CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
1 Functions
Teaching plan
Topic
Order in
chapter
Learning content
Resources
Mappings
and
functions
First
Understand the terms: function,
domain, range (image set), one-one
function, many-one function.
Coursebook:
Sections 1.1 and 1.2
Use the notation f (x) = sin x,
f : x ↦ lg x, (x . 0).
PowerPoints:
1.1a Vertical line test
1.2 Worked examples 1 & 2
Explain in words why a given
function is a function.
PDF files:
Chapter 1 Teacher notes
These resources are printable
and/or editable
Chapter 1 Lesson Plan
Chapter 1 Describing Mappings
Group Activities: Worksheets 1.1-1.4
Composite
functions
After
Mappings
and
functions;
composite
functions
and inverse
functions
could be
taught in
either order
Understand composition of
functions including knowing when a
composite function can be formed
and finding domains and ranges.
Use the notation gf (x) = g (f (x)) and
f 2 (x) = f (f (x)).
Coursebook:
Section 1.3
PowerPoints:
1.3a Bubble diagrams and
Worked examples 3 & 4
1.3b Existence of composite
functions and domains and ranges
1.3c Composite functions
further practice
PDF files:
Composite functions further practice
(Continued)
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Topic
Order in
chapter
Learning content
Resources
Inverse
functions
After
Mappings
and
functions;
composite
functions
and inverse
functions
could be
taught in
either order
Understand the term inverse
functions, including finding domains
and ranges.
Coursebook:
Sections 1.6 and 1.7
Explain in words why a given
function does or does not have
an inverse.
Use the notation f−1 (x).
Use sketch graphs to show the
relationship between a function and
its inverse.
PowerPoints:
1.6a Existence of inverse
functions and domains and ranges
using bubble diagrams
1.6b Finding the inverse function
including Worked example 7 using
swop & rearrange AND rearrange &
swop methods
1.1b Horizontal line test
1.7 Graphing a function and its
inverse and Worked example 8
1.8 Class discussion in section 1.7
PDF files:
Chapter 1 Exercise 1.7 student grids
Modulus
functions
Independent
functions
topic; could
be studied
later in the
course
Understand the relationship
between y = f (x) and y = |f (x)|,
where f (x) is linear.
Solve graphically or algebraically
equations of the type
| ax + b | = c and | ax + b | = cx + d.
Coursebook:
Sections 1.4 and 1.5
PowerPoints:
1.4 Class discussion and Worked
example 5 including graphs
1.5 Worked example 6 and
further examples, including Worked
example 5 with parts a & d
solved graphically
Learning plan
Learning intentions
Success criteria
Understand the terms: function,
domain, range (image set), one-one
function, inverse function and
composition of functions.
Students correctly interpret:
• the name of a function to be, for example, f
• what the function actually does (that is, its rule) to be,
for example, f (x).
They have a deep understanding of all other terms and are
able to use them, with correct language, in explanations and
to justify statements or results.
Use the notation f (x) = sin x, f: x ↦ lg x,
(x . 0), f −1(x) and f 2(x) [= f (f (x))].
Students can interpret and write, in any correct form,
a function, a composite function or an inverse function and
understand what is meant by each.
(Continued)
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Learning intentions
Success criteria
Understand the relationship between
y = f (x) and y = | f (x) |, where f (x)
is linear.
Solve graphically or algebraically
equations of the type | ax + b | = c and
| ax + b | = cx + d.
Students can:
• solve equations involving modulus functions
• draw graphs of these functions and understand the
relationship between the graphs and the solution of
an equation.
Explain in words why a given function
is a function or why it does or does not
have an inverse.
Students can use the correct language to give an appropriate
and complete explanation.
Find the inverse of a one-one function
and form composite functions.
Students are able to demonstrate, for many different
functions, that they can correctly apply the complete
technique to find:
• an inverse function, justifying the correct choice of sign,
for example, when the original function was quadratic
• the composition of functions, knowing also when it is
appropriate to compose functions.
Use sketch graphs to show the
relationship between a function and
its inverse.
Students can apply the knowledge that, when they are asked
to draw a function and its inverse, they draw the function
and then reflect it in the line y = x, rather than finding an
expression for the inverse function.
BACKGROUND KNOWLEDGE
• The following table details what knowledge it is assumed that students already have from studying
Cambridge IGCSE or O level mathematics. This knowledge should be at a much more basic level
than in Additional mathematics and is likely to be for simple polynomials only.
What your students should be able to do
Examples
Find an output for a given simple function.
When f (x) = 5x − 1, find f (3).
Find a simple composite function.
When f (x) = 3x − 2 and g (x) = 4 − x find fg (x).
Find the inverse of a simple function.
When f (x) = 3x + 5 find f −1 (x).
Sketch the graph of a linear and a
quadratic function.
a
Sketch the graph of y = 2x − 1.
b
Sketch the graph of y = x 2 + 1.
Solve linear and quadratic equations.
a
Solve 5 − 3x = 8.
b
Solve (x + 2) 2 = 16.
• In relation to the other topics in the syllabus, it is sensible to look at the material in this chapter
after looking at Chapter 2 Simultaneous equations and quadratics and Chapter 6 Straight-line
graphs. Even as soon as Worked example 2, students need to have a good understanding of
quadratic functions. It is helpful for graphing the modulus of a linear function if students have
revised the drawing of the graph of the function of which the modulus has been taken.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
CONTINUED
• This chapter starts with simple mappings and connects these to functions. The material then allows
you to consider domains and ranges as well as sketching graphs before moving on to composite
functions. Once students have a good understanding of these, they are then presented with the
concept of an inverse function. All this should reinforce skills students should already have and will
give you assessment for learning opportunities to check if your students have fully understood the
basic ideas. In this chapter, knowledge is extended to composites formed from and inverses of
more complex functions. Students will consider the existence of inverse and composite functions,
as well as the relationship between the graph of a function and its inverse.
LANGUAGE SUPPORT
The definitions of key words and phrases are
given in the glossary.
The word range is used in more than one way
in maths. It is a statistical measure of spread and
can also be used to represent a set of possible
solutions to an inequality, for example. This can
be confusing for students, and it is a good idea
to use diagrams, such as bubble diagrams, to
deal with this issue at the start. For a function,
the range is the set of outputs for its given set
of inputs.
Model this language and these ideas for
students as much as possible so that when they
need to explain a result or justify a result, which
is common in this topic, they will be familiar with
the language needed.
Links to Digital Resources
Composite functions
•
The Khan Academy has an interactive worksheet for composite functions which includes some bubble
diagram explanations.
Inverse functions
•
•
The Khan Academy has several short videos looking at finding inverse functions – the videos are
differentiated by ability. They could be used as a starter or part of a main teaching idea. Each video is about
ten minutes long, and there are five of them. The rearrange and swop method is used to find the inverse.
Geogebra has an activity that allows the graphical investigation of the formation of invertible functions by
restricting domains. It probably works best as a whole class teaching tool.
Absolute values
•
•
•
4
Geogebra has an activity that could be used as a whole class teaching aid or by individual students – it
shows the graph of a linear function and the modulus of the same function. The approach follows that in
the PowerPoint provided.
STEM also has several pages on functions and graphs. A login is required to use STEM resources, but it is
free to use.
The Maths is Fun website investigates functions and terminology.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
REAL-LIFE CONTEXT
Functions are very important to the
development of new ideas and to the
representation of existing relationships.
Functions can be used to translate what we think
about how the world works into mathematical
language. In the real world, functions can be
used to model relationships between sets of
different variables. For example, for the purpose
of research, scientists may use them to represent
relationships and make predictions. These
predictions may be about physical aspects
of the natural world, such as velocity and
displacement. A designer may model the shape
of a package using functions to determine
the shape that most efficiently uses resources.
Functions are an invaluable tool. The function
notation that we use today was first introduced
by the Swiss mathematician Leonhard Euler
(1707–1783). He solved the Königsberg Bridge
problem for fun, and the principles he used then
have been used in the development of
the internet.
Common Misconceptions and Issues
Misconception/issue
How to identify
How to avoid or overcome
Domain and range are often
muddled, and the notation used
is often poor.
Use starter activity 2 and check
students’ working.
Ensure students write these
correctly using the language of
mathematics. Domains should be
in terms of the input letter which
is often x. Ranges should be in
terms of y or the name of the
function, which is often f, or the
function rule f (x).
The mnemonic Dr Io may help
some students:
Domain
range
Input
output
The minimum value of the
domain always gives the
minimum value of the range.
Use starter activity 2.
Asking for a simple sketch of
the function to check this is a
very good idea.
This point can be demonstrated
using Worked examples 1 and
2 and is mentioned in the notes
in PowerPoint 1.2 where the
solutions to these examples
are given.
Misinterpretation of the function
notation f (x) or the composite
function notation gf (x),
for example.
Students confuse f (x) with f
times x or gf (x) as the product
g (x) × f (x) for example, in their
working, so this needs to
be checked.
Ensure students are clear about
how to read functions. Model the
correct reading of functions at
every opportunity.
It is better to read f (x) as ‘f of x’
and not ‘f x’.
It is better to read gf (x) as ‘g of f
of x’ or at worst ‘gf of x’ and not
‘gfx’.
Also, use of bubble diagrams
to aid understanding rather
than rote learning of terms is
highly advisable.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Misconception/issue
How to identify
How to avoid or overcome
Misinterpretation of the notation
for the inverse function.
Students confuse an inverse
function with the reciprocal of a
function in their working, so this
needs to be checked.
Model the correct reading of
functions at every opportunity.
Try to encourage students to
read f −1 as ‘f inverse’ not ‘f to the
minus 1’.
Also, use of bubble diagrams to
aid understanding rather than
rote learning of terms is highly
advisable.
Thinking that finding an inverse
function only involves swopping
the x and y.
For example, in their working
they may write x = 5y − 2 as
their answer for the inverse
function of y = 5x − 2, so this
needs to be checked.
This is rarely correct and indicates
that some students do not fully
understand the process.
Again, bubble diagrams should
help with this.
Student’s diagrams of functions
and their inverse functions are
very skew.
Starter activities such as ‘Mark
my work’ or assessment for
learning activities such as class
discussions where students
explain to each other which
diagram is suitable.
When graphing a function and
its inverse, students need to be
reminded to use ‘square’ scales
for their graphs – in other words,
the same scale horizontally and
vertically. This should make
sure that their reflections are
nicely symmetrical.
Thinking that for y = | f (x) |, the
values of x cannot be negative.
This is very common when
solving equations. Should a
value of x be negative, students
often think it should be
rejected and will indicate this
in their working.
This can be resolved by working
on the graphs of absolute value
functions so that students can
clearly see that x can have
negative values but that y cannot
and then linking the graphs back
to the equations they are solving.
Starter ideas
1 Mapping challenge
Description and purpose: This 5-minute quiz activity leads students to think about domains, when a function
is valid, function notation generally and the ways it is used. Most will be comfortable with the idea of a
mapping, so this language is used.
Resources:
PowerPoint 1 starter: Mapping challenge
Cards coloured red, yellow and green for students to hold up to show their answer
A timer
Activity:
•
•
•
•
If possible, put three cards (one of each colour) on the desk of each student at the start of
the lesson.
Give students about 30 seconds to read the question, and then click to reveal the three answers.
Tell students they have to be quick!
Look at the colours to see who is not correct, and use this assessment for learning information to help
select groups for activities.
Answers:
green; red; yellow; yellow; green
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
If you wish to use this activity to start other topic areas, you could allow your students to keep their cards,
if they have somewhere to keep them and they are not likely to be lost!
This activity could possibly lead into: more exploration on the validity of functions and the difference
between a mapping or relation and a formal function.
This activity could be adapted: You could print the PowerPoint slides, one per A4 page, for students to share,
one between two. Alternatively, you could print larger versions and pin them to a flip chart.
You could choose to do this with the answers included or with the answers removed. These printouts could
then be kept for use with other classes.
2 Complete the line
Description and purpose: A cloze sentence is one in which key words are omitted and students must
determine the missing words. This 5-minute cloze sentence activity can be used to model appropriate
language and notation, as well as check students’ recall of definitions and interpretation of the range.
Resources:
A flip chart with five pages or a display board or a simple whiteboard/blackboard
Activity:
•
Write the following five statements, one on each page of your flip chart, if possible.
The set of output values for a function is called the ____________ of the function.
A ____________ mapping or a ____________ mapping is called a function.
A many-one function is one where more than one element of the ___________ is mapped to a single
element in the ___________.
A function that has an inverse function must be a ____________ function.
The range of the function f (x) = x 2, where -5 , x < 5, is ____________.
•
Write or pin up the following key statements somewhere in the room where they are easy to see.
•
•
•
•
•
domain
0 , y < 25
input
output
25 , x < 25
one-one
one-many
25 , y < 25
inverse
mapping
−5 , x < 5
many-one
range
0 < f (x) < 25
composite
Ask students to suggest the best word or phrase to complete each blank.
Some words are needed more than once. You may wish to tell students this or let them deduce it.
When suggestions are made, discuss how good they are and whether there are any better options.
Use this as an assessment for learning activity to check knowledge.
The final statement will check that
• students know how to write a range correctly, using y or the function name f or the function
rule f (x)
• students are confident that the least value of the domain does not always give the least value
of the range. Ask students how they could check this. A simple sketch is the key, of course, but
encourage students to deduce this.
This statement is likely to lead to some good discussion points and be very valuable.
•
If support is needed with recalling the meaning of domain and range, the mnemonic Dr Io could be
mentioned here: Domain range Input output
Answers:
range; one-one, many-one (either order); domain, range; one-one; 0 < f (x) < 25
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
This activity could possibly lead into: a lesson on finding inverse functions, for example, after an initial lesson
about the types of mappings that are in fact functions.
This activity could be adapted: You could type the statements onto a page and print the pages for students
to share, one between two. These could then be kept for use with other classes. Alternatively, each question
could be typed into a PowerPoint presentation.
You could also omit the key statements and let students find these for themselves. This will increase the
length of time the activity takes and may be an option for a later lesson.
It could also have questions that include composite or modulus functions if a revision lesson.
Main teaching ideas
This topic could be taught with or without a calculator.
Mappings are a good introduction to this topic, but not essential to the syllabus. The mappings given in Exercise
1.1 could be done as a whole class or group exercise using technology such as Geogebra to graph the relations
given. Some printable worksheets and teacher notes have been provided with advice about how this can be done.
This is included as part of the Chapter 1 Lesson plan.
The use of bubble diagrams and graphs to consider domains, ranges and validity of functions is essential to
begin with. This approach is highly recommended.
Diagrammatic representations of functions should be used whenever possible. This will help students visualise
what is happening and make connections between the rule for a function and its graph. If possible, try to use
technology. The content of this chapter involves some graph sketching and drawing, but try to limit the amount
they do without technology. When trying to understand the set of values that are the range, for example, you
want them to be interpreting the correct graph and practising their interpretation skills, rather than practising
the drawing of the graph itself. Focus only on drawing graphs when that is the main aim of the lesson. Some of
these ideas will last for more than one lesson. All the suggestions made have assessment for learning activities
embedded within them.
1 Mappings and functions
Learning intention:
Understand the terms: function, domain, range (image set), one-one function, many-one function.
Use the notation f (x) =… , f : x ↦ … .
Explain in words why a given function is a function.
Resources:
•
•
•
•
•
PDF: Chapter 1 Lesson plan
PDF: Chapter 1 Describing Mappings Group Activities: Worksheets 1.1–1.4 and Chapter 1
Teacher notes
PowerPoint 1.1a: Vertical line test
PowerPoint 1.2: Worked examples 1 and 2
Coursebook Exercise 1.2
Description and purpose: The idea of a mapping is something students should have met before. They may
not realise that not all mappings are functions. The idea of a mapping or function needs to be considered
using graphs and the fundamental descriptions investigated. The Chapter 1 Lesson plan and worksheets are
printable/editable resources that have been provided to help with this. There are Teacher notes to support
the group activities. These activities are graded in terms of challenge and use the mappings in Exercise 1.1 of
the Coursebook.
Once students are confident, you can move onto the formal definition of a function. Bubble diagrams and
graphs are essential to their understanding, in most cases. Bubble diagrams to help explain the formal
definition of a function and the Vertical line test for a function are looked at in PowerPoint 1.1a. These
visual approaches are very helpful. PowerPoint 1.2 works through Worked examples 1 and 2 where more
focus is given to the domain and range of a function. This leads nicely into Exercise 1.2 in the Coursebook.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Differentiation:
Support:
•
•
•
•
Use diagrams and keep functions simple to start with.
Keep the concepts simple and methods clear.
Use technology such as Geogebra or Desmos for graphing where possible.
You may need to make sure that your students are using the correct notation in their answers when
finding the range.
Challenge:
•
•
The Describing mappings activity for group 4 is the most challenging.
Questions where the minimum or maximum value of the domain does not give the minimum or
maximum value of the range are more of a challenge. More of these could be devised for students
who have mastered the basics in good time.
Assessment for Learning: Group working and feeding back is expected in the Describing mapping group
activities, and class discussion should naturally arise when working through the worked examples,
with several discussion points indicated in PowerPoint 1.2. See Chapter 1 Lesson plan.
2 Composite functions
Learning intention:
Understand composition of functions including knowing when a composite function can be formed and
finding the domain and range of a composite function.
Use the notation gf (x) meaning g(f (x)) and f 2 (x) meaning f (f (x)).
Resources:
•
•
•
PowerPoint 1.3a: Bubble diagrams and Worked examples 3 and 4
Coursebook Exercise 1.3
PowerPoint 1.3b: Existence of composite functions including finding domains and ranges
Description and purpose: Students need to understand that forming a composite function is a very different
process to forming a product of functions. Again, using a simple bubble diagram to aid understanding is
a very good idea. PowerPoint 1.3a introduces the ideas needed and introduces domain and range before
working through Worked examples 3 and 4 from the Coursebook.
Once students have understood the basic idea of forming the rule for the composite function, they can
start to think about when this can or whether it cannot be done. PowerPoint 1.3b again starts with bubble
diagrams and demonstrates cases where composite functions can and cannot be formed. This should be
considered and discussed with the class as a whole. This then leads into an investigation. If students need
support at this stage, the first few functions of the investigation can be done as a whole class activity.
Students who need less support may be able to access the investigation in pairs or small groups without any
class discussion first. As usual, helpful teaching and learning points are made in the Teacher notes of each
PowerPoint slide. The solutions to the investigation are animated in the PowerPoint and this can then be
used as a useful reference for students for future work or revision. Each composite function is looked at in
turn to see if it exists. For those that exist, the rule, domain and range are then explored. You may wish to
ask your students to investigate the existence and then, together as a whole class, discuss the domain and
range. The answers should confirm the results found and allow the opportunity to discuss any issues.
The graphs of the functions are displayed to confirm the ranges.
Differentiation:
Support:
•
•
•
•
9
Use diagrams and keep functions simple to start with.
Keep the concepts simple and methods clear.
Use technology such as Geogebra or Desmos for graphing where possible.
You may need to make sure that your students are using the correct notation in their answers when
finding the range.
© Cambridge University Press & Assessment 2023
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Challenge:
•
•
•
•
These ideas could be extended into logs and exponential functions.
RISPs, RISP 18: When does fg equal gf ? consolidates/revises the composition of functions.
WolframAlpha has a composite function calculator.
Linking graphs of functions to real-life processes is a useful extension idea and helps students to
understand the fundamental importance of this topic. ‘What’s that graph’ examples can be found
at Nrich.
Assessment for Learning: Group working and feeding back is expected in the investigative task and class
discussion should naturally arise when working through the worked examples, with several discussion points
indicated in both PowerPoints. During the investigative task, the use of rich questions will give you feedback
about how well your students have understood this topic. Listen, too, to the questions your students ask
each other when group working. Is their thinking progressing in the correct direction? If not, can you help
them get back on track without telling them where their thinking is incorrect? Allow students time to think
and explore ideas, especially if you have not worked on some of the investigation as a whole class activity.
3 Inverse functions
Learning intention:
Understand the term inverse functions, including finding domains and ranges.
Explain in words why a given function does or does not have an inverse.
Use the notation f −1 (x).
Use sketch graphs to show the relationship between a function and its inverse.
Resources:
•
•
•
•
•
•
•
•
PowerPoint 1.6a: Inverse functions, existence, domains and ranges
PowerPoint 1.6b: Finding an inverse function and Worked example 7
Coursebook Exercise 1.6
PowerPoint 1.1b: Horizontal line test
PowerPoint 1.1c: Vertical and horizontal line test
PowerPoint 1.8: Class discussion
PowerPoint 1.7: Graphing a function and its inverse and Worked example 8
Coursebook Exercise 1.7
Description and purpose: Students should benefit from working through PowerPoint 1.6a which gives them
the basic idea behind an inverse function as the function that undoes f, for example. This is done using
bubble diagrams and is a short visual presentation, with plenty of discussion points, that also considers the
links between the domains and ranges of functions and their inverses. Students are usually happy with the
idea of an inverse function as being the ‘undoing’ function. They find the concept of the domain of one
being the range of the other difficult and bubble diagrams will help. It will help if you point out to students
that an inverse function must also obey the definition of a function. The name should suggest that an inverse
function is also a function. Some students do not notice this.
At this point, students should be ready to think about the process needed to find the inverse function.
PowerPoint 1.6b looks at this process using both ‘swop and rearrange’ and ‘rearrange and swop’ methods.
If you prefer to rearrange and then swop, you may wish to reinforce the final step with your students as
they often forget to swop variables and have an incomplete method. Encourage them to write ‘swop and
rearrange’ or ‘rearrange and swop’ for each question and then they have a point of reference to try to avoid
this. The Coursebook uses the swop and rearrange approach to find the inverse. The alternative order has
also been included in the relevant PowerPoints to allow you to choose which you prefer for your class. It is
recommended that you choose one approach for finding the inverse and use that without reference to the
other. This should avoid confusion. You may need to use the ‘Go to slide…’ or ‘See all slides’ function of the
PowerPoint to omit any unnecessary method slide. Exercise 1.6 will give students plenty of practice at the
method they have just explored. Some of these questions also require knowledge of composite functions.
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Once students understand this you could look at the horizontal line test for functions using PowerPoint
1.1b. It is important that this test is applied to functions and not mappings, so students may need to use the
vertical line test too to check that the relation they are working with is a function before checking to see if it
has an inverse function. The horizontal line test leads neatly into the class discussion in section 1.6. It can be
used as a starting point and the discussion should follow.
Please note that the horizontal and vertical line tests have also been included in a single PowerPoint, 1.1c,
as well as separately (the content for the tests is the same). This can be used for review at the end of the
chapter if you prefer, or as a student handout.
Once students have a good understanding of x becoming y and y becoming x for functions and inverse
functions, they should be ready to consider the relationship between the graph of a function and its inverse.
The ideas leading into Exercise 1.7 of the Coursebook are animated in PowerPoint 1.7, along with Worked
example 8. The class discussion in this section is animated in PowerPoint 1.8.
Differentiation:
Support:
•
•
•
Use diagrams and keep functions simple to start with.
Keep the concepts simple and methods clear.
Keep to one approach and avoid confusion.
Challenge:
•
Developing the idea of self-inverse functions mentioned in section 1.7 of the Coursebook,
using symmetry to generate such functions. Can they find, for example, two linear and one nonlinear self-inverse functions?
Assessment for Learning: It will be important to allow students the time to have meaningful discussions
about the concepts they are learning. The PowerPoints provided all offer the chance for meaningful class
discussion and this is important. The discussion points are a mixture of closed and open questions to allow
your students the chance to develop ideas for themselves. This will provide them with a better foundation for
later recall as their understanding should be deeper. Asking students to explain their reasoning can also give
you insight into their level of understanding and identify any issues that need to be addressed.
4 Modulus functions
Learning intention:
Understand the relationship between y = f (x) and y = | f (x) |, where f (x) is linear.
Solve graphically or algebraically equations of the type | ax + b | = c and | ax + b | = cx + d.
Resources:
•
•
•
•
PowerPoint 1.4: Class discussion and Worked example 5 including graphs
Coursebook Exercise 1.4
PowerPoint 1.5: Worked example 6 and further examples, including Worked example 5 with parts a and
d solved graphically
Coursebook Exercise 1.5
Description and purpose: Modulus functions can be looked at later if you wish. You may wish to return to it
after looking at Chapter 2, for example, if you have not covered that yet.
The class discussion in this section is considered in PowerPoint 1.4. This is an excellent starting point for
thinking about modulus functions.
PowerPoint 1.4 also includes a graphical check of solutions to the modulus equations in Worked example 5.
Students may find disregarding the correct extraneous solutions easier if this approach is used. It is offered
as an alternative to the numerical check given in the Coursebook. Even if students have not yet formally
studied the drawing of the graphs, looking at them and linking them to the equations should be valuable
and enrich their understanding of the connections between the graphical and algebraic approaches.
The graphical approach to solving equations involving modulus functions is mentioned in section 1.5 of the
Coursebook. You may wish to look at drawing the graphs of these functions before solving the equations,
as these topics can be studied in either order.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PowerPoint 1.5 includes an animation of the example in section 1.5 plus the derivation of the modulus
graphs needed for the graphical approach to the examples in section 1.4, with part a and part d being solved
using a graphical approach. It is important that students understand when to sketch and when to draw
accurately. This point is also emphasised in this PowerPoint. Part c is a sketch of a quadratic function.
This can be used if Chapter 2 has already been studied. This leads nicely into Exercise 1.5 of the Coursebook.
Differentiation:
Support:
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•
•
Thinking that for y = | f (x) | the values of x cannot be negative is a very common problem. This may
be prevented by starting with a graphical approach.
Use technology to start with to get this point across, but here the focus should be on the drawing of
the graphs, so try to limit technology and encourage students to draw here.
Plotting points may help students who need support to understand the need to reflect part of the
graph of y = f (x) in the x-axis.
Challenge:
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•
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Using graphs to solve inequalities.
These ideas could also be extended into the modulus of quadratic, cubic and trigonometric
functions.
The challenge question in Exercise 1.5 can be neatly drawn and demonstrated using Geogebra.
Students who wish to extend their studies beyond level 3 qualifications may benefit from the stretch
in thinking that they need to make here.
Assessment for Learning: Again, it will be important to allow sufficient time for students to take in the
information given to them and process it. They will have a better understanding of what they are doing if
they are allowed to work with graphs. There are many opportunities for discussion using the discussion
points in the PowerPoints provided.
Review activities
1 Mapping challenge revisited
Go back to the Mapping challenge starter and ask students to explain why each answer is what it is.
2 Complete the line 2
As the starter activity Complete the line but with five similar questions and no key statements given.
The new questions are:
The set of __________ values for a function is called the domain of the function.
A __________ mapping is a function that does not have an inverse.
A __________ function passes the horizontal line test.
A __________ function passes the __________ line test but does not pass the __________ line test.
The range of the function f (x) = x4, where -1 , x , 2, is __________.
Answers:
input (or other similar wording); many-one; one-one; many-one, vertical, horizontal; 0 < y , 16
This activity could possibly lead from: a lesson studying inverse functions.
This activity could be adapted: You could reuse the initial five statements from the starter activity here if you
wanted to. This would be a shorter activity and should help with modelling of language and reinforcement
of ideas. It could also have questions that included composite or modulus functions if a revision lesson.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Homework ideas
1 Coursebook: Mappings and functions, Exercise 1.2
Completion of this exercise should give students a good amount of practice of testing whether a mapping
is a function and of finding the range of a function. It also allows you the opportunity to check that the
correct notation is being used for the range.
2 PowerPoint 1.3c
Completion of the questions in this PowerPoint should measure the competence of students in checking
the existence of composite functions as well as finding the domain and range of composite functions they
have formed. This PowerPoint has two versions. The first version has no model answers included, but it does
have some hints in the Teacher notes for each slide. These can be removed if you do not want to give any
hints at all. The second version includes animated answers and is very supportive of those who need greater
modelling of what is needed. Again, hints, and details of what each animation will reveal, are included in the
Teacher notes for each slide. As well as a useful homework tool, this PowerPoint can be used as a revision
exercise, for self-study, for further practice in class or as part of a bank of resources students can access at
any point throughout the course when needed. This practice material is also available as a PDF file in case
technology is not available to your students.
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