Focus On Maths Grade 7 (Teacher's Guide)

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

Focus-on Mathematics Teacher’s Guide Grade 7 I Introduction The learning of Mathematics imparts many skills that contribute to the development of the human mind. It trains the learners to think methodically and rationally, analyse various types of situations, anticipate and plan, make decisions and solve problems. Mathematics also serves as a tool that facilitates the gaining of knowledge related to science and technology. Mathematical skills and knowledge are indeed essential to enhance our standard and quality of living in the modern area. Rationale Rasional Mata Pelajaran Matematika Matematika merupakan ilmu atau pengetahuan tentang belajar atau berpikir logis yang sangat dibutuhkan manusia untuk hidup yang mendasari perkembangan teknologi modern. Matematika mempunyai peran penting dalam berbagai disiplin ilmu dan memajukan daya pikir manusia. Matematika dipandang sebagai materi pembelajaran yang harus dipahami sekaligus sebagai alat konseptual untuk mengonstruksi dan merekonstruksi materi tersebut, mengasah, dan melatih kecakapan berpikir yang dibutuhkan untuk memecahkan masalah dalam kehidupan. Belajar matematika dapat meningkatkan kemampuan peserta didik dalam berpikir logis, analitis, sistematis, kritis, dan kreatif. Kompetensi tersebut diperlukan agar pembelajar memiliki kemampuan memperoleh, mengelola, dan memanfaatkan informasi untuk bertahan hidup pada keadaan yang selalu berubah, penuh dengan ketidakpastian, dan bersifat kompetitif. Mata Pelajaran Matematika membekali peserta didik tentang cara berpikir, bernalar, dan berlogika melalui aktivitas mental tertentu yang membentuk alur berpikir berkesinambungan dan berujung pada pembentukan alur pemahaman terhadap materi pembelajaran matematika berupa fakta, konsep, prinsip, operasi, relasi, masalah, dan solusi matematis tertentu yang bersifat formal-universal. Proses mental tersebut dapat memperkuat disposisi peserta didik untuk merasakan makna dan manfaat matematika dan belajar matematika serta nilai- nilai moral dalam belajar Mata Pelajaran Matematika, meliputi kebebasan, kemahiran, penaksiran, keakuratan, kesistematisan, kerasionalan, kesabaran, kemandirian, kedisiplinan, ketekunan, ketangguhan, kepercayaan diri, keterbukaan pikiran, dan kreativitas. Dengan demikian relevansinya dengan profil pelajar Pancasila, Mata Pelajaran Matematika ditujukan untuk mengembangkan kemandirian, kemampuan bernalar kritis, dan kreativitas peserta didik. Adapun materi pembelajaran pada Mata Pelajaran Matematika di setiap jenjang pendidikan dikemas melalui bidang kajian Bilangan, - 133 - Aljabar, Pengukuran, Geometri, Analisis Data dan Peluang, dan Kalkulus (sebagai pilihan untuk kelas XI dan XII). ©Praxis Publishing_Focus On Maths

II Focus-on Mathematics Teacher’s Guide Grade 7 Aim Tujuan Mata Pelajaran Matematika Mata Pelajaran Matematika bertujuan untuk membekali peserta didik agar dapat: 1. memahami materi pembelajaran matematika berupa fakta, konsep, prinsip, operasi, dan relasi matematis dan mengaplikasikannya secara luwes, akurat, efisien, dan tepat dalam pemecahan masalah matematis (pemahaman matematis dan kecakapan prosedural), 2. menggunakan penalaran pada pola dan sifat, melakukan manipulasi matematis dalam membuat generalisasi, menyusun bukti, atau menjelaskan gagasan dan pernyataan matematika (penalaran dan pembuktian matematis), 3. memecahkan masalah yang meliputi kemampuan memahami masalah, merancang model matematis, menyelesaikan model atau menafsirkan solusi yang diperoleh (pemecahan masalah matematis). 4. mengomunikasikan gagasan dengan simbol, tabel, diagram, atau media lain untuk memperjelas keadaan atau masalah, serta menyajikan suatu situasi ke dalam simbol atau model matematis (komunikasi dan representasi matematis), 5. mengaitkan materi pembelajaran matematika berupa fakta, konsep, prinsip, operasi, dan relasi matematis pada suatu bidang kajian, lintas bidang kajian, lintas bidang ilmu, dan dengan kehidupan (koneksi matematis), dan 6. memiliki sikap menghargai kegunaan matematika dalam kehidupan, yaitu memiliki rasa ingin tahu, perhatian, dan minat dalam mempelajari matematika, serta sikap kreatif, sabar, mandiri, tekun, terbuka, tangguh, ulet, dan percaya diri dalam pemecahan masalah (disposisi matematis). Learning Achievements Capaian Pembelajaran (Fase D) Pada akhir fase D, peserta didik dapat menyelesaikan masalah kontekstual peserta didik dengan menggunakan konsep-konsep dan keterampilan matematika yang dipelajari pada fase ini. Mereka mampu mengoperasikan secara efisien bilangan bulat, bilangan rasional dan irasional, bilangan desimal, bilangan berpangkat bulat dan akar, bilangan dalam notasi ilmiah; melakukan pemfaktoran bilangan prima, menggunakan faktor skala, proporsi dan laju perubahan. Mereka dapat menyajikan dan menyelesaikan persamaan dan pertidaksamaan linier satu variabel dan sistem persamaan linier dengan dua variabel dengan beberapa cara, memahami dan menyajikan relasi dan fungsi. Mereka dapat menentukan luas permukaan dan volume bangun ruang (prisma, tabung, bola, limas dan ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 7 III kerucut) untuk menyelesaikan masalah yang terkait, menjelaskan pengaruh perubahan secara proporsional dari bangun datar dan bangun ruang terhadap ukuran panjang, luas, dan/atau volume. Mereka dapat membuat jaring-jaring bangun ruang (prisma, tabung, limas dan kerucut) dan membuat bangun ruang tersebut dari jaring-jaringnya. Mereka dapat menggunakan sifat-sifat hubungan sudut terkait dengan garis transversal, sifat kongruen - 143 - dan kesebangunan pada segitiga dan segiempat. Mereka dapat menunjukkan kebenaran teorema Pythagoras dan menggunakannya. Mereka dapat melakukan transformasi geometri tunggal di bidang koordinat Kartesius. Mereka dapat membuat dan menginterpretasi diagram batang dan diagram lingkaran. Mereka dapat mengambil sampel yang mewakili suatu populasi, menggunakan mean, median, modus, range untuk menyelesaikan masalah; dan menginvestigasi dampak perubahan data terhadap pengukuran pusat. Mereka dapat menjelaskan dan menggunakan pengertian peluang, frekuensi relatif dan frekuensi harapan satu kejadian pada suatu percobaan sederhana. Syllabus Organisation The syllabus is organised within the scope of five content elements and five process elements. The content elements in the Mathematics subject are related to the view that mathematics is a subject matter that students must understand. Mathematical understanding is closely related to the formation of a comprehension flow towards mathematical learning materials, which include facts, concepts, principles, operations and formal-universal relations. 5 Content Elements + 1 Process Strand Number Bilangan Algebra Aljabar Measurement Pengukuran Geometry Geometri Statistics and Probability Analisis Data dan Peluang Mathematical Processes Number and Algebra Learning about real number system, the properties of numbers, ratio, percentage, estimation, problem-solving involving numbers, applying real numbers in real life, patterns, the relation of function and set, logic, expressions, monomial, polynomial, equation, equation system and inequality, graphs, the interest and value of money, sequences and series and applying the knowledge of numbers and algebra in various situations. ©Praxis Publishing_Focus On Maths

IV Focus-on Mathematics Teacher’s Guide Grade 7 Measurement and Geometry Learning about length, distance, mass, area, volume and capacity, money and time, measuring units, estimation for measurement, trigonometric ratio, geometric figures and properties, visualisation of geometric models, geometric theories, geometric transformation through translation, reflection and rotation, and applying the knowledge of measurement and geometry in various situations. Statistics and Probability Learning about statistical enquiry, data collection, statistic calculation, presentation and interpretation of qualitative and quantitative data, the fundamental counting principle, probability, applying the knowledge of statistics and probability in explaining various situations as well as for facilitating decision-making in real life. Key Skill Elements Relating to Mathematics Mathematical Skills and Processes Mathematical processes refer to the process skills involved in the process of acquiring and applying mathematical knowledge. This includes: Application is the ability to use the knowledge of mathematics as a tool in learning mathematics, other contents, other sciences and apply the knowledge in real life. Problem-solving is the ability to understand, analyse, plan and solve the problems, as well as choose the appropriate method by considering the reasoning and validity of the answers. Mathematical communication and connection are the abilities to use mathematical language and symbols in communication, representation, summary and presentation accurately and clearly. Reasoning is the ability to give reasons, provide and listen to the reasons to support or argue leading to the inferences underlined with the mathematical facts. Creative thinking skills and heuristics is the ability to enhance the previous concept that they have already known or create the new concepts to improve and develop the body of knowledge. ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 7 V How to use the Focus-on Maths series The series is written to cater for the needs of secondary school students in developing mathematical ideas, skills and attitudes. 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 series provides key mathematical concepts that make the learning of mathematics inquiry-based in order to develop critical thinking and logical reasoning. The series consists of a Textbook, Workbook, Teacher’s Guide, Teaching Aids and online resources. The Teacher’s Guide provides support for teachers to help students acquire the key knowledge they need in order to understand and build a solid foundation in maths. Textbook Each chapter begins with a highly colourful image to attract students to the exciting content that follows. The purpose of the opening page is to encourage as much discussion as possible with the probing questions. Students should develop a perspective on these big ideas, in a way that will facilitate learning of related ideas in the future. It also provides a good opportunity for teachers to assess students’ prior knowledge about the topic. It also tells students what they are aiming to study. The key facts and concepts are reinforced by thought-provoking questions and useful information in ‘Background Information’ and ‘Real World Connection’ . They help students understand key ideas better and develop their mathematical skills. Worked examples show systematic workings to facilitate the teaching process, as well as helping students develop proper methods of carrying out mathematical calculation and computation, which will be an essential skill as they progress to higher levels. Workbook The workbooks are written to complement the textbooks. The exercises provided in the workbook are closely linked to the respective topics in the textbook to facilitate the teaching and learning process. A wide variety of exercises are provided throughout each workbook to ensure sufficient practice, immediate reinforcement of concepts and consolidation of skills learnt. Enrichment exercises are included in the workbook to assess how well they have understood the topics taught. ©Praxis Publishing_Focus On Maths

VI Focus-on Mathematics Teacher’s Guide Grade 7 Teacher’s Guide The Focus-on Maths Teacher’s Guide is designed to support the teaching of secondary-level mathematics. It contains extensive guidance on all the topics and the activities presented in the textbook. Each chapter contains questions that teachers can use as a basis for class discussions. The emphasis at this stage is on linking what students know about everyday life to mathematical concepts. Throughout the Teacher’s Guide, teachers will also find ideas for practical activities which will help students to develop their mathematical skills as well as introducing them to the thrill of making maths meaningful and relevant. The Teacher’s Guide is a valuable resource for conducting lessons. Each chapter overview provides detailed guidance on recommended teaching time which is dependent on the learning objectives and types of activities of each lesson. It also provides information on the teaching of the topics in the textbook and workbook as well as suggestions for teachers on how to begin, build and conclude a lesson. Learning Objectives Determines what students will learn and understand at the end of the lesson. Bilingual Key Terms Provides translations or explanations of important mathematical terms in both languages. It helps students bridge the language gap and understand mathematical concepts more effectively. Start Up Background Information Provides information about the knowledge and key facts that students need to learn at the end of each lesson. Introduces clear mathematical concepts through thought-provoking questions to encourage critical thinking and develop analytical skill. Real World Connection - Assesses students’ prior knowledge about the chapter before entering into each specific topic. - Engages students with the topic and initiates whole-class discussion. - Applies the questioning technique that encourages discussion and finds out what students already know. ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 7 VII Teaching/Learning Activities - Sets the lesson presentation mode. - Explains the key concepts and ideas which students need to know and understand. - Helps students learn facts effectively and reinforces the main ideas for the lesson. - Provides information about the related exercises in the workbook so as to encourage students to practise independently and enable teachers to review and evaluate students’ knowledge and skills on the topics based on these exercises. Throughout the lesson, teachers can guide students to scan the QR codes of under each subtopic. The carefully selected websites and videos will help to engage students and maintain their interest in the subtopic so that they will learn more. Closing - Ends the chapter by guiding students to conclude the concept of each topic. - Asks the questions which are based on the concepts of all the topics covered in the chapter so as to summarise them and consolidate students’ mathematical understanding. Assessment - Focuses on students’ understanding and ability to apply what they know to solve problems. - Emphasises on processes such as reasoning and communication. Stem Activity - Encourages students to seek relevant information on the chapter and do the enrichment activity at home. - For further classroom activity, a STEM activity is introduced in selected chapters to develop thinking skill, creative reasoning, teamwork and investigative skill that students can use in their daily life. ©Praxis Publishing_Focus On Maths

VIII Focus-on Mathematics Teacher’s Guide Grade 7 Yearly Teaching Plan Mathematics Grade 7 7 chapters 105 hours Learning area Duration (hours) Chapter 1 Integers 1.1 Integers 1.2 Addition and Subtraction of Integers 1.3 Multiplication and Division of Integers 1.4 Combined Operations of Integers 1.5 Factors and Prime Factors 1.6 Lowest Common Multiple (LCM) and Highest Common Factor (HCF) 24 Chapter 2 Real Numbers 2.1 Rational Numbers 2.2 Irrational Numbers 2.3 Fractions 2.4 Decimals 2.5 Set of Real Numbers 2.6 Use of the Symbol <, >, ≤, ≥, =, ≠ 18 Chapter 3 Ratios, Rates and Proportions 3.1 Ratios 3.2 Rates 3.3 Proportions 3.4 Relationship between Ratios, Rates and Proportions with Percentages, Fractions and Decimals 3.5 Scale 10 Chapter 4 Algebraic Expressions 4.1 Algebraic Expressions 4.2 Addition and Subtraction of Algebraic Expressions 4.3 Multiplication of Algebraic Expressions 12 ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 7 IX 4.4 Division of Algebraic Expressions 4.5 Simplifying Algebraic Fractions 4.6 Expansions of Brackets 4.7 Factorisation of Algebraic Chapter 5 Linear Equation and Inequalities in One Variable 5.1 Equality 5.2 Linear Equations in One Variable 5.3 Solve Linear Equations in One Variable 5.4 Inequalities in One Variable 5.5 Solving Inequalities in One Variable 11 Chapter 6 Lines and Angles 6.1 Relationship between Lines 6.2 Dividing Line Segment into Multiple Parts 6.3 Angles 6.4 Relationship between Angles 6.5 Constructing Special Angles 6.6 Similarity 15 Chapter 7 Statistics 7.1 Data 7.2 Presenting Data in Table 7.3 Bar Chart 7.4 Line Graph 7.5 Pie Chart 15 Note: The hours needed for each subtopic can be changed when necessary. The above allocated hours are just a suggestion. Total hours for this subject are as prescribed in the basic learning time structure, while the learners must attain the standard as prescribed in the learning standards and outcomes. ©Praxis Publishing_Focus On Maths

X Focus-on Mathematics Teacher’s Guide Grade 7 INTEGERS 1 REAL NUMBERS 23 RATIOS, RATES AND PROPORTIONS 46 ALGEBRAIC EXPRESSIONS 72 LINEAR EQUATION AND INEQUALITIES IN ONE VARIABLE 94 LINES AND ANGLES 114 STATISTICS 138 ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 1 Time 24 hours Elemen Capaian Pembelajaran Bilangan Peserta didik dapat membaca, menulis, dan membandingkan bilangan bulat, bilangan rasional dan irasional, bilangan desimal, bilangan berpangkat bulat dan akar, bilangan dalam notasi ilmiah. Mereka dapat menerapkan operasi aritmetika pada bilangan real, dan memberikan estimasi/perkiraan dalam menyelesaikan masalah (termasuk berkaitan dengan literasi finansial). Peserta didik dapat menggunakan faktorisasi prima dan pengertian rasio (skala, proporsi, dan laju perubahan) dalam penyelesaian masalah. Learning Objectives Students will be taught to: • understand on how to compare numbers, 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 characteristic of prime numbers. • understand the characteristic and use the knowledge of factors of whole numbers. • solving problems regarding Least Common Multiple (LCM) and Highest Common Factor (HCF). Bilingual Key Terms Numbers Bilangan Number line Garis bilangan Positive numbers Bilangan positif Negative numbers Bilangan negatif Decimals Desimal Percentages Persentase Comparing numbers Membandingkan bilangan Prime Factors Faktor prima Least Common Multiple (LCM) Kelipatan Persekutuan Terkecil (KPK) Highest Common Factor (HCF) Faktor Persekutuan Terbesar (FPB) ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 2 Focus-on Mathematics Teacher’s Guide Grade 7 START UP Background Information Assess the students’ prior knowledge about integer by asking what the students know and what they want to know more about integer. Remind students that they have learnt about integers in primary school. Discuss with students about various types of numbers and how each of them is used in our daily life. Ask students if they encounter negative numbers in their daily life, e.g., in weather forecasts, owing and depositing money, profit and loss in financial transaction, losing and gaining points in games, etc. Real World Connection Refer the chapter opener (Textbook Grade 7: Page viii). Let students observe the picture. Ask the students what they see and choose one or two students to participate in briefing to the class about the picture they see and ask them if they know name of deserts around the world (e.g., Sahara Desert in Africa, Gobi Desert in China and Kalahari Desert in Africa). Arouse students’ interest in this topic by bringing in real-life applications. Discuss the examples of the use of negative numbers in the real world to bring across the idea of integers. Explain to the students about the usage of negative numbers in determining the depth of the sea (e.g., -40 metres below sea level), the use of Greenwich Mean Time (GMT) in determining the time of each place around the world and the use of negative as debts and positive as profits. (Textbook Grade 7: Page viii) Suggested answer(s): Due to a lack of humidity in deserts, the air cannot hold the heat radiated by the sand, which gets heated during the sunny daytime hours. Sand cannot hold the heat which turns the entire phenomenon hot. Sand in the deserts acts like a mirror to the Sun. During the daytime, it will stay warms and when the Sun is absent on night, this causes the temperature of deserts to fall rapidly at night. (Textbook Grade 7: Page 2) Ask students to complete the section and discuss about the answers with them. Answers: 1. (a) 1081 (b) 43 2. (a) 13 (b) 402 (c) 58 (d) 18 144 (c) 67 (d) 9 ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 3 (Textbook Grade 7: Page 2) Begin this section by explaining Greenwich Mean Time (GMT) to the class, as well as how and why Greenwich Mean Time was created, and how GMT is still used today. Let students to explore the diagram of World Map GMT, and give simple brief about the GMT. GMT is calculated based on when the Sun is exactly above the Prime Meridian, which is when time is 12:00 noon at Greenwich. The Prime Meridian is the imaginary line that splits the Earth to two equal parts, which are the Western Hemisphere and Eastern Hemisphere. This Prime Meridian is also use to determine the time zones for all other places around the world. Then, explain to the students about of the time zones around the world, for example Indonesia. Indonesia has 3 different time zones, which are GMT +7 for West Indonesia Time (WIB), GMT +8 for Central Indonesia Time (WITA) and GMT +9 for East Indonesia Time (WIT). Guide students to answer the questions given in the textbook. Teacher can select a few students to test their understanding, and present their answers in class. Suggested answer(s): (Accept all possible answers) • E.g., Let’s say you are now in Jakarta, Indonesia. Your time now is 12:00 noon. • The time in London is 5:00 a.m. • He/she might be sleeping as the time in London is 5:00 a.m. in the morning. (Textbook Grade 7: Page 2) The teacher may select some students to share their questions and encourage them to explain why they ask such questions. Suggested answer(s): (Accept all possible answers.) “Why the time in the East is earlier than the West?” The time in the East is earlier than the West because of the rotation of the Earth. As the Earth rotates from west to east, regions to the east experience sunrise and the start of a new day before those to the west. Time zones are used to standardise timekeeping and coordinate activities across different regions. Teaching/Learning Activities 1.1 Integers Teach students to read the negative number –2 as negative 2, not minus 2 (‘negative’ is a state while ‘minus’ is an operation). For example, if you have $5 and you owe your friend $2, how much money do you have left? Since nothing is mentioned about you returning money to your ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 4 Focus-on Mathematics Teacher’s Guide Grade 7 friend, you have $5 left. Thus $2 is a state of owing money. However, if you return $2 to your friend, you have $5 + (–$2) = $5 – $2 = $3 left, i.e., 5 minus 2 is an operation of returning money. Temperature problems are an ideal real-life example of negative numbers. Since it is a system that students already know from everyday life, it should make more sense to them intuitively. For example, negative numbers can be used to represent temperature during winter season. Four season countries will have winter season during December to March. During this period, the temperature will go down until below freezing temperature. Sometimes, when the temperature gets colder, it can be negative temperature (below zero). This temperature can be represented by using negative sign such as -5oC or -13oC. The teacher can provide a temperature scale with 0 degrees in the middle, the positive temperatures above 0 and the negative temperatures below 0. Students will use this as a vertical number line to figure out their answers, counting along the scale. Cash flow in bank statement can be negative when someone withdraw money over from the deposited amount into the account. The value would be in negative sign such as -$100. It means that someone is owing $100 to the bank. (Textbook Grade 7: Page 3) Suggested answer(s): Other uses of integers: • Goal scores in football tournaments. • Depth of the sea. • Data types in computer programming. A Understanding integers After learning about positive and negative numbers and how to differentiate them, teacher may introduce to the class that the set of whole numbers together with the set of negative integers is known as the set of integers. On a number line, 0 (zero) is called the origin. The origin separates the line into two horizontal sides, the positive side and the negative side. If the line is horizontal, the side to the right of the origin is taken to be the positive side and the side to the left as the negative side. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 5 Guide student to understand the usage of negative and positive sign when writing negative and positive value. When students want to write negative number, the negative sign should be in front followed by the number. For example, negative 5 should be written as -5. The sign is very important because it will determine the value of the number. This also applies to daily life applications when involving the depth of water or the temperature. However, for positive number, it can be written with or without positive sign in front of the number, because the sign will not affect the number. Guide students to study the Example 1 and Example 2 on page 4 to deepen the students’ understanding about negative and positive integers. Guide students to scan the QR codes of for more online resources. B Recognise positive and negative numbers Begin this section by revisiting the concept of positive and negative numbers. Explain that positive numbers represent values greater than zero, while negative numbers represent values less than zero. Give more real-life examples of situations where positive and negative numbers are used, such as temperatures, bank accounts, or elevations. (Textbook Grade 7: Page 5) Answers: (a) 16o C (b) –$15 (c) –4 cm (d) 3 m (e) –2 (f) –8 m (g) –7 km Guide students to study the Example 3 and Example 4 on page 5 to deepen the students’ understanding about negative and positive integers and how to use them in daily life applications. (Textbook Grade 7: Page 5) Suggested answer(s): To decide whether to use positive number or negative number, we need to differentiate either it is right or left, up or down, below zero or above zero. These conditions will be a help in determining ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 6 Focus-on Mathematics Teacher’s Guide Grade 7 whether they are positive or negative. Look for key words such as: Positive Negative Above Below Gain, Profit Loss, Owe Deposit Withdrawal Increase Decrease Earning Spending Credit Debit Rise Fall C Representing integers using a number line (Textbook Grade 7: Page 6) Divide the class in groups of four. Guide students to open the file Integer number line using GeoGebra. 4. Suggested answer(s): (a) The value of integer 8 as compared to zero means that 8 more than 0. (b) The positions of negative number are always at the left side of zero. (c) Draw a number line to represent 1, 2, 3, 4 and -1, -2, -3 and -4. Show to the students that positive and negative integers can be represented in a number line. Positive integers are integers that are more than zero whereas negative integers are integers that are less than zero. On a number line, negative integers are always on the left side of zero and positive integers are always on the right side of zero. Guide students to see Example 5 on page 7. Deepen the students understanding on the representations of negative and positive integers on a number line. D Comparing two integers (Textbook Grade 7: Page 7) Begin this section by explaining the different temperature or weather every country in this world might be, and what is the factor of the difference in temperature. Explain to the class, every capital city in the world would have different weather despite of the season and location of the city itself. The table in Let’s Explore: 2 shown 6 different temperatures for 6 capital cities depending on the current weather they have. Guide students to observe the table given in the textbook and try to answer the questions given. -4 -3 -2 -1 0 1 2 3 4 ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 7 Suggested answer(s): (a) The warmest city on that day is Jakarta (29∘C). (b) The coldest city is Moscow (-11∘C). (c) Warmest city = 29∘C (Jakarta), Coldest city = -11∘C (Moscow) The warmest city is 40∘C warmer than the coldest city. (d) Moscow, Beijing, London, Tokyo, Sydney, Jakarta. Guide students to see Example 6 on page 8. Deepen the students’ understanding on the representations of negative and positive integers on a number line. (Textbook Grade 7: Page 8) Suggested answer(s): (a) When two integers have different signs, the negative sign integers are always smaller than the positive sign integers. (b) When two integers have same sign, there would be two cases: (i) If both integers have positive sign, integer that has larger value is greater than the other. (ii) If both integers have negative sign, integer that has larger value is smaller than the other. E Arranging integers in order Guide students to see Example 7, Example 8 and Example 9 on page 8 and page 9. Deepen the students understanding on negative and positive integers on a number line by arranging them in order. Let students attempt Practice 1.1 (Textbook Grade 7: Page 9) and discuss about the answers with them. Practice 1.1 Answers: 1. (a) Negative seventeen (b) Positive twenty-three (c) Positive forty-eight (d) Negative sixty-nine (e) Negative two hundred and five (f) Positive four hundred and sixteen 2. (a) –42 (c) –68 (b) –9 (d) –270 3. –9, 4, –78 4. (a) (b) 5. (a) is less than (b) is greater than (c) is greater than (d) is less than (e) is greater than ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 8 Focus-on Mathematics Teacher’s Guide Grade 7 Independent Practice Assign students to complete Exercise 1.1 (Workbook Grade 7: Page 1 – 5). 1.2 Addition and Subtraction of Integers (Textbook Grade 7: Page 10) Divide the class in groups of four. Guide students to open file Addition and subtraction of integers in GeoGebra. Start the activity by clicking on both checkboxes ‘Show addition’ and ‘Show subtraction’. Allow students to explore about the addition and subtraction between positive and negative integers. Give students some time to explore the activity and learn the addition and subtraction operations. Give example by adding 2 whole numbers (without negative integers) first. Then, the teacher may proceed with addition and subtraction of one positive and negative integers. This activity proves that the addition on number line is moving towards right and subtraction on number line is moving towards left. (Textbook Grade 7: Page 10) Suggested answer(s): (a) The addition of +2 and –2 will give the result of zero. +2 + (–2) = 2 – 2 = 0 (b) Integers +2 and –2 are called opposite integers because both have different signs. 6. (a) –5, –4, –3, –1, 0, 2 (b) –9, –7, –5, 3, 6, 8 7. (a) 9, 7, 3, –6, –10, –12 (b) 8, 5, –3, –4, –6, –11 8. (a) Largest: 5 Smallest: –9 (b) Largest: 11 Smallest: –15 (c) Largest: –7 Smallest: –30 (d) Largest: 15 Smallest: –20 9. (a) 9, 5, 1, -3, -7, -11 (b) -12, -7, -2, 3, 8,13 (c) -28, -22, -16, -10, -4 (d) -32, -29, -26, -23, -20, -17 (e) -31, -22, -13, -4, 5 10. (a) –$80 (b) –42 m (c) +5 m (d) –8oC (e) +1 h 11. (a) –3 km (b) +4 ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 9 A Addition of integers Guide students to see Example 10 on page 11. Students will learn how to do the addition operation for positive and negative integers. Introduce the rules for adding integers: • When adding two positive numbers, simply add them together. • When adding two negative numbers, add them together and attach a negative sign to the sum. • When adding a positive number and a negative number, subtract the smaller absolute value from the larger one, and attach the sign of the number with the larger absolute value. The teacher can provide extra examples and guide students through each rule, explaining the reasoning behind them. Guide students to scan the QR codes of on page 10 for more online resources. B Properties of addition (I) Commutative property of addition Students have learnt about the properties of addition in primary school. Recap with the class that the commutative property of addition states that changing the order of the numbers being added does not affect the sum. In simpler terms, when adding two or more numbers, the order in which we add them does not change the result. Explain to the class that this property holds true for both positive and negative numbers. It allows us to rearrange the numbers and still obtain the same sum. Go through the example provided on page 12 with the class. (II) Associative property of addition Explain to the class that the associative property of addition states when adding three or more numbers, the grouping of the numbers being added does not affect the sum. In other words, we can regroup the numbers being added without changing the result. Explain how this property allow us to change the grouping of the numbers without altering the final sum. It is particularly useful when adding multiple numbers, as it provides flexibility in how we group them. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 10 Focus-on Mathematics Teacher’s Guide Grade 7 Go through Example 12 and 13 on page 12 with the class. Explain that both the commutative property and the associative property are fundamental properties of addition in mathematics. They allow us to manipulate and simplify addition expressions, making calculations more efficient and flexible. C Subtraction of integers Go through the notes provided on page 13 with the class. Introduce the rules for subtracting integers: • To subtract a positive number, it is equivalent to adding its opposite (negative) number. • To subtract a negative number, it is equivalent to adding its opposite (positive) number. • When subtracting a positive number from a negative number or vice versa, it is equivalent to adding the opposite of the positive number. Provide examples and guide students through each rule, explaining the reasoning behind them. Go through Examples 14, 15 and 16 on page 13 – 14 with the class. Encourage students to explore real-life situations where subtraction of integers is applicable and create their own word problems to solve. (Textbook Grade 7: Page 14) Suggested answer(s): (Accept all possible answers.) • (negative integer) − (negative integer) • (negative integer) − (positive integer) –4 – (–1) –2 – (–7) –5 – (+8) –1 – (+9) • (positive integer) − (positive integer) • (positive integer) − (negative integer) +9 – (+8) +6 – (+1) +5 – (–4) +7 – (–2) Guide students to create number line based on their respective answers. Make sure students follow the rule of constructing number line where zero should be at the middle between negative and positive integers. Let students attempt Practice 1.2 (Textbook Grade 7: Page 14) and discuss about the answers with them. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 11 Independent Practice Assign students to complete Exercise 1.2 (Workbook Grade 7: Page 6 – 9). 1.3 Multiplication and Division of Integers (Textbook Grade 7: Page 15) Suggested answer(s): • Sign of product: (+) / positive • Sign of product: (–) / negative • Sign of product: (+) / positive A Multiplication of integers Go to page 15. Explain the rules of multiplication of two integers to the class. When we multiply two positive integers, the product is always positive. When we multiply two Practice 1.2 Answers: 1. (a) 3 (b) –6 (c) –8 (d) –9 (e) 0 (f) –12 2. (a) 2 (b) –1 (c) –4 (d) 0 (e) –13 (f) –5 3. (a) –5 (b) –10 (c) 13 (d) 10 (e) –8 (f) –2 4. (a) –4 (b) 10 (c) –5 (d) –1 (e) 3 (f) –12 5. (a) 16 (b) (–4) (c) –18 (d) 18 (e) 20 (f) –6 6. 25 m below sea level or –25 m 7. 4oC 8. 9oC 9. 9 m 10. –6oC 11. 15 m below sea level ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 12 Focus-on Mathematics Teacher’s Guide Grade 7 integers where the first integer is positive integer, and the second integer is negative integer, and vice versa; the product is always a negative integer. When we multiply two negative integers, the product is always a positive integer. Guide students to see Example 17 on page 15. Students will learn and understand the rules of multiplication for two integers (between negative and positive integers). The teacher is encouraged to provide more examples to the class to ensure better understanding. Once the students mastered multiplication between two integers, go through Example 18 with them. This is an example on multiplication of three integers. Emphasise to the class that when multiplying three or more integers, always work from left to right. Proceed to Example 19 on page 15 to 16, which is about the application of multiplication of integers in daily lives. Explain the properties of multiplication to the class. Talk about the difference between commutative, associative and distributive properties. See Example 20 for further clarification of these properties. Guide students to scan the QR codes of on page 15 for more online resources. B Properties of multiplication Begin by discussing the concept of multiplication and its basic operation of combining groups of numbers. Explain that multiplication has certain properties that can help simplify calculations and manipulate expressions. (I) Commutative property of multiplication Introduce the commutative property of multiplication, which states that changing the order of the factors does not change the product. Go through the notes and example provided on page 16 with the class. Discuss how the commutative property allows us to rearrange the factors without changing the final result. (II) Associative property of multiplication Introduce the associative property of multiplication, which states that changing the grouping of the factors does not change the product. Go through the notes and example provided on page 16 with the class. Discuss how the associative property allows us to regroup the factors without altering the final product. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 13 (III) Distributive property of multiplication Tell the class that the distributive property of multiplication is a fundamental property that relates multiplication and addition. It states that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers separately and then adding the results together. Present the distributive property equation: . The teacher can break down the equation into simpler terms and explain the meaning of each part. Emphasise that the distributive property allows us to break down a multiplication expression and distribute the multiplication over the terms inside the parentheses. Guide students to see Example 20 on page 16. Discuss practical situations where the distributive property is applicable, such as calculating prices with discounts, expanding algebraic expressions, or factoring polynomials. Help students connect the concept to real-life examples and understand its usefulness in various contexts. (Textbook Grade 7: Page 17) Suggested answer(s): –7 + 3 – (–1) = –7 + 3 + 1 = –3 The player’s new statistic is -3. C Division of integers (Textbook Grade 7: Page 17) Suggested answer(s): • Sign of quotient: (+) / positive • Sign of quotient: (–) / negative • Sign of quotient: (+) / positive Explain the rules of division of two integers to the class. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 14 Focus-on Mathematics Teacher’s Guide Grade 7 When we divide two positive integers, the quotient is always positive. When we divide two integers where the first integer is positive integer, and the second integer is negative integer, and vice versa; the quotient is always a negative integer. When we divide two negative integers, the quotient is always a positive integer. Guide students to see Example 21 on page 17. Demonstrate the calculation for (a) and (b) while explaining the rules of division for two integers (between negative and positive integers). The teacher is encouraged to provide more examples to the class to ensure better understanding. Once the students mastered division between two integers, go through (c) and (d) with them. Both are examples on division of three integers. Emphasise to the class that when dividing three or more integers, always work from left to right. Proceed to Example 22 on page 17 to 18, which is about the application of division of integers in daily lives. Let students attempt Practice 1.3 (Textbook Grade 7: Page 18) and discuss about the answers with them. Independent Practice Assign students to complete Exercise 1.3 (Workbook Grade 7: Page 10 – 12). Practice 1.3 Answers: 1. (a) –10 (b) –40 (c) 0 (d) 54 2. (a) –40 (b) –144 (c) –72 (d) 150 3. (a) –3 (b) 4 (c) 20 (d) –12 (e) –7 (f) 6 4. (a) –5 (b) –3 (c) 7 (d) –4 5. (a) 2 (b) –7 (c) –3 (d) 2 6. 15 m below sea level or –15 m 7. –8oC 8. 7oC 9. 1.25 m ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 15 1.4 Combined Operations of Integers The students should be well-versed with the four different modes of operations (+, –, × and ÷). They have learnt the correct order of operations in primary school. Guide students to see Example 23 on page 18. Get the students to first identify the different operations involved in each sum. Elicit answers by pointing out the correct order of operations. Remind the class: • In a sum involving only multiplication and division, work from left to right. • In a sum involving only addition and subtraction, work from left to right. • If three or all four operations are involved in a sum, apply the BODMAS rule. Continue to Example 24 and Example 25 on page 19. Remind students to follow BODMAS rule when solving the combined operations of integers. Guide students to solve word problems involving the 4 operations. Refer Example 26, 27 and Example 28 in page 19 to 21. Guide students to scan the QR codes of on page 18 for more online resources. For problem solving questions, guide students to solve the problems by using the four-step Polya’s problem-solving method. Let students attempt Practice 1.4 (Textbook Grade 7: Page 22) and discuss about the answers with them. 4. Look back and reflect Work backward to check whether the answer is right or wrong. 3. Execute the plan Carry out the plan and solve the problems to get the best answer. 2. Make a plan Think of a plan on how to solve the problem using the facts and information listed in Step 1. 1. Understand the problem Try to understand the problems by carefully reading it, listing the facts in the question and state the problem that need to be solved in the question. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 16 Focus-on Mathematics Teacher’s Guide Grade 7 Independent Practice Assign students to complete Exercise 1.4 (Workbook Grade 7: Page 12 – 14). 1.5 Factors and Prime Factors Start the lesson by asking the students what do they know about factors. Students should understand that any whole number that can be divided exactly into another number with no remainder is called a factor. See the example given on page 22, where whole number 8 is divided exactly by 1, 2, 4 and 8. Emphasise that the number 1 is always a factor for all whole numbers. Go through Example 29 (page 23) with the class. The teacher can list all factors of the whole number 15 on the board. Give extra examples with other whole numbers for better understanding. For Example 30 (page 29), guide the class how to decide whether a given number is determined as a factor of a given number. Tell students to use division to test if a whole number is a factor. Focusing on the ‘remainder’, stress that if there is no remainder, the number is a factor. Otherwise, it is not. Revisit the definition of prime number with the class: A prime number is a whole number that only can be divisible by number 1 and itself. Bring students’ attention to the numbers in the diagram below. There are 25 prime numbers between 1 to 100. The purple highlighted numbers in the diagram show prime numbers between 1 – 100. Give the class some time to understand why those numbers are prime numbers. Practice 1.4 Answers: 1. (a) 1 (b) 6 (c) –12 (d) 4 2. (a) 16 (b) 68 (c) 12 (d) –72 3. (a) 25 (b) 5 (c) 12 (d) 12 (e) 9 (f) –35 (g) –3 (h) –4 4. 36oC 5. 10 marks 6. 12 m 7. (a) Melbourne is 9 hours ahead of Berlin. (b) Jakarta is 6 hours ahead of Berlin. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 17 Work with students on Example 31, 32 and 33 (page 23 to 24). explain to students that prime factorisation is used to express a number as a product of its prime factors. Show them the two methods: factor tree and division method. Emphasise that prime factorisation is done by dividing the number by the smallest prime factor until we obtain 1. Some students may think that 1 is a prime number. Explain to the class that although the number ‘1’ can be written as a product of two factors (1 × 1), the two factors are not different, therefore 1 is not a prime number. Guide students to scan the QR codes of on page 22 for more online resources. Let students attempt Practice 1.5 (Textbook Grade 7: Page 25) and discuss about the answers with them. Practice 1.5 Answers: 1. (a) Yes (b) No (c) Yes (d) Yes 2. (a) Yes (b) Yes (c) No (d) Yes 3. (a) No (b) Yes (c) No (d) Yes (e) Yes (f) No 4 (a) Yes (b) Yes (c) No (d) No (e) Yes (f) No (g) No (h) No 5. (a) 5 (b) 10 (c) 12 (d) 22 6. (a) Yes (b) No (c) Yes (d) Yes ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 18 Focus-on Mathematics Teacher’s Guide Grade 7 1.6 Least Common Multiple (LCM) and Highest Common Factor (HCF) Go to page 25. Start the lesson by revisiting the definition of multiples: Multiples are the product of a whole number is multiplied by a counting number. For example, the multiple of the whole number 2 are 4, 6, 8, 10 and so on. The counting number here is number 2, which makes it to be 4 after we multiply it with another 2. Discuss with the students about the example of multiples in the notes of textbook in page 25. Check students’ understanding of multiples using Example 35 and Example 36 (page 25 to 26). Once students are familiar with the concept of multiple, proceed to Example 37 and 38 (page 26 to 27). Guide the class to use the systematic listing method to find the common multiples of two 1-digit numbers. Highlight the terms: • common multiples • first common multiple • the first three common multiples Tell students that the multiple of a number can go on and on, therefore they only need to list the number of multiples required to solve the problem. Least Common Multiple (LCM) is defined as the smallest multiple that two or more numbers have in common. Use Example 39 to show how LCM is determined between two numbers. Discuss with the class how the least common multiple of the two numbers can be found. Assist the students by asking the following questions: • What are the multiples of 6 and 8 respectively? • What is the difference between a factor and a multiple? Explain to students that there are two methods of prime factorisation to find the LCM of two or more numbers. Lead students to see that: Method 1 is the systematic listing method, where all multiples are listed and students have to identify the lowest multiple that are common in both numbers. Method 2 is to find the prime factors of 6 and 8 respectively. In this case the multiples of 6 and 8 are listed respectively. Lead students to highlight the multiples that are common in both numbers. Explain to them that LCM is found by multiplying the common prime factors with the remaining prime factors. Method 3 is to use the repeated division method to find the prime factors by dividing both numbers each time. Lead them to see that in this method, both numbers are divided until they cannot be divided any further without any remainder. LCM of the two numbers can be found by multiplying all the prime factors found in the division method. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 19 Ask the class about their preferred method, let them share their reasons. Students should be well-versed with expressing a number as a product of its prime factors. In this lesson, students will learn that the highest common factor (HCF) is the greatest factor that is common between or among 2 or more numbers. In continuation of the earlier lesson in 1.5 Factors and Prime Factors, the division method of prime factorisation is employed to find the HCF. Guide students through Example 40 and Example 41. Work through Example 42 with the class and show the three methods to find the HCF of a set of numbers. The first method by listing the factors, second method by prime factorisation and the third method is by division method. Guide students to see problem solving examples, Example 43 and Example 44 in page 30 and 31. Guide students through the problem-solving examples, Example 43 and 44 (page 30 to 31). For problem solving questions, guide students to solve the problems by using the four-step Polya’s problem-solving method. 4. Look back and reflect Work backward to check whether the answer is right or wrong. 3. Execute the plan Carry out the plan and solve the problems to get the best answer. 2. Make a plan Think of a plan on how to solve the problem using the facts and information listed in Step 1. 1. Understand the problem Try to understand the problems by carefully reading it, listing the facts in the question and state the problem that need to be solved in the question. ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 20 Focus-on Mathematics Teacher’s Guide Grade 7 Let students attempt Practice 1.6 (Textbook Grade 7: Page32) and discuss about the answers with them. Independent Practice Assign students to complete Exercises 1.5 to 1.6 (Workbook Grade 7: Page 15 – 21). Closing Guide the whole class to conclude the concept of integers. Go to page 33, summarise the key points covered in the lessons. Check for students’ understanding by asking questions or having a brief class discussion. Let students attempt Mastery Practice 1 (P34 – 35). Then, discuss about the answers with them. Practice 1.6 Answers: 1. 552, 1752 2. (a) 20, 40 (b) 18, 36 3. (a) 6 (b) 45 (c) 36 (d) 72 4. (a) 20 (b) 84 (c) 24 (d) 240 5. (a) Yes (b) No (c) Yes (d) Yes 6. (a) 6 (b) 5 (c) 2 (d) 6 7. (a) 24 (b) 7 (c) 3 (d) 32 8. 15 plates 9. 36 seconds 10. 30 pieces 11. 24 days ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS Focus-on Mathematics Teacher’s Guide Grade 7 21 Independent Practice Assign students to complete Enrichment Exercises (Workbook Grade 7: Page 55 – 57). Mastery Practice 1 Answers: Section A 1. B 2. C 3. D 4. B 5. B 6. B 7. D 8. B 9. C 10. A Section B 1. (a) –12 (b) –6 (c) –16 2. (a) –9 (b) –4 (c) –27 3. 7 + 47; 11 + 43; 13 + 41; 17 + 37; 23 + 31 4. 5 5. (a) –23oC (b) –11oC 6. Below sea level = –50 m After 2 minutes = –50 + 120 5 ×2 = –2 m The diver had not reached the sea surface after 2 i t 7. (a) Cheque valued at $1730 (b) $1382 8. 6 and 60 9. 5:15 p.m. 10. 30 cm × 30 cm 11. The second Saturday 12. (a) 6 pages (b) 4 photographs and 7 newspaper ©Praxis Publishing_Focus On Maths

Chapter 1 INTEGERS 22 Focus-on Mathematics Teacher’s Guide Grade 7 Assessment At the end of this chapter, teacher needs to make sure that students should be able to • recognise and describe integers. • recognise positive and negative integers. • differentiate the position of negative and positive integers on a number line. • represent negative and positive integers on a number line. • substitute negative and positive integers and use them in real-life situations. • compare and arrange integers in order. • add and subtract integers using number lines or other appropriate methods. • multiply and divide integers using various methods. • perform computations involving combined basic arithmetic operations of integers. • describe the laws of arithmetic operations which are Commutative Law, Associative Law and Distributive Law. • solve problems involving integers. • determine and list the prime factors of a whole number. • determine the LCM of two or three whole numbers and solve problems involving LCM. • determine and list the factors of a whole number. • explain and determine common factors of a whole number. • determine the HCF of two or three whole numbers and solve problems involving HCF. Materials • GeoGebra App • Focus-on Mathematics Textbook Grade 7 • Focus-on Mathematics Workbook Grade 7 • Focus-on Mathematics Grade 7—PowerPoint ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 23 Time 18 hours Elemen Capaian Pembelajaran Bilangan Peserta didik dapat membaca, menulis, dan membandingkan bilangan bulat, bilangan rasional dan irasional, bilangan desimal, bilangan berpangkat bulat dan akar, bilangan dalam notasi ilmiah. Mereka dapat menerapkan operasi aritmetika pada bilangan real, dan memberikan estimasi/perkiraan dalam menyelesaikan masalah (termasuk berkaitan dengan literasi finansial). Peserta didik dapat menggunakan faktorisasi prima dan pengertian rasio (skala, proporsi, dan laju perubahan) dalam penyelesaian masalah. Learning Objectives Students will be taught to: • identify rational numbers and irrational numbers. • change fractions to decimals and vice versa. • use dot notation to write recurring decimals. • extend the concept of integers to fractions to solve problems. • extend the concept of integers to decimals to solve problems. • perform computation involving directed numbers (integers, fractions and decimals). • understand and use the symbols <, >, ≤, ≥, = and ≠. Bilingual Key Terms Integers Bilangan bulat Rational number Bilangan rasional Irrational number Bilangan tidak rasional Terminating decimal Desimal berakhir Recurring decimal Desimal berulang Real numbers Bilangan real Fractions Pecahan Percentages Persentase ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 24 Focus-on Mathematics Teacher’s Guide Grade 7 START UP Background Information Assess the students’ prior knowledge about integer by asking what the students know and what they want to know more about real numbers. Test students’ knowledge whether they have heard about real number and know what real number is. Remind students that they have learnt about integers in primary school and from integers we will know what is real numbers. Basically, most numbers that we work with every day are real numbers. Give simple explanations about real numbers and ask students if they know any real numbers in their daily lives, e.g., money in our wallets, statistics in sports, or measurement we see in the recipe books, etc. Real World Connection Refer the chapter opener (Textbook Grade 7: Page 36). Let students observe the picture. Ask the students what they see and understand from the picture, and how can they relate them to the topic discussed in this chapter. Arouse students’ interest in this topic by bringing in real-life applications. Emphasise that the applications of real numbers are common, we often use them in our daily life, such as paying for groceries at the market, paying bills via internet , and numbers shown in stock market in the news on television. Teacher may choose a few students and let them give examples of the use of real numbers in their daily life. Discuss the examples of the use of integers in the real world to bring across the idea of real numbers. (Textbook Grade 7: Page 36) Suggested answer(s): Negative and positive numbers play important roles in stock market. It is for financial purposes. Negative numbers act as numbers that represent loss, withdrawal, owing, and shows the stock market that goes up or down. Meanwhile positive numbers show profit, deposit or investment made by stock broker. (Textbook Grade 7: Page 38) Ask students to complete the section and discuss about the answers with them. Answers: 1. (a) –4 (b) 5 2. (a) 5 (b) 0.2 (c) –12 (d) 8 (c) 3 4 3. (a) 0.25 (b) 2.5 (c) 1.6 ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 25 (Textbook Grade 7: Page 38) Begin this section by briefly explaining the ASEAN Stock Markets to the students in the class. Guide students to see the table shown in the textbook and compare the stock prices at the end and start of the day, of companies P, Q, R and S. Ask students how can they figure out whether each company gain profit or suffer loss in their stock market. Suggested answer(s): (a) Positive and negative numbers represent gain and loss in the stock market. Normally, green colour is used to represent gain, while red colour is used to represent loss. (b) P Q R S (c) –0.43, +0.04, +0.7, +1.13 (d) (Textbook Grade 7: Page 2) Read the question and let students to discuss the answers among themselves. Select some students to share their answers and explain why. Further discuss about the answers and ask the opinions of the rest of the class. Suggested answer(s): P = 3.710 – 3.670 = +0.04 Q = 41.330 – 40.630 = +0.7 R = 65.970 – 64.840 = +1.13 S = 142.15 – 142.580 = 0.43 3.670 3.710 +0.04 40.630 41.330 +0.7 64.840 65.970 +1.13 142.15 142.580 –0.43 –0.43 +0.04 +0.7 +1.13 ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 26 Focus-on Mathematics Teacher’s Guide Grade 7 Teaching/Learning Activities 2.1 Rational Numbers (Textbook Grade 7: P39) Begin this section by discussing with the class about how some foods bought from supermarket need to be stored in refrigerator and freezer to avoid to be damaged. The teacher may ask students to give examples of food that need to be stored in freezer. Go through the situation in the question together and let students to attempt the questions. Suggested answer(s): (a) Possible temperatures: –22o C, –21o C, –20o C, –19o C, –18o C. (b) Build a number line, place the lowest number at the left side and the greatest number on the right side. Students should know how to determine which number is the lowest and which number is the greatest. Emphasise to students that the value of numbers is different if they have negative sign in front of them. (Textbook Grade 7: P39) Teacher may prepare simple number cards as follows before starting the lesson in the class. The teacher may prepare more than one set of number cards (each set should be able to cover the total number of groups in the class). Divide students in groups of four. Distribute number cards to each group of students. Guide students to form rational numbers using the cards using their creativity. Emphasise that rational numbers can be obtained when two numbers are written in the form , where a and b are integers and b ≠ 0. Give them 20 minutes to form as many as possible rational numbers and ask them to present their findings to the class. After this activity, recall back the definition of rational numbers. Rational number is any number that can be written in fractional form which ab such that a and b are integers, where b ≠ 0. Guide students to see the examples of rational numbers in the textbook and how to represent them in a number line. –22oC –21oC –20oC –19oC –18oC ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 27 (Textbook Grade 7: P39) Suggested answer(s): (a) A rational number is subjected to the condition q ≠ 0 because the value of q cannot be zero. If the denominator of a fraction is a zero, the expression is undefined. We cannot divide a whole into 0 equal parts. (b) Not all rational numbers are integers. Rational numbers can include fractions and decimals that are not integers. For example, 1 2 and 0.75 are rational numbers but not integers. Integers form a subset of rational numbers, specifically those rational numbers that have a denominator of 1. Guide students through Example 1, Example 2 and Example 3 on page 40 – 41 to deepen the students’ understanding about rational numbers. Guide students to scan the QR codes of on page 41 for more online resources. Let students attempt Practice 2.1 (Textbook Grade 7: P41) and discuss about the answers with them. 2.2 Irrational Numbers Start the lesson by giving the definition of irrational. Irrational means not rational. An irrational number is a real number that cannot be written as a simple fraction. To be precise, irrational numbers are numbers that cannot be written as a ratio of two integers. A number is said to be in irrational when a number that cannot be expressed in the form of fraction, , where a and b are integers, and b ≠ 0. Practice 2.1 Answers: 1. (a) All possible rational numbers are 7 2 , 4, 9 2 , 5 and 11 2 . (b) All possible rational numbers are 21 5 , 22 5 , 23 5 and 24 5 . 2. (a) – 1 2 is in the middle of 0 and –1. (b) 3 5 is in the middle of 1 5 and 1. (c) 3 6 is in the middle of 1 3 and 2 3 . (d) 7 16 is in the middle of 1 4 and 5 8 . 3. All the numbers given are rational numbers. –8 = −8 1 ; 3.15 = 63 20 ; 11 5 = 6 5 4. (a) –7.2 (b) 2.58 (c) –0.3125 (d) –3.905 5. 33 8 kg 6. 1.805 m ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 28 Focus-on Mathematics Teacher’s Guide Grade 7 (Textbook Grade 7: P41) Suggested answer(s): (Accept all possible answers.) Three examples of irrational numbers: Pi (π) = 3.14159265……. √2 √3 Go through Example 4 on page 42 with the class to deepen the students’ understanding about irrational numbers. Guide students to scan the QR codes of on page 42 for more online resources. Let students attempt Practice 2.2 (Textbook Grade 7: P42) and discuss about the answers with them. Independent Practice Assign students to complete Exercise 2.1 (Workbook Grade 7: Page 26 - 29). 2.3 Fractions A Understanding fractions Start the lesson by accessing students’ prior knowledge about fractions. Ask students what they know about fractions. Go to page 42, guide students to recall what they have learnt about fractions in primary school. Explain the concept of fraction using the provided examples. • What are the top and bottom numbers on fractions? (Allow pupils to answer and discuss the question. As appropriate, ask some of the follow-up questions.) Practice 2.2 Answers: 1. (a) 0.2 is a rational number. (b) 1.04 is a rational number. (c) 0.333 is a rational number. (d) √9 is a rational number. (e) √3 is an irrational number. (f) √6 is an irrational number. 2. √10 0.838383… √11 1 √2 ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 29 • Which part is the numerator and denominator? (Make sure pupils know the numerator is the top number and the denominator is the bottom number.) • What does the numerator represent? (Make sure pupils know the numerator represents how many parts there are.) • What does the denominator represent?" (Make sure pupils know the denominator represents what size the parts are – how many parts make a whole). B Positive and negative fractions (I) Representing fractions with diagrams Remind students that diagrams are always useful to represent fractions visually, making it easier to understand and compare different fractions. The teacher can draw an example of a whole shape on the board and divide the shape into equal parts to represent fractions. For example, divide a circle into four equal sections. Then, use the divided shape to explain the numerator and denominator. Shade a specific number of sections to represent the numerator. Emphasise that the denominator indicates the total number of equal parts in the whole shape. Go through the examples given in the textbook (p43) with the class. (II) Writing fractions for given diagrams When writing fractions for a given diagram, students need to know where the location of the numerator and the denominator in a fraction. Remind students again about this before proceed with writing of a fraction. Go through the examples in page 43 with the class. The teacher is encouraged to display a variety of diagrams divided into equal parts, then explain that each diagram represents a whole object or a group, which can be divided into parts. For more practice, ask students to write the corresponding fractions for the diagrams. ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 30 Focus-on Mathematics Teacher’s Guide Grade 7 (III) Representing fractions as decimals The teacher can begin this lesson by reviewing the concepts of decimals that students learned in primary school. To convert a fraction to a decimal, we divide the numerator by its denominator. Guide students to see Examples 5 to 7 on page 44 to 45 to give the ideas of representing fractions as decimals. For Example 6 (b), teacher may emphasise to students that when converting mixed numbers to decimals, students may convert it to improper fraction first and convert them into decimals or stays the whole number to the left and only change the fraction part to the decimal. Summarise the process of converting fractions into decimals, highlighting the importance of understanding and utilizing both representations. For something extra, the teacher can also emphasise the connection between fractions and decimals in real-world contexts. (Textbook Grade 7: P45) Teacher may start this section by briefing the students about The New York Stock Exchange (NYSE). NYSE is an American stock exchange in New York city. It is known as the world’s largest stock exchange. The NYSE is a marketplace where stocks (shares of ownership in companies) are bought and sold. Its main purpose is to facilitate the trading of securities and provide a platform for investors and businesses to raise capital. In the past, stock prices were quoted and traded in fractions like eighths or sixteenths. However, in 2001, the NYSE transitioned to decimal pricing, adopting a system where changes in stock values are expressed as decimals. This change allowed for greater transparency, efficiency, and easier comparison with other financial instruments. Today, stock prices on the NYSE are quoted and traded in decimal form, typically to four decimal places. Suggested answer(s): Since 1 2 5 = 1.4, − 3 16 = –0.1875, 3 4 = 0.75, −1 7 16 = –1.4375, −1 1 2 = –1.5, From the least to the greatest: −1 1 2 , −1 7 16 , − 3 16 , 3 4 , 1 2 5 (IV) Representing fractions as percentages Remind students that they have learnt about percentages in primary school. • The symbol use for percentage is ‘%’ . • Fractions can be converted into percentages, which show a part out of 100. ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 31 • To convert a fraction to a percentage, multiply it by 100. • Use equivalent fractions with a denominator of 100 to simplify the conversion. Connect the concept to real-life situations where percentages are commonly used, such as grades, discounts, or statistics. Go through the notes on page 45 with the class and guide students through Examples 8 to 10 on page 46. Explain to the class how to convert a fraction to a percentage and vice versa. Guide students to scan the QR codes of on page 42 for more online resources. B Positive and negative fractions (I) Represent positive and negative fractions on number lines (Textbook Grade 7: Page 47) Divide the class in groups of four. Guide students to open file Fraction number line in GeoGebra. Guide students through this activity by following the instructions in the textbook. Ask them what they see and let them discuss their observations in groups. Let each group select a representative and present their findings to the class. Then, make conclusions from their observations. Representing negative and positive fractions on a number line is the same as representing integers on number line. Emphasise that positive fractions are located on the right side of zero and negative fractions are located on the left side of zero. Guide students to see Example 11 on page 48 to learn how to represent positive and negative fractions on number line. (II) Compare and arrange fractions in order After students have mastered ordering positive and negative fractions on number line, teacher may teach students on how to arrange fractions in ascending and descending order. See Example 12 on page 48 to learn on how to determine and compare the least or greatest ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 32 Focus-on Mathematics Teacher’s Guide Grade 7 fractions. Note that the value of negative fractions is always lesser than the positive fractions. In the case of mixed numbers, emphasise to students to see the whole number before the fractions. For example, –4 is always smaller than –1. Guide students to see Example 13 on page 49 to learn how to arrange positive and negative fractions in ascending or descending order. Remind the class: To arrange fractions, first we must change the denominator to the same value. Represent them on a number line to determine the value of each fraction. If we want to arrange in ascending order, list the fractions from left to right and vice versa. Discuss about the alternative methods that can be used to obtain the final answers. The teacher may also display other methods to the class (if any) for extra information. (III) Perform computations involving combined basic arithmetic operations of positive and negative fractions (Textbook Grade 7: Page 50) Begin this section by explaining to students that every food served have their own recipe. Guide students to see diagram in Let’s Explore 3 and ask them what they observed. Encourage students to write down the important points before working on the questions. Answer(s): • Amisha uses 7 8 cup of orange juice, 2 3 cup of pineapple juice, 3 4 cup of lime soda, 1 4 cup of cranberry juice and 1 3 cup of ice cubes. • 7 8 + 2 3 + 3 4 + 1 4 + 1 3 = 27 8 cups Amisha made 27 8 cups of punch. • 27 8 − 3 4 = 21 8 Amisha has 21 8 cups of punch left. • To keep the flavour the same, Amisha needs 7 16 cup of orange juice, 3 8 cup of lemon lime soda, 1 8 cup of cranberry juice, and 1 6 cup of ice cube. • She will need 1 4 cup of soda. From Let’s Explore 3, we can see that the order of operations involving addition, subtraction, multiplication, division and brackets for fractions is the same as the order of operations involving integers. When solving addition, subtraction, multiplication and division of fractions, always solve from left to right. Guide students to see Example 14 on page 50 to solve operations of fractions. Emphasise to students to always use the BODMAS rules when solving the operations. ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 33 (IV) Solve problems involving fractions See Example 15 in page 51 about using operations of fractions in daily life applications. For problem solving questions, guide students to solve the problems by using the four-step Polya’s problem-solving method. (Textbook Grade 7: Page 52) Answer(s): Jolin received 126 sweets. Let students attempt Practice 2.3 (Textbook Grade 7: P53) and discuss about the answers with them. 4. Look back and reflect Work backward to check whether the answer is right or wrong. 3. Execute the plan Carry out the plan and solve the problems to get the best answer. 2. Make a plan Think of a plan on how to solve the problem using the facts and information listed in Step 1. 1. Understand the problem Try to understand the problems by carefully reading it, listing the facts in the question and state the problem that need to be solved in the question. Practice 2.3 Answers: 1. (a) (b) (c) (d) 2. (a) 3 4 (b) 5 8 (c) 7 16 (d) 9 20 3. (a) 3% (b) 47% (c) 201% ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 34 Focus-on Mathematics Teacher’s Guide Grade 7 Independent Practice Assign students to complete Exercises 2.2 – 2.5 (Workbook Grade 7: P29 – 45). 2.4 Decimals Teacher may start the lesson by guiding the class to recall what they have learnt about decimals previously. Ask the class for the definition of decimal. Prompt the class for answers before telling them a decimal is a number that contain of a whole number and a fractional part separated by a decimal point. The dot that in between the whole number and the fractional part is called decimal point. 4. (a) 7 100 (b) 29 100 (c) 121 100 5. (a) 60% (b) 28% (c) 120.4% 6. (a) 1 50 (b) 8 25 (c) 13 5 7. (a) 0.008 (b) 0.96 (c) 1.75 8. (a) (b) (c) (d) 9. (a) 4 5 (b) – 1 1 2 (c) – 2 3 5 (d) – 2 1 9 10. (a) (i) − 1 2 , − 3 8 , − 1 8 , 1 2 , 5 8 (b) (i) − 7 10, − 2 5 , 1 10, 1 2 , 3 5 (ii) 5 8 , 1 2 , − 1 8 , − 3 8 , – 1 2 (ii) 3 5 , 1 2 , 1 10, − 2 5 , − 7 10 (c) (i) −1 1 3 , −1 1 6 , − 5 6 , 4 9 , 7 9 (d) (i) −1 7 8 , −1 1 2 , − 5 8 , 1 1 4 , 3 2 (ii) 7 9 , 4 9 , − 5 6 , −1 1 6 , −1 1 3 (ii) 3 2 , 1 1 4 , − 5 8 , −1 1 2 , −1 7 8 11. (a) −1 3 8 (b) 3 (c) −1 1 18 (d) − 3 4 (e) −4 1 8 12. 2250 m below sea level 13. Not enough. She is still short of 7 10 kg of sugar. 14. 1 3 4 ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 35 If needed, provide an example to the class to remind them the idea of decimal number. Move on to page 54. Get students to look at the place value table of decimal. The table shows the place value in the decimal system. Explain to students that decimal numbers can be classified into two types: Terminating Decimals and Recurring Decimals. Terminating decimals have a finite number of digits and do not continue indefinitely. In other words, they do not repeat. Recurring decimals, on the other hand, are decimal numbers that have an infinite number of repeating digits. Encourage students to explore Examples 16 and 17, which will help them grasp the concept of terminating and recurring decimals more effectively. While examining Example 17, draw attention to the notation used to indicate a repeating decimal. Emphasise that a repeating portion of a decimal is denoted by placing a single dot above the repeating digits. Proceed to Examples 18 and 19, which demonstrate how to express recurring decimals in standard notation and convert fractions into decimal form. C Positive and negative decimals (I) Represent positive and negative decimals on number lines (Textbook Grade 7: P56) Divide the class in groups of four. Guide students to assess the Decimal number line activity in GeoGebra. Allow students a few minutes to explore the GeoGebra platform and familiarise themselves with its features and functions. Instruct students to follow the guidelines provided in the textbook and use the activity to observe and interact with the number line. Encourage them to discuss their observations within their groups. Facilitate a discussion based on the presentations and encourage students to draw conclusions from their observations. Emphasise that representing negative and positive decimals on a number line follows the same principles as representing integers on a number line. Decimal point Whole number part Fractional part ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 36 Focus-on Mathematics Teacher’s Guide Grade 7 Emphasise that positive decimals are located on the right side of zero and negative decimals are located on the left side of zero. (II) Compare and arrange decimals Deepen students’ understanding of representing positive and negative decimals on a number line by referring to Example 20. Once students have become proficient in ordering positive and negative decimals on a number line, the teacher can proceed to teach them how to arrange decimals in ascending and descending order. Refer to Example 21 to learn how to determine and compare the least or greatest decimals. Note that negative decimals always have a lesser value than positive decimals. Comparing the first digit of each decimal makes it easier to determine which one is larger. Explore Example 22 to understand how to arrange positive and negative decimals in ascending or descending order. When arranging in ascending order, create two separate categories for positive and negative decimals. Compare the first digit within each category. This method simplifies the process of determining the value of each decimal before arranging them in order. (III) Perform computations involving combined basic arithmetic operations of positive and negative decimals (Textbook Grade 7: P58) Begin this section by explaining to students that in certain countries such as the United States of America, England, Australia, and Russia, the winter season occurs from December until March every year, spanning four seasons. During this time, the temperature of surrounding will drop below freezing, requiring the use of heaters to keep homes warm. Guide students to read through the provided paragraph and ask them to answer the questions. Facilitate a discussion to discuss the answers with the students. Suggested answer(s): (a) The Lucas Family: $171.23 + $134.35 + $123.21 = $428.79 The Nabila Family: $134.25 + $103.27 + $98.66 = $336.18 (b) $428.79 (c) 1. Utilise natural heat sources: Open curtains during the day to let sunlight warm the home, and close them at night to trap heat. ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 37 2. Programmable thermostat: Use a programmable thermostat to schedule temperature adjustments based on the family's needs. 3. Energy saving habits: Turn off lights, appliances and electronic devices when not in use. From Let’s Explore, it is evident that the order of operations for decimals, involving addition, subtraction, multiplication and division follows the same rules as those for integers. Remind students when solving operations with decimals, always work from left to right. Guide students to refer to Example 23 on page 59 for a practical illustration of solving operations involving decimals. Emphasise to students to always use the BODMAS rules when solving the operations. (Textbook Grade 7: Page 59) Suggested answer(s): A common way to multiply decimals is to treat them as whole numbers, and then position the decimal point in the product. The number of digits after the decimal points in the factors determines where the decimal point is placed in the answer. For example, 0.3 x 0.8 = 0.24. (IV) Solve problems involving decimals Refer to Example 24 on page 59 to understand how operations with decimals can be applied in real-life situations or daily life applications. When encountering problem-solving questions involving decimals, guide students to utilise Polya's four-step problem-solving method. 4. Look back and reflect Work backward to check whether the answer is right or wrong. 3. Execute the plan Carry out the plan and solve the problems to get the best answer. 2. Make a plan Think of a plan on how to solve the problem using the facts and information listed in Step 1. 1. Understand the problem Try to understand the problems by carefully reading it, listing the facts in the question and state the problem that need to be solved in the question. ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS 38 Focus-on Mathematics Teacher’s Guide Grade 7 (V) Perform computations involving combined basic arithmetic operations of rational numbers (Textbook Grade 7: Page 60) Suggested answer(s): Yes, it is possible for the product of a positive rational number multiplied by a negative rational number to be closer to 0 than either of the numbers being multiplied. For example, positive rational number = 1 3 , negative rational number = − 1 2 . 1 3 ⨉ (− 1 2 ) = − 1 6 In this case, the product, − 1 6 , is closer to 0 than either of the numbers being multiplied. Both 1 3 and − 1 2 are greater in magnitude than the product − 1 6 , which is closer to 0 on the number line. It is important to note that the specific values of the rational numbers being multiplied will determine whether the product is closer to 0 or farther away. In some cases, the product may be farther from 0 than either of the numbers being multiplied, while in other cases, it may be closer to 0. Guide students to see Example 25 on page 61 to understand how to solve operations involving decimals. Emphasise to students the importance of using the BODMAS rules (Brackets, Order, Division and Multiplication, Addition and Subtraction) when solving these operations. Explore Example 26 on page 61 with the class to discuss about how the combination of operations with decimals and fractions can be applied in real life situations or daily life applications. Let students attempt Practice 2.4 (Textbook Grade 7: P62) and discuss about the answers with them. Practice 2.4 Answers: 1. (a) 5.29, 5.38, 5.4, 5.8 (b) 0.56, 0.6, 0.63, 0.7 (c) 2.02, 2.12, 2.21, 2.22 (d) 1.816, 1.826, 18.162, 18.216 2. (a) 4.8 (b) 0.72 (c) 1.625 (d) 0.3125 (e) 0.53125 (f) 0.140625 3. (a) 3. 3̇ (b) 22.7̇ 2̇ (c) 00.2̇ 98̇ (d) 8.5̇ 71428̇ 4. (a) 0.666… (b) 0.090909… (c) 0.401401401… (d) 0.0123456789 0123456789 0123456789… 5. (a) 0. 1̇ (b) 0. 6̇ 3̇ (c) 0. 1̇ 23̇ (d) 0.06 (e) 0.83̇ 6̇ (f) 0.31̇ 8̇ ©Praxis Publishing_Focus On Maths

Chapter 2 REAL NUMBERS Focus-on Mathematics Teacher’s Guide Grade 7 39 Independent Practice Assign students to complete Exercises 2.6 to 2.9 (Workbook Grade 7: P46 – 54). 2.5 Set of Real Numbers The teacher may start the lesson by introducing the concept of the set of real numbers. Explain that the set of real numbers is a combination of set of rational numbers and the set of irrational numbers. Emphasise that the set of real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Direct students to see Example 27 to deepen their understanding of the set of real numbers, its components, and how these numbers are classified. 6. (a) (b) 7. (a) 2.9 (b) –5.6 (c) –1.11 8. (a) (i) –6.7, –3.31, –1.4, 3.87, 4.5 (ii) 4.5, 3.87, –1.4, –3.31, –6.7 (b) (i) –5.2, –3.0, –0.4, 0.9, 1.4 (ii) 1.4, 0.9, –0.4, –3.0, –5.2 (c) (i) –4.11, –3.22, –1.44, 1.55, 2.33 (ii) 2.33, 1.55, –1.44, –3.22, –4.11 (d) (i) –5.44, –5.42, –2.9, 0.03, 0.3 (ii) 0.3, 0.03, –2.9, –5.42, –5.44 9. (a) 8.985 (b) –6.665 (c) –19.35 (d) –5.647 (e) –3.072 10. –4oC 11. –6.43 m or –14.13 m below the sea level. 12. 23.3 × 40.2 = 936.66 km 13. Tharishini scored the highest marks. Emilia: 5 questions answered correctly, 4 questions answered incorrectly and 1 question not answered. The marks obtained is 8.5. Ker Er: 5 questions answered correctly, 5 questions answered incorrectly and 0 question not answered. The marks obtained is 7.5. Tharishini: 4 questions answered correctly, 0 questions answered incorrectly and 6 questions not answered. The marks obtained is 9. ©Praxis Publishing_Focus On Maths