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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 3: Functions (Extension)
Recommended prior knowledge
All of Core and particularly Core 3. Only those parts of the learning objectives or notes and exemplars not included in the core units are itemised, so this document
should be read alongside the core document. It is also important that Core 1, 2, 3 and 7 and Extension 1, 2 and 8 (8.2) have been completed and understood.
Context
There are two Core algebra units and this is the second of two Extension Algebra units. Once Core units 1, 2 and 3 and the other prior experience for Core 3, are
completed this unit can be slotted in at any point. It is probably best taught in parts as it would provide a very length spell of algebra to complete. There are links to
Extension unit 7 and 8 and both could be taught before this unit. Extension 8 (8.2) must be taught first.
Outline
The unit extends the knowledge of Core 3 so be aware that examination questions that relate to aspects of Core 3 may have a greater degree of challenge as they
combine with other areas of mathematics. This unit covers plotting non linear graphs, comparing the properties of two functions when one is plotted, recognition of
cubic, exponential and trigonometric functions, average rates of change and estimated rate of change of a graph, characteristics of exponential growth or decay.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
1
Syllabus ref
3.2
CCSS:
A-REI10
A-REI11
F-IF7
Learning objectives
Construct tables of
values and construct
graphs of functions of
the form axn where a
is a rational constant
and n = –2, –1, 0, 1, 2,
3 and simple sums of
not more than three of
these and for
functions of the type
ax where a is a
positive integer.
Solve associated
equations
approximately by
graphical methods
Suggested teaching activities
Learning resources
The odd pages of the document listed have some useful problems that could
be slotted into this unit.
General guidance
The advise for setting up the tables of values is the same as Core 3.2
www.counton.org/resources/ks3framework
/pdfs/graphs.pdf
Past Paper 43 June 2011 Q5
(syllabus 0580)
Past Paper 41 June 2011 Q7
(syllabus 0580)
Past Paper 42 June 2011 Q4
(syllabus 0580)
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 or f(x).
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x2
25
16
9
4
1
0
1
4
9
16
25
2x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
y
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.
In addition students need to look at solving other problems from related
functions on the graph. This needs to be linked to Core 3.12.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
2
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Although this objective is about plotting graphs, students could use graphing
packages or graphics calculators to check that the plot is reasonable and that
the associated solving problems are correct.
3.4
CCSS:
F-IF9
Compare properties of
two functions each
represented in a
different way
(algebraically,
graphically,
numerically in tables,
or by verbal
descriptions)
Notes and exemplars
e.g. given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
General guidance
The knowledge for this skill is acquired in other sections of this unit and
means that good questioning about the graphs plotted and language used is
honed in those sections.
http://nrich.maths.org/5966
http://nrich.maths.org/6990
Past Paper 43 June 2011 Q9d
(syllabus 0580)
Students need to
1. look at the patterns of rise and fall in the function values in tables
and to describe these to know when turning points, asymptotes, etc
are occurring in the table
2. know the shape to expect from a function when it is plotted
3. have good use of the vocabulary of graphs
3.5
CCSS:
F-IF4
F-IF7
F-BF3
F-LE5
Recognition of the
following function
types from the shape
of their graphs:
cubic f(x) = ax3 + bx2 +
cx + d
exponential f(x) = ax
with 0 < a < 1 or
a>1
trigonometric f(x) =
asin(bx); acos(bx);
tanx
Interpret the key
features of the
graphs—to include
intercepts; intervals
where the function is
v1 2Y01
Notes and exemplars
Some of a, b, c and d may be 0.
http://nrich.maths.org/6506
http://nrich.maths.org/6427
Including period and amplitude.
General guidance
Students need to plot families of graphs using graphing packages or graphics
calculators to note the effect of changing a, b, c and d. Initially keep b, c, d at
zero. Then leep a=1 and two of b, c, d =0 and change the third a step at a
time. Note the effects.
Teaching activities
Using the general guidance ask students to compare families f(x) = ax3, then
f(x)= x3 + bx2, f(x) = x3 + cx and f(x) = x3 + d. using graphing packages or
graphics calculators and to report back on their findings. (Some advice about
scale may be needed). Discuss afterwards the number of turning points so
that they realise that f(x) = ax3 is the special case where the turning points
are all together but that a cubic normally has two turning points (maxima and
Cambridge IGCSE Mathematics (US) 0444
3
Syllabus ref
Learning objectives
Suggested teaching activities
increasing,
decreasing, positive,
negative; relative
maxima and minima;
symmetries; end
behaviour and
periodicity
minima) and a point of inflexion. The effects of the values of a, b, c and d
should be predictable from the work in Core 3.
Learning resources
Complete similar processes for other types of function that are required by
the syllabus.
Use tables of values possibly produced in a spreadsheet to explore these
phenomena in a different way.
Use card matching games of functions, tables and plots.
Repeat a similar exercise with other functions in this objective. When looking
at the Trig functions increase the domain to -360° to 720° or higher to
explore periodicity and amplitude within the discussion about other features.
Discuss this by referring to the turning circle graph in Extension 8 unit 8 (8.2)
which should already have been covered.
The work can be split between groups, so that each group reports on one
element or type of graph.
Use a final card match or true false type activity with a mixture of all the
functions, and tables and algebraic expressions; include the same function
with different range and domain values.
3.7
CCSS:
F-IF6
Calculate and
interpret the average
rate of change of a
function (presented
symbolically or as a
table) over a specified
interval.
Estimate the rate of
change from a graph.
Notes and exemplars
e.g. average speed between 2 points
e.g. use a tangent to the curve to find the slope
General guidance
Relate the average speed between two points as the gradient of the line
joining those two points for a distance time graph. However, if the graph is
the distance from home rather than distance travelled this can become
nonsense. This needs discussing and the distinction being made. See the
definition in the ‘gcseguide’ link.
www.gcseguide.co.uk/travel_graphs.htm
http://nrich.maths.org/6428
http://nrich.maths.org/4851
Past Paper 21 June 2011 Q19
(syllabus 0580)
Rates of change as tangents have to be explained as a concept first.
Clearly there is a link to calculus here but, calculus is not in the IGCSE so the
skill is being treated in a pre-calculus way.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
4
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Choose a quadratic curve with a pronounced change that then slows and
explore the gradients of lines between two points where one end is fixed at
the point and the other slides along the curve until it comes closer and closer
to the point. Do this from both directions. Then look at slopes of lines when
the x values for the two points are equidistant from the x value of the point
and again reduce the gap. Note the way the first two sets of lines change
gradient but that the third produces parallel lines.
Use this as a guide for asking students to plot tangents on a quadratic curve,
and an opportunity to practice ruler and set square constructions of parallel
lines.
Repeat the exercise for a graph like f(x) =x3 + 3x2 for the points -4 to 1 to
show this doesn’t work. Discuss how this might help to estimate the slope of
the tangent at the point.
Clearly there is a link to calculus here but, calculus is not in the IGCSE so the
skill is being treated in a pre-calculus way.
Give students a number of examples to try and then to check with graphing
packages or graphics calculators.
3.8
CCSS:
F-IF8
F-LE1
F-LE3
Behaviour of linear,
quadratic, and
exponential functions
linear f(x) = ax + b
quadratic f(x) = ax2 +
bx + c
exponential f(x) = ax
with 0 < a < 1 or a > 1
Notes and exemplars
Observe, using graphs and tables, that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
Use the properties of exponents to interpret expressions for exponential
functions,
e.g., identify percent rate of change in functions
such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
y = (1.2)t/10, and classify them as representing exponential growth or decay.
http://nrich.maths.org/2677
General guidance
Clearly there is a link to calculus here but, calculus is not in the IGCSE so the
skill is being treated in a pre-calculus way
Link this to the explorations in 3.7 and teach the specific requirements for
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
5
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
percentage rate of change. i.e. by looking at the percentage change from one
gradient to the next whether found from chords or tangents and looking to
see if this is changing in a particular manner. Again students will need
experience of a variety of cases to distinguish them.
3.9
CCSS:
F-LE2
3.10
Construct linear and
exponential functions,
including arithmetic
and geometric
sequences, given a
graph, a
description of a
relationship, or two
input-output pairs
(include reading these
from a table).
Simplification of
formulae for
composite functions
such as f(g(x)) where
g(x) is a linear
expression.
Notes and exemplars
e.g. find the function or equation for the
relationship between x and y
x
–2 0
2
4
y
3
5
7
9
General guidance
Link to Extension unit 2 (2.13) and the recognition of shapes of functions.
Teaching activities
This could be a quiz game ‘What’s my function’ i.e. each team starts with 5
points and as increasing clues to a function are revealed the points scored
for a right answer decreases. Play teams off against one another or get
teams to challenge one another. When using this version if clues are wrong
the other team receives double the points they would have had at this point.
Notes and exemplars
e.g., f(x) = 6 + 2x, g(x) = 7x,
f(g(x)) = 6 + 2(7x) = 6 + 14x
CCSS:
F-BF4
Inverse function f –1.
Notes and exemplars
Find an inverse function.
Solve equation of form f(x) = c for a simple function that has an inverse.
Read values of an inverse function from a graph or a table, given that the
function has an inverse.
Teaching activities
v1 2Y01
www.purplemath.com/modules/fcncomp3.
htm
http://nrich.maths.org/6959
General guidance
Some students become confused that ‘x’ becomes g(x). So when writing out
include the intermediate step f(g(x)) = 6 + 2(g(x).
Ensure that students know that order matters. e.g. in the example above
g(f(x) = 7(6 +2x) = 42 + 14x.
3.11
http://nrich.maths.org/6141
Cambridge IGCSE Mathematics (US) 0444
Past Paper 21 June 2011 Q20
(syllabus 0580)
Past Paper 43 June 2011 Q9a and c
(syllabus 0580)
Past Paper 22 June 2011 Q19
(syllabus 0580)
www.counton.org/resources/ks3framework
/pdfs/sequences.pdf pages 161 and 163
Past Paper 21 June 2011 Q20
(syllabus 0580)
Past Paper 43 June 2011 Q9b
(syllabus 0580)
6
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
The mapping diagrams in the ‘framework document page 161 and 163
provide a route into this topic, as do reversing function machines (see core
unit 2 (2.3)).
3.13
CCSS:
A-REI12
v1 2Y01
Graph the solutions to
a linear inequality in
two variables as a
half-plane (region),
excluding the
boundary in the case
of a strict inequality.
Graph the solution set
to a system of linear
inequalities in two
variables as the
intersection of the
corresponding halfplanes.
4. Geometry—
Extended
Notes and exemplars
e.g. identify the region bounded by the
inequalities
y > 3, 2x + y < 12, y ≤ x.
www.kutasoftware.com/FreeWorksheets/Al
g1Worksheets/Systems%20of%20Inequali
ties.pdf
http://nrich.maths.org/7021
General guidance
Students should plot the equality case using only three points (see Extension
7 (7.5)) and be given guidance on how to code for the line included or
excluded.
Past Paper 41 June 2011 Q9
(syllabus 0580)
Shading the correct side of the line can faze some students who find above
and below the line inadequate descriptions. They should choose a point on
one side of the line but not on it and substitute the x value into the inequality
and compare to their y value to see if they are in the correct region or the
opposite one.
Cambridge IGCSE Mathematics (US) 0444
7