface area.
Notes:
*Add 23f - Convert between units of area and volume e.g. square metres to square centimetres
London Reference:
Chapter 19 p.384-406
Other references:
Ray (H) p.188
Discussion opportunities:
Where do the sphere formulae come from? Can you prove them?
Pair / Group Work:
Practical investigation into the scale factor for mass of similar solids.
ICT Links:
Use of excel to input figures into area/volume formulae.
Investigation:
Investigation based on the cylinder e.g. A closed cylinder, half filled with water, is placed on a
flat surface. Mark the position of the water level. Repeat when the cylinder is a quarter full.
Generalise for any fraction.
Spiritual, Moral and Citizenship Links:
The notion of approximation similarity can be discussed, for example between infants and adults.
Time:
9 lessons
AUTUMN TERM A
Topic:
TOPIC 23
Simplifying Algebraic
Expressions
Target Grade: C/B/A
Edexcel Content:
NA5d:
NA5b:
NA5b:
NA3a:
Simplifying expressions using the rules of indices
Simplifying expressions involving algebraic fractions
Factorising of quadratic expressions
Understanding and using reciprocals
Prior Knowledge:
HCF, fractions, removing and factorising with one pair of brackets. An appreciation that if the
product of two numbers is zero then one of the numbers must be zero.
Learning Objectives:
Simplify algebraic expressions using the rules of indices
Simplify any algebraic expression involving fractions
Use factorising methods to simplify algebraic fractions
Understand the term reciprocal, and use this in calculations involving powers
Differentiation & Extension:
Set up quadratics from practical situations (e.g. area of a rectangle with both edges expressed in
terms of x)
Notes:
There may be a need to remove the HCF of a quadratic to make factorising easier.
London Reference:
Chapter 20 p.407-427
Other references:
Ray (H) p.261
Discussion opportunities:
Discuss if the students already know how to factorise quadratics.
Pair / Group Work:
Domino/Bingo/Snap games – match expressions with their factorised or simplified equivalent
ICT Links:
Autograph or graphics calculators could be used to show that, for example, x2 – 5x + 6 is
equivalent to (x – 6)(x + 1) or www.waldomaths.com
Investigation:
Try to find a method for factorising quadratics before being told.
Spiritual, Moral and Citizenship Links:
Sometimes a complicated algebraic fraction can be reduced to a much simpler expression – this
can be very satisfying.
Time:
8 – 10 lessons
AUTUMN TERM B
Topic:
Quadratic Equations
TOPIC 24
Target Grade: B/A/A*
Edexcel Content:
NA5k:
NA5k:
NA5k:
NA5b:
NA5l:
NA5l:
NA6e:
NA6h:
Solving quadratic equations by factorising
Solving quadratic equations by using the difference of two squares
Solving quadratic equations by using the quadratic formula
Simplify expressions by cancelling common factors
Solving by substitution a pair of simultaneous equations (one non-linear)
Use simultaneous equations to calculate where a straight-line graph meets a circle
Using graphs to solve a pair of simultaneous equations (one non-linear)
Using graphs to show where a straight-line graph intersects a circle
Prior Knowledge:
Topics 2, 9, 21 and 23. Drawing quadratic graphs.
Use BIDMAS for complicated examples, and substitute values into a complex formula.
Learning Objectives:
Factorise a trinomial
Use a factorised trinomial in one variable to solve a quadratic equation
Factorise using the difference of two squares and use this to solve quadratic equations
Use the formula to solve quadratic equations
Solve by substitution a pair of simultaneous equations (one non-linear)
Use simultaneous equations to calculate where a straight line graph meets a circle
Using graphs to solve a pair of simultaneous equations (one non-linear)
Use graphs to show where a straight line graph intersects a circle
Solve simple inequalities involving squares
Differentiation & Extension:
Significance of whether discriminant is positive, negative or zero.
Apply difference of two squares to Pythagoras’ Theorem.
Area of an annulus. Evaluation of calculations e.g. 982 - 22.
Proof of quadratic formula and the relevance of completing the square to graph sketching.
Notes:
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.
London Reference:
Other references:
Chapter 21 p.428-453
Ray (H) p.263
Discussion opportunities:
Which is the best way to solve a quadratic equation?
Sing the song “We know how to complete the square…”
Pair / Group Work:
ICT work (below) can be done in pairs.
ICT Links:
Using graphical calculators or Autograph to solve quadratic equations (see ICT folder) and draw
inequality graphs.
Algebra Foundations/ Algebra Tutor (Outware Education) on department laptop.
The Open Box Problem (see Edexcel Teachers’ guide – ICT)
Investigation:
Investigate the discriminant of a quadratic equation.
Lots of investigations give rise to quadratic sequences. Such as Bad Tomatoes, Borders, Lines,
Regions and Crossovers and those on p.441)
Spiritual, Moral and Citizenship Links:
Can population growth be (simplistically) modelled using quadratic expressions?.
Quadratics are used in mechanics (e.g. projectiles)
Time:
8 - 10 lessons
AUTUMN TERM B
Topic: Advanced Trigonometry
TOPIC 25
Target Grade: A/A*
Edexcel Content:
SSM2g:
SSM2g:
SSM2g:
SSM2f:
SSM2g:
SSM2g:
Using the sine rule
Area of a triangle using sin C
Using the cosine rule
Using Pythagoras’ theorem in 3-D problems
Using trigonometric relationships in 3-D problems
Angle between a line and a plane
Prior Knowledge:
Names and properties of 3-D shapes. The three trigonometric ratios. Pythagoras’ Theorem.
Ability to substitute and rearrange complex formulae.
Learning Objectives:
Use the sine rule to find the size of an angle or side in a non-right-angled triangle
Use the cosine rule to find the size of an angle or side in a non-right-angled triangle
Find the area of a triangle using sin C
Solve problems (including those involving bearings) using the sine and cosine rules
Use Pythagoras theorem , trigonometric ratios, sine rule and cosine rule to solve problems
in 3-D
Differentiation & Extension:
The ambiguous case for sine rule.
Find a side using the cosine rule, when the quadratic formula is required.
Notes:
Apply these principles to bearings (past exam paper questions)
Remind students that the sine rule and cosine rule should not be used on right-angled triangle.
Reminders of simple geometrical facts may be helpful, e.g. angle sum of a triangle, the smallest
side is opposite the smallest angle.
London Reference:
Other references:
Chapter 22 p.454-467
Ray (H) p.220
Discussion opportunities:
Discuss why the area formula works (explained on p.454)
Pair / Group Work:
Practical experiments involving bearings
ICT Links:
Cabri Geometry 2 plus – check answers or initial investigations.
Programming in Excel or graphics calculators
Investigation:
Investigate the area formula before being told.
Can the students find an example where the sine rule doesn’t work?
Spiritual, Moral and Citizenship Links:
Finally scalene triangles can join in with all the trigonometric fun that the right-angled triangles
have been enjoying.
Time:
6 lessons
MOCK EXAMINATIONS
Try to ensure that the class has attempted at least one old examination paper before the mock
examinations. They could, for example, complete a non-calculator paper in lessons and finish the
calculator paper over Christmas.
SPRING TERM A
Topic:
Exploring Numbers 2
TOPIC 26
Target Grade: A/A*
Edexcel Content:
NA2f:
NA2f:
NA3n:
NA3n:
NA3q:
NA3q:
Terminating and recurring decimals
Finding a fraction equivalent to a recurring decimal
Using surds and  in exact calculations
Rationalising a denominator – simple cases only
Finding upper and lower bounds when adding, subtracting, multiplying and dividing values
Absolute and percentage errors
Prior Knowledge:
Topics 1 and 5. Evaluating powers, and the rules of indices. Fractions.
Learning Objectives:
Change between fractions and decimals, including those that recur
Simplify numeric calculations by manipulating surds
Rationalise a denominator*
Find upper and lower bounds when adding and multiplying values
Find upper and lower bounds when subtracting and dividing values
Calculate absolute and percentage error
Differentiation & Extension:
Rationalise/simplify more complicated surds.
Notes:
Understanding that a surd is more accurate than a rounded answer.
Where appropriate, pupils will need to move between power and surd or power and reciprocal
forms.
*Simple cases only
London Reference:
Other references:
Chapter 23 p.468-481
Ray (H) p.36
Discussion opportunities:
Discuss why a surd is more accurate than a rounded answer.
Pair / Group Work:
Domino/Bingo/Snap games – match surds with their simplified equivalent
ICT Links:
Graphics calculators can be used to check the surd solutions to quadratic equations
Investigation:
Investigate how to change a recurring decimal into a fraction.
Spiritual, Moral and Citizenship Links:
Irrational numbers such as surds have decimals which go on to infinity. What does infinity really
mean? Is double infinity still infinity?
Time:
6 lessons
SPRING TERM A
Topic:
TOPIC 27
Applying Transformations
to Sketch Graphs
Target Grade: A/A*
Edexcel Content:
NA6g: Applying transformations to the function y = f(x):
y = f(x) + a,
y = f(ax)
y = f(x + a)
y = af(x)
(for linear, quadratic, sine and cosine functions)
Prior Knowledge:
Knowledge of sin, cos and tan. Experience of substituting values into a function.
Learning Objectives:
Given the graph of y = f(x), be able to sketch the graphs of y = f(x) + a, y = f(ax), y = f(x
+ a), and y = af(x) by applying transformations
Draw such graphs as y = a + b sin x
Differentiation & Extension:
A-level questions, e.g. the graph of y = -3(x + 1)4 has been produced from the graph of y = x4 by
three successive transformations. Define each transformation clearly.
Notes:
This is a difficult chapter as much of the work is A-level standard. Try to make it as investigative
as possible using ICT.
London Reference:
Chapter 24 p.482-505
Other references:
Ray (H) p.273
Discussion opportunities:
Plenty of opportunity when investigating the effect of different transformations
Pair / Group Work:
ICT work could be done in pairs.
ICT Links:
Graphical calculators (see ICT folder) and Autograph (see ICT folder for excellent ideas on
‘Transformation of Quadratic Graphs’ and ‘Trigonometric Graphs’)
www.waldomaths.com
Investigation:
All initial work on transformations could be investigative (see ICT links)
Investigation of curves which are unaffected by particular transformations.
Spiritual, Moral and Citizenship Links:
Sine and cosine graphs can model people’s emotions (up and down).
Time:
8 lessons
SPRING TERM A
Topic:
TOPIC 28
Target Grade: A*
Vectors
Edexcel Content:
SSM3f:
SSM3f:
SSM3f:
SSM3f:
SSM3f:
SSM3f:
SSM3f:
Understanding and using vector notation
Calculating the sum and difference of two vectors
Calculating a scalar multiple of a vector
Calculating the resultant of two vectors
Representing graphically the sum and difference of two vectors
Representing graphically a scalar multiple of a vector
Solving simple geometrical problems in 2-D using vector methods
Prior Knowledge:
The ability to use the four rules confidently with fractions and negative.
Learning Objectives:
Understand and use vector notation
Calculate the sum and difference of two vectors
Calculate a scalar multiple of a vector
Calculate the resultant of two vectors
Represent graphically the sum and difference of two vectors
Represent graphically a scalar multiple of a vector
Solving simple geometrical problems in 2-D using vector methods
Differentiation & Extension:
Find the modulus of a vector, or its angle to the horizontal.
Notes:
Students often find the pictorial representation of vectors more difficult than the manipulation of
column vectors.
London Reference:
Chapter 25 p.507-523
Other references:
Ray (H) p.208
Discussion opportunities:
What are the uses of vectors? How can they be used to give directions on a map?
Pair / Group Work:
Any map work could be done in pairs.
ICT Links:
Translation and vectors (see Edexcel Teachers’ guide – ICT)
Investigation:
Investigate how to find the modulus of a vector and its angle to the horizontal.
Spiritual, Moral and Citizenship Links:
Are vectors used in the army to launch missiles or find positions on maps?
Time:
7 lessons
SPRING TERM A
Topic:
Circle Theorems
TOPIC 29
Target Grade: B/A
Edexcel Content:
SSM2h: Understanding and using right angles between tangent and radius
SSM2h: Understanding and using tangents of equal length
SSM2h: Explaining why the perpendicular from the centre of a chord bisects the chord
SSM2h: Prove and use circle theorems
SSM2h: The angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the
circumference
SSM2h: The angle subtended at the circumference by a semi-circle is a right angle
SSM2h: Angles in the same segment are equal
SSM2h: Opposite angles of a cyclic quadrilateral add up to 180 degrees
SSM2h: Explain why the perpendicular from the centre of a circle bisects the chord
SSM2h: Prove and use the alternate segment theorem
Prior Knowledge:
Topics 19 and 22. Nomenclature of a circle. Angle facts for triangles, quadrilaterals,
interior/exterior angles of polygons, parallel lines, angles on a straight line, and angles at a point.
Learning Objectives:
Calculate angles within circles using rules relating to tangents and radii
Explain why the perpendicular from the centre of a circle bisects the chord
Prove and use circle theorems
Understand that the angle subtended by an arc at the centre of a circle is twice the angle
subtended at any point on the circumference
Understand that the angle subtended at the circumference by a semi-circle is a right angle
Angles in the same segment are equal
Opposite angles of a cyclic quadrilateral add up to 180 degrees
Explain why the perpendicular from the centre of a circle bisects the chord
Prove and use the alternate segment theorem
Differentiation & Extension:
The most able students could investigate radians.
Notes:
You could get the students to discover the circle theorems themselves by LOCI or by the ICT
links below.
London Reference:
Other references:
Chapter 26 p.524-547
Ray (H) p.226
Discussion opportunities:
What constitutes a proof? Why do we need proofs in mathematics?
Pair / Group Work:
Investigative work could be done in pairs.
ICT Links:
Cabri Geometry.
www.waldomaths.com
Investigation:
LOCI questions may be used to discover circle theorems (see p.137 of Edexcel GCSE Higher (old
edition).
Spiritual, Moral and Citizenship Links:
What can and can’t be proved?
Time:
7 lessons
SPRING TERM A
Topic:
TOPIC 30
Introducing Modelling
Target Grade: A*
Edexcel Content:
NA6f: Using powers to explore exponential growth and decay
NA6f: Graphs of the form y = pqx
Prior Knowledge:
Topics 1, 5, 21 and 26.
Learning Objectives:
Understand exponential growth and decay, and use it to make predictions
Use the graph of y = pqx to find the values of p and q
Differentiation & Extension:
Investigating the constant e.
Notes:
The only sections of the textbook that appear to be on the syllabus are 28.1 and 28.5. Do not do
the other sections in this chapter unless you have extra time.
London Reference:
Chapter 28 p.561-576
Other references:
Ray (H) p.245
Discussion opportunities:
This topic has many real-life applications (see p.563) to discuss.
Pair / Group Work:
Any research into financial investment could be done in pairs.
ICT Links:
The Exponential Function – Autograph (see ICT folder)
Graphical calculators
Investigation:
Research into real-life modelling such as population growth, mechanics, financial investment or
radioactivity (half life)
Investigating the constant e.
Spiritual, Moral and Citizenship Links:
Population growth in nature can be modelled by exponential graphs (see p.562)
Time:
2 - 3 lessons
SPRING TERM B
TOPIC 31
Topic: Calculations and Computers
Target Grade: A/A*
Edexcel Content:
NA3o: Using a calculator effectively and efficiently for complex calculations
NA3o: Using an extended range of calculator function keys
Prior Knowledge:
Revision of many topics
Learning Objectives:
Use a calculator effectively and efficiently for complex calculations
Use an extended range of calculator function keys
Differentiation & Extension:
Using a graphics calculator effectively (graphs, solving simultaneous and quadratic equations
etc.)
Notes:
This topic is a good way to revise many key areas. It covers many more objectives than those
listed above.
London Reference:
Other references:
Chapter 29 p.577-589
Ray (H) p.65
Discussion opportunities:
Students will already understand how to use many functions on their calculator. You could begin
the topic by discussing what they already know.
Discuss the benefits of using Excel for trial and improvement.
Pair / Group Work:
Any ICT work could be done in pairs.
ICT Links:
Scientific/graphics calculators
Excel for spreadsheet work (see ICT folder and Edexcel Teachers’ guide – ICT)
Autograph
Investigation:
See the Fencing Problem example on p.587-588
Spiritual, Moral and Citizenship Links:
Are GCSEs easier now, as you can use calculators in one paper, compared to 50 years ago?
Time:
3 - 4 lessons
COURSEWORK 3: Choose a task from the Edexcel coursework folder (in the maths staff room)
Time: 2 weeks
REVISION FOR GCSE EXAMINATIONS
Revise individual topics and Examination papers (there are practice papers at the back of the
textbook).