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
1A
Secondary
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ŠSHINGLEE 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 4539 6
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MINISTRY OF BY E
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Printed in Singapore
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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 and E imation Approx
ER
3
n stimatio
instances e many there ar s and ily lives, imation In our da e approx us to need when we s. on ow ati kn estim nted to gi urist wa m Chan ple, a to nds is fro For exam a Bay Sa in ar M how far , ied pl ,” Vani re Airport. t 10 km it is abou “I think n be ation. ca tim ich es an e, wh making ated valu e. an estim tual valu 10 km is m, the ac or far fro at the close to, found th d an e to lin y Sands ecked on arina Ba Albert ch from M km. e .6 nc 18 sta di en to be driving giv s ng wa ivi Airport at the dr gi Changi states th to Chan e website y Sands “An onlin arina Ba ,” Albert from M ly 19 km distance ate im is approx t or rp Ai values, tourist. oximate told the are appr e actual and 19 km from th 18.6 km unded off is correct. ro ite e bs ar we which that the suming to make value, as will learn d apter, we ations an In this ch im ox pr te ap appropria s. on ati estim
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.
Learning Outcomes help students to be aware of what they are about to study so as to monitor their progress.
es utcom ing O apter? Learn in this ch aces and on are we learn cimal pl ll wi t d estimati ber of de Wha ation an ired num approxim e to a requ rs be • What m in real lif nu plications round off useful ap • How to t figures on have d estimati significan an on proximati • Why ap
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
ry
and a breadth th of 64 cm et with a leng vanguard she guard sheet. Albert has a leftover van e, without any are? big as possibl squ each the length of er? (i) What is he cut altogeth y squares can (ii) How man
of 48 cm. He
it into wants to cut
squares that
are as
Recap 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.
est F) and the low
factor (HC est common bers, the high s. ut prime num ds of problem g to learn abo , we are goin solve these kin us pter cha help this can In which ltiple (LCM), common mu
bers
1.1
Prime num
cap) bers below? factors (Re missing num mbers and What are the s and factors. whole number know about we t wha on , 6, … Let us recap , 4, , 12, 14, …. , 8, s are 0, 1, 2, 2, 4, by 2, e.g. 0, whole number , 15, 17, …. are divisible Examples of , 9, 11, 3, 5, numbers that s are whole e by 2, e.g. 1, Even number are not divisibl that s ber num s are whole ber num Odd 18? the factors of Can you find
A. Whole nu
18 = 1 × 18 =2× ×6 =
are 1, 2, factors of 18 Therefore, the its factors? e by each of Is 18 divisibl
, 6,
ving Tip Problem-sol is divisible by A number n e is ber p if ther another num n n is divided whe er no remaind by p.
and 18.
Important Results summarise important concepts or formulae obtained from Investigation, Class Discussion or Thinking Time.
bers. mbers s and odd num ng whole nu even number two groups: have. classified) into of factors they divided (or by the number bers can be them up Fro gro Whole num to m is the Invest le numbers igation on to classify who page 106, Another way we
B. Classifyi
observe the
The Distr ibutive La w a (b + c) = ab + ac
P A G E
2
CHAPTER
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. Similar and Further Questions follow after Practise Now to help teachers select appropriate questions for students’ self-practice. P A G E
IV
KEY FEATURES
: Big Idea
This is cal led the Dis tributive multiplied Law bec separately e the Mult , to each of Lowestaus firstiple In particu Common factor a is lar, Factor andthe two ter distribute ms, b and est Common d, or Primes, High c, in the sec ond factor (b + c). Negative of (x + y) and negati −(x + y) = ve of (x − −x − y y) and −(x − y) = −x +y
1
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.
following
The Distrib utive Law can
be visualise P
d using the b
rectangle T
PQRS (se e Fig
c
a
Equivalenc e Two express ions are equ if the valu ivalent e of both expression is the sam s e for any valu substitute into the var e we iables, e.g. a(b + c) = ab of a, b and + ac for any values c. its equival Writing a(b + c) in ent form ab + ac can help us to sim such as tho plify expressions se in Worke Example d 5.
4.15).
Q
S U R Since area Fig. 4.15 of rectangle PQ then RS = are a of rectan a(b + c) gle PTUS which is the = + area of Distributi rectangle ab ve Law. TQRU, + ac,
Expanding
exp
ressions Expand eac using Distr h of the fol ibutive La lowing exp (a) 6(x – w ressions. 2) (b) –3(4x – y) (c) 7 − a(− 5x + z) (a) 6(x – 2) = 6x – 12
4
(b) –3(4x – y) = (c) 7 − a(− 5x
Similar and
4
Further Que stions Exercise 4B Questions 1(a)–(h), 2
Basic Alg ebra and
–12x + 3y −3 × (−y) = 3y − 12x = +3y rearrange order of ter + z) = 7 + 5ax − az ms
Expand eac h of the fol lowing exp (a) 2(x − ressions. 7) (c) 8 − a(− x + 2z) (b) –5(3x – 4y) (d) 6 – b(− 3y −
Algebraic Manipul
Problem-s olving Tip (b) We usu ally leave our ans as 3y – 12x , which loo wer simpler tha ks n –12x + 3y.
z)
ation
CHAPTER
4
107
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.
r tive facto d a nega sitive an
2C
Explanation Questions require students to communicate their explanations in writing and are spread throughout the textbook.
of 0.
a po e down 11. Writ of the of each plain. e factors tiv er n? Ex ga ne any integ ive and true for the posit 2 n always (d) 1 = rs. 3. Find n be e values m Is nu 12. e possibl (c) 16 3 following find all th (b) −2 198 are integers, le of (a) 12 e multip and n2 n tiv If ga . ne 13 and a a positive of n. e down mbers. wing nu 4. Writ why. the follo , explain llowing. each of (d) 0 e to do so bl 7 ssi ate the fo −1 po t (c) ÷ (–6) 14. Evalu If it is no −8 (–2) × 5 ) (b ) 24 × (a × (–5) ng (a) 5 10 – 13 e followi ch of th (b) 4 × 3 ÷ ( –5 ) ots of ea –8 – 20 ro ÷ re 0 ua (c) 16 ÷2 the sq 2 3 (d) 100 57 −77 ) 2 5. Find 25 5 −3 − ( ) rs. ] × (–4) (c be ) m (d 4)2 nu 3 – (–1) (b) 16 ÷ (−2 + 2 – 18) ÷ it + (−7)] (e) [(1 (a) 81 ssible. If 3 × 2 + [−4 po e 3 (−3) er 2] + 3 } ÷ 2 (5 − 2) wing wh 3) llo (f) fo (− e 2 ] +5 of th − [12 + −2) −8 × y. ate each (g) {−10 −64 −3) +8 × ( 6. Evalu ssible, explain wh 7 ) − [−2 ( 9 (d) 3 −2 × (−3 (c) − is not po er ) 4 (h sw is ) rrect an ble. 81 (b piad, a co arded (a) ere possi wing wh atics Olym ong answer is aw er the follo mathem wr of a a no answ t ch In If ea . bu . 15 y. marks, value of ducted) e 3 wh de rt d th ain be de nd pl ll ar aw e, ex rt took pa 7. Fi ark wi 3 t possibl (i.e. 1 m ded. Albe (b) −4 If it is no . He be awar −1 mark ll ns wi tio es ks 3 3 8 0 mar re 30 qu ns (d) is given, (a) (−4) . There we ly and 5 questio 3) lympiad 3 −216 correct in this O (f) (c) −(−4 score? questions ks did he 3 −1000 3 125 swered 17 ar m an y an − (h) (e) . How m wrongly 3 −64 to answers (g) − llowing. eck your of the fo or to ch e of each a calculat e lu va Us e . 16 late th . r 14 lcu Ca on 12 − sti ) 8. e numbe Que 5 + (−10 s a prim +3 (a) −5 If it show − (−2)] a prime is rolled. it shows 2 − [(−8) 20 ded die (b) −1 If it arded. If + (−5) + -si ) d. aw six e 45 cte A ar (− ts 17. 00 + be dedu d, 5 poin (c) −1 ints will arded. that is od en, 9 po + 3 × 15 will be aw that is ev (d) −2 is 0 points number number number, − 2)(−3) es each + 32) y other 12 tim (e) (−5 an y s (− ÷ an ow sh times. how m 2 5 × (−4) e shows the die 20 (f) −2 2 − (7 − 2) The tabl en rolls 3) uf (− Sh × en 5 6 wh (g) 3 10] 3 4 obtained × (−2) – 1 2 5 (h) 5[3 2 0 1 n on die (−2)] 4 7 2 ÷ [2 − ber show 3 m −1 Nu (i) es −2) r of tim 10 −3 × ( Numbe e? or (j) sc to al s ’s fin answer Es ufen Shtim ation of eck your What isEs area or to ch timate the a calculat ratio of the 9. Use unshade area of the 8. d region shaded reg in the fig Question ion to tha ure on the of e t of the e lu lu right. va e va the lating th whether out calcu 1), explain 32 (− 10. With ÷ (−654) × 87 e. −9 or negativ Numbers is positive and Real
8
Numb Rational Integers,
Performance Task consists of mini-projects designed5to 2 develop research and presentation skills of students, through writing a report and/or giving an oral presentation. P A G E
R2
CHAPTE
∴ ratio of Similar and Further Questions Exercise 3B Questions 9, 15, 16
8
area of sha
ded region
Estimate the pe
rcentage
to that of
of the fig
Problemsolving Tip To estima te the shaded the area, we divide region (us dotted line ing s) approxima into areas that are tely equal of the uns to the are haded reg a ion. Since the unshaded the right area is on side of the dividing figure, star the shaded t the right region fro side to obt m ain a mo accurate re estimate.
ers
unshade d reg
ure on the
ion ≈ 3 :
right tha t is shade
1
d.
Recall the estimation strategy In group that you s, select on have learnt e task be 1. Estim low and in this sec ate the tot write a de tion. al numb tailed rep 2. Estim er of hours ort on ho ate the am your classm w you ob ount of piz 3. Estim tain your ates spent za needed ate the am estimation on social to fee ou . media pla d all the nt of mo 4. Estim ney colle students tforms in ate the nu cted by the in your sch a month mber of As a class, . ool durin drinks sta pa sse ng compare ers on a g an excurs ll in your packed tra the estim sch ion. oo l canteen ates for eac in during on a weekd h task an peak hours ay. d discuss . the estim ation strate gies used by the var ious group s.
Introductory Problem Revisited revisits an application-based Introductory Problem later in the chapter. This is absent if the Introductory Problem leads directly to the development of a concept. Introduct Problem ry Revisited
P A G E
Buzz has 1 parent – the female queen two parents bee. The fem – a female and ale queen bee a male. The was hatched refore, Buzz His grandfath from a fertilis has 2 grandp er has only ed egg and arents – a fem 1 parent and female and so she has ale and a ma his 1 male. grandmoth le. er has 2 par ents. So Buzz Since every will have 3 male bee has great-grandpa only 1 female grandparents rents – 2 parent and – 3 female and every female 2 male. It also and 3 male. bee has 2 par follows that ents, Buzz wil Buzz will hav l have 5 gre e 8 great-grea We can extend at-g reatt-great-gran a similar rea dparents – soning to Bet 5 female ty’s family tree Organising our observatio . ns in a table will help us to recognise Number the pattern Number of better. of parents Number of grandparent gre Buzz s Number of grandparent at1 gre s Number of gre 2 at-grandpar atBetty gre ents 3 2 great-grand at-great3 parents 5 5 8 8 Can you see 13 Table 7.8 that the fam ily tree of bee interesting s follows the as it is not me Fibonacci seq rely a theore of a flower and uence exactly tical sequen the growth ? The Fibona ce but is also of bees. Thi cci sequence prevalent in s is indeed a is nature such fascinating as the number application of petals of number patterns in real life! Relationshi p between The following Fibonacci shows a seq sequence uence of rec The number and Golde tangles made n Ratio in each squ up of Fibona are is the len cci squares. gth of the squ are.
1
1 1
Figure 1 Fig ure 2
5 1 1 2 Figure 3
3
1 1 2
3
5 1 1 2
3
1 1 2
8
Figure 4 1. Figure 6 Figure 5 is a rectangle made up of 1, 1, 2, 3, 5 Figure 6 6 squares. The and 8. lengths of the In this Investi squares are gation, we wil the first 6 Fib l take the len onacci num Find the len bers: gth of the rec gth and the tangle to be breadth of the longer than rectangle in its breadth Figure 6. Are (exception: they consec Figure 1). utive Fibona cci number s?
P A G E
194
1. What do I alread y know ab 2. How out estim is estimati ation in on differe the real wo 3. What nt from ap rld that co have I lea proximati uld guide rnt in thi on? my learn s section ing in thi or chapter s section? that I am still uncle ar of?
86
CHAPTER
3
Looking Back complements the Chapter Opener and helps students internalise the big ideas that they have learnt in the chapter. Approxima tion
and Estimat
ion
inue to will cont many which we used to model nction – en lators atics – fu and thus are oft m calcu m fro he s, at s m nction machine y variable fu ncept in r e an co ea or m m nt e lin porta two or trated by tput. Th to an im linear between n be illus ding ou duced us follow a . ionship output ca uce a correspon ter intro bles that constant e the relat nction’s od This chap two varia adient, which is ns describ iqueness of a fu ific inputs to pr gr describes Functio ugh the un n ec its ro e sp is th tio e Th e nc . us us explore. lin aight rtant fu ttleship ssible to achines situations str po ba ce m a e e im rld ac th of e es is e wo ty ad lik realnes. Th atics. Th ps. Just or proper eas are m g machi mathem tween lationshi measure e other. These id to vendin est function in tions be portant sualise re pl e connec p. An im e variable over th ms to help us vi is the sim us to mak lationshi m helps ge of on -line) re of diagra ste an ht us sy ch ig es te ra of (st mak dina rate ents the sian coor m, which It repres nate syste oblem, the Carte n coordi Pr Cartesia ductor y the Intro ! ra game in eb and alg geometry
. n O(0, 0) the origi angles at at right x the secting We call . ter y) in , (x is, ir ax d the ydered pa stem by an or x-axis an dinate sy scribed axes, the , y). sian coor can be de s of two P are (x n plane dinates e consist 1. Carte the coor a Cartesia sian plan y. . i.e rte on P P, Ca t of A e output poin dinate exactly on ion of a e y-coor The posit produces and y th input x nate of P at every th x-coordi . ch y) su , and y d pair (x riables x h. an ordere d a grap n two va itten as p betwee tion values an can be wr lationshi 2. Func a table of function n is a re equation, ut y of a an tp s, A functio ou rd e g wo t x and th ted usin 3. The inpu represen y = 2x + n can be nction is A functio linear fu nstant c nctions. ple of a d the co linear fu of es An exam e line an pl r exam ient of th two othe the grad • Give tant m is ns co e th e c, where aight lin = mx + line is y tion of str straight 3. Equa tion of a The equa . pt interce is the yepness: e of its ste aight lin measure ient of str line is a straight 4. Grad rise t ient of a nge or n . ha e gradien l c The grad ca ru verti Negativ ge t gradien t = horizontal chan ive. Positive Gradien t is posit its gradien t is negative. , ht rig ien m left to , its grad wards fro t to right slopes up s from lef If a line wnward e object. slopes do eed of th If a line of the sp measure aph is a tion. e graph e-time gr nc m nc fu -ti r sta ce ea di an h of a lin line in a 5. Dist of a grap ient of a example The grad real-life another • Give 6
Summary compounds the key concepts taught in the chapter in a succinct manner. Questions are included to help students reflect on their learning.
7
Number Patt erns
P A G E
V
KEY FEATURES
CHAPTER
R
CHAPTE
173
P A G E
9.
× 375 is a to explain why 15 Use prime factors perfect square. n that whole number. Give (ii) k is a non-zero the ct cube, write down 15 × 375 × k is a perfe smallest value of k. bers. Find the num e prim both p (iii) p and q are a that 15×375× q is values of p and q so
10. (i)
the products and 810, written as 2 The numbers 504 3 × 7 and rs, are 504 = 2 × 3 of their prime facto 4 810 = 2 × 3 × 5. Find n for zero whole number nonlest smal the (i) iple of 810, which 504n is a mult 504 h e number m for whic m (ii) the smallest whol is a factor of 810.
perfect cube.
1.
2.
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.
Review Exercise at the end of each chapter helps students consolidate their learning. 6
Plot each set of the given points on a sheet of graph paper. Join identify each geometrica the points (in order l shape obtained. ) with straight lines and (a) (–2, 2), (–2, 6), (4, 6), (4, 2) (c) (2, –4), (8, 4), (b) (2, –2), (6, 2), (6, 8), (–2, 4) (2, 6), (–2, 2) (d) (0, 7), (2, 7), (2, 5), (–4, 1) The figure shows a circle.
on page 205. Yourself are provided . Hints for Challenge numbered 1 to 1000 1000 closed lockers 1000 students, and has ol scho new 1. A rs. locke s all the 1000 The first student open iples of 2. numbers that are mult r if it is closes all lockers with - he closes the locke The second student are multiples of 3 with numbers that rs locke all rses’ ‘reve The third student d. close all the locker if it is and so forth until open, and opens the are multiples of 4, with numbers that reverses all lockers The fourth student rs. se the relevant locke rever nts stude 1000 end? be left open in the Which lockers will
C
2 1
A –5
–4
–2
0
–1
1
2
3
4
x
5
–2 –3
J
Stage 4: Look back er is correct? check if your answ (e) How can you
G
–4
I
H
–5
(a) Write down the coordinates of each of the points shown (b) State the point in the figure. on the circle that has (i) the same x-coo rdinate as E, (ii) the same y-coo rdinate as J.
3.
The equation of a funct
ion is y = 4x −1 1 . Find the value of y when 2 (b) x = 2 1 , (c) x = − 1 . 2 2
(a) x = 12,
squares. 4.
5.
The equation of a funct ion is y = 250 – 20x. Find the value of x (a) y = 150, when (b) y = 450, (c) y = –1150. On a sheet of graph paper, using a scale P of 2 cm to represent 1 unit on the x-axis A 1 1 on the y-axis and 1 cm to represent CHAPTERunit G , draw the graph of the linear function E y = 2 1 x + 3 for value (ii) The points (–2, s of x from –3 to 3. 2 a) and (b, 3) lie on the graph in part (i). Find the value of a and of b. (i)
29
le
t Common Multip
on Factor and Lowes
–3
–1
the plan open in the end? Stage 3: Carry out numbers that are left e about the locker (c) What do you notic Is there a pattern? 1000 lockers? for true be will that the pattern (d) How do you know
Primes, Highest Comm
F
3
a plan Stage 2: Think of say, 10 lockers? lify the problem for, (b) Can you simp
three identical shape made up of The figure shows a identical parts. Divide it into four
E
4
B
Model) Problem Solving (based on Pólya’s Guiding questions nd the problem Stage 1: Understa rses’ the locker? ‘reve term the by rstand (a) What do you unde
2.
Hints for Challenge Yourself are provided at the end of the textbook to guide students where necessary.
y D 5
P A G E
174
CHAPTER 6 Linear Functions and
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
Graphs
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.
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 and the plastic bags, from either cookies, chip 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.
ercise Revision Ex
7% GST) Cost of ingredients (exclusive of Supermarket A
B1
Item g) H/Family all-purpose flour (454 H/Family all-purpose flour (1 kg)
$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
Supermarket B $0.90 $1.90 Not on offer
Invitatio
n to Cod
e
Part 1
= 7.5 $1.15 0.3) – 1.25x ations. (b) 5(x + following equ Kings baking soda (150 g) $4.80 each of the – 5x) 1. Solve Explanatio 5 − 7 =4 7) = 5 – 8(4 TST butter (250 g) n: What is Buy 2 and get $2.65 off (d) y −1 2− 2 y (a) 7(x – happening It is a sim y 2 − here? ple progra $1.65 3y +9 = 17 m which is a breakd 3 calculate (c) 7 Caster sugar (500 g) own of the $1.50 thinking beh s and outputs the are ns is area of a circ 5b . a of a circ ind the pro (10 per pack) of all the coi Eggs e le) of valu for $3.85 l e le when you any input gram. Can 8a ns. If the tota (in 2a −3b = 7 , find the valu key in (or coi you see tha t thi pack) per s cen (30 cas Eggs e, the rad input) the $7.15 t there is exa 2. If a+ b 11 t coins or 50ius of the radius. He per pack) ctly one out circle)? Th either 20-cen re Main Clear plastic bag for packaging (100 put (in thi d at $6.40 is is the ide . ns which are roa box coi of the 54 s case, the tch s in a of a fun coins contain ng a stre (Main) Th ction. Chocolate chips (350 g) of 20-cent 3. A box the rest alo is ber is and the num /h the “start” of the spends on the progra way at 95 km he $20.70, find ress ich exp wh of an m. t [2] journey on is twice tha ns clearly. /h Rea calculatio -km your km Show l 375 rad 65 a from? flour ius t of d at your class buy the ist travels par stretch of roa (b) Which supermarket should Here, we nds on the declare the 4. A motor rney. of the time he spe variable “ra his entire jou that w the graphs 65 km/h. The dius”, which Output “Pl e taken for chip cookies. They must make sure h axes, dra ease ent is a real num . Find the tim selling price for one packet of chocolate unit on bot at least $1000 tothe val Your class needs to decide on the expressway ber. represent 1 and the plastic bags, and to make ue of the rad er ts to ingredien the of cm costs 1 the We ask for ius.” g a scale of they charge enough money to cover usin 6. the er, to val pap –4 ue of “radius” from et of graph term: working to donate to charity. us values of x Show her vio cookies. for chip e. 6 pre 5. On a she + chocolate of 1 x +3 and y = –x ber from the [7] Input rad for your class to charge for a packet amount num e sensible a Suggest sam = (c) y ius ting the 2 functions The number by subtrac justify your decision. ce is found keyed in by 52, … ing sequen the user wil p, 73, q, r, in the follow Output Pi* l be stored radius^2 6. Each term as the value Given the sequence. radius, the of “radius”. th term of the r. P and Contexts n q Real-World program the in p, Problems of of the are for the A n outputs the values a of a circ terms of n, in , G ion (i) Find or prints the le, ress using the E End ce. formula for value down an exp uen ite seq Wr the the area of (ii) (End) This is a term of a circle. –81 if is lain the end of the (iii) Exp Task 2: Ex 2 program. 4=2 ploration pattern: 2 In this tas ing number k, you are 4 + 12 = 4 r the follow going to “co 7. Conside = 62 1. Down de your firs 4 + 12 + 20 load, install t pro 2 gram”. and run a + 28 = 8 Alternativ copy of Flo ely, you ma 4 + 12 + 20 wgorithm ⋮ y just dra (www.flow 2. Const w boxes to gorithm.o ruct the flow represent rg) on you chart for a – 4) = 1024 your code. example as r comput computer + … + (4m er. a starting pro ⋮ 4 + 12 + 20 gram that point. (To To add or calculates insert a box change pro the area of in Flowgori perties ins a rectangle 3. (Challe thm ide , a box, dou click on the . You may nge) Think tern. ble click on use the abo arrow bet of a difficul . th line in the pat the side ween Main ve it.) flowchart t or challen t-hand for the com and End. down the 7 ging calcul on the righ 4. (Just (i) Write puter pro with 2020 ation that m. gram that for fun) Co the pattern you want the value of will do the nstruct the the comput l be a line in (ii) Find the user. computatio flowchart er they wil er eth to wh do for you. Co for a com n for you. (iii) Explain puter pro nst ruc t gram that will have Exposition a simple con versation with In this sim ple introd uction to program computatio that will hel nal thinki p you per more adv ng, you hav form som anced pro e the opport e comput grams tha ations. In function — unity to con t will enable Invitation the relatio struct a flow you to do to Code Par nship betwe chart of the coding. more com ts 2 and 3, en the inp plex tasks. you will exp ut and the More impor lore unique out tantly, we put — is nec see how the P essary for idea of us A to explore the world G cise B1 of E Revision Exer
236
Revision Exercise helps students revise and assess their learning after every few chapters.
203
Invitation to Code are sections which help students get started on coding. The free software Flowgorithm (www.flowgorithm.org) allows students to learn the basic computational thinking behind coding without the use of complex programming languages. P A G E
P A G E
VI
KEY FEATURES
178
Invitation to Code
(Part 1)
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 students’ 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 ‘Recap’ 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) to get started.
KEY FEATURES
VII
P A G E
CONTENTS Primes, Highest Common Factor and Lowest Common Multiple
CHAPTER 1
1
1.1
Prime numbers
1.2
Square roots and cube roots
10
2
1.3
Highest common factor and lowest common multiple
16
Summary
27
Review Exercise 1
28
CHAPTER 3
CHAPTER 2
CHAPTER 4
Integers, Rational Numbers and Real Numbers
31
Basic Algebra and Algebraic Manipulation
91
2.1
Negative numbers
32
4.1
Basic algebraic concepts and notations
92
2.2
Addition and subtraction involving negative integers
36
4.2
Addition and subtraction of linear terms
99
2.3
Multiplication, division and 46 combined operations involving negative integers
2.4
Fractions and mixed numbers 53
2.5
Decimals
58
2.6
Rational, irrational and real numbers
Approximation and Estimation
71
3.1
Rounding and significant figures
73
4.3
Expansion and factorisation of linear expressions
105
3.2
Approximation and approximation errors in real-world contexts
80
4.4
Linear expressions with fractional coefficients
112
61
3.3
Estimation and estimation errors in real-world contexts
83
Summary
67
Summary
89
Review Exercise 2
68
Review Exercise 3
89
CHAPTER 5
Summary
118
Review Exercise 4
119
Revision Exercise A1
121
Revision Exercise A2
122
CHAPTER 7
CHAPTER 6
Method of calculating a quantity, multiplied by 1 1 added 4 it has come to 10. 2 What is the quantity that says it? Then you calculate the difference of this 10 to this 6. Then 6 results. Then you divide 1 by 1 1 . Then 2 results. 3 2
Linear Equations 5.1 Linear equations 5.2 Linear equations with fractional coefficients and fractional equations 5.3 Applications of linear equations in real-world contexts 5.4 Mathematical formulae Summary Review Exercise 5 P A G E
VIII
CONTENTS
123 124 129 133 136 139 139
Linear Functions and Graphs 6.1 Cartesian coordinates 6.2 Functions 6.3 Linear functions 6.4 Applications of linear graphs in real-world contexts Summary Review Exercise 6
141 143 145 152 165 173 174
Invitation to Code (Part 1)
177
Number Patterns
179
7.1
Number sequences
180
7.2
Number sequences and patterns
188
Summary
200
Review Exercise 7
201
Revision Exercise B1
203
Revision Exercise B2
204
Hints for Challenge Yourself
205
Answer Keys
207
