Mathematical Studies Standard Level for the IB Diploma
Scheme of work – Topic 6: Mathematical models
Coursebook chapters 17–19
Introduction
This scheme of work offers an example route through the specification with suggestions of activities and discussion points that you could consider along the way. For each
chapter there are references to the associated PowerPoint file and interactive GeoGebra files, as well as to relevant websites and video clips. The PowerPoint files contain a
number of discussion slides that raise Theory of Knowledge questions; students should be encouraged to talk about these both in maths lessons and during specific Theory of
Knowledge lessons. The scheme of work, while not intended to cover the syllabus in full, does aim to provide a framework that you can supplement and adapt with your own
activities and ideas.
Note: Italic text describes suggested uses of the materials referenced.
The GeoGebra files referenced in this scheme were created using GeoGebra version 4.2.31.0. Some functions, such as check-boxes, were introduced in
this version; so if you have an older version of the software, you will need to download the latest version from the GeoGebra website to be able to use
these resources to their full potential.
Key
[ppt]
[tt]
[ggb]
[V]
[www]
[TOK]
[GDC]
PowerPoint activity
textbook exercise
GeoGebra activity
video link
useful website
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Theory of Knowledge activity
GDC question and worked answer
Copyright Cambridge University Press 2014. All rights reserved.
Page 1 of 8
Mathematical Studies Standard Level for the IB Diploma
Topic 6:
Mathematical models
20 hours
Aims: To understand and identify the characteristics of some common functions and be able to confidently analyse them, sketch their
graphs and solve problems using them
To learn how to use a function to model a situation and analyse the function to make predictions about the situation
Phase
Chapter 17:
Functions
and graphs
Estimated
time
allocation
5 hours
Focus from
the text
What is a
function?
Learning objectives
Activities
Links
Understand the
meaning of a
function given in
different forms and
what the domain and
range are for a
function
[ppt] [TOK] Chapter 17 slides 2–3: Mathematics in nature
This activity could be used in a number of places in the
course but, since it raises the question of whether maths
governs reality or the other way around, it fits particularly
well with this topic on mathematical models. Images
showing random events as well as more regular patterns
and structures are used to generate discussion – no
definite answers here but should stimulate some
interesting discussions.
[www] http://www.mathsisfun.com/sets
/function.html
[tt] Page 495 Exercise 17.1 ‘Determining the validity of a
function’
Functions in
more detail
Be able to find
values relating to
functions;
be able to determine
the domain and range
of a function
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 498 Exercise 17.2 ‘Evaluating functions for
different values of x’
[www] http://functions.wolfram.com/
The Wolfram functions site
[tt] Page 503 Exercise 17.3 ‘Finding the domain and range
of a function’
[www] http://www.khanacademy.org
[ggb] ch17 domain and range
This interactive file allows you to set a function which is
either linear or quadratic and then determine the domain
and range for the function. Use the check-boxes to choose
[www] http://www.mathsisfun.com/sets
Page 2 of 8
/math/algebra/algebra-functions
/domain-range-codomain.html
Page explaining domain and range
Mathematical Studies Standard Level for the IB Diploma
the type of domain: left, right or between; then either one
or two blue points will appear on the curve, and you can
move them along the curve to see the corresponding
domain and range. The domain will be shown as a red
(left), blue (between) or green (right) line segment on the
x-axis; the range will be shown as a violet line segment on
the y-axis. You can toggle the domain and range on and
off independently.
Rational
functions
Drawing
graphs and
diagrams
Understand the
nature of rational
functions and be able
to find the vertical
asymptote of such
functions
Understand the
difference between a
sketch, a plot and a
drawing;
be able to sketch,
plot and draw graphs
[tt] Page 508 Exercise 17.4 ‘Finding asymptotes of
rational functions’
/Asymptote.html
Page explaining asymptotes
[ggb] ch17 rational
This interactive file shows the general equation of a simple
rational function as defined in the chapter. Each
coefficient can be changed using a slider to explore its
effect on the graph. This would be suitable for use as a
short discovery activity, where students could be asked to
explore and deduce the effect on the graph of changing
each of the numbers a, b and c.
[GDC] Function
[ppt] Chapter 17 slides 7–8: Key points of a function
This summarises the important features that students need
to label on their sketches and can be used to remind
students whenever they need to sketch graphs.
[tt] Page 512 Exercise 17.5 ‘Drawing or sketching graphs
and using them to determine domains and ranges’
[tt] Page 516 Exercise 17.6 ‘Finding values for functions
and using their graphs to solve equations’
[ppt] [TOK] Chapter 17 slides 4–6: Do graphs have
meaning in themselves?
Copyright Cambridge University Press 2014. All rights reserved.
[www] http://mathworld.wolfram.com
Page 3 of 8
Mathematical Studies Standard Level for the IB Diploma
This activity asks students to suggest context for a graph
without any markings or labels. What could it be about?
The ‘Show’ button reveals the scale and the quantities
measured along the axes, but it raises an interesting
question about mathematics in real context as opposed to
pure maths.
Review of
Chapter 16
1 hour
Geometry of
threedimensional
solids
Secure the skills of
solving problems
involving 3D shapes
by undertaking past
paper-style questions
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 488 Mixed examination practice
Page 4 of 8
Mathematical Studies Standard Level for the IB Diploma
Phase
Chapter 18:
Linear and
quadratic
models
Estimated
time
allocation
5 hours
Focus from
the text
Linear
models
Quadratic
functions and
their graphs
Learning objectives
Activities
Understand the
nature of linear
functions and be able
to use them to
analyse situations
modelled by a linear
function
[tt] Page 526 Exercise 18.1 ‘Using and understanding
linear models’
Know the properties
of a quadratic
function and its
graph;
be able to find
characteristics of a
quadratic function
from either the graph
or the equation
Links
[ggb] ch18 linear
In this simple interactive file, you can use sliders to
change the values of the gradient and y-intercept to see
how this affects the equation in both forms. It is the
reverse version of the GeoGebra file ‘ch14 equation of a
line’ in that it allows you to control the parameters
directly. It might be useful to highlight this difference by
showing both files side by side.
[ppt] [TOK] Chapter 18 slides 2–3: The nature of curves
This provides a visual stimulus to discuss the difference
between straight lines and curves. In addition, it
challenges students to think about what makes a curve
curved. This links to rates of change in Topic 20 and can
be used again at that point.
[ggb] ch18 quadratic
This is a straightforward interactive file which allows you
to manipulate the coefficients of the quadratic function
and see what effect this has on the graph’s position and
shape. Like many of these GeoGebra files, it has been
designed to be used primarily as a student discovery tool;
thus you could set up a structured investigative task for
students to explore the effect of each coefficient. Equally,
though, you could use this activity to summarise the effects
for the class.
[ggb] ch18 maxima and minima
Like ‘ch18 quadratic’, this file allows the coefficients of
Copyright Cambridge University Press 2014. All rights reserved.
Page 5 of 8
[www] http://www.mathsisfun.com
/quadratic-equation-solver.html
[www] http://www.mathsisfun.com
/algebra/functions-maxima-minima.html
Mathematical Studies Standard Level for the IB Diploma
the quadratic function to be changed, but in addition to
drawing the graph it identifies the associated maximum or
minimum point. It can be used to demonstrate how
negative values of ‘a’ (the coefficient of x2) give maximum
points while positive values give minimum points; but
students could also investigate how changing ‘b’ or ‘c’
affects the position of the maximum or minimum point.
One question for students to explore would be to relate the
values of a, b and c to the coordinates of the turning point.
[ggb] ch18 symmetry
This is an adaptation of ‘ch18 quadratic’ that displays the
equation of the line of symmetry. The line of symmetry is
shown as a red line, which remains visible as you adjust
the coefficients of the quadratic function using the sliders.
[tt] Page 532 Exercise 18.2A ‘Understanding quadratic
functions and their characteristics’
Quadratic
models
Be able to use
quadratic models to
solve problems in
context and make
predictions
[tt] Page 535 Exercise 18.2B ‘Finding intercepts of
quadratic functions’
[ppt] Chapter 18 slides 4–5: Quadratic functions
This summarises the key features of quadratic functions
and would provide a useful way of reminding students of
previous learning prior to its application.
[tt] Page 540 Exercise 18.3 ‘Using and understanding
quadratic models’
Review of
Chapter 17
1 hour
Functions
and graphs
Secure the skills of
using functions and
their graphs by
undertaking past
paper-style questions
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 519 Mixed examination practice
Page 6 of 8
Mathematical Studies Standard Level for the IB Diploma
Phase
Chapter 19:
Exponential
and
polynomial
functions
Estimated
time
allocation
6 hours
Focus from
the text
Exponential
functions and
their graphs
Learning objectives
Activities
Links
Understand the
nature and general
form of exponential
functions;
be able to identify
the horizontal
asymptote from the
equation or by using
your GDC
[ppt] Chapter 19 slides 2–3: Exponential functions
This summarises the properties of exponential functions
with reference to the graph, highlighting the y-intercept,
horizontal asymptote and growth of the curve. Also shown
is the effect of a negative power on the graph. This
summary can be used after students have explored and
identified these key features using the interactive
GeoGebra file.
[V] http://youtu.be/Pt2gqODYuAU
14-minute video on exponential
functions
[ggb] ch19 exponential
This interactive file allows students to vary the coefficients
of an exponential function and observe the changes to the
graph of the function. The horizontal asymptote is shown,
and there is a check-box which can be used to switch the
function and graph to the negative-power version.
Exponential
models
Polynomial
functions
Be able to use
exponential models
to analyse situations
and make predictions
Understand how
quadratic functions
can be extended to
higher-power
functions;
understand the
concepts of maxima,
minima and turning
points in relation to
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 551 Exercise 19.1 ‘Finding values and key
features of exponential functions’
[tt] Page 556 Exercise 19.2 ‘Using and understanding
exponential models’
[GDC] Population growth
[ppt] Chapter 19 slides 4–5: Polynomial functions
This shows a sequence of simplified images of higherpower polynomial graphs, to prompt students to reflect on
the impact the power has on the shape of the curve.
Although there are obviously many variations of curve
shape, this activity intentionally focuses on helping
students appreciate that the number of turns in the curve
is, at most, one less than the power.
Page 7 of 8
[www] http://www.mathsisfun.com
/algebra/polynomials.html
Mathematical Studies Standard Level for the IB Diploma
the graphs of
polynomials
Review of
Chapter 18
1 hour
Modelling
with
polynomial
functions
Be able to use your
GDC to analyse
polynomial models
and make predictions
Linear and
quadratic
models
Secure the skills of
using linear and
quadratic functions
to solve problems by
undertaking past
paper-style questions
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 561 Exercises 19.3 ‘Finding the key features of
polynomial functions’
[GDC] Multiple functions
[tt] Page 564 Exercise 19.4 ‘Using and understanding
polynomial models’
[tt] Page 543 Mixed examination practice
Page 8 of 8
[www] http://www.mathsisfun.com
/algebra/mathematical-models.html