Download Scheme of work – Topic 5: Geometry and trigonometry

Mathematical Studies Standard Level for the IB Diploma
Scheme of work – Topic 5: Geometry and trigonometry
Coursebook chapters 14–16
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 9
Mathematical Studies Standard Level for the IB Diploma
Topic 5:
Geometry and trigonometry
18 hours
Aims: To learn to analyse lines in two-dimensional space and identify the relationships between different lines
To be able to calculate lengths in two- and three-dimensional space using trigonometry and Pythagoras’ theorem
To be able calculate measurements such as areas and volumes in two- and three-dimensional space
Phase
Chapter 14:
Equation of a
line in two
dimensions
Estimated
time
allocation
5 hours
Focus from
the text
Learning objectives
The gradient of Be able to determine
a line
the gradient of a line
from coordinates of
points on the line or
from the graph
Activities
Links
[ppt] [TOK] Chapter 14 slides 2–3: Descartes
Providing some historical background as well as a
TOK link, slide 3 raises the question of whether
graphical analysis has the same level of rigour as
algebraic analysis. It offers an opportunity for students
to discuss algebra and for the formality of written
mathematics to be reinforced.
[www] http://plato.stanford.edu/entries
[ggb] ch14 gradient two points
In this simple but useful interactive file, students can
move two points around, and it will show the changes
in x- and y-coordinates between the points as well as
the right-angled triangle thus formed, enabling
students to quickly calculate the gradient. A ‘show
gradient’ button toggles the gradient on and off so that
students can check their calculations. This is most
suitable for use as a discovery activity, in which
students could explore the link between the
coordinates, the change in coordinates and the
gradient to derive a method for working out the
gradient directly from the coordinates of two points.
Copyright Cambridge University Press 2014. All rights reserved.
Page 2 of 9
/descartes-works/
Site dedicated to René Descartes
[www] http://www-history.mcs.st-and.
ac.uk/Mathematicians/Descartes.html
History of maths page for René
Descartes
Mathematical Studies Standard Level for the IB Diploma
[tt] Page 415 Exercise 14.1 ‘Finding and
understanding gradients of straight lines’
The y-intercept
Finding the
equation of a
straight line
Know how to find
the y-intercept
graphically
Be able to find the
equation of a line
both manually and
using the GDC
[ggb] ch14 equation of a line
This interactive file facilitates exploration of the
meaning of gradient and y-intercept. The line can be
adjusted by moving the point marked with a blue dot.
As the line changes, students can attempt to find the
gradient and y-intercept manually. Buttons can be
used to reveal the gradient and y-intercept coordinates
along with two equation forms of the line. This is a
flexible resource that could be adapted to suit many
investigations, especially student discovery tasks. If
necessary, you could provide more structure to the
activity, for example by asking students to look at
particular equations and then find the gradient and
intercept to discover the link between the equation and
the values.
[tt] Page 419 Exercise 14.2 ‘Finding the equation of a
line’
The equation
of a straight
line
Know the different
forms of the equation
of a straight line;
be able to find the
equation of a line
when presented with
information about
the line
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 423 Exercise 14.3 ‘Further questions on
finding the equation of a line, including equations of
parallel and perpendicular lines’
[ggb] ch14 parallel lines
This is an extended version of the ‘equation of a line’
GeoGebra file which contains two parallel lines, so
that the relationship between the equations, gradients
and intercepts can be explored. Again, the blue dot is
the point that can be moved around. The ‘Show values’
button can be used to toggle the information on the
Page 3 of 9
[www] http://www.mathsisfun.com/al
gebra/line-parallel-perpendicular.html
Web page summarising how to find
equations of lines, including parallel
and perpendicular lines
[V] http://youtu.be/0r2XxCwgzSA
10-minute video showing examples of
finding the equation of a line parallel
to another line
Mathematical Studies Standard Level for the IB Diploma
screen, so you could start without the equations and
values. Again, students could be encouraged to
discover the relationships for themselves, or you could
use the file for a demonstration displayed on an
interactive whiteboard or by a projector.
Review of
Chapter 13
1 hour
Drawing a
straight line
graph from an
equation
Know how to draw a
straight line from its
equation, both
manually and with a
GDC
Chi-squared
hypothesis
testing
Secure the skills of
using χ2 hypothesis
tests by undertaking
past paper-style
questions
Copyright Cambridge University Press 2014. All rights reserved.
[ggb] ch14 perpendicular lines
This interactive file enables exploration of
perpendicular lines to deduce the relationship between
their gradients. The lines can be rearranged by moving
the blue dot, and clicking on the ‘Show values’ button
displays the equations together with the gradients and
y-intercepts. The link between the equations is less
obvious in this case, but it can be seen more easily
when one gradient is integer-valued. It would be
sensible to allow sufficient time for students to
discover the gradient relationship for themselves, and
you could set up structured exploration tasks to guide
them along.
[tt] Page 431 Exercise 14.4 ‘Drawing a line given the
[V] http://youtu.be/EUzl8_aQ3xc
equation’
6-minute video demonstration of how
to draw the graph of a line from its
equation
[tt] Page 405 Mixed examination practice
Page 4 of 9
Mathematical Studies Standard Level for the IB Diploma
Phase
Chapter 15:
Trigonometry
Estimated
time
allocation
6 hours
Focus from
the text
Trigonometric
ratios
Learning objectives
Activities
Links
Understand the
relationships
between the sides
and angles in a rightangled triangle;
be able to use these
relationships to find
out missing
information
[ppt] Chapter 15 slides 2–3: Origins
This short discussion activity highlights the historical
significance of right-angled triangles in mathematics.
It could be used as a short research task.
[www] http://www.mathsisfun.com
[ggb] ch15 similarity
This file shows an image of two similar overlapping
triangles and the ratios between their side lengths; it is
intentionally drawn with the same labels for the
common angle and sides as the image on the first page
of Chapter 15, so that students or the teacher can refer
to it. A slider can be used to vary the angle, and by
moving the blue points you can change the size and
orientation of the triangles; this demonstrates that the
three ratios stay equal throughout such changes.
[ggb] ch15 ratios
This interactive file allows you to select the position of
the angle within the triangle in relation to the right
angle; it then shows the trigonometric ratios sin, cos
and tan. You can move the corners of the triangle to
see how this affects the ratios. Tick both A and C to
view both sets of ratios simultaneously. The key point
to notice is that the sides O and A switch around and
that this swaps the values of sin and cos and inverts
tan.
[tt] Page 438 Exercise 15.1 ‘Using right-angled
trigonometry’
Copyright Cambridge University Press 2014. All rights reserved.
Page 5 of 9
/pythagoras.html
This is a summary page which could
be used to discuss the history of how
the relationships between the sides of a
right-angled triangle developed, from
the earliest examples in China and
Egypt through to its naming as
Pythagoras’ theorem.
[www] http://www.mathsisfun.com
/sine-cosine-tangent.html
[www] http://www.mathsisfun.com
/algebra/trig-inverse-sin-cos-tan.html
Mathematical Studies Standard Level for the IB Diploma
Angles of
elevation and
depression
Understand what
these terms mean
and be able to solve
problems involving
angles given in this
context
Harder
Be able to answer
trigonometry
any question
problems
involving rightangled triangles
The sine rule
Be able to use the
sine rule to calculate
missing angles and
sides
The cosine rule Be able to use the
cosine rule to
calculate missing
angles and sides
Area of a
triangle
Be able to find the
area of a triangle
using the sine
formula
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 442 Exercise 15.2 ‘Solving problems
involving angles of elevation and depression’
[tt] Page 445 Exercise 15.3 ‘Further two-dimensional
right-angled trigonometry problems’
[tt] Page 449 Exercise 15.4 ‘Finding sides and angles
using the sine rule’
[www] http://www.mathsisfun.com
[tt] Page 452 Exercise 15.5 ‘Finding sides and angles
using the cosine rule’
[www] http://www.mathsisfun.com
/algebra/trig-sine-law.html
/algebra/trig-cosine-law.html
[ppt] [TOK] Chapter 15 slides 4–5: Generalisation
This highlights the similarity between Pythagoras’
theorem and the cosine rule and can be used together
with the following GeoGebra file to explore the
concept of mathematical generalisation.
[ggb] ch15 cosine vs pythagoras
This interactive file demonstrates the relationship
between the cosine rule for a non-right-angled triangle
and Pythagoras’ theorem for an associated rightangled triangle. The blue corners of the non-rightangled triangle can be moved, and the calculations
will adjust accordingly.
[tt] Page 456 Exercise 15.6 ‘Finding areas of triangles
using the sine formula’
[GDC] Land management
Page 6 of 9
[www]
http://www.mathsisfun.com/algebra/trig
-area-triangle-without-right-angle.html
Mathematical Studies Standard Level for the IB Diploma
Review of
Chapter 14
1 hour
Constructing
labelled
diagrams
Be able to solve
more complex word
problems requiring
the drawing of a
diagram
[tt] Page 460 Exercise 15.7 ‘Further trigonometry
problems, some requiring interpretation and
construction of diagrams’
Equation of a
line in two
dimensions
Secure the skills of
finding and using
equations of straight
lines by undertaking
past paper-style
questions
[tt] Page 433 Mixed examination practice
Copyright Cambridge University Press 2014. All rights reserved.
Page 7 of 9
Mathematical Studies Standard Level for the IB Diploma
Phase
Chapter 16:
Geometry of
threedimensional
solids
Estimated
time
allocation
4 hours
Focus from
the text
Finding the
length of a line
within a threedimensional
solid
Learning objectives
Activities
Links
Be able to identify
right-angled triangles
within a three
dimensional solid
and use these to find
missing lengths
[ppt] [TOK] Chapter 16 slides 2–3: Axiomatic systems
This discussion is nearly an investigation or research
project and could easily be adapted for this purpose.
Slide 3 defines the rules for three-point geometry to
introduce students to axioms and to stimulate a
general discussion about this aspect of mathematics.
The idea could be taken further by reviewing the
properties of four-point geometry and quadrilaterals
and then moving on to other polygons.
[www] http://www.mathsisfun.com
/geometry/solid-geometry.html
[www] http://www.beva.org/math323
/asgn5/nov5.htm
Extension of the idea for the TOK
activity ‘Axiomatic systems’, this
could be used to support further
investigation
[ppt] Chapter 16 slides 6–11: Right-angled triangles in
three-dimensional objects
These slides provide a visual summary of ways of
creating right-angled triangles in solids. They could be
used to introduce or review the properties of these
shapes, and the diagrams are the same as those on the
revision sheet for this chapter.
[tt] Page 473 Exercise 16.1 ‘Finding lengths using
Pythagoras’ theorem in 3D solids’
Finding the
size of an
angle in a
threedimensional
solid
Be able to find
missing angles in
three-dimensional
solids
Copyright Cambridge University Press 2014. All rights reserved.
[ppt] [TOK] Chapter 16 slides 4–5: Mathematical fact?
This presents a challenging concept for students,
which may surprise them as it suggests that the
internal angles of a triangle don’t necessarily add up
to 180°. On slide 5 are two images, one of which
shows a normal triangle on the plane and the other
showing a triangle on the surface of a sphere. The
triangles are drawn accurately, and the angles given
are the actual angles measured electronically
(students will suspect it is a trick, but you can reassure
Page 8 of 9
[www] http://en.wikipedia.org/wiki
/Spherical_geometry
Wikipedia page on spherical geometry
Mathematical Studies Standard Level for the IB Diploma
them that it isn’t). This could lead into a research
project about distances of plane journeys around the
globe and the shortest distance between two points.
[ppt] Chapter 16 slide 12–15: Angle between a line
and a plane
These slides provide a visual summary of two methods
for constructing a right-angled triangle in order to
find the angle between a line and a plane. They could
be used to introduce or review this situation, and the
diagrams are the same as those on the revision sheet
for this chapter.
Review of
Chapter 15
1 hour
Calculating
volumes and
surface areas
of threedimensional
solids
Become familiar
with the formulas for
volume and surface
area of solids and be
able to use them as
required
Trigonometry
Secure the skills of
problem solving
using trigonometry
by undertaking past
paper-style questions
Copyright Cambridge University Press 2014. All rights reserved.
[tt] Page 477 Exercise 16.2 ‘Finding lengths and
angles in 3D solids using trigonometry and
Pythagoras’ theorem’
[tt] Page 485 Exercise 16.3 ‘Finding volumes and
surface areas of 3D solids, particularly cylinders and
spheres’
[tt] Page 463 Mixed examination practice
Page 9 of 9
[www] http://www.learner.org
/interactives/geometry/area.html
Website of interactives for finding
volumes and surface areas