Scheme of Work - Level 2 Award and Level 3 Award in ...
Mathematics A (1MA0)December 2010UG025517Sharon Wood and Ali MelvilleIssue 2
Contents
Introduction
Level 2 scheme of work
Level 2 course overview
Level 2 modules
Level 2 concepts and skills (AAL20)
Resources Table
Level 2
Introduction
This scheme of work has been designed for teachers delivering the Edexcel Award in Algebra. This scheme of work is based upon a course model which can be taught over a single year, or more years if desired, for both Level 2 and Level 3 tier students.
The scheme of work is structured so each topic contains:
• Module number
• Recommended teaching time, though of course this is adaptable according to individual student needs
• Level
• Contents, referenced back to the specification
• GCSE specification references
• Prior knowledge and links with the alternative level
• Objectives for students at the end of the module
• Links to the GCSE scheme of work modules
• At level 3 links to (C1) GCE Mathematics
• Ideas for differentiation and extension activities
• Notes for general mathematical teaching points and common misconceptions
Updated versions of this scheme of work will be available via a link from the Edexcel Mathematics Awards website (mathsawards).
This course can be taught as a stand-alone course.
Alternatively it can be taught as an introduction to GCSE Mathematics, with differentiation extending the content from an Awards course into GCSE topics where possible. The links to the GCSE schemes of work are given for this reason. The level 2 course can also stand alongside GCSE Mathematics. The level 3 course can be used as an introduction to A(S) Mathematics, or even as a bridge between GCSE Mathematics and A(S) Mathematics.
This course can be supported by a range of resources. At the end of the document we have provided a table which can be populated with the resources and links of choice to support delivery.
The Edexcel Award
in Algebra
Level 2 (AAL20)
Scheme of work
Level 2 course overview
The table below shows an overview of modules in the Level 2 scheme of work.
Teachers should be aware that the estimated teaching hours are approximate and should be used as a guideline only.
|Module number |Title |Estimated teaching hours |
|1 |Roles of symbols |3 |
|2 |Algebraic manipulation |6 |
|3 |Formulae |5 |
|4 |Linear equations |4 |
|5 |Graph sketching |5 |
|6 |Linear inequalities |4 |
|7 |Number sequences |4 |
|8 |Gradients of straight line graphs |6 |
|9 |Straight line graphs |4 |
|10 |Graphs for real-life situations |7 |
|11 |Simple quadratic functions |6 |
|12 |Distance-time and speed-time graphs |6 |
| |Total |60 |
Module 1 Time: 4 – 5 hours
Awards Tier: Level 2
Contents: Roles of symbols
|1.1 |Distinguish between the roles played by letter symbols in algebra using the correct notation |
|1.2 |Distinguish in meaning between the words equation, formula and expression |
|1.3 |Write an expression to represent a situation in ‘real life’ |
GCSE SPECIFICATION REFERENCES
|A a |Distinguish the different roles played by letter symbols in algebra, using the correct notation |
|A b |Distinguish in meaning between the words ‘equation’, ‘formula’, and ‘expression’ |
|A c |Manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out |
| |common factors, multiplying two linear expressions, factorise quadratic expressions including the difference of two |
| |squares and simplify rational expressions |
PRIOR KNOWLEDGE
Experience of using a letter to represent a number
Ability to use negative numbers with the four operations
Recall and use BIDMAS
Writing simple rules algebraically
OBJECTIVES
By the end of the module the student should be able to:
• Use notation and symbols correctly
• Write an expression
• Select an expression/equation/formula from a list
LINKS TO LEVEL 3 (extension work)
Module 1 Roles of symbols
LINKS TO GCSE SCHEME OF WORK (2-year)
A/Module 4 B/Module 2-8
DIFFERENTIATION & EXTENSION
This topic can be used as a reminder of the KS3 curriculum and could be introduced via investigative material, eg frogs, handshakes, patterns in real life, formulae
Use examples where generalisation skills are required
Extend the above ideas to the equation of the straight line, y = mx + c
Look at word formulae written in symbolic form, eg F = 2C + 30 to convert temperature (roughly) and compare with F = [pic]C + 32
NOTES
Emphasise good use of notation, eg 3n means 3 × n
Present all working clearly
Module 2 Time: 5 – 7 hours
Awards Tier: Level 2
Contents: Algebraic manipulation
|2.1 |Collect like terms |
|2.2 |Multiply a single term over a bracket |
|2.3 |Factorise algebraic expressions by taking out all common factors |
|2.4 |Use index laws for multiplication, division and raising a power to a power |
GCSE SPECIFICATION REFERENCES
|A c |Manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking |
| |out common factors, multiplying two linear expressions, factorise quadratic expressions including the difference of two |
| |squares and simplify rational expressions |
|A c |Simplify expressions using rules of indices |
PRIOR KNOWLEDGE
Ability to use negative numbers with the four operations
Recall and use BIDMAS
OBJECTIVES
By the end of the module the student should be able to:
• Recall and manipulate algebraic expressions by collecting like terms
• Multiply a single term over a bracket
• Simplify expressions using index laws
• Use index laws for integer powers and powers of a power
• Factorise algebraic expressions by taking out common factors
LINKS TO LEVEL 3 (extension work)
Module 2 Algebraic manipulation
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 4 B/module 2-9, 3-5
DIFFERENTIATION & EXTENSION
Practise factorisation where the factor may involve more than one variable
NOTES
Avoid oversimplification
Ensure cancelling is only done when possible (particularly in algebraic fractions work)
Module 3 Time: 4 – 6 hours
Awards Tier: Level 2
Contents: Formulae
|3.1 |Substitute numbers into a formula |
|3.2 |Change the subject of a formula where the subject only appears once |
GCSE SPECIFICATION REFERENCES
|A f |Derive a formula, substitute numbers into a formula and change the subject of a formula |
PRIOR KNOWLEDGE
Experience of using a letter to represent a number
Ability to use negative numbers with the four operations
Recall and use BIDMAS
OBJECTIVES
By the end of the module the student should be able to:
• Derive a formula
• Use formulae from mathematics and other subjects
• Substitute numbers into a formula
• Substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3
• Change the subject of a formula where the subject only appears once
LINKS TO LEVEL 3 (extension work)
Module 3 Formulae
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 14 B/module 2-10, 3-7
DIFFERENTIATION & EXTENSION
Consider more complex expressions
Changing the subject of a formula including cases where the subject is on both sides of the original formula or where a power of the subject appears
NOTES
Break down any manipulation into simple steps all clearly shown
Module 4 Time: 3 – 5 hours
Awards Tier: Level 2
Contents: Linear equations
|4.1 |Solve linear equations with integer coefficients where the variable appears on |
| |either side or on both sides of the equation |
|4.2 |Solve linear equations which include brackets, those that have negative signs occurring anywhere in the equation, those |
| |with negative and fractional solutions and those with fractional coefficients |
GCSE SPECIFICATION REFERENCES
|A d |Set up and solve simple equations |
PRIOR KNOWLEDGE
Experience of finding missing numbers in calculations
The idea that some operations are the reverse of each other
An understanding of balancing
Experience of using letters to represent quantities
Understand and recall BIDMAS
OBJECTIVES
By the end of the module the student should be able to:
• Set up linear equations from word problems
• Solve simple linear equations
• Solve linear equations, with integer coefficients, in which the unknown variable appears on either side or on both sides of the equation
• Solve linear equations that include brackets, those that have negative signs occurring anywhere in the equation and those with a negative and fractional solution
• Solve linear equations in one unknown with fractional coefficients
LINKS TO LEVEL 3 (extension work)
No direct link
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 14 B/module 3-6
DIFFERENTIATION & EXTENSION
Use negative numbers in formulae involving indices
Use investigations to lead to generalisations
Apply changing the subject to y = mx + c
Derive equations from practical situations (such as finding unknown angles in polygons or perimeter problems)
NOTES
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
Remind students 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 important
Students can leave their answers in fractional form where appropriate
Module 5 Time: 4 – 6 hours
Awards Tier: Level 2
Contents: Graph sketching
|5.1 |Sketch graphs of quadratic functions, considering orientation and labelling the point of intersection with the y-axis, |
| |considering what happens to y for large positive and negative values of x |
GCSE SPECIFICATION REFERENCES
|A t |Generate points and plot graphs of simple quadratic functions, and use these to find approximate solutions |
PRIOR KNOWLEDGE
Recognising quadratic graphs as parabolas
The link between linear equations as graphs and intersection with the y-axis
Substitution into quadratic expressions
Four rules of negative numbers
OBJECTIVES
By the end of the module the student should be able to:
• Understand the form of a quadratic function
• Understand the relationship between a quadratic function and its graph
• Sketch graphs for simple quadratic functions
• Understand what happens to function values when values of x get large
LINKS TO LEVEL 3 (extension work)
Module 10 Graphs of functions
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 26 B/module 3-12
DIFFERENTIATION & EXTENSION
Sketch graphs for reciprocal or exponential functions
NOTES
Build up the algebraic techniques slowly
Link the graphical solutions with quadratic graphs and changing the subject and use practical examples, eg projectiles
Use of graph-plotting software would enhance investigation of graphs
Module 6 Time: 3 – 5 hours
Awards Tier: Level 2
Contents: Linear inequalities
|6.1 |Show inequalities on a number line, using solid circles to show inclusive inequalities and open circles to show exclusive |
| |inequalities |
|6.2 |Write down an inequality shown on a number line |
|6.3 |Solve simple linear inequalities in one variable |
GCSE SPECIFICATION REFERENCES
|A g |Solve linear inequalities in one variable, and represent the solution set on a number line |
PRIOR KNOWLEDGE
Experience of finding missing numbers in calculations
The idea that some operations are the reverse of each other
An understanding of balancing
Experience of using letters to represent quantities
Understand and recall BIDMAS
Ordering numbers
Knowledge of inequality signs , ≤, ≥
OBJECTIVES
By the end of the module the student should be able to:
• Solve simple linear inequalities in one variable and represent the solution set on a
number line
• Use the correct notation to show inclusive and exclusive inequalities
LINKS TO LEVEL 3 (extension work)
Module 7 Inequalities
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 14 B/module 3-6
DIFFERENTIATION & EXTENSION
Derive inequalities from practical situations (such as finding unknown angles in polygons or perimeter problems)
NOTES
Students can leave their answers in fractional form where appropriate
Module 7 Time: 3 – 5 hours
Awards Tier: Level 2
Contents: Number sequences
|7.1 |Generate terms of a sequence using term-to-term definition or using position-to-term definition |
|7.2 |Find and use the nth term of a linear arithmetic sequence |
GCSE SPECIFICATION REFERENCES
|A i |Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence |
|A j |Use linear expressions to describe the nth term of an arithmetic sequence |
PRIOR KNOWLEDGE
Experience of using a letter to represent a number
Recall and use BIDMAS
Recognise simple number patterns, eg 1, 3, 5
Writing simple rules algebraically
OBJECTIVES
By the end of the module the student should be able to:
• Generate simple sequences of numbers, squared integers and sequences derived
from diagrams
• Describe the term-to-term definition of a sequence in words
• Identify which terms cannot be in a sequence
• Generate specific terms in a sequence using the position-to-term and
term-to-term rules
• Find and use the nth term of a linear arithmetic sequence
LINKS TO LEVEL 3 (extension work)
Module 8 Arithmetic series
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 10 B/module 2-12
DIFFERENTIATION & EXTENSION
Sequences and nth term formula for triangle numbers, Fibonacci numbers etc
Prove a sequence cannot have odd numbers for all values of n
Extend to quadratic sequences whose nth term is an2 + bn + c
NOTES
Emphasise good use of notation, eg 3n means 3 × n
When investigating linear sequences, students should be clear on the description of the pattern in words, the difference between the terms and the algebraic description of the nth term
Module 8 Time: 5 – 7 hours
Awards Tier: Level 2
Contents: Gradients of straight lines
|8.1 |Find the gradient of a straight line graph |
|8.2 |Interpret the gradient of real-life graphs |
GCSE SPECIFICATION REFERENCES
|A l |Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients |
|A m |Understand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value |
| |of the y-intercept |
|A n |Understand the gradients of parallel lines |
|A s |Interpret graphs of linear functions |
PRIOR KNOWLEDGE
Substitute positive and negative numbers into algebraic expressions
Rearrange to change the subject of a formula
OBJECTIVES
By the end of the module the student should be able to:
• Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane
• Find the gradient of a straight line from a graph
• Analyse problems and use gradients to interpret how one variable changes in relation to another
• Interpret and analyse a straight-line graph
• Understand that the form y = mx + c represents a straight line
• Find the gradient of a straight line from its equation
• Use the fact that when y = mx + c is the equation of a straight line then the gradient of a line parallel to it will have a gradient of m
LINKS TO LEVEL 3 (extension work)
Module 9 Coordinate geometry
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 15 B/module 2-13
DIFFERENTIATION & EXTENSION
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
Students should explore lines that are not only parallel, but also perpendicular to given lines
NOTES
Careful annotation should be encouraged; Students should label the coordinate axes and origin and write the equation of the line
Students need to recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, eg science, geography
Link conversion graphs to converting metric and imperial units
Module 9 Time: 3 – 5 hours
Awards Tier: Level 2
Contents: Straight line graphs
|9.1 |Recognise, plot and draw graphs of the form y = mx + c |
|9.2 |Given a straight line graph, find its equation |
GCSE SPECIFICATION REFERENCES
|A l |Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding |
| |gradients |
|A m |Understand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value|
| |of the y-intercept |
|A s |Interpret graphs of linear functions |
PRIOR KNOWLEDGE
Substitute positive and negative numbers into algebraic expressions
Rearrange to change the subject of a formula
OBJECTIVES
By the end of the module the student should be able to:
• Draw, label and scale axes
• Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane
• Plot and draw graphs of straight lines with equations of the form y = mx + c
• Understand that the form y = mx + c represents a straight line
• Find the equation of a straight line from its graph
LINKS TO LEVEL 3 (extension work)
Module 9 Coordinate geometry
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 15 B/module 2-13
DIFFERENTIATION & EXTENSION
Students should find the equation of the line through two given points
Students should find the equation of the perpendicular bisector of the line segment joining
two given points
Use a graphical calculator or graphical ICT package to draw straight-line graphs
NOTES
Careful annotation should be encouraged; students should label the coordinate axes and origin and write the equation of the line
Students need to recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, eg science, geography
Module 10 Time: 6 – 8 hours
Awards Tier: Level 2
Contents: Graphs for real-life situations
|10.1 |Understand that straight and curved graphs can represent real-life situations |
|10.2 |Draw, and interpret information from graphs of real-life situations |
GCSE SPECIFICATION REFERENCES
|A l |Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients |
|A r |Construct linear functions from real-life problems and plot their corresponding graphs |
|A s |Interpret graphs of linear functions |
PRIOR KNOWLEDGE
Scaling and labelling graphs
Drawing graphs from tabular information
Work out points for plotting from conversion or foreign currency rates
OBJECTIVES
By the end of the module the student should be able to:
• Draw and interpret straight-line graphs for real-life situations such as:
– Ready reckoner graphs
– Conversion graphs
– Fuel bills, eg gas and electric
– Fixed charge (standing charge) and cost per unit
• Analyse problems and use gradients to interpret how one variable changes in relation
to another
• Interpret and analyse a straight-line graph
LINKS TO LEVEL 3 (extension work)
None
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 15 B/module 2-13, 3-8
DIFFERENTIATION & EXTENSION
Use Functional Elements in terms of mobile phone bills
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 to scatter graphs and correlation
NOTES
Careful annotation should be encouraged; students should label the coordinate axes and origin and write the equation of the line
Students need to recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, eg science, geography
Link conversion graphs to converting metric and imperial units
Module 11 Time: 5 – 7 hours
Awards Tier: Level 2
Contents: Simple quadratic functions
|11.1 |Plot graphs of simple quadratic functions |
|11.2 |Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function |
GCSE SPECIFICATION REFERENCES
|A e |Solve quadratic equations |
|A r |Construct linear, quadratic and other functions from real-life problems and plot their corresponding graphs |
|A t |Generate points and plot graphs of simple quadratic functions, and use these to find approximate solutions |
PRIOR KNOWLEDGE
An introduction to algebra
Substitution into expressions/formulae
Linear functions and graphs
Solving equations
OBJECTIVES
By the end of the module the student should be able to:
• Generate points and plot graphs of simple quadratic functions, then more general quadratic functions
• Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function
LINKS TO LEVEL 3 (extension work)
Module 10 Graphs of functions
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 26 B/module 3-12
DIFFERENTIATION & EXTENSION
Use graphical calculators or ICT graph package where appropriate
Draw graphs of reciprocal or exponential functions
NOTES
Build up the algebraic techniques slowly
Link the graphical solutions with linear graphs and changing the subject and use practical examples, eg projectiles
Emphasise that inaccurate graphs could lead to inaccurate solutions; encourage substitution of answers to check they are correct
In problem-solving, one of the solutions to a quadratic equation may not be appropriate
Module 12 Time: 5 – 7 hours
Awards Tier: Level 2
Contents: Distance-time and speed-time graphs
|12.1 |Draw distance-time graphs and speed-time graphs |
|12.2 |Interpret distance-time graphs and speed-time graphs |
|12.3 |Understand that the gradient of a distance-time graph represents speed |
|12.4 |Find speed and distance from information on a travel graph |
GCSE SPECIFICATION REFERENCES
|A r |Construct linear functions from real-life problems and plot their corresponding graphs |
|A s |Interpret graphs of linear functions |
PRIOR KNOWLEDGE
A basic understanding of speed
Interpret the slope of a graph as its gradient
Interpret the gradient within a real life context
OBJECTIVES
By the end of the module the student should be able to:
• Draw distance-time and speed-time graphs
• Interpret distance-time and speed-time graphs
• Understand that the gradient of a distance-time graph represents speed
• Calculate speed using distance-time graphs
LINKS TO LEVEL 3 (extension work)
Module 12 Distance-time and speed-time graphs
LINKS TO GCSE SCHEME OF WORK (2-year)
A/module 15 B/module 2-13, 3-8
DIFFERENTIATION & EXTENSION
Draw distance-time graphs of journeys of several stages
Consider the link between distance-time and speed-time graphs
NOTES
Consider the importance of understanding zero gradient in both types of graph
Use of different scales in accurate reading and drawing of graphs
Accuracy of plotting is important
Calculating the gradient from given information regarding speed and/or acceleration is worth practising
Level 2 concepts and skills
What students need to learn:
| |Topic |Concepts and skills |
|1. |Roles of symbols |Distinguish between the roles played by letter symbols in algebra using the correct |
| | |notation |
| | |Distinguish in meaning between the words equation, formula and expression |
| | |Write an expression to represent a situation in ‘real life’ |
|2. |Algebraic manipulation |Collect like terms |
| | |Multiply a single term over a bracket |
| | |Factorise algebraic expressions by taking out all common factors |
| | |Use index laws for multiplication, division and raising a power to a power |
|3. |Formulae |Substitute numbers into a formula |
| | |Change the subject of a formula where the subject only appears once |
|4. |Linear equations |Solve linear equations with integer coefficients where the variable appears on either|
| | |side or on both sides of the equation |
| | |Solve linear equations which include brackets, those that have negative signs |
| | |occurring anywhere in the equation, those with negative and fractional solutions and |
| | |those with fractional coefficients |
|5. |Graph sketching |Sketch graphs of quadratic functions, considering orientation and labelling the point|
| | |of intersection with the y-axis, considering what happens to y for large positive and|
| | |negative values of x |
|6. |Linear inequalities |Show inequalities on a number line, using solid circles to show inclusive |
| | |inequalities and open circles to show exclusive inequalities |
| | |Write down an inequality shown on a number line |
| | |Solve simple linear inequalities in one variable |
|7. |Number sequences |Generate terms of a sequence using term-to-term definition or using position-to-term |
| | |definition |
| | |Find and use the nth term of a linear arithmetic sequence |
|8. |Gradients of straight line |Find the gradient of a straight line graph |
| |graphs |Interpret the gradient of real-life graphs |
|9. |Straight line graphs |Recognise, plot and draw graphs of the form |
| | |y = mx + c |
| | |Given a straight line graph, find its equation |
|10. |Graphs for real-life |Understand that straight and curved graphs can represent real-life situations |
| |situations |Draw, and interpret information from graphs of real-life situations |
|11. |Simple quadratic functions |Plot graphs of simple quadratic functions |
| | |Find approximate solutions of a quadratic equation from the graph of the |
| | |corresponding quadratic function |
|12. |Distance-time and |Draw distance-time graphs and speed-time graphs |
| |speed-time graphs |Interpret distance-time graphs and speed-time graphs |
| | |Understand that the gradient of a distance-time graph represents speed |
| | |Find speed and distance from information on a travel graph |
Resources Table
Level 2
|Module number |Title |Resources |Web resources |
|1 |Roles of symbols | | |
|2 |Algebraic manipulation | | |
|3 |Formulae | | |
|4 |Linear equations | | |
|5 |Graph sketching | | |
|6 |Linear inequalities | | |
|7 |Number sequences | | |
|8 |Gradients of straight line graphs | | |
|9 |Straight line graphs | | |
|10 |Graphs for real-life situations | | |
|11 |Simple quadratic functions | | |
|12 |Distance-time and speed-time graphs | | |
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Acknowledgements
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