Focus On Maths Grade 7

FOCUS-ON TEXTBOOK MATHS 7 Singapore Maths Approach Progressive Practices 21st Century Learning Skills Formative & Summative Assessments Digital Resources Enhancements ©Praxis Publishing_Focus On Maths

FOCUS-ON MATHS TEXTBOOK 7 First Published 2023 6001 Beach Road, #14-01 Golden Mile Tower, Singapore 199589. E-mail: [email protected] ISBN 978-981-17293-0-0 Distributed by PT. Penerbitan Pelangi Indonesia Ruko the Prominence, Block 38G No. 36, Jl. Jalur Sutera, Alam Sutera, Tangerang, 15143, Indonesia. Tel: [021]29779388 Fax: [021]30030507 Email: [email protected] Printed in Malaysia by Pelangi FormPress Sdn. Bhd. No. 16, Jalan Bukit 2, Kawasan MIEL Seri Alam, 81750 Masai, Johor Darul Takzim. 2023 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, photocopying, mechanical, recording or otherwise, without the prior permission of ©Praxis Publishing_Focus On Maths

II FOCUS-ON MATHS is an exciting new series that has been developed to match the latest Indonesian Mathematics syllabuses (Phase D) for Grade 7 to Grade 9. The topic coverage in each grade is arranged to address all the learning achievements (Capaian Pembelajaran) as prescribed by the Indonesian Ministry of Education. The series adopts the Singapore Maths method which is a world-class maths teaching approach. FOCUS-ON MATHS is a complete mathematics programme that comprises the Textbook, Workbook, Teacher’s Guide and online resources. It is a comprehensive, task-based and learner-centred programme designed to cultivate students’ interest in the learning of Mathematics, equip them with an in-depth understanding of Mathematical concepts and help them to achieve their fullest potential in mathematics. The content is structured to develop 21st Century Skills and Higher Order Thinking Skills in students. Students are challenged with problem-solving tasks and problems in real-word contexts. This enables them to think independently and build foundations for many of the advanced applications of mathematics that are relevant to today’s world. A collection of STEM activities is also integrated in this series to foster inquiring minds, logical reasoning and collaboration skills in students. FOCUS-ON MATHS aims to develop a greater awareness of the nature of mathematics and bring greater connections between different topics in mathematics, as well as between mathematics and other subjects, creating a deeper and more robust understanding of mathematics. PREFACE 21st Century Learning Skills Indonesian Maths Syllabuses s Singapore Maths Method FOCUS-ON MATHS Formative and Summative Assesment Digital Resources Enrichment Project - Based Learning STEM Activities ©Praxis Publishing_Focus On Maths

III KEY FEATURES • Introduces the chapter in a real-world setting, allow students to realise the relevance and utility of mathematics in our daily lives. • Challenges students with questions that promote critical and reflective thinking. • Presents the important mathematical terms in the chapter. • Outlines the learning outcomes that will be covered in the upcoming lesson. • Provide an overview of the chapters in the book which sets the learning pace. • Recounts the history and development of mathematics and the contributions of great mathematicians. APPLICATION OF THIS CHAPTER CONCEPT MAP LEARNING OUTCOMES KEY TERMS MATHS HISTORY CRITICAL THINKING ©Praxis Publishing_Focus On Maths

IV • Provides students opportunities to work in small groups to explore mathematical concepts. • Poses mind-stimulating questions for students to think creatively, logically and analytically. • Prepares various types of activities aimed at involving students individually, in pairs or in groups inside or outside the classroom. • A quick assessment of students’ prior knowledge of the concepts learnt previously. • Demonstrates the steps in solving mathematical problems and techniques in answering questions. LET’S EXPLORE FLASHBACK WORKED EXAMPLE CREATIVE THINKING INTERACTIVE ZONE ©Praxis Publishing_Focus On Maths

V • Direct students to websites with extra learning materials or resources to enhance the learning experience. LET’S INVESTIGATE • Encourages students to explore the learning contents by themselves and involves them in active discussions during the lesson. SCAN ME ©Praxis Publishing_Focus On Maths

VI • Summarise important terms, definitions and mathematical properties to be used as helpful reinforcements. • Provides extra mathematical, scientific or historical facts and information related to the topic. • Remind students of key information or support that is useful to tackle an exercise, or simply useful to know. MATHS TIPS MATHS INFO ©Praxis Publishing_Focus On Maths

VII • Let students explore, model and solve non-routine prolems in real-world contexts to devlop students’ problem-solving skills with prompts to guide students using Polya’s 4-step approach. PROBLEM SOLVING ©Praxis Publishing_Focus On Maths

VIII ALTERNATIVE METHOD • Presents different methods of working in a practical and easy-to-follow way. CALCULATOR CORNER • Shows the use of scientific calculators in calculations HOT CHALLENGE • Helps pupils practise answering questions that promote higher order thinking skill. ©Praxis Publishing_Focus On Maths

IX MATHS LINK • Provides opportunities to students to explore how certain mathematical concepts can be connected or related to other concepts. TEAM WORK • Prepares various types of activities aimed at involving students individually, in pairs or in groups inside or outside the classroom. ©Praxis Publishing_Focus On Maths

X • Provides questions with different difficulty levels that allow students to practise methods that have just been introduced. These range from simple ‘recall and drill’ activities to applications and problem-solving tasks. SUMMARY • Sums up and highlights important concepts and formulae for quick revision. PRACTICE ©Praxis Publishing_Focus On Maths

XI MASTERY PRACTICE • Provides a further exercise to test students’ mastery of the concepts and skills learnt in each chapter. Students can download the free GeoGebra Classic software program to open the related files in . https://www.geogebra.org/download ©Praxis Publishing_Focus On Maths

XII CONTENTS CHAPTER 1 Integers XIV 1.1 Integers 3 1.2 Addition and Subtraction of Integers 10 1.3 Multiplication and Division of Integers 15 1.4 Combined Operations of Integers 18 1.5 Factors and Prime Factors 22 1.6 Lowest Common Multiple (LCM) and Highest Common Factor (HCF) 25 CHAPTER 2 Real Numbers 36 2.1 Rational Numbers 39 2.2 Irrational Numbers 41 2.3 Fractions 42 2.4 Decimals 54 2.5 Set of Real Numbers 63 2.6 Use of the Symbol , , , , =, ≠ 64 CHAPTER 3 Ratios, Rates, and Proportions 68 3.1 Ratios 71 3.2 Rate 82 3.3 Proportions 89 3.4 Relationship between Ratios, Rates and Proportions with Percentages, Fractions and Decimals 100 3.5 Scale 105 CHAPTER 4 Algebraic Expressions 114 4.1 Algebraic Expressions 117 4.2 Addition and Subtraction of Algebraic Expressions 125 4.3 Multiplication of Algebraic Expressions 129 ©Praxis Publishing_Focus On Maths

XIII 4.4 Division of Algebraic Expressions 136 4.5 Simplifying Algebraic Fractions 140 4.6 Expansions of Brackets 148 4.7 Factorisation of Algebraic Expressions 153 CHAPTER 5 Linear Equations and Inequalities in One Variable 160 5.1 Equality 163 5.2 Linear Equations in One Variable 166 5.3 Solve Linear Equations in One Variable 170 5.4 Inequalities in One Variable 181 5.5 Solving Inequalities in One Variable 186 CHAPTER 6 Lines and Angles 196 6.1 Relationship between Lines 199 6.2 Dividing Line Segment into Multiple Parts 203 6.3 Angles 210 6.4 Relationship between Angles 215 6.5 Constructing Special Angles 225 6.6 Similarity 229 CHAPTER 7 Statistics 236 7.1 Data 239 7.2 Presenting Data in Table 244 7.3 Bar Chart 249 7.4 Line Graph 256 7.5 Pie Chart 262 ©Praxis Publishing_Focus On Maths

Applications of this chapter There are numerous numbers directly or indirectly connected to our lives. Some of their applications in real life are in engineering, budget calculation, shopping, baking and recipes, weight measurement, use of lifts or elevators, ways of telling the time, temperature measurement, etc. The temperature in some deserts is very high during the day and very low at night. The average temperature in most deserts reaches 38ºC during the day. Whereas in some deserts, the temperature decreases to – 4ºC at night. This temperature varies depending on the locations of the deserts. Why do some deserts have high temperatures during the day and low temperatures at night? XIV 1INTEGERS ©Praxis Publishing_Focus On Maths

Integers Factors Integers Concept Map • Numbers • Number line • Positive numbers • Negative numbers • Decimals • Percentages • Comparing numbers • Prime factors • Lowest Common Multiple (LCM) • Highest Common Factor (HCF) Key Terms Learning Outcomes Learning Outcomes • Understand on how to compare numbers, fractions, decimals and factors. • Perform computations involving addition, subtraction, multiplication and division of integers to solve problems. • Perform computations involving combined operations of integers to solve problems. • Understand the characteristics of prime numbers. • Understand the characteristics and use the knowledge of factors of whole numbers. • Solving problems regarding Lowest Common Multiple (LCM) and Highest Common Factor (HCF). Operations Lowest Common Multiple (LCM) Highest Common Factor (HCF) Addition Subtraction Multiplication Division As early as 200 B.C., the Chinese used bamboo rod system to record use of negative numbers. They used red rods to represent positive numbers and black rods to represent negative numbers, especially when dealing with economic situations. The amount sold was positive and the amount purchased was negative. Negative numbers did not appear in Europe until the 15th century. Maths History Negative Integers Positive Integers Common multiples Common and Prime Factors Multiples 132 5089 –704 – 6027 1 ©Praxis Publishing_Focus On Maths

2 CHAPTER 1 Integers Flashback 1. Calculate the following. (a) 434 + 635 + 12 = (b) 143 – 31 – 69 = (c) 812 ÷ 14 = (d) 567 × 32 = 2. Evaluate. (a) 40 ÷ 10 × 2 + 5 = (b) 214 × 2 – (676 ÷ 26) = (c) 516 + 310 – 759 = (d) 608 ÷ 16 – 812 ÷ 28 = What other questions can you ask about this map? Thinking Greenwich Mean Time (GMT) is a standard for setting time zones. There are 24 time zones. All the places in the same time zone have the same time. The map shown is the world’s time zone map. Many land time zone lines vary so that each country can have a manageable time system. (a) What time is it where you are now? (b) You want to call a friend in London, United Kingdom. What time is it there? (c) What might your friend be doing now? 0-1-2-3-4-5-6-7-8-9-10-11 +12+11+10+9+8+7+6+5+4+3+2+1 0-1-2-3-4-5-6-7-8-9-10-11 +12+11+10+9+8+7+6+5+4+3+2+1 5:00 AM 2:00 AM 3:00 PM 12:00 NOON Nuuk Washington, D.C. Brasilia 7:00 AM Moscow New Delhi Jakarta Canberra Beijing Tokyo London 10:30 AM 1:00 PM 2:00 PM 8:00 AM 2:00 AM 12:00 MIDNIGHT Cairo 1 ©Praxis Publishing_Focus On Maths

3 Integers CHAPTER 1 1.1 Integers In our daily lives, we often come across numbers with the plus sign ‘+’ or the minus sign ‘–’. We use them to represent quantities with opposite directions or meanings. The concept of owing money can be expressed as a number less than zero or a negative number. For example, Chris borrows $10 from his mother. The number of dollars that Chris owes his mother is –$10. Negative numbers are also often used to describe temperatures. The temperature –5°C is ‘negative five degrees’ and it means 5 degrees below zero. The bank statement shows that Chris had withdrawn more than he had deposited so he was $100 in the red. He had –$100 or he owed the bank $100. The screen shows data about the financial crisis because of the Coronavirus disease. What other uses of integers do you know? Thinking 280122 050222 160222 230422 270822 190922 SAL ITR CAM BNK SAL INS 3800.00 1000.00 500.00 750.00 2200.00 1450.00 ******3800.00 ******3300.00 ******2550.00 *******350.00 ******1350.00 *******100.00 – A Understanding integers An integer is a whole number that has a positive sign (+) or a negative sign (–), including zero. For example, –7, –4, 0, 3, 5 and 8 are integers. A positive integer is a whole number with or without a positive (+) sign. For example, +1, +8, 9, 12. A negative integer is a whole number with a negative (–) sign. For example, –3, –5, –10, –42. ©Praxis Publishing_Focus On Maths

4 CHAPTER 1 Integers EXAMPLE 1 (a) Write +72 in words. (b) Write negative fifty-six in figures. Solution: (a) Positive seventy-two (b) –56 EXAMPLE 2 – 4, 3 5, +3.5, 6, +90, 1 1 2 State the integers from the list. Solution: The integers are –4, 6 and +90. Set of integers, Z = {... , –3, –2, –1, 0, 1, 2, 3, ...} Z+ = {1, 2, 3, ...} Z– = {–1, –2, –3, ...} 0 is also called the neutral integer. Hence N  Z. B Recognise positive and negative numbers We usually use positive and negative numbers in our daily life. A positive number is a number that is greater than 0. A positive number is written with or without the (+) sign. For example, +1, +3.5, + 1 2 or 1, 3.5, 1 2 are positive numbers. A negative number is a number that is less than 0. A negative number is written with (–) sign. For example, –1, – 3.5, – 1 2 are negative numbers. The situations that can be represented by positive and negative numbers: The movement to the north is represented by a positive number and the movement to the south is represented by a negative number. For example, if 8 km to the north is represented by +8 km, 10 km to the south is represented by –10 km. The upward movement is represented by a positive number and the downward movement is represented by a negative number. For example, 36 m above sea level is written as +36 m. 50 m below sea level is written as –50 m. The temperature that is higher than 0°C is represented by a positive number and the temperature that is lower than 0°C is represented by a negative number. For example, 8°C above 0°C is written as +8°C 5°C above 0°C is written as –5°C We must read the sign in front of an integer first before the number. For example, +5 or 5 is read as ‘positive five’ or ‘five’. –23 is read as ‘negative twenty-three’. Zero Integers ..., –3, –2, –1, 0, 1, 2, 3,... Negative integers Positive integers ©Praxis Publishing_Focus On Maths

5 Integers CHAPTER 1 Profit is represented by a positive number and loss is represented by a negative number. The increase in the price is represented by a positive number and the decrease in the price is represented by a negative number. For example, the price of an egg increased by 2 cents can be written as +2 cents. The price of an egg decreased by 1 cent can be written as –1 cent. EXAMPLE 3 Represent each of the following by using a positive number or a negative number. (a) A loss of $750. (b) The price of a packet of Nasi Lemak has increased by 50 cents. (c) Price of share market of Company P has increased by $2.10 within 1 month. Solution: (a) – $750 (b) +50 cents (c) +$2.10 EXAMPLE 4 Complete the following sentences. (a) 400 m to the right is written as +400. Thus, 250 m to the left is written as . (b) 56 km to the west is written as –56. Thus, 65 km to the east is written as . (c) 35 m above the sea level is written as 35 m. Thus, 12 m below the sea level is written as . Solution: (a) – 250 (b) +65 or 65 (c) –12 Discuss with your classmates, how did you decide to use a positive number or a negative number? What clues did you look for to help you decide? INTERACTIVE ZONE Critical Thinking Write an integer to represent each of the following: (a) A temperature 16 degrees above zero. (b) Owing $15. (c) A depth of 4 cm. (d) 3 m above sea level. (e) Two floors below ground level. (f ) 8 m to the left of the starting point. (g) 7 km south of the starting point, if north is positive. ©Praxis Publishing_Focus On Maths

6 CHAPTER 1 Integers C Representing integers using a number line Objective: To explore the representation of integers on a number line. Instruction: Do this activity in groups of four. 1. Open the file Integer number line using GeoGebra. 2. Identify the red dot on number line. Click and drag the red dot on the number line to define an integer on the number line. 3. Observe the position of the defined integer on the number line in relation to the position of zero. 4. Discuss the questions below with your group members. (a) Describe the value of integer 8 as compared to zero. (b) How do you determine the positions of negative numbers –4, –3, –2, and –1 on a number line? (c) How do you represent numbers 1, 2, 3, 4 and –1, –2, –3, –4 on a number line? 5. Present your findings in class. 1 From the findings in Activity 1, it is found that (i) positive integers are integers more than zero, whereas (ii) negative numbers are integers less than zero. Integers can be represented using a horizontal or a vertical number line. –3 –1–2–4 0 1 2 3 4 Negative integers Zero Positive integers Horizontal number line Vertical number line –4 –3 –2 –1 0 1 2 3 4 Zero Positive integers Negative integers A thermometer is a vertical number line. Scan or click the above QR code to download this activity file. GeoGebra ©Praxis Publishing_Focus On Maths

7 Integers CHAPTER 1 EXAMPLE 5 (a) Use a number line to represent the integers from –5 to 3. (b) Mark 6, – 3, –1 and 4 on a number line. Solution: (a) –5 –4 –3 –2 –1 0 1 2 3 (b) –3 –2 –1 0 1 2 3 4 5 6 D Comparing two integers City Maximum temperature (ºC) Beijing –4 London 3 Sydney 28 Moscow –11 Tokyo 10 Jakarta 29 At the end of CNI news, temperatures are given for many of the world’s capital cities. The table above shows the temperatures in six cities on a particular day in February. (a) Which city was the warmest on that day? (b) Which city was the coldest? (c) How many degrees warmer was the warmest city than the coldest city? (d) Arrange these cities according to their temperatures in increasing order. On a horizontal number line, an integer is always greater than the integers to its left and less than the integers to its right. For example, –3–4 –2 –1 0 1 2 3 –2 is greater than –4 but is less than 1. • –2 is greater than –4 can be written as ‘–2  – 4’. • –2 is less than 1 can be written as ‘–2  1’. 2 ©Praxis Publishing_Focus On Maths

8 CHAPTER 1 Integers EXAMPLE 6 (a) Which integer is smaller, –5 or 3? (b) Which integer is greater, –2 or –8? Solution: (a) –5 –4 –3 –2 –1 0 1 2 3 –5 is smaller than 3. (b) –8 –7 –6 –5 –4 –3 –2 –1 0 –2 is greater than – 8. Discuss with your classmates. (a) When two integers have different signs, how can you tell which is greater? (b) When two integers have the same sign, how can you tell which is greater? INTERACTIVE ZONE E Arranging integers in order We can arrange integers in increasing or decreasing order using a number line. EXAMPLE 7 (a) Arrange – 4, 6, –3, 5, 0 and 1 in increasing order. (b) Arrange 6, 0, 4, – 2 and –4 in decreasing order. Solution: (a) –4 –3 –2 –1 0 1 2 3 4 5 6 Increasing order: –4, –3, 0, 1, 5, 6 (b) –4 –3 –2 –1 0 1 2 3 4 5 6 Decreasing order: 6, 4, 0, –2, – 4 We can identify the largest integer and the smallest integer by arranging the given integers in order. EXAMPLE 8 Determine the largest integer and the smallest integer from the following set of integers. 1, –2, 3, 0, –5 Solution: –4–5 –3 –2 –1 0 1 2 3 The largest integer is 3 and the smallest integer is –5. ©Praxis Publishing_Focus On Maths

9 Integers CHAPTER 1 If the pattern of a sequence of integers is determined, we can find the missing terms in the sequence. 1 Write each of the following integers in words. (a) –17 (b) +23 (c) +48 (d) –69 (e) –205 (f) +416 B Write each of the following integers in figures. (a) Negative forty-two (b) Positive nine (c) Positive sixty-eight (d) Negative two hundred and seventy C State the integers from the list below. –9, 1 2 , 6.2, 4, –78, – 4 5 , –9.6 D Use a number line to represent the integers from (a) –3 to 4, (b) –20 to –15. E Copy and complete each of the following with ‘is greater than’ or ‘is less than’. (a) –4 +2 (b) +8 –9 (c) – 3 –7 (d) –10 –6 (e) +9 –20 F Arrange each of the following sets of integers in increasing order. (a) – 5, – 3, 0, –1, 2, –4 (b) 8, –7, –5, 6, – 9, 3 G Arrange each of the following sets of integers in decreasing order. (a) 9, –12, –6, 3, 7, –10 (b) –11, – 4, 8, –3, 5, –6 H Determine the largest integer and the smallest integer from each of the following sets of integers. (a) –7, 5, –9, 3, 0, –2 (b) 8, –12, 13, –15, 7, 11 (c) –20, –15, –19, –7, –30 (d) 5, –15, –20, 15, 10, –5 I Copy and complete each of the following sequences of integers. (a) 9, 5, 1, , , (b) –12, , , 3, 8, (c) , , –10, –4 (d) –32, , , –23, –20, (e) , , –13, – 4, J Use a positive or a negative number to represent each of the following word descriptions. (a) A loss of $80 (b) 42 m below sea level (c) An increase of 5 m (d) 8°C below freezing point (e) 1 hour after take-off K Copy and complete each of the following. (a) If 2 km to the east is written as +2 km, 3 km to the west is written as . (b) If going up 4 steps is written as , going down 5 steps is written as –5. EXAMPLE 9 Copy and complete the following sequence of integers. , –5, 0, 5, , Solution: –10 , –5, 0, 5, 10 , 15 +5 +5 +5 +5 +5 Practice 1.1 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths

10 CHAPTER 1 Integers 1.2 Addition and Subtraction of Integers Objective: To explore addition and subtraction of integers on a number line. Instruction: Do this activity in groups of four. 1. Open the file Addition and subtraction of integers using GeoGebra. 2. Click on the checkbox ‘Show addition’. 3. On the displayed screen, click and drag the red slider and blue point. 4. Observe the movement of other points in relation of addition and subtraction of integers. 5. Discuss the findings with your group members. Make a generalisation of addition and subtraction of integers. 6. Present your findings in class. 2 From the findings in Activity 2, it is found that on a number line, the addition of positive integers is represented by moving towards the right and the addition of negative integers is represented by moving towards the left. We also know that the subtraction of positive integers is represented by moving towards the left and the subtraction of negative integers is represented by moving towards the right. Discuss with your classmates. (a) What happens when you add +2 and −2? (b) Why do you think integers such as +2 and −2 are called opposite integers? INTERACTIVE ZONE A Addition of integers Addition of integer is a process of finding the sum of two or more integers. Adding an integer to a positive integer can be represented using a number line by a movement towards the positive direction, which means from left to right. Scan or click the above QR code to download this activity file. GeoGebra ©Praxis Publishing_Focus On Maths

11 Integers CHAPTER 1 For example, – 2 + 4 = 2. Start from –2, move 4 steps to the right. –2 –1 0 +4 1 2 Positive integers Adding an integer to a negative integer can be represented using a number line by a movement towards the negative direction, which means from right to left. For example, 2 + (–4) = –2. Start from 2, move 4 steps to the left. –2 –1 0 –4 1 2 Negative integers Integers with like signs are integers with the same sign. For example, 2 and 5, – 8 and –12. Integers with unlike signs are integers with different signs. For example, – 4 and 10, 3 and –9. EXAMPLE 10 Simplify each of the following. (a) 5 + (–2) (b) –3 + 5 Solution: (a) 5 + (–2) = 3 –2 123456 Start from 5, move 2 steps to the left. Coloured tiles method Yellow tiles, + , represent positive integers and red tiles, – , represent negative integers. For example, 5+(–2) Represent zero + + + + + – – Thus, 5 + (–2) = 3. (b) – 3 + 5 = 2 Start from –3, move 5 steps –3 –2 –1 0 to the right. +5 1 2 ©Praxis Publishing_Focus On Maths

12 CHAPTER 1 Integers EXAMPLE 11 Evaluate 6 + (–4) + (–5). Solution: 6 + (–4) + (–5) = –3 1 Start from 6, move 4 steps to the left. 2 Then move another 5 steps to the left. –3 –2 –1 0 –5 –4 123456 2 1 EXAMPLE 12 Fill in the blanks. (a) 32 + 15 = 15 + (b) [13 + (–6)] + 3 = 13 + ( + 3) Solution: (a) 32 + 15 = 15 + 32 (commutative property of addition) (b) [13 + (–6)] + 3 = 13 + ( –6 + 3) (associative property of addition) EXAMPLE 13 The initial temperature of a cold storage was –2°C. Two hours later, the temperature dropped by 3°C. When the cold storage was switched off, the temperature rose by 7°C. What was the new temperature of the cold storage? Solution: –2 + (–3) + 7 = 2 –5 –4 –3 –2 –3 +7 –1 0 1 2 2 1 Therefore, the new temperature of the cold storage was 2°C. B Properties of addition (I) Commutative property of addition This rule says that we can add two numbers in any order. For example, 38 + 16 = 16 + 38 –12 + 10 = 10 + (–12) (II) Associative property of addition Here it says that we can group numbers in a sum in any way we want and still get the same answer. For example, (3 + 10) + 6 = 3 + (10 + 6) (–4 + 8) + 2 = –4 + (8 + 2) a + b = b + a (a + b) + c = a + (b + c) ©Praxis Publishing_Focus On Maths

13 Integers CHAPTER 1 C Subtraction of integers Subtraction of integers is a process of finding the difference between two integers. The difference between two integers is the number of steps required to move from the second integer to reach the first integer on a number line. • If you move to the right, you will get a positive integer. For example, –3 – (–8) = 5. From –8, we need to move 5 steps to the right (positive direction) to reach –3. Thus, the answer is +5 or 5. • If you move to the left, you will get a negative integer. For example, –8 – (– 3) = – 5. From –3, we need to move 5 steps to the left (negative direction) to reach –8. Thus, the answer is –5. –8 –7 –6 –5 –4 –3 Second integer First integer +5 Positive integers First integer Second integer –8 –7 –6 –5 –4 –3 First integer Second integer –5 Negative integers First integer Second integer EXAMPLE 14 Simplify each of the following. (a) 4 – (–3) (b) –6 – 2 Solution: (a) 4 – (– 3) = 7 Move 7 steps to the right. –3 –1–2 0 1 2 4 3 +7 (b) – 6 – 2 = – 8 Move 8 steps to the left. –3–4–5–6 –1–2 0 1 2 –8 To perform a subtraction involving three integers, always work out from left to right. EXAMPLE 15 Simplify 8 – (– 4) – 3. Solution: 8 – (– 4) – 3 = 8 + 4 – 3 = 12 – 3 = 9 Simplifying addition and subtraction of integers: • a + (+b) = a + b • a + (–b) = a – b • a – (+b) = a – b • a – (–b) = a + b Press 8 – (–) 4 – 3 = 9 Work out from left to right. ©Praxis Publishing_Focus On Maths

14 CHAPTER 1 Integers EXAMPLE 16 In the morning, the temperature of a city was – 3°C. Its temperature then dropped by 5°C in the afternoon. At night, its temperature dropped by another 4°C. Find the temperature of the city at night. Solution: – 3 – 5 – 4 = – 8 – 4 = –12 Therefore, the temperature of the city at night was –12°C. –3 –1 31 5 –8 –4 40–8 –12 Practice 1.2 Basic Intermediate Advanced 1 Calculate each of the following. (a) –4 + 7 (b) –9 + 3 (c) 5 + (–13) (d) –7 + (–2) (e) 6 + (–6) (f) –4 + (–8) B Simplify each of the following. (a) 5 + (–7) + 4 (b) 3 + 6 + (–10) (c) –7 + 1 + 2 (d) –4 + 9 + (–5) (e) –6 + (–4) + (–3) (f) –8 + (–7) + 10 3 Evaluate each of the following. (a) –7 – (–2) (b) –4 – 6 (c) 8 – (–5) (d) 9 – (–1) (e) –2 – 6 (f) –5 – (–3) 4 Simplify each of the following. (a) 4 – (–4) – 12 (b) 6 – (–3) – (–1) (c) 8 – 15 – (–2) (d) –2 – 5 – (–6) (e) –6 – (–7) – (–2) (f) –9 – (–5) – 8 5 Fill in the correct numbers. (a) 18 + 16 = + 18 (b) –4 + 90 = 90 + (c) –6 + (–18) = + (–6) (d) (–8 + 18) + (–6) = –8 + [ + (–6)] (e) (20 + 14) + 14 = 20 + (14 + ) (f) (–6 + 8) + 20 = + (8 + 20) 6 A submarine was 40 m below sea level. Three hours later, it rose by 15 m. What was the new position of the submarine? 7 The temperature of a town was –3°C in the morning. Its temperature rose by 7°C at noon. What was the temperature of the town at noon? 8 In an experiment, the temperature of a solution was –2°C. When it was heated, its temperature increased by 7°C. One hour later, its temperature increased by another 4°C. What was the final temperature of the solution? I An eagle is flying directly above a diver. The eagle is 7m above sea level while the diver is 2m below sea level. Find the vertical distance between the eagle and the diver. J The initial temperature of a cold storage was –1°C. 30 minutes later, its temperature dropped by 5°C. What was the new temperature of the cold storage? K A submarine is 8 m below sea level. It descends 4 m and then descends another 3 m. Find its final position. team work Here are 4 types of subtraction questions: • (negative integer) − (negative integer) • (negative integer) − (positive integer) • (positive integer) − (positive integer) • (positive integer) − (negative integer) Write a question for each type of subtraction. Show how your team uses number line to solve each question. ©Praxis Publishing_Focus On Maths

15 Integers CHAPTER 1 1.3 Multiplication and Division of Integers EXAMPLE 17 Find the product of the following. (a) –4 × 3 (b) – 8 × (–7) Solution: (a) – 4 × 3 (b) –8 × (–7) = – (4 × 3) = +(8 × 7) = –12 = 56 When multiplying three integers, we work from left to right. EXAMPLE 18 Calculate the following. (a) –2 × 3 × (–5) (b) – 6 × (– 4) × (– 3) Solution: (a) –2 × 3 × (– 5) (b) –6 × (–4) × (– 3) = – 6 × (–5) = 24 × (–3) = 30 = –72 EXAMPLE 19 The temperature in a refrigerator decreases 2°C every hour. If the temperature now is 0°C, find its temperature (a) 3 hours later, (b) 4 hours earlier. A Multiplication of integers Rules for multiplication of two integers: (+) × (+) = (+) (–) × (–) = (+) (+) × (–) = (–) (–) × (+) = (–) 1 Determine the sign of the product. 2 Multiply the whole numbers. Discuss with your classmates. What is the sign of the product when you multiply 2 integers: • if both integers are positive? • if one integer is positive and the other integer is negative? • if both integers are negative? INTERACTIVE ZONE ©Praxis Publishing_Focus On Maths

16 CHAPTER 1 Integers Solution: (a) –2 × (+3) = –6°C Therefore, the temperature 3 hours later is –6°C. (b) –2 × (–4) = 8°C Therefore, the temperature 4 hours earlier was 8°C. Decrease 2°C = –2 3 hours later = +3 Decrease 2°C = –2 4 hours before = –4 B Properties of multiplication (I) Commutative property of multiplication a × b = b × a This rule says we can multiply numbers in any order we want without changing the result. For example, 2 × 3 = 3 × 2 (–6) × 9 = 9 × (–6) (II) Associative property of multiplication (a × b) × c = a × (b × c) Here it says we can group numbers in a product in any way we want and still get the same result. For example, (5 × 10) × 2 = 5 × (10 × 2) [(–6) × 8] × 3 = (– 6 × 3) × (8 × 3) (III) Distributive property of multiplication a × (b + c) = (a × b) + (a × c) This property involves multiplication of a number by the sum of two addends. We can first add and then multiply or multiply first and then add. Either way, the multiplication is distributed over all the terms inside the brackets, and gives the same answer. For example, 6 × (2 + 3) = (6 × 2) + (6 × 3) (–5) × (7 + 6) = (–5 × 7) + (–5 × 6) EXAMPLE 20 Fill in the blanks. (a) 6 × (–2) = –2 × (b) (8 × 1) × 10 = 8 × (1 × ) (c) 2 × (5 + 6) = (2 × 5) + ( × 6) Solution: (a) 6 × (–2) = –2 × 6 (commutative property of multiplication) (b) (8 × 1) × 10 = 8 × (1 × 10 ) (associative property of multiplication) (c) 2 × (5 + 6) = (2 × 5) + ( 2 × 6) (distributive property of multiplication) ©Praxis Publishing_Focus On Maths

17 Integers CHAPTER 1 Rules for division of two integers: 1 Determine the sign of the quotient. 2 Divide the whole numbers. EXAMPLE 21 Calculate the following. (a) 10 ÷ (–5) (b) –16 –8 (c) 18 ÷ (–2) ÷ (–3) (d) –54 ÷ 3 ÷ (–9) Solution: (a) 10 ÷ (–5) = –(10 ÷ 5) (b) –16 –8 = +1 16 8 2 = 2 = –2 (c) 18 ÷ (–2) ÷ (–3) (d) –54 ÷ 3 ÷ (–9) = –9 ÷ (–3) = –18 ÷ (–9) = 3 = 2 Maths LINK Sport In hockey, every player has a plus/minus statistic. A player's plus/minus statistic increases by 1 when his team scores a goal while he is on the ice. A player’s plus/minus statistic decreases by 1 when his team is scored against while he is playing on the ice. For example, a player begins the game with a plus/minus statistic of –7. During the game, his team scores 3 goals while he is on the ice and the opposition team scores 1 goal. What is the player's new plus/minus statistic? C Division of integers (+) ÷ (+) = (+) (–) ÷ (–) = (+) (+) ÷ (–) = (–) (–) ÷ (+) = (–) Discuss with your classmates. What is the sign of the quotient when you divide 2 integers: • if both integers are positive? • if one integer is positive and the other integer is negative? • if both integers are negative? INTERACTIVE ZONE When we perform division involving three integers, we work from left to right. EXAMPLE 22 Ake, Sak and Wach started a business together. In the first month, they made a loss of $42 480. Find the loss of each person if the loss is shared equally among them. Solution: –$42 480 ÷ 3 = –$14 160 ‘–’ means a loss. Therefore, each person made a loss of $14 160. ©Praxis Publishing_Focus On Maths

18 CHAPTER 1 Integers Practice 1.3 Basic Intermediate Advanced 1 Calculate the following. (a) 2 × (–5) (b) –5 × 8 (c) – 40 × 0 (d) –9 × (–6) B Solve the following. (a) –2 × 4 × 5 (b) 6 × 3 × (–8) (c) –9 × (–2) × (–4) (d) –10 × (–5) × 3 3 Fill in the blanks. (a) 2 × (–3) = × 2 (b) [6 × (–2)] × 4 = 6 × [(–2) × ] (c) (25 × 20) + (25 × 10) = 25 × ( + 10) (d) [(–4) × (–12)] × 2 = (–4) × [ × 2] (e) (–6) × (–7) = × (–6) (f) 32 × (6 + 5) = (32 × ) + (32 × 5) 4 Calculate the following. (a) 15 ÷ (–3) (b) –21 ÷ 7 (c) –56 ÷ (–8) (d) –36 9 5 Solve the following. (a) 28 ÷ (–7) ÷ (–2) (b) –56 ÷ (–2) ÷ (–4) (c) –72 ÷ 8 ÷ (–3) (d) –84 ÷ (–6) ÷ 7 6 A submarine dived 3 m each minute. How deep was the submarine after 5 minutes? 7 The temperature at a highland resort drops by 2°C every hour. Find the total drop in temperature after 4 hours. 8 The temperature of a cold storage room drops constantly by 28°C in 4 hours. Calculate the drop in temperature each hour. I The water level in a reservoir decreases by 5 m in 4 days. Find the average decrease in the water level per day. 1.4 Combined Operations of Integers A Combined operations of integers Combined operations are also known as mixed operations. Computations involving combined operations of addition and subtraction should be done from left to right. EXAMPLE 23 Evaluate the following. (a) –3 + 7 – 5 (b) –9 – (–4) + (–3) Solution: (a) –3 + 7 – 5 = 4 – 5 = –(5 – 4) = –1 (b) –9 – (–4) + (–3) = –9 + 4 – 3 = –5 – 3 = –8 a – b = –(b – a) if b . a Computations involving combined operations of multiplication and division should also be done from left to right. ©Praxis Publishing_Focus On Maths

19 Integers CHAPTER 1 EXAMPLE 24 Calculate the following. (a) –4 × (–6) ÷ 3 (b) –104 ÷ (–8) × (–5) Solution: (a) –4 × (–6) ÷ 3 = 24 ÷ 3 = 8 (b) –104 ÷ (–8) × (–5) = 13 × (–5) = –65 Combined operations involving addition, subtraction, multiplication and division of integers including the use of brackets are performed by following the BODMAS rule as follows: 1 Operations in brackets should be done first. 2 Followed by × or ÷ from left to right. 3 Followed by + or – from left to right. EXAMPLE 25 Evaluate the following. (a) 4 + (–6) × 2 – 9 (b) –14 + (8 – 24) ÷ (–4) Solution: (a) 4 + (–6) × 2 – 9 = 4 + (–12) – 9 = –8 – 9 = –17 (b) –14 + (8 – 24) ÷ (–4) = –14 + (–16) ÷ (–4) = –14 + 4 = –10 EXAMPLE 26 The initial temperature in a freezer is 5°C. If the temperature decreases by 4°C every minute, find its temperature after 6 minutes. Solution: The change of temperature in the freezer = –4°C 5 + 6 × (–4) = 5 + (–24) = –19°C Therefore, the temperature in the freezer after 6 minutes is –19°C. ©Praxis Publishing_Focus On Maths

20 CHAPTER 1 Integers EXAMPLE 27 The outside temperature at 11:00 p.m. is –10°C. The weather forecaster predicts that it will drop 2°C each hour. Then the temperature will rise 3°C each hour after 2 a.m. What will the temperature be at 6:00 a.m.? Solution: Stage 1: Understand the problem List the facts and the question. Facts: At 11:00 p.m. the temperature is –10°C. The temperature drops 2°C each hour until 2:00 a.m. The temperature rises 3°C each hour after 2:00 a.m. Question: What will the temperature be at 6:00 a.m.? Stage 2: Think of a plan Make a table to record the hourly temperature change. When the temperature drops, use a negative integer. When the temperature rises, use a positive integer. Stage 3: Carry out the plan Complete the table. Add –2°C (or subtract 2°C) for each hour until 2:00 a.m. Add 3°C for each hour until 6:00 a.m. Time 11:00 12:00 1:00 2:00 3:00 4:00 5:00 6:00 Temperature –10°C –12°C –14°C –16°C –13°C –10°C –7°C –4°C –2 –2 –2 +3 +3 +3 +3 At 6:00 a.m. the temperature will be –4°C. Stage 4: Look back Work backwards to check. Add –3°C and add +2°C. (– 4°C) + (– 3°C) + (– 3°C) + (– 3°C) + (– 3°C) + (+2°C) + (+2°C) + (+2°C) = –10°C B Problem-solving ©Praxis Publishing_Focus On Maths

21 Integers CHAPTER 1 EXAMPLE 28 Affendi’s credit card account showed a balance of debts of $250 at one time. He used his credit card to pay for three books each costing $130. A week later, his credit card account was charged an interest of $3 and Affendi made a payment of $400 to his account. Explain whether Affendi had cleared his debts. Solution: Stage 1: Understand the problem List the facts and the question. Facts: Total account balance = −$250 Cost of a book = $130 Interest charged = $3 Payment made = $400 Question: Explain whether Affendi had cleared his debts. Stage 2: Think of a plan Account balance means debt, use a negative integer. Total payment for books, use a negative integer. Interest charged, use a negative integer. Payment to account, use a positive integer. Stage 3: Carry out the plan Total account balance = –250 Total payment for books using credit card = 3 × (–130) = –390 Interest charged = –3 Payment to account = +400 Final credit card account balance = –250 + (–390) + (–3) + 400 = –250 – 360 – 3 + 400 = –243 Affendi had not cleared his debts because his credit card account still showed a balance of debts of $243. Stage 4: Look back Work backwards to check. Account balance = 250 + 390 + 3 – 400 = $243 ©Praxis Publishing_Focus On Maths

22 CHAPTER 1 Integers 1.5 Factors and Prime Factors A Listing the factors of whole numbers The factors of a given whole number are the numbers that can divide the given whole number exactly. For example, 8 ÷ 1 = 8 8 ÷ 2 = 4 8 ÷ 4 = 2 8 ÷ 8 = 1 8 can be divided exactly by 1, 2, 4 and 8. Therefore, 1, 2, 4 and 8 are the factors of 8. We can divide the number by itself and by numbers which are smaller than itself to list the factors of a given number. • The number 1 is a factor of all whole numbers. • Every number is a factor of itself. Practice 1.4 Basic Intermediate Advanced 1 Evaluate the following. (a) 4 – (–3) + (–6) (b) –5 + 7 – (–4) (c) –9 + (–6) – (–3) (d) –7 – (–5) + 6 B Calculate the following. (a) –6 × (–8) ÷ 3 (b) –51 ÷ 3 × (– 4) (c) 4 × (–15) ÷ (–5) (d) –56 ÷ (–7) × (–9) C Evaluate the following. (a) –2 × (–10) – (–5) (b) 9 ÷ (–3) – (–8) (c) 27 ÷ (–9) × (–4) (d) 48 ÷ [12 + (–8)] (e) 11 – 20 ÷ (–4) + (–7) (f) 5 × (–8) – 10 ÷ (–2) (g) –4 × [11 + (–8)] – (–9) (h) 8 × [–2 + (–4)] ÷ 12 D A chemical compound with an initial temperature of 29°C was heated until its temperature rose by 15°C. Then the compound was put in a beaker. The temperature of the compound decreased by 8°C when it was cooled. Find the final temperature of the chemical compound. E In a mathematics quiz, each correct answer is given +3 points and each wrong answer is given –2 points. If Elin gave 6 correct answers and 4 wrong answers, find the total score for Elin. F Jill is climbing up a steep and slippery path to fetch a bucket of water. When she is 6 m above her starting point, she slips back 1 m, grasps some bushes by the side of the path and climbs 7 m more to a flat section. When she reaches the resting place, how far is she from the starting point? G Local time in Melbourne is 3 hours ahead of Jakarta (Indonesia) time, which is 5 hours behind Auckland (New Zealand) time. Auckland is 11 hours ahead of Berlin (Germany) time. What is the time difference between: (a) Melbourne and Berlin? (b) Jakarta and Berlin? ©Praxis Publishing_Focus On Maths

23 Integers CHAPTER 1 B Determining the factors of whole numbers We can determine whether a number is the factor of a given number by finding out whether the given number is divisible by the number. EXAMPLE 30 Determine whether 9 is a factor of each of the following numbers. (a) 63 (b) 129 Solution: (a) 63 ÷ 9 = 7 Therefore, 9 is a factor of 63. (b) 129 ÷ 9 = 14 remainder 3 Therefore, 9 is not a factor of 129. An exact division Not an exact division C Identifying prime factors The prime factors of a given whole number are the factors of the given whole number which are also prime numbers. For example, 2 and 3 are factors of 6 which are also prime numbers, therefore 2 and 3 are the prime factors of 6. EXAMPLE 31 Determine whether each of the following factors of 24 is a prime factor. (a) 2 (b) 6 Solution: (a) 2 is a prime number. Therefore, 2 is a prime factor of 24. (b) 6 is not a prime number. Therefore, 6 is not a prime factor of 24. EXAMPLE 29 List all the factors of 15. Solution: 15 ÷ 1 = 15 15 ÷ 5 = 3 15 ÷ 3 = 5 15 ÷ 15 = 1 Therefore, the factors of 15 are 1, 3, 5 and 15. We can also find the factors of a given whole number by expressing the number as a product of two numbers. For example, 15 = 1 × 15 = 3 × 5 Therefore, the factors of 15 are 1, 3, 5 and 15. ©Praxis Publishing_Focus On Maths

24 CHAPTER 1 Integers D Finding the prime factors of whole numbers The prime factors of a given whole number can be found by using repeated division. EXAMPLE 32 Find all the prime factors of 30. Solution: 2 30 3 15 5 5 1 Therefore, the prime factors of 30 are 2, 3 and 5. EXAMPLE 33 Find the sum of all the prime factors of 56. Solution: The prime factors of 56 are 2 and 7. Therefore, the sum of all the prime factors of 56 is 2 + 7 = 9. The factor tree method can also be used to find the prime factor of a given whole number. For example, 30 3 × 10 3 × 2 × 5 From the factor tree method, we have 30 = 2 × 3 × 5. Therefore, the prime factors of 30 are 2, 3 and 5. Keep dividing by the smallest prime factor until you get 1. 2 56 2 28 2 14 7 7 1 E Determining the prime factors of whole numbers EXAMPLE 34 Determine whether 13 is a prime factor of the following numbers. (a) 195 (b) 228 Solution: (a) 195 ÷ 13 = 15 An exact division Therefore, 13 is a factor of 195. Since 13 is also a prime number, 13 is a prime factor of 195. (b) 228 ÷ 13 = 17 remainder 7 Not an exact division Therefore, 13 is not a prime factor of 228. ©Praxis Publishing_Focus On Maths

25 Integers CHAPTER 1 Practice 1.5 Basic Intermediate Advanced 1 Determine whether 6 is a factor of each of the following numbers. (a) 42 (b) 51 (c) 96 (d) 138 B Determine whether 8 is a factor of each of the following numbers. (a) 24 (b) 104 (c) 210 (d) 344 C Determine whether 9 is a factor of each of the following numbers. (a) 126 (b) 243 (c) 288 (d) 325 D Determine whether each of the following factors of 84 is a prime factor. (a) 2 (b) 3 (c) 4 (d) 6 (e) 7 (f) 12 (g) 14 (h) 21 E Find the sum of all the prime factors of each of the following numbers. (a) 18 (b) 30 (c) 42 (d) 57 F Determine whether each of the following numbers is a prime factor of the number in brackets. (a) 3(27) (b) 4(36) (c) 5(60) (d) 7(196) 1.6 Lowest Common Multiple (LCM) and Highest Common Factor (HCF) A Listing the multiples of whole numbers A multiple of a given whole number is the product of itself and another non-zero whole number. For example, multiples of 4: 4 × 1, 4 × 2, 4 × 3, 4 × 4, 4 × 5, … 4, 8, 12, 16, 20, … The list of multiples of a given number is also a number sequence. For example, multiples of 7: 7, 14, 21, 28, … A number sequence 7×1 7 × 2 7 × 3 7 × 4 EXAMPLE 35 List the first five multiples of 9. Solution: 9, 9 × 1 18, 9 × 2 27, 9 × 3 36, 9 × 4 45 9 × 5 Whenever a number is multiplied by 1, the product will be the number itself. For example, 2 × 1 = 2 , 5 × 1 = 5 Therefore, every number is a multiple of itself. ©Praxis Publishing_Focus On Maths

26 CHAPTER 1 Integers B Determining the multiples of whole numbers The multiples of a given whole number are divisible by the given whole number. For example, 2, 4, 6, 8, … are multiples of 2. Therefore, 2, 4, 6, 8, … are divisible by 2. We can determine whether a number is a multiple of another number by working out the division. EXAMPLE 36 Determine whether 30 is a multiple of each of the following numbers. (a) 6 (b) 7 Solution: (a) 30 ÷ 6 = 5 Therefore, 30 is a multiple of 6. (b) 30 ÷ 7 = 4 remainder 2 Therefore, 30 is not a multiple of 7. An exact division Not an exact division C Finding the common multiples of whole numbers A number that is a multiple of two or more whole numbers is called the common multiple of these whole numbers. For example, 21 is a multiple of 3 and 7 respectively. Therefore, 21 is a common multiple of 3 and 7. We follow the steps below to find the common multiples of two or more whole numbers. 1 List the multiples of each given whole number. 2 Identify the common multiples. EXAMPLE 37 Find the first three common multiples of 4 and 6. Solution: Multiples of 4: 4, 8, 12 , 16, 20, 24 , 28, 32, 36 , … Multiples of 6: 6, 12 , 18, 24 , 30, 36 , … Therefore, the first three common multiples of 4 and 6 are 12, 24 and 36. ©Praxis Publishing_Focus On Maths

27 Integers CHAPTER 1 D Determining the common multiples of whole numbers EXAMPLE 38 Determine whether 30 is a common multiple of (a) 3 and 7, (b) 5, 6 and 10. Solution: (a) 30 ÷ 3 = 10 30 ÷ 7 = 4 remainder 2 30 is a multiple of 3 but not a multiple of 7. Therefore, 30 is not a common multiple of 3 and 7. (b) 30 ÷ 5 = 6 30 ÷ 6 = 5 30 ÷ 10 = 3 30 is a multiple of 5, 6 and 10 respectively. Therefore, 30 is a common multiple of 5, 6 and 10. We can also obtain the first three common multiples of 4 and 6 by following the steps below: 1 Find the first common multiple of 4 and 6. Multiples of 4: 4, 8, 12 , 16,... Multiples of 6: 6, 12 , 18, ... 2 Determine the next two common multiplies by working out (12 × 2) and (12 × 3). Therefore, the first three common multiples of 4 and 6 are 12 , 24 , 36 The list of common multiples is also a number sequence. E Determining the common multiples of whole numbers The lowest common multiple (LCM) of two or more numbers is the smallest common multiple of these numbers. EXAMPLE 39 Find the lowest common multiple of 6 and 8. Solution: Method 1: List all multiples Multiples of 6: 6, 12, 18, 24 , 30, 36, 42, 48 , … Multiples of 8: 8, 16, 24 , 32, 40, 48 , … Therefore, the LCM of 6 and 8 is 24. Method 2: Prime factorisation 6 = 2 × 3 8 = 2 × 2 × 2 2 × 2 × 2 × 3 Therefore, the LCM of 6 and 8 is 2 × 2 × 2 × 3 = 24. Ensure all the factors are prime numbers. ©Praxis Publishing_Focus On Maths

28 CHAPTER 1 Integers Method 3: Repeated Division 2 6 8 2 3 4 2 3 2 3 3 1 1 1 Therefore, the LCM of 6 and 8 is 2 × 2 × 2 × 3 = 24. When you use repeated division to find the LCM, select a divisor that can divide as many dividends as possible for every division. Keep dividing by the smallest prime factor until you get 1. Carry down the numbers that are not divisible by the prime factors. Multiply all the divisors. F Finding the common factors of whole numbers A number that is a factor of two or more numbers is called the common factor of these numbers. For example, 2 is a factor of 4 and 10 respectively. Therefore, 2 is a common factor of 4 and 10. EXAMPLE 40 Find the lowest common factors of 8 and 12. Solution: Factors of 8 : 1 , 2 , , 4 , 8 Factors of 12 : 1 , 2 , 3 , 4 , 6 , 12 Therefore, the common factors of 8 and 12 are 1, 2 and 4. The number 1 is a common factor of all whole numbers. G Determining the common factors of whole numbrs EXAMPLE 41 Determine whether 4 is a common factor of (a) 24 and 38, (b) 36, 52 and 68. Solution: (a) 24 ÷ 4 = 6 38 ÷ 4 = 9 remainder 2 4 is a factor of 24 but not a factor of 38. Therefore, 4 is not a common factor of 24 and 38. ©Praxis Publishing_Focus On Maths

29 Integers CHAPTER 1 (b) 36 ÷ 4 = 9 52 ÷ 4 = 13 68 ÷ 4 = 17 4 is a factor of 36, 52 and 68 respectively. Therefore, 4 is a common factor of 36, 52 and 68. H Determining the common factors of whole numbrs The highest common factor (HCF) of two or more numbers is the largest common factors of these numbers. EXAMPLE 42 Find the highest common factor of 36 and 48. Solution: Method 1: List all factors Factors of 36: 1 , 2 , 3 , 4 , 6 , 9, 12 , 18, 36 Factors of 48: 1 , 2 , 3 , 4 , 6 , 8, 12 , 16, 24, 48 Therefore, the HCF of 36 and 48 is 12. Method 2: Prime factorisation 36 = 2 × 2 × 3 × 3 48 = 2 × 2 × 2 × 2 × 3 2 × 2 × 3 Therefore, the HCF of 36 and 48 is 2 × 2 × 3 = 12. Method 3: Repeated division 2 36 48 2 18 24 3 9 12 3 4 Therefore, the HCF of 36 and 48 is 2 × 2 × 3 = 12. Multiply all the divisors. Identify the prime factors which are common factors of both numbers. Stop dividing when there are no more common factors except 1. ©Praxis Publishing_Focus On Maths

30 CHAPTER 1 Integers EXAMPLE 43 The Boy Scouts of a school held a charity activity. A total of 252 shirts, 180 pairs of trousers and 108 pairs of shoes were donated by members to an orphanage. All the items were divided equally in each pack. What would be the maximum number of packs that were prepared? Solution: Stage 1: Understand the problem List the facts and the question. Facts: 252 shirts, 180 pairs of trousers and 108 pairs of shoes were divided equally in each pack. Question: Find the maximum number of packs that were prepared. Stage 2: Think of a plan Find the HCF of 252, 180 and 108. Stage 3: Carry out the plan 252 = 2 × 2 × 3 × 3 × 7 180 = 2 × 2 × 3 × 3 × 5 108 = 2 × 2 × 3 × 3 × 3 The HCF of 252, 180 and 108 is 2 × 2 × 3 × 3 = 36. The maximum number of packs that were prepared would be 36. Stage 4: Look back Work backwards to check. 252 ÷ 36 = 7 180 ÷ 36 = 5 108 ÷ 36 = 3 I Problem - solving ©Praxis Publishing_Focus On Maths

31 Integers CHAPTER 1 EXAMPLE 44 Canned coffee is sold in 6 cans per box and canned tea is sold in 9 cans per box. Aimy wishes to buy the same number of canned coffee and canned tea for her sister’s birthday party. What is the minimum number of boxes of each type of canned drinks she needs to buy? Solution: Stage 1: Understand the problem List the facts and the question. Facts: Number of cans of coffee = 6 cans per box Number of cans of tea = 9 cans per box Question: What is the minimum number of boxes of each type of canned drinks she needs to buy? Stage 2: Think of a plan Find the LCM of 6 and 9 to determine the same number of cans. Use division to find the number of boxes of canned coffee and canned tea. Stage 3: Carry out the plan Multiples of 6 : 6, 12, 18 , 24, 30, … Multiples of 9 : 18 , 27, 36, 45, … Thus, LCM of 6 and 9 = 18. Number of boxes of canned coffee = 18 ÷ 6 = 3 Number of boxes of canned tea = 18 ÷ 9 = 2 Therefore, the minimum number of boxes Aimy needs to buy is 3 boxes of canned coffee and 2 boxes of canned tea. Stage 4: Look back Work backwards to check. Number of canned coffee = 3 × 6 = 18 Number of canned tea = 2 × 9 = 18 ©Praxis Publishing_Focus On Maths

32 CHAPTER 1 Integers Practice 1.6 Basic Intermediate Advanced 1 552, 1752, 2948, 13 692 Which of the numbers above are multiples of 8? B Find all the common multiples that are less than 50 for each of the following sets of numbers. (a) 4 and 5 (b) 3, 6 and 9 C Find the LCM of each of the following sets of numbers using prime factorisation. (a) 2 and 3 (b) 9 and 15 (c) 3, 9 and 12 (d) 6, 18 and 24 D Find the LCM of each of the following sets of numbers using repeated division. (a) 4 and 5 (b) 12 and 28 (c) 6, 8 and 12 (d) 16, 24 and 40 E Determine whether each of the following numbers is a common factor of the set of numbers in brackets. (a) 3 (9 and 24) (b) 7 (30 and 42) (c) 5 (10, 55 and 70) (d) 4 (12, 24 and 60) F Find the HCF of each of the following sets of numbers by listing all the factors. (a) 6 and 30 (b) 20 and 35 (c) 2, 6 and 8 (d) 12, 30 and 54 G Find the HCF of each of the following sets of numbers using prime factorisation. (a) 24 and 48 (b) 56 and 91 (c) 6, 9 and 18 (d) 32, 64 and 96 H During Hari Raya, Mina prepared 45 pieces of Kuih Wajik, 75 pieces of Kuih Wingko and 90 pieces of curry puffs. She wants to serve all the food on some plates. What would be the maximum number of plates needed if all the food was distributed equally on all the plates? I A neon light blinks every 9 seconds and another neon light blinks every 12 seconds. If both the lights blink together when the switch is on, after how many seconds will both the neon lights blink together again? J Mia wants to pack some souvenirs, including a key chain and a refrigerator magnet in each bag for her friends. Key chains are sold in packs of 10 whereas refrigerator magnets are sold in packs of 6. What is the minimum quantity of key chains that Mia must buy so that the quantity of refrigerator magnets she buys can be packed exactly with all the key chains? K Faiz loves sports. He climbs a mountain every 12 days, then goes swimming every 8 days. If he goes climbing and swimming today, how many days later will he go climbing and swimming on the same day again? ©Praxis Publishing_Focus On Maths

33 Integers CHAPTER 1 Factors Numbers that can divide the given number exactly. Common factors Numbers that are the factors of two or more given numbers. Highest common factor (HCF) The greatest common factor of two or more given numbers. Prime numbers Numbers that can only be divided exactly by itself and the number 1. For example, 2, 3, 5, 7, 11, … Prime factors Factors of a given number which are also prime numbers. Multiples Product of itself and another non-zero whole number. Common multiples Numbers that are the multiples of two or more given numbers. Lowest common multiple (LCM) The smallest common multiple of two or more given numbers. Summary Summary Summary Addition of integers • Adding a positive integer can be represented on a number line by moving to the right. For example, –5 + (+4) = –1. –6 –5 –4 –3 –2 –1 0 +4 • Adding a negative integer can be represented on a number line by moving to the left. For example, –2 + (–3) = –5. –6 –5 –4 –3 –2 –1 0 –3 • a + (+b) = a + b • a + (–b) = a – b Subtraction of integers • Subtraction of integers can be represented on a number line by moving from the second integer to the first integer and count the number of steps required. For example, –1 – (–6) = +5. Move to the right, the difference is positive. –6 –5 –4 –3 –2 –1 0 +5 For example, –5 – (–1) = –4. Move to the left, the difference is negative. –6 –5 –4 –3 –2 –1 0 –4 • a – (+b) = a – b • a – (–b) = a + b Positive integers Whole number with a ‘+’ sign or without any sign. For example, +5, +9, 24, 32. Zero 0 is an integer but neither positive nor negative. Negative integers Whole number with a ‘–’ sign. For example, –2, –7, –49. Integers Integers Multiplication and division of integers • The product/quotient of two integers with the same sign is a positive integer. • The product/quotient of two integers with different signs is a negative integer. • (+) × (+) = (+) • (–) × (+) = (–) • (+) ÷ (+) = (+) • (–) ÷ (–) = (+) • (–) × (–) = (+) • (+) × (–) = (–) • (–) ÷ (+) = (–) • (+) ÷ (–) = (–) Combined operations of integers To perform computations involving combined operations, • calculate within the brackets first, • followed by division or multiplication, • then addition or subtraction, from left to right. ©Praxis Publishing_Focus On Maths

34 CHAPTER 1 Integers 1 Section A 1. How many integers are there between – 4 and 2? A 4 B 5 C 6 D 7 2. –15 P Q 90 The diagram shows a number line. What are the values of P and Q? P Q A –12 6 B –9 6 C –12 3 D –9 3 3. Arrange –8, –10, 7, –2, 0 and 3 in increasing order. A 3, 7, 0, –2, –8, –10 B 0, –2, 3, 7, –8, –10 C –2, –8, –10, 0, 3, 7 D –10, –8, –2, 0, 3, 7 4. –7 –6 –5 –4 –3 –2 –1 0 –5 The diagram is a number line which represents an operation for integers. Which of the following is represented by the number line? A –5 + (–2) = –7 B –2 + (–5) = –7 C –7 – (–5) = –2 D –(–2) – 7 = –5 5. The temperature of a town was 2°C in the afternoon. Its temperature dropped by 6°C in the evening and dropped by another 5°C at night. What was the temperature of the town at night? A –13°C B –9°C C –8°C D –3°C 6. 17, 19, , 29, 31, 37 are prime numbers arranged in increasing order. must be filled with A 21 B 23 C 25 D 27 7. 2 7 5 10 x The diagram shows some factors of 350. The possible value of x is A 3 B 8 C 11 D 14 8. Which of the following are the prime factors of 110? A 2 and 3 B 2 and 11 C 3 and 7 D 7 and 11 9. 81 is a multiple of p. p is a multiple of 3. Based on the above statement, which of the following is not a value of p? A 3 B 9 C 18 D 27 10. Which of the following pairs of numbers having 8 as its highest common factor? A 24 and 40 B 24 and 36 C 20 and 32 D 40 and 52 ©Praxis Publishing_Focus On Maths

35 Integers CHAPTER 1 Section B 1. Evaluate each of the following. (a) –6 + (–11) + 5 (b) 7 + (–5) + (–8) (c) –9 + (–4) + (–3) 2. Calculate the value of each of the following. (a) –10 – 7 – (–8) (b) 5 – (–3) – 12 (c) –21 – 13 – (–7) 3. Write the number 54 as the sum of two prime numbers. Give all the possible answers. 4. Find the difference between the largest value and the smallest value for the prime factors of 84. 5. The temperature in Ciwidey, Bandung at a certain time was 12°C. The temperature dropped until – 6°C. The temperature then rose by 3°C and finally dropped by 8°C. Determine (a) the change in temperature of the town, (b) the final temperature of the town. 6. A diver was at 50 m below sea level. The diver swam up 2 m every 5 seconds. Explain whether the diver would have reached the sea surface after 2 minutes. 7. The current account of Mr Arif showed a balance of $1238. He signed two payment cheques of $890 and $1730 respectively. (a) Determine whether the $890 cheque or the $1730 cheque would bounce when the cheques were credited. (b) How much would Mr Arif have to top up in his account so that both cheques that he signed would not bounce when they are credited? 8. The LCM of two numbers is 60 and the HCF of the two numbers is 6. Find the two possible numbers. 9. A clock is set to ring every 15 minutes whereas another clock is set to ring every 25 minutes. If both the clocks ring together at 4:00 p.m., find the time at which both clocks will subsequently ring together again. 10. A room has a measurement of 7.5 m × 9.6 m. If Mr Zagy wishes to lay square tiles on the floor of the room, what is the biggest size, in cm, of the tiles so that the tiles will fully cover the entire floor? 11. A cafeteria serves groundnut cakes every 4 days and burgers every 6 days. If groundnut cakes and burgers are served on Monday, on which day will both types of food be served on the same day again at the cafeteria? 12. Anissa wants to produce a history scrapbook using 24 photographs and 42 newspapers cuttings. She wishes to use all the photographs and newspaper cuttings such that every page of the scrapbook contains the same number of photographs and newspaper cuttings. (a) What is the maximum number of pages she can have for the scrapbook? (b) For each page of the scrapbook, how many photographs and newspaper cuttings will there be? ©Praxis Publishing_Focus On Maths