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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 3: Functions (Core)
Recommended prior knowledge
Unit 1, and most of Unit 2 and unit 7 (7.1 and 7.5) and the symmetry work of unit 5. Students should also have some experience of dropping positive and negative
integers into function machine rules and completing a table of values to plot a linear function.
Context
This is the second algebra unit. Since symmetries are required teach after unit 5. Students who are following the extended syllabus will move through this faster but
need to have all these skills in place before working on the extended units, or applying them in other areas of mathematics. This unit can be split into three blocks
• Block 1 – 3.1,3.2, 3.3, 3.6 deal with the function notation and the plotting of functions
7.5 of unit 7 needs to be taught before block 2
• Block 2 – 3.5 general characteristics of functions
• Block 3 – 3.12 the effect of a constant on an existing function
Block 2 and 3 can be taught later in the course to provide variety and to allow one set of ideas to embed and then to be revisited at the beginning of the next block to
consolidate.
Outline
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. The learning
resources give both teaching ideas, summaries of the skills and investigative problems to develop the problem solving skills and a depth of understanding of the
mathematics, through exploration and discussion. This unit covers the vocabulary of functions, plotting functions, recognising types of functions from their graphs
and translating functions. It also links functions to real life problems. The unit could be taught as a whole, but could equally well be taught in small bites scattered
throughout the course, so long as the links are drawn between this and other functions.
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
General guidance
These resource gives developmental steps and ideas for underpinning the
learning which also link functions and mapping diagrams to sequences..
www.counton.org/resources/ks3framework
/pdfs/graphs.pdf
www.counton.org/resources/ks3framework
/pdfs/sequences.pdf
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
3.1
Use function notation
www.purplemath.com/modules/fcns2.htm
CCSS:
F-IF1
F-IF2
Knowledge of domain
and range
Mapping diagrams
General guidance
Define domain and range – ensure this meaning of range is distinguished
from range in statistics.
Distinguish between functions that have infinite domains and ones that only
have meaning for positive domains (e.g. problems relating to hire purchase
turned into functions).
Also note when it makes sense to join points with a line/ curve to indicate all
possible values, and those which should be left as points only because the in
between values have no meaning i.e. Car hire where you can only hire for
complete days.
Mapping diagrams can be a means of creating values for co-ordinate pairs or
arranging information for pattern spotting to find functions.
2x
3.2
CCSS:
A-REI11
A-REI11
Understand that the
graph of an equation
in two variables is the
set of all its solutions
plotted in the coordinate plane
Construct tables of
values for functions of
the form ax + b,
± x 2 + ax + b,
a (x≠0) where a and b
x
are integral constants;
draw and interpret
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The function notation, vocabulary and mapping diagrams should be used
throughout the unit.
General guidance
When setting up tables of values to plot (pupils should understand this often
only gives enough information to plot the function or to look at the most
interesting aspects of a function) some students get lost in moving from x to y
in one step so either encourage them to use mapping diagrams or to add
additional rows below the ones given on the exam paper. They can then sum
back to get y.
x
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-4
-3
-2
-1
0
1
2
3
4
5
25
16
9
4
1
0
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9
16
25
www.purplemath.com/modules/solvquad5.
htm
Past Paper 32 June 2011 Q7
(syllabus 0580)
Past Paper 31 June 2011 Q5
(syllabus 0580)
y
x2
Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
Learning objectives
such graphs
Solve associated
equations
approximately by
graphical methods
Suggested teaching activities
2x
-10
-8
-6
-4
-2
Learning resources
0
2
4
6
8
10
-3
-3
-3
-3
-3
-3
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-3
-3
Encourage students to check there plots or coordinates if the plot does not
produce a line or smooth curve.
Ensure students know that only linear functions are joined with a line and that
curves are joined with a smooth curve, points as crosses not blobs.
Give sufficient opportunities for students to see the symmetry properties of
quadratics and to try for additional data to get an exact maxima and minima
by finding the line of symmetry and hence the x value and substituting to find
the y value.
Explore the case where the function is equal to a value that enables the value
to be read from the graph by intersecting with y=constant.
In the special case when y=0 this can be used to solve a quadratic and
explain why there are two answers.
Also look at cases where a non-linear is intersected by a linear and link to
simultaneous equations.
3.3
CCSS:
A-REI10
F-IF8
Write a function that
describes a
relationship between
two quantities
Notes and exemplars
e.g., C(x) = 50,000 + 400x models the cost of producing x wheelchairs. Write
a function that represents the cost of one wheelchair.
Past Paper 32 June 2011 Q4
(syllabus 0580)
Teaching activities
Obtain some different mobile phone tariffs.
e.g. Number of texts against cost, number of calls against cost, monthly
rentals.
Create functions with t for number of texts and m for number of minutes etc.
Compare different components on graphs and against a student’s likely
usage to decide the best tariff for the individual.
Include in discussion this would be a good deal ...........for but not for....
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Syllabus ref
Learning objectives
Suggested teaching activities
3.5
Recognition of the
following function
types from the shape
of their graphs:
linear f(x) = ax + b
quadratic f(x) = ax2 +
bx + c
reciprocal f(x) = a
x
Notes and exemplars
Some of a, b, c may be 0.
CCSS:
F-IF4
F-IF7
F-LE5
Interpret the key
features of the
graphs—to include
intercepts; intervals
where the function is
increasing,
decreasing, positive,
negative; relative
maxima and minima;
symmetries; end
behaviour
3.6
CCSS:
F-IF5
Relate the domain of a
function to its graph
and, where applicable,
to the quantitative
relationship it
describes
Learning resources
General guidance
This has a link to unit 7 (7.5) and should be taught after 7.5.
Teaching activities
Use graphics calculators or graphing packages to generate many linear,
quadratic and inverse graphs so that pupils can discuss similarities and
differences and discuss, where they are increasing, decreasing, the maxima,
minima and symmetries, and end behaviour.
Make up packs of graphs, tables and function names and ask pupils to match
them (put some rogue ones in so that they have to work out all the intended
solutions and don’t finish off the difficult ones by guesswork).
Notes and exemplars
e.g., if the function h(n) gives the number of person-hours it takes to
assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.
www.royalmail.com/portal/rm/content1?cat
Id=400036&mediaId=400347
Teaching activities
Plotting UK parcel weights for international mail against cost gives a graph
that is different from the functions they have been working with so opens up
the debate. i.e. the plot will be a series of non overlapping ascending
horizontal lines. Discuss how you would code the non-inclusive end of the
lines.
Look at cases where particular ranges are more important than others or
where particular domains are invalid even though the function as a series of
algebraic terms could extend to larger domains and ranges.
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Plotting the base of a rectangle against its area could lead to discussion
about the validity of the point (0,0) i.e. is it a rectangle?
Discuss the cases where a sequence from a tiling pattern has only integer
values for x but the nth rule is like a function an infinite set of values, whereas
for the specific case only positive integers make sense.
3.12
CCSS:
F-BF3
Description and
identification, using
the language of
transformations, of the
changes to the graph
of
y = f(x) when y = f(x) +
k, y = k f(x), y = f(x +
k) for f(x) given in
section 3.5
Look at cases where time can be negative if in the particular case could
describe before a given time or after it and when distance can be negative if it
goes backwards.
Notes and exemplars
Where k is an integer.
http://nrich.maths.org/6961&part=
http://nrich.maths.org/7120
Teaching activities
Use graphic calculators or a graphing package to explore the effects and ask
students to generalise.
Give them pre-drawn graphs and ask them to model the function, either
graphically or from the table of values obtained from the coordinates read
from the graph.
Once again unit 7(7.5) should precede this element of work.
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