New Syllabus Mathematics 8th Edition
Nautilus Shell
Consultant Dr Yeap Ban Har PhD, MA, MEd, PGDE (Dist), BSc Consultant and Author Dr Joseph B. W. Yeo PhD, MEd, PGDE (Dist), BSc (Hons) Authors Dr Choy Ban Heng PhD, MA, BSc (Hons) • Teh Keng Seng BSc, Dip Ed Wong Lai Fong PGDE, MEd, BSc • Sharon Lee PGDE, BSc Ong Chan Hong PGDE, BSc (Hons)
Textbook
1B
Secondary
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©SHINGLEE PUBLISHERS PTE LTD All rights reserved. No part of this book 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 permission of the Publishers. First Published 1982 Reprinted 1983, 1984, 1985, 1986 Second Edition 1987 Reprinted 1987, 1988, 1989, 1990, 1991 Third Edition 1992 Reprinted 1992, 1993, 1994, 1995 Fourth Edition 1997 Reprinted 1997, 1999 Fifth Edition 2001 Reprinted 2002, 2003, 2004, 2005, 2006 Sixth Edition 2007 Reprinted 2007, 2008, 2009, 2010, 2011, 2012 Seventh Edition 2013 Reprinted 2013, 2014, 2015, 2016, 2017, 2018, 2019 Eighth Edition 2020 Reprinted 2020
ISBN 978 981 32 4540 2
ACKNOWLEDGEMENTS We are grateful to Mr Tan Teck Hock for designing the interactive geometry templates. Images and links produced from www.geogebra.org used with permission from GeoGebra The Geometer’s Sketchpad® name and images used with permission of Key Curriculum Press, www.keycurriculum.com/sketchpad All licensed images purchased under standard license agreement with www.shutterstock.com While every effort has been made to trace the copyright holders and obtain permission for material reproduced in this book, if there has been an oversight, the Publisher will be grateful to hear from anyone who has not been appropriately acknowledged and to make the corrections in future reprints or editions of this book.
MINISTRY OF BY E
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Printed in Singapore
–2 e fr o m 2020
0
PREFACE think! Mathematics is an MOE-approved textbook specially designed to provide students valuable learning experiences by engaging their minds and hearts as they learn mathematics. The features of this textbook series reflect the important shifts towards the development of 21st century competencies and a greater appreciation of mathematics, as articulated in the Singapore mathematics curriculum and other international curricula. Every chapter begins with a Chapter Opener and an Introductory Problem to motivate the development of the key concepts in the topic. The Chapter Opener gives a coherent overview of the big ideas that will frame the study of the topic, while the Introductory Problem positions problem solving at the heart of learning mathematics. Two key considerations guide the development of every chapter – seeing mathematics as a tool and as a discipline. Opportunities to engage in Investigation, Class Discussion, Thinking Time, Journal Writing and Performance Tasks are woven throughout the textbook to enhance students’ learning experiences. Stories, songs, videos and puzzles serve to arouse interest and pique curiosity. Real-life examples, applications and Problems in Real-World Contexts (PRWC) serve to influence students to appreciate the beauty and usefulness of mathematics in their surroundings. Underpinning the writing of this textbook series is the belief that all students can learn and appreciate mathematics. Worked Examples are carefully selected, questions in the Reflection section prompt students to reflect on their learning, and problems are of varying difficulty levels to ensure a high baseline of mastery, and to stretch students with special interest in mathematics. The use of ICT helps students to visualise and manipulate mathematical objects with ease, hence promoting interactivity. Coding opportunities are included to cater to students with coding knowledge. To help students who are new to coding, we have included three sections on Invitation to Code. We hope you will enjoy the subject as we embark on this exciting journey together to develop important mathematical dispositions that will certainly see you through beyond the examinations, to appreciate mathematics as an important tool in life, and as a discipline of the mind.
PREFACE
III
P A G E
KEY FEATURES
CHAPT
12
ER
res
igu Plane F Area of d n a r te Perime
rs . Designe any of us yoh iliar to m c Toa Pa ms are fam The iconi body shapes. d trapeziu a g an d in s an m est ra head inter d, parallelog dragon to create nstructe s, circles, th a large operties d was co ature wi the as triangle ial geometric pr playgroun to know tures a cre ures such spec hen the needed figure 1979, fea Plane fig es with pillars. W rtion to ers likely built in ed po ese figur ild re, th ap ch bu e po e ea -sh us ga th often ed. pezium d in Sin d area of tra uir an un r by ro req s ete yg ted pla perim aterial wa dragon gs suppor m h rin uc el m rful ste res of out how of colou o measu ea are tw rimeter r and ar figure: pe Perimete a plane dary dary of the boun of the boun gth of space es the len amount quantifi tifies the ea quan undary. while ar in the bo th wi d enclosed how to fin will learn ms apter, we rallelogra In this ch area of pa eter and the perim ziums. and trape
Introductory Problem provides students with a more specific motivation to learn the topic, using a problem that helps develop a concept, or an application problem that students will revisit after they have gained necessary knowledge from the chapter. Introduct Problem
Learning Outcomes help students to be aware of what they are about to study so as to monitor their progress.
es , utcom as squares ing O es (such chapter? Learn ne figur s) rn in this area of pla and trapezium ll we lea es eter and ms What wi ne figur ra rim pla pe log of e lle para find th and area s, circles, • How to rimeter e s, triangle ng the pe in real lif rectangle s involvi cations problem pli ap lve l so es) ve usefu site figur • How to figures ha g compo of plane (includin ea ar d an rimeter • Why pe
ry
B A s. of parallel side least one pair figure with at is a four-sided of D. A trapezium ber ABC num m the nting ws a trapeziu ABCD by cou Fig. 12.1 sho of trapezium find the area (a) Can you C ezium ABCD. of a square units? the area of trap find to ing the area D hod find ther met ways of (b) Use ano the different Fig. 12.1 you learn from use you can (c) What can = BC. How , where AD trapezium? n trapezium? is a special one ezium given area of any give (d) The trap ve to find the e previously e learnt abo g what we hav what you hav eziums usin 2 to m2 and ams and trap to convert cm s of parallelogr find the area d to know how learn how to first, we nee will But we , les. rectang In this chapter triangles and the areas of learnt about vice versa.
12.1
Conversion
Chapter Opener gives students an overview of the topic. It includes rationales for learning the chapter to arouse students’ interest and big ideas that connect the concepts within the chapter or with other chapters.
Two angles are supplementa ry when they Fig. 10.7 show add up to 180° s two examples . of supplementa ry angles.
45° (a)
10.2
ut measures
Big idea abo
Important Results summarise important concepts or formulae obtained from Investigation, Class Discussion or Thinking Time.
Attention ‘Complementar y angles’ and ‘supplementary angles’ are terminologies that describe only two angle s that 90° or 180°. They add up to are not angle properties.
73° (b)
Fig. 10.7
Similar and Further Ques tions Exercise 10A Questions 3(a)– (d), 4(a)–(d)
of units
analyse that we can al objects so (m). or mathematic is the metre rld property ts. its base unit ntify a real-wo in measuremen sures to qua ic measure and d bas mea use a as is are d th that s are use mple, leng ntists n base units 1. Number ties. For exa tes why scie e up of seve these proper your classma units) is mad with (SI s its or compare cus Un Dis of e units. tional System and their bas The Interna ic measures other six bas value? Find out the e the any negative system. can combin value, zero or example, we developed this any positive ths in take s sure measure. For sures of leng mea obtain a new bine the mea seven basic to the com ties per can Can binations of pro we 2. at other com of speed; or sures of two Wh mea . sure e area mea bin of the com sure 3. We can time to obtain ain the mea obt and e to ) anc dist and breadth . Find measures of s (i.e. length measure area t dimension and acres to two differen sure length, you think of? are used to mea units. s and miles measures can ted to the SI , inches, yard they are rela feet how es, Stat and ts ited men plane figures 4. In the Un ts of measure sure area of ut these uni used to mea out more abo 2 Other units 2 ). (km ). are metres (m are kilometres sured in squ the 2 n to say that sroom is mea (mm ) and squ s mo clas a etre com of lim re 2 are mil The floor area mple, it is mo s (cm ), squ ther. For exa are centimetre t of area to ano2 include squ y? we say t from one uni 000 m . Wh m2 . Instead, d to conver 2 000 269 nee 0 720 we of 0.00 es, is ead Sometim 720 km inst a 10-cent coin Singapore is ss section of land area of area of the cro say that the do not usually cm2 . Why? Similarly, we area is 2.69 l res ona ecti of Plane Figu eter and Area that its cross-s
107°
135°
Properties of angles formed (Recap) by intersecti ng
A. Adjacent Fig. 10.8 show
Recap
lines
angles on a str
revisits relevant prerequisites at the beginning of the chapter or at appropriate junctures so that students are ready to learn new knowledge built on their existing schema.
aight line
s two examples of adjacent angl • share a com es on a straight mon vertex, line. Adjacent angles are angl • have a com es that mon side, • lie on opp osite sides of Attention the common side. ∠a and ∠b are not adjacent common common angles in the following exam side vertex ples. • C W a O x A b X a z y B b O common Z no common side vertex common • (a) side shared Y (b) by ∠x and ∠y Fig. 10.8 In primary scho ol, we have lear a b nt the followin g angle property do not lie on opposite sides : Adjacent ang of common side les on a straight line The sum of adja cent angles on a straight line (Abbreviation: is 180°. adj. ∠s on a str. line)
Perim
P A G E
144
CHAPTER
Adjacent ang
les on a
12
1
straight line Given that AOB is a straight line , calculate (a) the value of a, C A
P A G E
80
a° O
49°
(b) ∠COD. O
B
B
°
31.6
A 57.2°
Exterior
D
(a)
Similar and Further Que stions Exercise 11A Questions 3, 4, 7, 8, 13
Practise Now consists of questions that help students achieve mastery of procedural skills. Puzzles are sometimes used for consolidation to make practice motivating and fun.
x° = 76° + 69° (ext. ∠ of ) = 145° ∴ x = 145
2
1.
In the dia gram, AC E and BC are straig D ht lines. (a) Find the (b) Calcu value of y. late ∠CED . A
75°
14°
E
Exterior
In the dia
2.
Similar and Further Que stions Exercise 11A Questions 9–11
3
1.
angle of tri
angle, an
// DE. Ca
d by paral
E.
KEY FEATURES
C
B
gles forme
lculate DA
In the dia gram, BC // DE. Calculate DAE. A B
lel lines an
d transver
sal
A
B
C
67° D
E
138°
Reflection
2.
C 147°
E
R
and Geomet
D 93°
41°
BDE = 67° (vert. opp. ∠s) ABC = BD E (corr. ∠s , BC // DE = 67° ) BCE = 138 ° (alt. ∠s, BC // DE ∴ DAE = ) BCE – AB C (ext. ∠ of ) = 138° – 67° = 71°
72° D
Polygons
In the fig ure, ABC, ADF and are straig BDE ht lines. (a) Calcu late ∠CBD . (b) Find the value of b. E 33° b° F
D A
gram, BC
D 107°
Reflection (a) What is anothe r way to solve for x? Which way do you prefer?
y° C
x° C
E
34°
3
76° 69°
B
(b) ∠CED + ∠CDE = x° (ext. ∠CED + ∠ of ) 107° = 145 ° ∠CED = 145° – 107 ° = 38°
B
Similar and Further Questions follow after Practise Now to help teachers select appropriate questions for students’ self-practice.
IV
A
2
Worked Example shows students how to present their working clearly when solving related problems. In more challenging worked examples, Pólya’s Problem Solving Model is used to help students learn how to address a problem.
P A G E
angle
of triangle In the dia gram, AC E and BC (a) Find D are straig the value ht lines. of x. (b) Calcu late ∠CED . Basic Geometry
C
CHAPTER 10
What is ano ther way to DAE? Wh ich way do solve for you prefer?
In the fig ure, PQ // RT. Calculate QST. P Q 97° 63° 36° S
T
rical Con
structions
CHAPTER
11
103
P A G E
Exercise questions are classified into three levels of difficulty – Basic, Intermediate and Advanced. Questions at the Basic level are usually short-answer items to test basic concepts and skills. The Intermediate level contains more structured questions, while the Advanced level involves applications and higher order thinking skills.
Open-ended Problems are mathematics problems with more than one correct answer. Solving such problems expose students to real-world problems.
at a Li Ting sells it to sells the hine and Ting then fax mac Li a . ys ice bu pr the price e cost 12. Siti 25% on % on th 60 a loss of e sells gain of 25 a pays $3 Nadia at zen. If sh i. If Nadi hine to tage ac 8 per do Sit en m rc $1 for it? m pe at fax fro s it rose Siti pay ss as a she buys uch did rist buys ess her lo at which , how m ch, expr 2. A flo ne ea hi ac .20 m $1 x. There them at for the fax . 8 per bo lling price price. ples at $2 of her se ples are the cost xes of ap of the ap is 35% of ys 200 bo s to box. 15% gerator Raju bu he want fri . if ch re e ea 13 a pl in ples per ap , profit on are 60 ap ng price lling price 80, find 3. The the selli price. ) the se ofit is $2 nd st (ii pr Fi e co n. e th If rotte , t on th cost price tal 80% profi (i) the les at a to earn an r. tical artic frigerato at is 300 iden of the re s 12% on a price th eper buys ticles at bert lose ke ar g Al op 0 in 0, sh 26 . ok sells r $16.5 14. A e remain of the bo 500. He book fo ch of th selling a cost price cost of $1 e selling price. Ea e th st th of co nd e 4. By % at is 50 price. Fi above th the th % e nd ice 20 th Fi pr s . the cost a Expres for $464 . is sold at articles. 80 is sold cost price articles the 260 ge of his iced at $5 each of percenta cklace pr price of ofit as a nt. 5. A ne eper’s pr ge discou een $10 ke ta tw op en be sh rc s pe book a ices his g table at % a selling pically pr a foldin gives a 12 ice and price of blisher ty e cost pr rmarket ofit as a marked 15. A pu of the a possibl The hype isher’s pr 6. The t is $45. sale price the publ $20. Give e ke d at th ar th an rm nd ch hype ok su %. sale. Fi a bo 30 g a is of rin ice du price cost pr discount ount ge of the table. er a disc percenta folding r $700 aft 7% on a is sold fo ount of nditioner e is a disc d co er fin th air 9, iti le, nd oner. 16. An nt is $4 ing a sa , the air co . e discou 7. Dur sale price price of ked price of 12.5% set. If th (ii) the marked the mar television Find the given on d price, is ke ar (i) nt ld at a m ou so % disc re it is (i) the 10 fo t. a be se If er le price (ii) evision condition %, would the sa of the tel of the air 2.5 ly. fore GST. scount of ing clear ount $270 be rther di ur work total am en costs fu e yo ov th e ow d av , fin . 00? Sh icrow ave oven T is at 7% 8. A m still be $7 e microw ers a 16% g that GS th in off r m A fo su d y As pa e , Bran and A ar Ali has to th niversary sive of bers of Br of money its 16 an 391 inclu ms. Mem 17. On . ld for $1 on all ite dget. discount dget is so discount of the ga for $420 nal 14% m ronic ga ice tio ct ite di pr d ele ad an at marke 9. An given an ice of th r bought Find the on arked pr n-membe 7% GST. n of 2.5% nd the m (i) A no mmissio e sale. Fi during th arges a co m if a agent ch ite 0, ty e er 00 m e. op the sa r $650 item. of a hous 10. A pr price of house fo ng price the sale t sells a the selli receives. the agen (ii) Find ission he buys it. ven that of comm e at a (a) Gi member ives a amount fried ric e ce th re d ice he fin seafood arked ng pr 6. of occasion, The Goldescount. The m e plate the selli another orders on % di n Ratio is ve 000. Find 25 ao 2 a H n that de s $1 (b) On Yi er of 18. ion .50. Gi noted by Useeisa cal $9cu which off (a) ric t , the symbol Φ (pr an commiss ur lat or to ed is at 7% resta onounced od fri(b) use. and GST find the values of Φ 2 as ‘phi’). Us10 the seafo e% of the ho a cal per kg It has som y. culator and Φ + price of costs $8 charge of s to to pa e interesti 1. What find the ey he ha do you no a service ng prope value of 1 lls tter which on is se e e m bu H er of of . rties. tice? th amount Φ . What do you thi ixes 2 kg l $6 per kg m ta as sts r to t e co de 1 nk it is eq find th his profi r which s 11. A tra ua es tte = l pr to? bu Φ – Ex of Φ r 250 g. pe with 3 kg .55 ture at $2 price. e the mix Percentag s selling tage of hi a percen
8C
Explanation Questions require students to communicate their explanations in writing and are spread throughout the textbook.
Performance Task consists of mini-projects designed to develop 30 research and presentation skills of students, through writing a report and/or giving an oral presentation.
Search the Internet to find ou Great Py t what oth ramid) or er man-m natural oc in Fig. 9.4 ade struc currences ) have to tures (e.g. (e.g. the do with the an A4-si the nautilus, zed poste Golden Ra a sea cre r. Rememb tio. Presen ature, as er to inc shown t your fin lude som dings on e photos.
R8
CHAPTE
P A G E
Fig. 9.4 1. What do I alread y know ab 2. What out ratios new know that could ledge of ratio have guide my I learnt in learning in this sec this sectio tion? n?
4A
In the figure, AB // CD. (a) List (i) one pair of equal corresponding angles, (ii) one pair of equal alternate angles, (iii) one pair of interior angles which are supplementary.
Similar and Further Questions Exercise 10B Questions 1(a)–(c)
(b) Explain your answer to the following questions. (i) Is ∠e = ∠a? (ii) Is ∠g = ∠i? (iii) Is ∠h + ∠c = 180°?
g
e
b
h
c
i
j
C
k
l
9A
R
P f
A
1.
a
B
d n
m
o
p
Q
D
S
Introduct ry Problem Revisited x
y
Fig. 10.17 When drawing the white lines to define each lot, how do painters ensure that the lines are parallel? Hint: What can you say about ∠x and ∠y?
Corresponding angles, alternate angles and interior angles In the figure, AB // CD. Calculate the values of a, b and c.
R P 48° 61°
B Introductory Problem 4 Looking a°Back b° Revisited D C c° a° = 48° (corr. ∠s, AB // CD) complements the Q S revisits an application-based ∴ a = 48 Chapter Opener and To find value of b: Introductory Problem Let the point of intersection of the lines AB and RS be T. Method 1: helps students internalise later in the chapter. ThisP is R Then ∠BTS = 61° (vert. opp. ∠s) 48° 61° the big ideas that they // CD) ∠s, AB 180° (int. b° + ∠BTS = B absent if the Introductory A T b° + 61° = 180° have learnt in the chapter. b° = 180° − 61° Problem leads directly a°to the b° = 119° D C c° ∴ b = 119 development of a concept. Q S A
Method 2: Then ∠ATS = 180° − 61° (adj. ∠s on a str. line) = 119° b° = 119° (alt. ∠s, AB // CD) ∴ b = 119
P A G E
88
CHAPTER 10
Summary compounds the key concepts taught in the chapter in a succinct manner. Questions are included to help students reflect on their learning. Basic Geometry
There are 14 boys an d 25 girls badmint in a schoo on team. Find the l (i) the ratio of number of boys to (ii) the the numb number er of girls, of girls to players in the total number the team. of 2. Sim plify eac h of the fol lowing rat (a) 3 : 9 ios. 8 4 (b) 1 : 3 (c) 0.45 : 0.85 7 (d) 1.6 :4 3. Find the ratio of (a) 1.5 m to 300 cm, (b) (c) 50¢ 600 ml to to $1.25, 1.2 l, (d) 2.4 kg to 4000 4. (a) g. Find the val ue of a if (b) Given a : 400 = that 4 2 : 3 2 6 : 25. = 8 : 3b, find the value of Ratio and Rate b.
5.
A certain amount of mone Weiming y is shared and between If Weiming Kumar in the rat io 5 : 9. gets $44 less than total amou Kumar, fin nt of mo d the ney that the two bo is shared ys. between 6. Given that a : b : c = 75 : (i) sim 120 : 132, plify a : b : c, (ii) find b : a, (iii) find b : c. 7.
Simplify each of the following 2 3 ratios. 5 3 :2 : 8 (b) 2 : 7 7 (c) 0.33 : 0.63 : 1.8 6 :9 (d) 1.4 ): 7an: d 6.3 (a)
d angle t, line an ith zero .g. poin point (w school (e ts like a see primary al objec actually ric learnt in ot et ve nn al om ha but ge at we ometric i.e. we ca CHAP th u, , ge s TER yo ds te ea e 9 in ea id P m ris cr me key in our ay surp We can so ly . m ts.45 GA d It on en ite jec s. se ist these ob dth) ex e the un ng angle , we revis ea of lis lvi ter E br es ua vo ap ns ut in no vis io ch e attrib esentat In this to help gth but operties th pr cs pr len re re ati w e su th es ne ea m me ject wi mathem apes. Th ideas. ms, and learnt so e (an ob beauty of es and sh abstract al diagra n) or a lin wer and ample, ints, lin to these ometric . For ex dimensio ms of po tes the po insights using ge ts in ter pare them is illustra and gain angle ent them ical objec and com rk with n of an them! Th ys se d, repres wo tio ph in aly to e no m an e r in ou int. Th we can abling us autiful in to describ po at y en be , th ed is wa objects ns a so fix cs e situatio about a operties athemati s provid rld es pr M . am wo lin of s gr o rld altw wo Dia ent re measure the real between to repres e up with res.” of turn tions in are used the sphe also com amount d revolu aticians acing of re of the tations an l worlds! Mathem in the sp a measu ings, ro physica ng turn used as is music tal and e is lvi en . er m vo cs gle th r in , emati an an tions e strings nnects ou ty of math er applica ing of th way it co the beau has furth e humm d in the preciate tions an etry in th ber to ap is geom its applica us remem , “There id let ”, sa gs ce in as on the str Pythagor ming of “the hum arvel at As we m
es s of angl 1. Type gle Acute an
Obtuse
gle
Right an
x° x°
x° 0° < x° < Straight
90° angle
x°
angle
x° = 90°
90° < x° x angle
Refle
x°
180° < x°
< 180°
x°
< 360°
x° = 180°
gles 90°. entary an s. d up to supplem s that ad tary angle to 180°. tary and two angle pplemen plemen at add up gles are pair of su 2. Com s and a entary an are two angles th gle em an pl m ry s ta (a) Co ry angle plemen lementa ir of com e). (b) Supp example of a pa es a str. lin an ecting lin 180° (adj. ∠s on • Give by inters ed line is rm ht gles fo a straig an t). on of in s s po ertie t angle s at a adjacen 3. Prop is 360° (∠ opp. ∠s). sum of a point s. (a) The l (vert. angles at propertie are equa sum of ove angle e angles (b) The R 10 of the ab opposit ch lly CHAPTE ea ca illustrate (c) Verti to e ur a fig • Draw ometry Basic Ge
KEY FEATURES
V
93
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P A G E
Challenge Yourself problems are included at the end of each chapter to extend the learning of students. In most chapters, the first problem includes guiding questions based on Pólya’s Problem Solving Model.
1.
rectangle ABCD. s a point E inside a The diagram show of the rectangle. a fraction of the area A E
Express the sum of
d triangles as
the area of the shade
Name
B
• Think of real-world
1.
about using algebra? Stage 4: Look back the problem? How er method to solve (e) Is there anoth figure on of the area of the d triangle(s) as a fracti shade the of area the express following figures, 2. For each of the answers? the by (iii) ised B surpr ABCD. Are you A (ii) B E A (i) B A E E
3.
a diameter of s a semicircle with (i) The figure show of the figure. Find the perimeter
2πr2 + 2πrh
use these formulae
3 cm
3 cm 3 cm
70 cm.
2 cm 2 cm
1 cm
(d)
4 cm
The figure shows a sketch of the world ’s largest gold bar that is 45.5 cm long. It is a solid prism with uniform cross section of a trapezium. 20 cm 17 cm 45.5 cm
22.5 cm
(i) Find its volum e. (ii) The mass of the gold bar is 250 kg. Find the volume of a gold bar with a mass of 200 g, leaving your answe r in mm3. (iii) Suppose the manu facturer of the gold bar decides to mould it into smaller identical pieces, each weighing 200 g and with dimensions as show n.
2 cm 5 cm
1 cm
20 mm
A rectangular brick measures 18 cm by 9 cm by 6 cm. Find the numb er of bricks required to build a rectangular wall 4.5 m wide, 18 cm thick and 3.6 m high.
15 mm
50 mm
x mm
Find the value of x.
of Plane Figures Volume and Surface
CHAPTER 12
Find the total surfac e area of a cube that has a volume of 343 cm3.
5.
1 cm
5 cm
1 cm 1 cm
A cuboid is 256 mm long, 152 mm wide and 81 mm high. (a) If the cuboid is melted to form a cube, find the length of each side of the cube. (b) The cuboid is cut into small cubes of length 30.5 mm. Find the maxim um number of cubes that can be obtain ed.
3 cm
1 cm 4 cm
2.
and surface areas of.
4.
6 cm
2 cm (c)
3.
3 cm
2 cm
(b)
70 cm
to find the volumes
13
1 cm
Perimeter and Area
168
objects that you can
3 cm
B different diameters. A s five semicircles of 70 cm (ii) The figure show figure. the perimeter of the If AB = 70 cm, find you conclude? can what (ii), and ers in (i) that its (iii) From your answ positive integers, such , where l and b are you have a breadth of b units do you prove that a length of l units and ble rectangles. How 4. A rectangle has dimensions of all possi the Find area. its to perimeter is equal found all of them?
P A G E
πr2h
Each of the following figures is made up of two or more rectangular prisms. For each of the following prisms, find (i) its volume, (ii) its total surfac e area. (a) 6 cm 2 cm
C
C
D
C
Total surface area
r
Model) Problem Solving (based on Pólya’s Guiding questions any point inside d the problem rectangle? Can it be Stage 1: Understan point E inside the the by d rstan (a) What do you unde the diagram? it fixed as shown in the rectangle, or is a plan to solve the problem? Stage 2: Think of ter that you may use learnt in this chap to solve the problem? (b) What have you way that may help rectangle in some two lines to cut the (c) Can you draw plan the does it help? Stage 3: Carry out rectangle help? How the two lines in the (d) Does drawing
D
Volume
h
C
D
Hints for Challenge Yourself are provided at the end of the textbook to guide students where necessary.
Figure
Closed cylinder
Area of Prisms and
Cylinders
Review Exercise 195 at the end of each chapter helps students consolidate their learning. CHAPTER 13
Problems in Real-World Contexts (PRWC) are authentic problems that happen in the real world which are spread throughout the entire textbook. In particular, more structured PRWC are placed in a separate section at the end of Textbook 1B.
P A G E
xts Problems in Real-World Conte Problem 4: Cookies for fun fair
a fun fair. Your class decides to needy, your school is organising In an effort to raise funds for the The ingredients are as follows: chip cookies to sell at the fun fair. chip cookies (makes 48 cookies) List of ingredients for chocolate
make chocolate
350 g all-purpose flour 1 teaspoon baking soda (7 g) 130 g butter, softened 300 g caster sugar 300 g chocolate chips 1 egg . Each plastic bag (or packet) contains plastic bags used for food packaging The cookies are packed into clear of chocolate chip cookies. they will be able to sell 480 packets [1] 6 cookies. Your class estimates that kilograms. class need? Give your answer in (a) How much flour does your from either chip cookies, and the plastic bags, chocolate the make to ts in the Your class decides to buy the ingredien ts from each supermarket are shown is cheaper. The cost of the ingredien Supermarket A or B, whichever table below.
1.
2.
4.
. Find its
cm icircle is 144
ter of a sem The perime 22 π = 7 .) area. (Take
7% GST) Cost of ingredients (exclusive of Supermarket A
D1
ercise Revision Ex
3 cm
m
5c
5c
m
4 cm
8 cm
. haded region ter of the uns rect to the perime region, cor (a) Find the shaded the area of (b) Find ce. pla l ima 1 dec h an s triangle wit is an isoscele half, figure, BDE s △BDE into 3. In the 2 If EC divide . a of the , find the are area of 24 cm cm 14 = and EF AF = 8 cm ABEF. trapezium E 14 cm F 8 cm A
B
l sur
Kings baking soda (150 g) TST butter (250 g)
6 cm
large wn using a of design dra m shows a al semicircles The diagra , two identic meter 8 cm ngle. tria s cele circle of dia isos cm and an diameter 3
3 cm
tota ume and the Find the vol the solid.
Item g) H/Family all-purpose flour (454 H/Family all-purpose flour (1 kg)
face area of
C
D
12 cm
12 cm
Caster sugar (500 g) Eggs (10 per pack) Eggs (30 per pack) per pack) Clear plastic bag for packaging (100 Chocolate chips (350 g)
3 cm 6 cm
18 cm
Others
54° tel x° Ho
G E
$1.15 $4.80 Buy 2 and get $2.65 off $1.65 $1.50 $3.85 $7.15 $6.40
Invitatio
n to Cod
Explanatio n: What is
KEY FEATURES
231
e
Part 2
happening here?
It is a sim ple progra m which behind the decides wh program. ether a giv Do you un en number derstand is even or what is hap odd. Here pening her is a breakd e? own of the Main thinking Integer testnumb er Output “Pl ease enter a natural number.” Input testnumb
er
testnumb er mod 2 = 0
True Output tes tnumb & “is even!” er
236
Travel cost that was tage of the the percen (a) Find spent on (i) food, el, (ii) hotel. spent on trav 1 % of the total cost was (b) If 11 3 e of x. find the valu
cise D1 Revision Exer
VI
Supermarket B $0.90 $1.90 Not on offer
al h an extern of container wit [2] cm is made n cylindrical your calculations clearly. ght of 120 5. An ope your class buy the flour from? Show cm and a hei (b) Which supermarket should diameter 50 thick. Find litres, -cm in er 1.5 s tain glas They must make sure that con er, packet of chocolate chip cookies. one the contain acity of the for price ke cap selling to ma the on the $1000 to (i) s used Your class needs to decide plastic bags, and to make at least ume of glas if the density the costs of the ingredients and the (ii) the vol tainer, in kg, they charge enough money to cover 3 ss of the con g/cm . (iii) the ma donate to charity. Show working to False used is 2.7 for a packet of chocolate chip cookies. of the glass [7] amount for your class to charge sensible a Suggest (c) of akdown the cost bre justify your decision. Output tes chart shows tnumber 6. The pie & “is odd a holiday. !” Food P Problems in Real-World Contexts A 144°
Revision Exercise helps students revise and assess their learning after every few chapters.
P A G E
$0.95 $1.85 2 for $3.50 $0.80 $4.80 2 for $7.00 $1.65 $1.65 Not available $8.43 $6.50
Declare the variable “te stnumber which is an ”, integer. Ask for the
number to be
tested.
The number keyed in by the user wil be stored as the value l of “testnum ber”. Mod is a fun ction that gives the rem when the number is ainder divided by e.g. 3 mod the diviso 2 = 1. r, The progra m displays whether the number is given odd or eve n.
End
Task 2: Ex
ploration
P A G E
In this tas k, you are going to “co true or fals de a progra e. m” that sho 1. Down ws two diff load, install erent result , and run s depending to represent a copy of on wheth Flowgorithm your code. er someth on your com 2. Const ing is ruct the flow puter. Alt ernatively chart for a You may , you may computer use the exa just draw program mple provid boxes that determ that you do ed as a start ines wheth not know point. (If er a given of, e.g. mo 3. (Just you number is d, go to ww need certain for fun) Co a multiple w.flowgor nstruct the operators of 3. ithm.org user depend or functio flowchart and look ns in Flowg ing on the for a com under the puter pro user’s gen orithm tab Docum gram that der. entation.) will have Exposition a different conversat ion with the In this seg ment, you have the opp procedure ortunity to s depending construct on the tru possible out a flowcha th-value of puts — Tru rt of the pro a statement e or False. logic is oft gram that . This is an Boolean log en used in performs example of ic is useful programm different a Boolean ing. for testing function, whether som which has ething is tru P two e or false, A and this Invitation G to Cod
Invitation to Code are sections which help students get started on coding. The free software Flowgorithm (www.flowgorithm.org) allows 74 the basic computational thinking behind students to learn coding without the use of complex programming languages. E
e (Part 2)
Guided investigation provides students the relevant learning experiences to explore and discover important mathematical concepts. It usually takes the Concrete-Pictorial-Abstract (C-P-A) approach to help students construct their knowledge meaningfully. The connections between concrete experiences (manipulative or examples), different pictorial representations and symbolic representations are explicitly made. Some investigations may also involve the use of Information and Communication Technology (ICT).
Questions are provided to engage students in discussion, with the teacher acting as the facilitator. Class discussions provide students the relevant learning experiences to think and reason mathematically, enhance their oral communication skills, and learn new concepts and skills.
Key questions are included at appropriate junctures to provide students the relevant learning experiences to think critically on their own before sharing their thoughts with their classmates. Mathematical fallacies are sometimes included to check and test studentsâ&#x20AC;&#x2122; understanding.
Journal writing provides opportunities for students to reflect on their learning and to communicate mathematically in writing. It can also be used as a formative assessment for the teacher to provide feedback for their students.
Students are usually required to reflect on what they have learnt at the end of each section so as to monitor and regulate their own learning. The reflection questions provided can be generic prompts or specific to the topics in the section or chapter, to check if students have understood the key ideas.
MARGINAL NOTES Big Idea This provides additional details of the big idea mentioned in the main text.
Information This includes information that may be of interest to students.
Internet Resources This guides students to search the Internet for valuable information or interesting online games for their independent and self-directed learning.
Recall Unlike the key feature â&#x20AC;&#x2DC;Recapâ&#x20AC;&#x2122; in the main text, this contains justin-time recall of prerequisite knowledge that students have already learnt.
Reflection This guides students to think about different methods used to solve a problem.
Just For Fun This contains puzzles, fascinating facts and interesting stories about mathematics as enrichment for students.
Attention This contains important information that students should know.
Problem-solving Tip This guides students on how to approach a problem in Worked Examples or Practise Now.
Coding This provides coding opportunities for students who know how to code. Students new to coding can refer to the section Invitation to Code (Part 1) in Textbook 1A to get started.
KEY FEATURES
VII
P A G E
CONTENTS CHAPTER 8
CHAPTER 9
CHAPTER 10
Ratio and Rate
35
2
9.1
Ratio
36
8.2
Percentage change, percentage 15 point and reverse percentage
9.2
Rate
47
9.3
Speed
64
8.3
Percentage in real-world contexts
Percentage
1
8.1
Percentage
23
Summary
32
Review Exercise 8
33
CHAPTER 11
Summary
70
Review Exercise 9
71
Invitation to Code (Part 2)
73
Polygons and Geometrical Constructions
97
11.1 Triangles
98
11.2 Quadrilaterals
Basic Geometry
75
10.1 Basic geometrical concepts and notations
76
10.2 Properties of angles formed by 80 intersecting lines (Recap) 10.3 Properties of angles formed by two parallel lines and transversal
85
Summary
93
Review Exercise 10
94
CHAPTER 14
106
11.3 Geometrical constructions of 114 triangles and quadrilaterals
CHAPTER 12
11.4 Polygons
122
Summary
136
Review Exercise 11
138
Revision Exercise C1
141
Revision Exercise C2
142
CHAPTER 13 200
14.2 Pictogram
201
14.3 Bar graph
203
14.4 Pie chart
209
14.5 Line graph
213
13.1 Conversion of units
170
13.2 Three-dimensional solids
172
14.6 Evaluation of statistical representations
218
14.7 Statistical investigation
220
Summary
227
Review Exercise 14
227
13.3 Volume and surface area of cubes and cuboids
174
12.1 Conversion of units
144
13.4 Volume and surface area of prisms
180
12.2 Perimeter and area of basic plane figures
146
13.5 Volume and surface area of cylinders
185
12.3 Perimeter and area of parallelograms
151
13.6 Volume and surface area of composite solids
191
12.4 Perimeter and area of trapeziums
158
Summary
194
Summary
165
Review Exercise 13
Review Exercise 12
166
Invitation to Code (Part 3)
CONTENTS
14.1 Frequency table
169
143
VIII
199
Volume and Surface Area of Prisms and Cylinders
Perimeter and Area of Plane Figures
P A G E
Statistical Data Handling
Revision Exercise D1
231
Revision Exercise D2
232
Problems in Real-World Contexts
233
195
Hints for Challenge Yourself
241
197
Answer Keys
243