Download Chapter 12: Trigonometry

Chapter 12: Trigonometry
Stages 5.1/5.2 –
Year 9
Unit length: 4 weeks
Strand:
Measurement and Geometry
Substrand:
Rationale:
Trigonometry is a branch of geometry that is very important in fields such as
navigation, surveying, engineering, astronomy and architecture. Students will
use basic trigonometry to find unknown sides and angles in right-angled
triangles.
Teacher:
Outcomes:
Review of earlier work from Stage 4 and selections from:
Right-Angled Triangles (Trigonometry) [Stages 5.1, 5.2◊]
Dates
taught:
A student:
Content
statements:
•
selects and uses appropriate strategies to solve problems
(MA5.1-2WM)
•
provides reasoning to support conclusions that are appropriate to the
context (MA5.1-3WM)
•
selects appropriate notations and conventions to communicate
mathematical ideas and solutions (MA5.2-1WM)
•
applies trigonometry, given diagrams, to solve problems, including
problems involving angles of elevation and depression
(MA5.1-10MG)
•
applies trigonometry to solve problems, including problems involving
bearings (MA5.2-13MG)
•
Use similarity to investigate the constancy of the sine, cosine and tangent
ratios for a given angle in right-angled triangles (ACMMG223)
•
•
Apply trigonometry to solve right-angled triangle problems (ACMMG224)
Solve right-angled triangle problems including those involving direction
and angles of elevation and depression (ACMMG245)
Right-Angled Triangles
(Trigonometry)
________ to ________
Teacher reflection
Strengths
Weaknesses
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1
1
Resources:
Literacy:
•
•
Foundation worksheets
• Drag-and-drop activities
GeoGebra activities
adjacent
bearing
opposite side
tangent ratio
angle of
depression
cosine ratio
similar triangles
trigonometric ratios
hypotenuse
sine ratio
trigonometry
Other
angle of elevation
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1
2
Student book / eBook
12:01 Right-angled
triangles
12:02 Similar
right-angled
triangles:
The ratio of sides
12:03 Tangent ratio
Content dot points
•
identify the hypotenuse, adjacent sides
and opposite sides with respect to a
given angle in a right-angled triangle in
any orientation
•
label sides of right-angled triangles in
different orientations in relation to a
given angle
•
label the side lengths of a right-angled
triangle in relation to a given angle
•
define the sine, cosine and tangent
ratios for angles in right-angled triangles
•
use similar triangles to investigate the
constancy of the sine, cosine and
tangent ratios for a given angle in rightangled triangles
•
•
Register
Technology
Drag-and-drop
Maths terms 12
Resources and suggestions
Foundation worksheet 12:01 – Sides of
right-angled triangles
Review labelling sides according to the
angle given. Use triangles in different
orientations.
GeoGebra
Ratios of the sides
of similar rightangled tringles
use trigonometric notation
find the size in degrees and minutes of
unknown angles in right-angled
triangles
Use Questions 1 and 2 of Exercise 12:02
to discover the relationship between the
sides of a triangle for a 30 angle and 50
angle. Use this to generalise the pattern.
Preview the GeoGebra activity on p. 360.
Read through pp. 361–362 to understand
why tan  = o/a.
Show students how to draw the commonly
used Greek letters and liken them to other
pronumerals previously used.
Foundation worksheet 12:03 – The
tangent ratio
Investigation 12:03 – Tangent ratio
(p. 365)
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1
3
12:04 Finding an
unknown side
•
use a calculator to find the values of the
trigonometric ratios, given angles
measured in degrees and minutes
GeoGebra
Finding an unknown
side using the
tangent ratio
Foundation worksheet 12:04 – Finding
unknown sides using the tangent ratio
Help students draw diagrams representing
the problems described.
Preview the GeoGebra activity on p. 368.
12:05 Finding an
unknown angle
•
•
•
12:06 Sine and cosine
ratios
•
12:07 Finding unknown
sides with sine
and cosine
A Finding a
short side
•
use a calculator to find approximations of
the trigonometric ratios for a given angle
measured in degrees
use a calculator to find an angle correct
to the nearest degree, given one of the
trigonometric ratios for the angle
use a calculator to find the size in
degrees and minutes of an angle, given a
trigonometric ratio for the angle
Drag-and-drop
find the lengths of unknown sides in
right-angled triangles where the given
angle is measured in degrees and
minutes
Drag-and-drop
select and use appropriate trigonometric
ratios in right-angled triangles to find
unknown sides, including the hypotenuse
B Finding the
hypotenuse
12:08 Using sine and
cosine to find an
unknown angle
Finding angles
Ensure students know how to use their
calculators when finding unknown angles.
Foundation worksheet 12:05 – Finding
unknown angles using the tangent ratio
The trigonometric
ratios
Use memory strategies to remember the
relationship between the trigonometric
ratios, e.g. Soh Cah Toa or ‘Some Old
Hare Came A Hopping Through Our
Area’, p. 373.
Drag-and-drop
Finding sides
Foundation worksheet 12:07A – Finding
unknown sides using sine and cosine
GeoGebra
Finding an unknown
side using the sine
and cosine ratios
Preview the GeoGebra activity on p. 378.
Foundation worksheet 12:07B – Finding
the hypotenuse
Preview the GeoGebra activity on p. 381.
Finding the
hypotenuse using
the sine and cosine
ratios
•
select and use appropriate trigonometric
ratios in right-angled triangles to find
unknown angles correct to the nearest
degree
GeoGebra
Using the trig ratios
to find an angle
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1
Preview the GeoGebra activity on p. 384.
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12:09 Miscellaneous
exercises
A Angles of
elevation and
depression
B Compass
bearings
C Other topics
•
solve a variety of practical problems
involving angles of elevation and
depression, including problems for which
a diagram is not provided
•
draw diagrams to assist in solving
practical problems involving angles of
elevation and depression
•
interpret three-figure bearings and
compass bearings
•
interpret directions given as bearings and
represent them in diagrammatic form
•
solve a variety of practical problems
involving bearings, including problems
for which a diagram is not provided
•
draw diagrams to assist in solving
practical problems involving bearings
•
check the reasonableness of solutions to
problems involving bearings
Drag-and-drop
Bearings 1
Bearings 2
Review
Foundation worksheet 12:09B –
Compass bearings
Use compasses as a class to navigate
around the school. Come back to the
classroom and draw the route taken
including the relevant bearings.
Make use of Maths terms 12. (p. 390)
Diagnostic test 12 – Use the right-hand
column to assist in remediation when
errors occur. (p. 391)
Assignment 12A – Exam-style questions
for revision (p. 392)
Assignment 12B – Working
mathematically problems (p. 393)
Assignment 12C – Use this cumulative
revision to review previous topics (p. 394)
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1
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