Macmillan
Series titles
www.macmillan.com.au
Active Maths 7
Australian Curriculum edition
ISBN 978 1 4202 3063 5
Active Maths 8
Australian Curriculum edition
ISBN 978 1 4202 3065 9
Active Maths 9
Australian Curriculum edition
ISBN 978 1 4202 3067 3
Active Maths 10
Australian Curriculum edition
ISBN 978 1 4202 3069 7
Active Maths 7 Teacher book
Australian Curriculum edition
ISBN 978 1 4202 3064 2
Active Maths 8 Teacher book Active Maths 9 Teacher book Active Maths 10 Teacher book
Australian Curriculum edition Australian Curriculum edition Australian Curriculum edition
ISBN 978 1 4202 3066 6
ISBN 978 1 4202 3068 0
ISBN 978 1 4202 3070 3
GE
S
AM
The program has a simple and flexible design, enabling it to successfully
complement any other Mathematics resource being used. It is written
by teachers for teachers, and has been successfully trialled and used
throughout Australia.
PA
The teacher book features:
££introductory material on the topics in all books, showing alignment
with the Australian Curriculum
££simple suggestions on ways the program can be used in your school
££a quick teacher reference guide that includes a list of:
− skillsheets and a breakdown of concepts covered
− investigations, with descriptions and problem-solving behaviours
highlighted
− technology tasks, with descriptions and details of the
technologies used in each task
££all student tasks from the student book, with answers included for
easy marking
££a simple record sheet that can be photocopied to record individual
student progress
££four certificate templates that can be photocopied and modified to
your school needs.
8
E
Australian
Curriculum
edition
Australian Curriculum edition
Each student book covers the core Mathematics topics taught at that
level, and incorporates skill sheets, investigations and technology tasks.
Active
Maths 8
Active
Maths
S
program featuring a student book and teacher resource book at Years 7
to 10. The series has been revised to address all the content descriptions
and achievement standards contained in the Mathematics Australian
Curriculum.
Active Maths 8 Australian Curriculum edition Homework Program
Active Maths Homework Program is a topic-based homework
Australian Curriculum
PL
Macmillan
m
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Monique Miotto
Tracey MacBeth-Dunn
Australian Curriculum
GE
S
PA
E
PL
AM
Australian Curriculum edition
ACTIVE
MATHS
S
8
200998&201002 ActiveMaths8 ACE ttl pg-CS5.indd 1
200998 AM8 SB.indb 1
m
a
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k
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Series editor and lead author
Monique Miotto
Author
Tracey MacBeth-Dunn
Contributing authors
Ingrid Kemp
Jessica Murphy
Jim Spithill
18/05/12 12:01 PM
22/05/12 12:24 PM
First published 2012 by
MACMILLAN EDUCATION AUSTRALIA PTY LTD
15–19 Claremont Street, South Yarra, VIC 3141
Visit our website at www.macmillan.com.au
Associated companies and representatives
throughout the world.
Copyright © Macmillan Education 2012
The moral rights of the author have been asserted.
All rights reserved.
Except under the conditions described in the
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no part of this publication may be reproduced,
stored in a retrieval system, or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise,
without the prior written permission of the copyright owner.
GE
S
Educational institutions copying any part of this book
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Copyright Agency Limited (CAL) licence for educational institutions
and must have given a remuneration notice to CAL.
Licence restrictions must be adhered to. For details of the CAL licence contact:
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Telephone: (02) 9394 7600. Facsimile: (02) 9394 7601. Email: [email protected]
Author:
Title:
ISBN:
PA
Publication data
Miotto, Monique and others
Active Maths 8: Homework Program (Australian Curriculum edition)
9781420230659
Printed in Malaysia
SA
M
PL
E
Publisher:
Colin McNeil and Peter Saffin
Project editor:
Eve Sullivan
Liz Waud
Editor:
Illustrators:
Paul Lennon (cartoons). Andy Craig and Nives Porcellato (technical diagrams)
Dim Frangoulis
Cover designer:
Text designer: Sunset Digital
Production control: Loran McDougall
Permissions clearance:Elizabeth Sim
Typeset in Times Ten Roman 10/13pt by Sunset Digital and Nikki M Group
At the time of printing, the internet addresses appearing in this book were correct. Owing to the dynamic nature of the internet,
however, we cannot guarantee that all these addresses will remain correct.
The author and publisher are grateful to the following for permission to reproduce copyright material:
Extract from ‘Bottled water—the facts’, Australasian Bottled Water Institute (ABWI), 2008, http://www.bottledwater.org.au. Adapted
with permission from ABWI, 18; Extract from ‘Did you know—bottled water facts’, http://www.gotap.com.au, © Do Something!
2009. Adapted with permission from Do Something!,17; The Geometer’s Sketchpad® name and images used with permission of Key
Curriculum Press, 1-800-995-MATH, www.keypress.com/gsp., 44, 81, 82.
The author and publisher would like to acknowledge the following:
Microsoft Excel screenshots © 2012 Microsoft Corporation. All rights reserved, 8, 9, 19, 20, 31, 51, 69, 89.
While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental
infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful
owner in each case.
200998 AM8 SB.indb 2
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Record sheet
Name: ....................................................... Teacher: ....................................................... Class: ...........
TEACHER FEEDBACK
4
Geometry
5
Indices
6
Integers
53
7
Linear equations and graphs
61
8
Measurement
71
9
Ratio and rates
83
11
21
E
PL
M
SA
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
200998 AM8 SB.indb 3
GE
S
Fractions, decimals and
percentages
PA
3
Technology task
Chance and data
Investigation
2
Skill sheet 4
1
Skill sheet 3
Algebra
Skill sheet 2
Skill sheet 1
1
33
45
Permission is granted for this page to be
photocopied by the purchasing institution.
22/05/12 12:24 PM
GE
S
PA
SA
M
PL
E
(This page intentionally left blank)
200998 AM8 SB.indb 4
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Skill sheet
Macmillan Active Maths 8
Geometry 1
Name: .............................................................................................. Due date: .........../.........../...........
For 1–19, find the value of the pronumerals, giving
reasons.
[Find angle]
72°
[Triangle
properties]
x
10°
3x
p
65°
5
133º
70°
x
g
z
h
x
32°
8.6 m
y
11
115°
110°
[Find angle]
m
65°
SA
[Find angle]
7 cm
M
[Find angle]
10
PL
4
70°
[Triangle
properties]
PA
[Find angle]
E
3
9
y
35°
2
[Find angle]
q
100° x
[Find angle]
GE
S
1
8
12
[Find angle in
parallel lines]
x
z
155°
y
6
[Find angle]
c 75°
a
40°
b
13
[Find angle]
105°
46°
x
7
[Find angle in
parallel lines]
z
x
y
15°
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
33
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14
19
[Find angle in
parallel lines]
m
n
n
[Quadrilateral
properties]
45°
p
p
For 20–24, answer true or false.
15
c
b
The diagonals of a rhombus meet at
20
a
[Quadrilateral right angles.
properties]
4 cm
21
The diagonals of a rectangle meet at
[Quadrilateral right angles.
7 cm
properties]
22
[Quadrilateral
properties]
y
50°
Both pairs of opposite angles in a kite
[Quadrilateral are equal.
24
11 cm
properties]
17
[Quadrilateral
properties]
y
z
x = 26
y = 27
z = [Find angle]
For 28–30, find the value
of the unknown angles in
the following diagram,
giving reasons.
x
25
[Find angle]
[Find angle]
SA
66°
M
x
PL
[Quadrilateral
properties]
For 25–27, find the unknown
angles in the diagram below,
giving reasons.
E
18
Opposite angles in a trapezium are
[Quadrilateral always equal.
properties]
x
115°
a
23
7 cm
PA
16
[Quadrilateral
properties]
The angles of a trapezium add up to 360°.
GE
S
[Quadrilateral
properties]
m
70°
28
110°
[Find angle]
y
29
[Find angle]
30
z
y
x
25°
a
40°
60°
85°
c
b
a = b = c = [Find angle]
Student comment
Guardian comment/signature
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
Teacher feedback
MEASUREMENT AND GEOMETRY
Geometry
34
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Skill sheet
Macmillan Active Maths 8
Geometry 2
Name: .............................................................................................. Due date: .........../.........../...........
For 1–3, answer true or false.
1
[Congruent
figures]
6
[Congruent
figures]
120°
120° 110°
60°
Congruent figures have exactly the same
shape and size.
70°
70°
15 cm
15 cm
[Congruent
figures]
If two shapes are congruent then their
corresponding angles will be equal.
For 7–8, find the value of x in each pair of congruent
figures.
PA
2
GE
S
7
4 cm
8 cm
7 cm
x cm
10 cm
PL
[Congruent
figures]
[Congruent
figures]
If two shapes are congruent then their
areas do not need to be equal.
E
3
M
For 4–6, are the following shapes congruent—yes or no?
8
[Congruent
figures]
SA
4
[Congruent
figures]
1 cm
7
3 cm
3 cm
1 cm
55°
3
7
x
3
5
[Congruent
figures]
6 cm
6 cm
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
35
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For 9–10, refer to the following diagram. The figure
ABCD has undergone transformations to create the
two images.
a On the graph draw the figure
ABCDEF after it has been rotated
90° clockwise around A and label it
A′B′C′D′E′F′.
11
[Transformations
and congruency]
y
5
C
C′
4
b Is ABCDEF ≡ A′B′C′D′E′F′? C′′
3
2
B
D 1
B′
-5 -4 -3 -2 -1 0 1
-1
9
2
3
-2
A
B′′
D′
4
5
6
A′
-3
D′′
7
x
8
12
[Transformations
and congruency]
A′′
b Explain why ABCDEF is congruent
to A″B″C″D″E″F″.
a What transformation of ABCD
produced the image A′B′C′D′?
[Transformations
and congruency]
b Is A′B′C′D′ congruent (≡) to
ABCD? 13
figures.
A″ D″ PL
b What transformation of A′B′C′D′
produced the image A″B″C″D″?
Is A′B′C′D′ ≡ A″B″C″D″? SA
For 11–12, refer to the following diagram.
y
14
Triangles are congruent if they satisfy
[Transformations one of which four tests?
and congruency]
M
c
Answer true or false: Performing the
E
PA
A′ D′ [Transformations transformations, translation, reflection
and congruency] and rotation, always creates congruent
a State the coordinates of the
following points.
[Transformations
and congruency]
GE
S
10
a On the grid above, draw the final
figure after it has been rotated 90°
clockwise around A and then
translated 4 units up.
15
[Congruent
triangles]
8
Name the congruent triangles.
B
5 cm
7
6
5
4
5 cm
40° 7 cm
Y
2
1
-1 0 1
-1
C
2
5 cm 40°
A
F
3
4
5
6
7
8
40°
7 cm
X
3
P
7 cm
A
E
D
Q
C
R
x
Z
B
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
36
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For 16–19, why are each of the pairs of triangles
congruent? Use the reasons SSS, SAS, AAS and RHS.
16
[Congruent
triangles]
For 21–22, refer to the following diagram.
A
5 cm
3 cm
5 cm
α
α
B
D
C
3 cm
Complete the proof to show that
triangles ACB and ACD are congruent.
21
[Congruent
triangle
proofs]
In D and D :
1 ∠ACB = ∠ = 90°
( )
17
[Congruent
triangles]
2 ∠ = ∠ADC ( )
3 is common
\ D ≡ D ( )
GE
S
If AB = 10 cm and ∠BAC = 70°,
complete the following.
22
[Congruent
triangle
proofs]
18
PA
[Congruent
triangles]
b ∠DAC = (corresp. angles in congruent D equal)
PL
E
a AD = (corresp. sides in congruent D equal)
19
[Congruent
triangle
proofs]
5 cm
SA
20
B
M
[Congruent
triangles]
For 23–25, refer to the following diagram.
A
23
[Congruent
triangle
proofs]
D
A
B
4 cm
C
2 cm
D
m
40°
C
c
8 m
75°
5 cm
E
Complete the proof to show that DABC
is congruent to triangle DDEC.
4 cm
7c
2 cm
Complete the proof to show that
DABC ≡ DDEC.
In D and D :
1 ∠ACB = ∠ ( )
2 ∠ = ∠CED
( )
3 = ( )
E
In DABC and DDEC:
1 AC = (given)
\ D ≡ D ( )
2 = CE ( )
3 ∠ACB = ∠DCE
( )
\ D ≡ D (SAS)
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
37
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24
[Congruent
trianges]
Complete the following.
28
[Congruent
triangles
proof]
a CD = (corresp. sides in ≡ D equal)
b CE = ( )
25
[Congruent
trianges]
Find ∠DCB and ∠DBC, giving reasons.
Complete the following.
29
a ∠BAC = (angle sum D)
[Construct
congruent
triangles]
b ∠CDE = ( )
Use a ruler and protractor to construct
a triangle with angles 40° and 35° and a
side 4 cm long.
A
C
D
B
26
Mark the following information onto
the diagram:
∠BDA = 31°, ∠BAD = 42°, AD = CB
and AB = CD
[Construct
congruent
triangles]
Complete the proofs below to show that
DABD and DCDB are congruent.
M
[Congruent
triangles
proofs]
In DABD and DCDB:
1 2 SA
27
PL
E
[Interpret
information]
Use a ruler and a pair of compasses to
contruct a triangle with side lengths
3 cm, 5 cm, and 7 cm.
PA
30
GE
S
For 26–28, refer to the following diagram.
3 \
Student comment
Guardian comment/signature
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
Teacher feedback
MEASUREMENT AND GEOMETRY
Geometry
38
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Skill sheet
Macmillan Active Maths 8
Geometry 3
Name: .............................................................................................. Due date: .........../.........../...........
For 1–5, prove that opposite sides of a parallelogram
are equal.
1
[Interpret
diagram]
8
[Congruent
triangles
proof]
On the parallelogram ABCD below
draw in the diagonal AC.
B
Prove DEFG ≡ DEHG.
In DEFG and DEHG:
1 FG = (given)
2 3 EG C
\ D
[Congruent
triangles
proof]
9
Prove triangles ABC and CDA are
congruent.
[Interpret
diagram]
In DABC and DCDA:
1 ∠BCA = ∠CAD (alternate angles)
Draw rectangle EFGH again, draw in
diagonal FH and EG. Label the equal
sides and the right angles.
PA
2
For 9–13, show that the diagonals of a rectangle are
congruent.
D
GE
S
A
F
G
E
H
2 ∠BAC = ∠ACD (alternate angles)
3 AC is common
\ DABC ≡ DCDA (AAS)
4
[Conclusion]
BC = What does this prove about the sides of
a parallelogram?
[Conclusion]
Prove DEFH ≡ DHGE.
In DEFH and DHGE:
1 2 11
If ∆EFH ≡ ∆HGE, EG must be equal to
FH because they are sides of congruent triangles.
12
The diagonals of a rectangle are
.
13
What has been proven?
F
E
[Interpret
proof]
[Interpret
proof]
G
[Conclusion]
On the rectangle EFGH above draw in
the diagonal EG.
7
Label the equal sides and the right
angles in the rectangle.
For 14–21, show that the following quadrilateral has
opposite sides parallel.
H
6
[Interpret
diagram]
[Congruent
triangles
proof]
What does it prove about the angles in a
parallelogram?
For 6–8, prove that DEFG = DEHG.
[Interpret
diagram]
10
3 \ D
SA
5
E
AB = (corresponding sides)
PL
[Interpret
proof]
M
3
14
[Interpret
diagram]
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
K
L
J
M
On the quadrilateral JKLM draw in the
diagonal JL and label JK = LM and
∠JKL = ∠JML = 90°
MEASUREMENT AND GEOMETRY
Geometry
39
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[Congruent
triangles
proof]
Prove that DJKL ≡ DJML.
In D D :
1 2 [Interpret
information]
Mark pairs of equal angles in the
congruent triangles on the diagram
above.
17
∠KJL = (corresponding angles)
[Conclusion]
[Interpret
proof]
\ KL is parallel to 27
[Congruent
triangles
proof]
What has been proven?
[Conclusion]
From the given information, what type
of quadrilateral is JKLM?
PL
If you were also told that KL = JK, what
type of quadrilateral is JKLM?
M
[Conclusion]
For 22–30, show that one of the diagonals of a kite
bisects the other.
[Interpret
diagram]
1 2 So, ∠XWY = Draw the diagonal XZ on the kite and
label the point of intersection M. Label
the pair of equal angles on the diagram.
Prove that DWXM ≡ DWZM.
:
In 1 2 28
XM = (corresponding sides)
29
What has been proven?
[Interpret
proof]
[Conclusion]
SA
22
:
3 \
E
20
[Interpret
diagram]
∠KLJ = (corresponding angles)
21
25
26
In PA
19
Prove DWXY ≡ DWZY.
3 \
\ KL is parallel to [Interpret
information]
24
[Congruent
triangles
proof]
16
18
Label WX = WZ and XY = YZ.
[Interpret
information]
3 \ D
[Interpret
proof]
23
GE
S
15
30
[Conclusion]
On the kite WXYZ below, draw in the
diagonal WY.
∠XMW = ∠ZMW (corresponding
angles) = 90° (supplementary angles)
What else does this prove about the
diagonals of a kite?
X
Y
W
Z
Student comment
Guardian comment/signature
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
Teacher feedback
MEASUREMENT AND GEOMETRY
Geometry
40
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Investigation
Macmillan Active Maths 8
Quadrilateral properties
Name: .............................................................................................. Due date: .........../.........../...........
1
Follow the steps below to create an origami kite.
• If you need to make origami squares: Take an A5 sheet and fold
the right corner down to the bottom and crease the paper. Cut
the paper carefully along the edge of the fold as shown.
PA
[Make a
model]
GE
S
The family of quadrilaterals includes a number of special
quadrilaterals: the trapezium, the rhombus and the kite.
These shapes are used in many origami designs. In this
investigation you will use paper-folding to investigate the
special properties of these shapes. You will need some
origami squares or A5 paper (A4 cut in half).
• Take a square and make a triangle fold, folding along one of the
diagonals of the square. If you made your own squares, you will
already have a folded diagonal.
PL
E
• Unfold the paper. Now fold the right and left sides so that their
corners meet the crease line as shown below. You now have a kite.
SA
M
• Fold the kite in half along the diagonal again so that the opposite
vertices meet and then unfold.
a Look at your folded kite. What is true of the length of adjacent sides?
b What is true of angles of the opposite vertices?
c
If you look at the back of your kite you will see a small triangle
formed at the bottom. Fold this triangle over as shown below and
then unfold. What appears to be true of the diagonals of the kite?
d How many axes of symmetry does the kite have?
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
41
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2
[Make a
model]
To make a rhombus, start by making a kite as described previously.
• Now fold the top right edge to meet the crease in the centre as shown below. Fold the other
edge in the same way. Turn over your design. You now have a rhombus.
• Fold the opposite vertices of the rhombus together as shown below,
and then unfold again.
a Based on your folding of the rhombus, what is true of the lengths of
its sides?
GE
S
b Based on your folding of the rhombus, what is true of the opposite
angles of a rhombus?
c
Do the diagonals bisect each other (cut each other in half)? At what angle? PA
d How many axes of symmetry does the rhombus have? e True or false: Opposite sides of a rhombus are parallel. E
To make a trapezium, start by making a kite as described previously.
• Now fold the bottom vertex up as shown below. Then fold the top triangle down as before.
You now have an isosceles trapezium. (Base angles are equal.)
• Fold the trapezium in half and unfold. Now fold the top right vertex down as shown below.
SA
M
[Make a
model]
Based on your answer to e, what must be true of the adjacent angles of a rhombus?
PL
3
f
a True or false: A trapezium has only one pair of parallel sides. b Complete: There are pairs of supplementary angles in a trapezium.
c
Draw a diagram of your trapezium below, showing the pairs of
supplementary angles.
Student comment
Guardian comment/signature
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
Teacher feedback
MEASUREMENT AND GEOMETRY
Geometry
42
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Macmillan Active Maths 8
Technology task—The Geometer’s Sketchpad
Transformations and congruence
Name: .............................................................................................. Due date: .........../.........../...........
The Geometer’s Sketchpad can be used to transform triangles and other shapes on the Cartesian plane. In
this task, you will transform a triangle by reflecting, rotating and translating it and use transformations to
demonstrate that two triangles are congruent.
1
Follow these steps to use Geometer’s Sketchpad to reflect, rotate and translate ∆ABC on the
[The default Cartesian plane.
direction for
• Open a new file, name and save it. Open the Graph menu and select Show Grid. Then select
rotation is
Snap Points on the Graph menu.
anticlockwise.]
GE
S
• Use the Line Segment tool to draw a triangle ∆ABC with coordinates A(–8, 2), B(–6, 6),
C(–2, 4). Label the points using the Text tool.
• Use the Arrow tool to click on each vertex of the triangle and then open the Construct
menu and select Construct Triangle Interior.
PA
• To reflect ∆ABC in the y-axis, double-click the y-axis to mark it as the mirror line. Then select
∆ABC, open the Transform menu and select Reflect. The reflected image should appear.
a What are the coordinates of the reflected image? E
• To rotate ∆ABC anticlockwise about point C by 90°, click on point C to mark it as the
centre of rotation. Select the triangle. Open the Transform menu and select Rotate.
Check Rotate By: Fixed Angle and enter 90. Then click on Rotate.
PL
b What are the coordinates of the rotated image? What are the coordinates of the translated image ? SA
c
M
• To translate ∆ABC +3 units horizontally and +4 vertically, select the triangle. Open the
Transform menu, click on Translate, check Translation Vector: Rectangular, and enter 3
for horizontal fixed distance and 4 for vertical fixed distance. (The default grid is a 1 cm
grid.) Then click on Translate.
∆ABC and its reflected, rotated and translated images are congruent triangles. Paste a copy
of your sketch in the space below.
© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
MEASUREMENT AND GEOMETRY
Geometry
43
200998 AM8 SB.indb 43
22/05/12 12:24 PM
Follow these steps to demonstrate that ∆ABC and ∆DEF are congruent triangles.
2
[An image
•
can be
selected in
the same way
•
as a
constructed •
shape.]
Open the File menu and click on Document options. Click on Add Page, Duplicate: 1. This
should create a new page 2 in your document, which is a duplicate of the first page. On the
new page, delete the transformed images. You should be left with ∆ABC.
Draw a triangle ∆DEF with coordinates D(0, 0), E(6, –2), F(2, –4). Label the points.
Reflect ∆ABC in the x-axis.
• Now experiment with translating the reflected image so that it coincides with ∆DEF.
(You should see a faint image before you click on Translate.)
• Describe the combination of transformations required to demonstrate that ∆ABC and
∆DEF are congruent.
On a new document page, use a combination of reflection and rotation to transform ∆ABC.
Then select the final image and reverse the transformations to return the image to original
location of ∆ABC. Save your file.
M
Try
this!
PL
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PA
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© Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9
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MEASUREMENT AND GEOMETRY
Geometry
44
200998 AM8 SB.indb 44
22/05/12 12:24 PM