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 a r g o r P k r o w e m Ho 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 r g o r P k r o w e m Ho 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 Copyright Act 1968 of Australia (the Act) and subsequent amendments, 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 for educational purposes under the Act must be covered by a 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: Copyright Agency Limited, Level 15, 233 Castlereagh Street, Sydney, NSW 2000. 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 22/05/12 12:24 PM 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 22/05/12 12:24 PM 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 200998 AM8 SB.indb 33 22/05/12 12:24 PM 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 200998 AM8 SB.indb 34 22/05/12 12:24 PM 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 200998 AM8 SB.indb 35 22/05/12 12:24 PM 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 200998 AM8 SB.indb 36 22/05/12 12:24 PM 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 200998 AM8 SB.indb 37 22/05/12 12:24 PM 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 200998 AM8 SB.indb 38 22/05/12 12:24 PM 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 200998 AM8 SB.indb 39 22/05/12 12:24 PM [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 200998 AM8 SB.indb 40 22/05/12 12:24 PM 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 200998 AM8 SB.indb 41 22/05/12 12:24 PM 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 200998 AM8 SB.indb 42 22/05/12 12:24 PM 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 E PA GE S SA Student comment Guardian comment/signature © Macmillan Education Australia 2012 ISBN: 978 1 4202 3065 9 Teacher feedback MEASUREMENT AND GEOMETRY Geometry 44 200998 AM8 SB.indb 44 22/05/12 12:24 PM
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