Focus On Maths Grade 8 (Teacher's Guide)

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Focus-on Mathematics Teacher’s Guide Grade 8 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 8 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 ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 8 III dan 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 8 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. 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 ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 8 V 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. 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. ©Praxis Publishing_Focus On Maths

VI Focus-on Mathematics Teacher’s Guide Grade 8 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. 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. ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 8 VII 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 8 Yearly Teaching Plan Mathematics Grade 8 7 chapters 114 hours Learning area Duration (hours) Chapter 1 Sets 1.1 Sets 1.2 Properties of Sets 1.3 Operations of Sets 18 Chapter 2 Algebraic Expressions 2.1 Simplifying Algebraic Expressions 2.2 Determining the Values of Algebraic Expressions 2.3 Applying Algebraic Expressions 14 Chapter 3 Linear Equations in Two Variables 3.1 Linear Equations in Two Variables 3.2 Simultaneous Linear Equations in Two Variables 14 Chapter 4 Linear Functions 4.1 Relations 4.2 Functions 4.3 Linear Functions 4.4 Graphing Linear Functions 4.5 Equations of Straight Lines 4.6 Solving Problems Involving Linear Functions 20 Chapter 5 Polygons and Congruency 5.1 Polygons 5.2 Interior Angles and Exterior Angles of Triangles 5.3 Interior Angles and Exterior Angles of Polygons 5.4 Congruence 18 ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 8 IX Chapter 6 Triangles and Quadrilaterals 6.1 Triangles and Quadrilaterals 6.2 Geometric Properties and Types of Triangles 6.3 Perimeter and Area of Triangles 6.4 Geometric Properties and Types of Quadrilaterals 6.5 Perimeter and Area of Quadrilaterals 6.6 Solving Problems Involving Perimeter and Area 6.7 Perimeter and Area of Irregular Geometrical Plane 18 Chapter 7 Statistics 7.1 Mean, Mode and Median 7.2 Measures of Dispersion 7.3 Measures of Central Tendency 12 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

Focus-on Mathematics Teacher’s Guide Grade 8 SETS 1 REAL NUMBERS 24 RATIOS, RATES AND PROPORTIONS 44 ALGEBRAIC EXPRESSIONS 60 LINEAR EQUATION AND INEQUALITIES IN ONE VARIABLE 88 LINES AND ANGLES 109 STATISTICS 132 X ©Praxis Publishing_Focus On Maths

Focus-on Mathematics Teacher’s Guide Grade 8 1 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: • explain what a set is. • define a set using set notation and description. • represent sets using Venn diagrams. • recognise universal sets and empty sets. • identify the cardinality of sets. • list and state the number of elements in a set. • understand and use the concept of subset, universal set and complement of a set. • understand intersection between sets. • understand the union of sets. • perform operations on sets. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 2 Focus-on Mathematics Teacher’s Guide Grade 8 Bilingual Key Terms Set Himpunan Set notation Set notasi Universal net Himpunan semesta Empty set Himpunan kosong Venn Diagram Diagram Venn Cardinality Kardinalitas Element Anggota Subset Himpunan bagian Complement Pelengkap Power Set Himounan Kuasa Equality Kesamaan Intersection Beririsan Disjoint Terpisah Union Gabungan Difference Perselisihan START UP Background Information Assess the students’ prior knowledge about sets by asking what the students know and what they want to know more about sets. To arouse students' interest in sets, it's important to present the concept in a relatable and engaging manner. Start by providing a clear definition of what sets are: a collection of well-defined objects and elements. Emphasise that sets are used to categorise various things around us in our daily lives. During the discussion, encourage students to identify examples of sets they encounter in their everyday experiences. For instance, they may sort different items into categories like fruits and vegetables, fish and meat, living things and non-living things, etc. This real-life application helps students connect the abstract concept of sets to tangible situations, making the learning experience more meaningful. Real World Connection Begin by referring to the chapter opener in Textbook Grade 8 (Page xiv) and directing students' attention to the picture. Encourage them to share their observations, selecting one or two students to briefly present to the class about what they see. Prompt them to consider other things that can be differentiated into different categories, fostering critical thinking and discussion. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 3 To arouse students' interest further, draw parallels between sets and real-life applications. For instance, discuss examples of sets in everyday life that relate to the topic at hand. One relevant example is the organisation of a kitchen. Encourage students to think about how their mothers might separate bowls, cups, and different sets of utensils in the kitchen arrangement, effectively using sets to categorise items for practical purposes. Another real-life scenario could be the layout of a shopping mall. Encourage students to imagine a famous shopping mall and how it is divided into distinct sections. For instance, the clothing shops might be on one floor, while the food court is located in another part of the mall. This example highlights the practical use of sets to categorise and organise different types of stores for the convenience of shoppers. (Textbook Grade 8: Page XIV) Suggested answer(s): There are groups of vegetables and fruits. Within those groups, they can further be categorised based on their colors, such as red, orange, green, and so on. (Textbook Grade 8: Page 2) Ask students to complete the section and discuss about the answers with them. Suggested answer(s): 1. a, e, i, o, u 2. 1, 2, 3, 4, 5, 6, 7, 8, 9 3. Red, Orange, Yellow, Green, Blue, Indigo and Violet. 4. (Accept any possible answers) T-shirts, blouses, pants, scarves, glasses, watches, shoes, sandals, pyjamas, jacket, skirts, etc. 5. White and red 6. 2, 3, 5, 7, 11, 13, 17. (Textbook Grade 8: Page 2) Begin this section by explaining the concept of recycling and its significant benefits to our daily lives today. Describe recycling as the process of collecting and processing materials that would otherwise be discarded as trash, transforming them into new products. Highlight how recycling positively impacts the community, economy, and, most importantly, the environment. Encourage students to actively participate in recycling by properly disposing of recyclable items like papers, plastic, aluminium, glass, and more. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 4 Focus-on Mathematics Teacher’s Guide Grade 8 Next, elaborate on the idea that recycling involves converting waste materials into useful resources, forming an essential part of green technology. Provide examples, such as how used paper can be recycled to create paper plates or toilet paper. Furthermore, emphasise that recycling plays a crucial role in reducing greenhouse gas emissions, water pollutants, and conserving energy. Guide students in answering the questions provided in the textbook to reinforce their understanding of the topic. To assess comprehension, select a few students to share their responses with the class. Suggested answer(s): (a) (b) These categories of materials can be represented using sets notation. (Textbook Grade 8: Page 2) Suggested answer(s): (Accept any possible answers) To carry out the effective classification of solid waste: 1. Raise public awareness and educate about waste classification. 2. Provide clear guidelines and proper waste collection infrastructure. 3. Monitor and enforce waste sorting regulations. 4. Collaborate with waste handlers and recycling facilities. 5. Use communication channels for reminders and updates. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 5 Teaching/Learning Activities 1.1 Sets A Representing sets (Textbook Grade 8: Page 3) Let students carry out this activity in groups. Suggested answer(s): 2. Land transport: car, lorry, van, bus Water transport: ship, boat, ferry, rowboat Air transport: hot air balloon, airplane, rocket, helicopter 3. Modes of transport. (I) Introduction to sets Teach students the concept of sets and their elements. A set is a collection of objects, and the objects within it are called elements or members. These elements can be of various types, including other sets. They don't have to be of the same kind. Explain that sets are organised collections of objects and can be represented in two forms: setbuilder form or roster form. Typically, sets are denoted using curly brackets{}. For example, we can represent a set A with elements 1, 2, 3, and 4 as A = {1, 2, 3, 4}. Deepen students’ understanding by exploring Example 1 and Example 2 on page 4. (II) Identifying the elements of a set Elements of a set are denoted using small letters, such as x, y, or z. We can describe a set by listing all its elements within braces. For instance, if set A contains the numbers 2, 4, 6, and 8, we write it as A = {2, 4, 6, 8}. Emphasise the symbols '∈' and '∉', which represent 'belongs to' and 'not belongs to,' respectively. Go through Example 3 on pages 4 to 5 with the class. Encourage students to explore additional online resources by scanning the provided QR codes of on page 3. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 6 Focus-on Mathematics Teacher’s Guide Grade 8 B Universal sets and empty sets (I) Universal sets Begin by guiding students to understand the concept of universal sets. Explain that a universal set (often denoted by ξ) is a set that contains all the elements from all the related sets, without any repetition. Illustrate with an example: If we have two sets, A = {1, 2, 3} and B = {1, a, b, c}, then the universal set associated with these two sets is given by ξ = {1, 2, 3, a, b, c}. Reinforce the concept by directing students to Example 4 on page 5. (II) Empty sets Provide a brief explanation of empty sets. An empty set, also known as the null set or void set, is a set that contains no elements. Offer an example for better understanding: Consider the set of possible outcomes when rolling a die to get a number greater than 6. As we know, there are no outcomes greater than 6, resulting in an empty set. Introduce the symbol used to denote an empty set: ∅ = { }. Encourage students to explore Example 5 and Example 6 on pages 5 – 6 to gain a clear understanding of the concept of empty sets. C Venn diagram After covering the concepts of sets, elements, universal sets, and empty sets, the teacher can introduce the class to an essential tool for visualising and understanding set relationships - the Venn Diagram. Explain to the students that a Venn Diagram consists of two large circles that intersect, forming a space in the middle. Each circle represents a set of items or data that we want to compare and contrast. Where the circles intersect, we can write traits or elements that are common to both sets. Elaborate further on the Venn Diagram's usefulness in illustrating relationships between two or more data sets. It is particularly valuable for highlighting similarities and differences between different sets of data. Each circle in the Venn Diagram represents a distinct data set. To deepen their understanding, encourage students to explore Example 7, Example 8, and Example 9 on pages 7 to 8. These examples will help them grasp how to use the Venn Diagram effectively to represent data and analyse relationships between sets. Let students attempt Practice 1.1 (Textbook Grade 8: Page 8) and discuss about the answers with them. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 7 Independent Practice Assign students to complete Exercise 1.1 (Workbook Grade 8: Pages 4 – 7). 1.2 Properties of Sets A Cardinality of sets (I) The number of elements of a set To introduce the concept of cardinality of sets, the teacher can begin by providing a clear definition. Explain that the cardinality of a set refers to the total number of unique elements it contains. For instance, if we have a set A = {i}, its cardinality is 3, as it consists of three elements. Emphasise to the students that cardinality represents the size or the number of elements in a set. It is denoted using vertical bars, similar to absolute value signs. For example, the cardinality of set A is represented as |A|. Practice 1.1 Answers: 1. (a) The set of the first five prime numbers. (b) The set of the first six even numbers. (c) The set of days in the week. (d) The set of vowels. 2. (a) A = {21, 22, 23, 24, 25, 26} (c) C = {red, orange, green} (b) B = {January, June, July} (d) D = {31, 33, 35, 37, 39} 3. (a) ∈ (c) ∉ (b) ∈ (d) ∉ 4. (a) P = Ø (c) R = Ø (b) Q ≠ Ø (d) S ≠ Ø 5. (a) ξ = {1, 2, 3, 4, 5, 6, 7, 8} (b) P = {1, 3, 5} (c) Q = {2, 4, 6, 8} 6. (a) (c) (b) (d) ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 8 Focus-on Mathematics Teacher’s Guide Grade 8 Additionally, introduce the notation n(A) to denote the number of elements in set A. For example, if A = {1, 3, 5}, then n(A) = 3, indicating that the set A has three elements. (Textbook Grade 8: Page 9) Suggested answer(s): If A is an empty set, then the cardinality of set A, denoted as n(A), is equal to 0. This is because an empty set has no elements, and the number of elements in an empty set is zero. In set theory, the cardinality of an empty set is always zero, regardless of the context or the elements that could have potentially been in the set. So, n(A) = 0. Guide students to study the Example 10 and Example 11 on page 9 – 10 to deepen the students’ understanding about the cardinality of sets. Encourage students to explore additional online resources by scanning the provided QR codes of on page 9. B Subsets (Textbook Grade 8: Page 10) Divide the class into groups of four. Carry out this activity with students in class to introduce the concept of subsets. Provide each group with manila cards to create number cards, as shown on page 10. Engage in the activity together with the students and observe the results. Do this activity together with students and observe the result from this activity. Explain that a set A is considered a subset of set B if every element in set A is also an element in set B. The symbol ⊂ is used to denote 'is a subset of'. For example, if set A is a subset of set B, we write A ⊂ B. Conversely, the symbol ⊄ is used to denote 'is not a subset of'. To deepen their understanding, direct students to explore Example 12 on page 11. This example will illustrate the concept of subsets in a real-world context, further solidifying their comprehension. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 9 (Textbook Grade 8: Page 11) Suggested answer(s): In set theory, if two sets, F and H, are both subsets of another set G (i.e., F ⊂G and H ⊂ G ), it does not necessarily mean that F and G are equal. They can be distinct sets even though they share a common subset, G. The diagram shows that the areas of circles F and H do not overlap, indicating that set F and set H have no elements in common. In this case, F and H are distinct sets even though they are both subsets of G. The equality of sets is determined by whether they have exactly the same elements, not just by whether they share a common subset. Guide students to see Example 13 on page 11 to learn about the concept of subsets and how it can be represented in a Venn diagram. (Textbook Grade 8: Page 12) Suggested answer(s): (a) (b) (i) B ⊂ A (ii) C ⊄ D (c) E ⊂ A’ and E ⊂ D’. Set E is contained in the universal set but outside the set A and set D. Go through Example 14 on page 12 with the class and explain about the concept of subsets and how to list subsets of a set. C Complement of sets Teacher may start this topic by giving the definition of complement. A complement is something that completes or perfects. The complement of a set is the set of all elements that are not in the given set but are part of the universal set. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 10 Focus-on Mathematics Teacher’s Guide Grade 8 Explain that if A is a subset of the universal set ξ, the complement of set A, denoted as A', is the set containing all elements of ξ that are not in A. Emphasise the symbol used to denote the complement of a set, which is an apostrophe ('), so A' represents the complement of set A. Provide a simple example to illustrate the concept. For instance, let A be the set of even numbers less than 10, so A = {2, 4, 6, 8}. If the universal set ξ represents all natural numbers less than 10, then the complement of A, denoted as A', would be the set of odd numbers less than 10, which is A' = {1, 3, 5, 7, 9}. For better understanding, the teacher can use a Venn diagram to visually represent the concept of the complement of sets. Draw two overlapping circles, with one representing set A and the other representing its complement, A'. Show that the elements in A' are all the elements outside of set A but within the universal set ξ. Discuss some essential properties of complements, such as: • The complement of the universal set is the empty set (∅). • The complement of the empty set is the universal set (A' = ξ). (Textbook Grade 8: Page 12) Suggested answer(s): Empty set (∅) means there are no elements in the set, so the complement of an empty set or a null set is the universal set (ξ) containing all the elements. Guide students to see Example 15 – Example 18 on page 13 to learn about the concept of complement of a set. D Relationship between sets Start by guiding students to understand and revise the relationships between different sets. Remind the class that these relationships can be effectively illustrated using Venn diagrams, which visually represent the connections between sets, subsets, the universal set, and complements of sets. Explain that if set A is a subset of set B (A ⊂ B), it means that all the elements in A are also present in B. Use a Venn diagram to demonstrate this relationship, with circle A being entirely contained within circle B. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 11 Discuss the concept of the universal set (ξ), which represents the collection of all elements under consideration. Show that all sets under consideration are subsets of the universal set. Use a Venn diagram to illustrate this, with multiple circles representing different sets, all of which are enclosed within the universal set. Revisit the concept of complement of a set, denoted as A', which consists of all elements that are not part of set A but belong to the universal set. Use a Venn diagram to visually represent the complement of set A outside of circle A within the universal set. Direct students to explore Example 19 on page 14 to deepen their understanding of the relationships between sets and how to represent these relationships using Venn diagrams. Work through the example together to analyse the connections between the given sets. E Power Sets Start by introducing the concept of power sets. Explain that the power set of a given set is the set that contains all possible subsets of the original set, including the empty set and the set itself. Define the power set of set A as P(A). Emphasise that P(A) contains all the possible combinations of elements that can be formed from set A. Explain that the cardinality of the power set |P(A)| is the number of elements in the power set. The power set is formed by considering all possible combinations of elements from the original set. Discuss the formula for calculating the cardinality of the power set. If set A has 'n' elements, then the power set P(A) will have 2n elements. Provide several examples to illustrate the cardinality of the power set for different sets. Encourage students to construct power sets and count the number of elements to verify the formula. Introduce the properties of the power set to the class, which are characteristics that govern the behavior of power sets. Discuss the properties of the power set related to the empty set (∅) and the universal set (ξ). • The power set of the empty set (∅) is a set containing only the empty set itself, which has one element: P(∅) = {∅}. • The power set of the universal set (ξ) is a set containing two sets: the empty set (∅) and the universal set itself (ξ). • The power set of an infinite set has an infinite number of subsets. For example, if set Y has infinite numbers of elements, this set will have infinite numbers of subsets. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 12 Focus-on Mathematics Teacher’s Guide Grade 8 • Thus, the power set exists for both finite set and infinite set. Guide students through Example 20 on page 15 to deepen their understanding about power set. F Equality of sets Begin the lesson by introducing the concept of equality of sets. Explain that two sets are considered equal if and only if they contain precisely the same elements, regardless of the order in which the elements are listed. Additionally, two sets are said to be equivalent if they have the same cardinality or the same number of elements. Emphasise that equality of sets means that both sets have precisely the same elements. Use examples to illustrate that the order of elements in the sets does not matter. For instance, {1, 2, 3, 4} and {3, 4, 2, 1} are equal sets because they contain the same elements, even though the order is different. Provide additional examples to reinforce the concept of equality of sets. Use sets with different elements and orders to demonstrate that sets can be equal despite their arrangement. Directstudents to explore Example 21 on page 16 to deepen their understanding of the concept of equality of sets. Work through the example together, discussing how to determine if sets are equal or not based on their elements. (Textbook Grade 8: Page 16) Suggested answer(s): (a) True, because two equal sets must have the same elements and the same number of elements. (b) False, because any two sets with the same number of elements do not necessarily have the same elements. Guide students to see Example 22 and Example 23 on page 16 to deepen their understanding about equality of sets. Let students attempt Practice 1.2 (Textbook Grade 8: Page 17) and discuss about the answers with them. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 13 Independent Practice Assign students to complete Exercise 1.2 (Workbook Grade 8: Pages 7 – 9). Practice 1.2 Answers: 1. (a) A = {11, 13} (b) B = {a, b, c, d, e} (c) C = {red, blue, yellow, green} (d) D = {3, 4, 5, 6, 7} 2. (a) 5 (b) 6 (c) 0 (d) 8 3. (a) True (b) False (c) False (d) True 4. (a) { }, {3}, {4}, {3, 4} (b) { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z} (c) { }, {11}, {13}, {11, 13} 5. (a) P' = {e, g} (b) Q' = {a, b, e, g} 6. (a) {3, 5, 7, 9, 11, 12} (b) {4, 5, 7, 8, 10, 11} (c) {4, 6, 8, 9, 10, 12} (d) {5, 7, 8, 9, 10, 11} (e) {3, 4, 5, 7, 9, 11, 12} 7. (a) If the number of elements in a set ‘n’, there will be 2n elements in the power set. Since an empty set does not contain any elements, the power set will contain 20 elements or 1 element. Therefore, the power set of the empty set is an empty set, P (E ) = {}. (b) Power set of a set with ‘n’ elements is given by P (A ) = 2n. it is given that a set has ‘k + 1’ elements, therefore the power set of the set will contain 2k + 1 elements. 8. (a) P ≠ Q (b) M = N (c) V = W (d) C ≠ D 9. 7 10. (a) (b) 11. (a) (i) R' ⊂ ξ (ii) R ⊂ Q (iii) P ⊂ Q' (iv) R ⊂ P' (v) R ⊂ Q ⊂ ξ (b) ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 14 Focus-on Mathematics Teacher’s Guide Grade 8 1.3 Operations of sets A Intersection of sets (I) Determining and describing the intersection of sets Introduce the concept of the intersection of sets to students. Explain that the intersection of two sets consists of all the elements that are common to both sets. Define the intersection of sets A and B as the set that contains elements present in both A and B. Symbolically, the intersection of A and B is represented as A ∩ B. Use Venn diagrams to visually represent the intersection of sets A and B. Draw two circles representing sets A and B, and show the overlapping region, which represents the elements that are common to both sets. Emphasise that the intersection of two sets follows the commutative law, meaning A ∩ B = B ∩ A. Explain that the order in which the sets are intersected does not change the result. Provide examples to further illustrate the concept of the intersection of sets. Use different sets and Venn diagrams to help students understand the process of finding the common elements. The teacher can provide extra practice exercises for students to determine the intersection of various sets. Encourage them to apply the commutative law and use Venn diagrams to verify their answers. Guide students to see Example 24 and Example 25 on page 19 – 20 to deepen their understanding about intersection of sets. (II) Complement of the intersection of sets Introduce the complement of the intersection of sets to students. Explain that the intersection of sets A and B contains elements that are in both A and B, while the complement of a set contains all the elements that are not in that set. Define the intersection of sets A and B as A ⋂ B, and the complement of set A as A'. Use examples and Venn diagrams to illustrate the concept of the complement of the intersection of sets. Show how to find the elements that are not part of the intersection. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 15 Use Venn diagrams to visually represent the complement of the intersection of sets. Show the regions outside the intersection, which represent the elements that are not common to both sets. Direct students to explore Example 26, Example 27, and Example 28 on pages 20 – 21. Work through these examples together to deepen their understanding of the complement of the intersection of sets. B Disjoint of sets Teacher may begin the lesson by introducing the concept of disjoint sets. Explain that two sets are considered disjoint if they have no elements in common, meaning their intersection is the empty set (∅). Define disjoint sets as sets that do not share any elements. Emphasise that the intersection of two disjoint sets is always the empty set. For example, set A = {2, 3} and set B = {4, 5} are disjoint sets. Set C = {3, 4, 5} and set D = {3, 6, 7} are not disjoint as both the sets C and D are having 3 as a common element. Use Venn diagrams to visually represent disjoint sets. Draw two circles representing two sets, and show that there is no overlapping region between them, indicating that they are disjoint. On a Venn diagram, A and B are represented by two non-intersecting circles. Guide students through Example 29 on page 22 to deepen their understanding about disjoint sets. C Union of sets (I) Determining and describing the union of sets Go to page 23 to explain about union of sets. Explain that the union of two sets is a new set that contains all the elements from both sets, without any repetitions. Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 16 Focus-on Mathematics Teacher’s Guide Grade 8 Explain that the union of two sets can be represented using a Venn diagram. The shaded region in the Venn diagram below represents set A ∪ B. (Textbook Grade 8: Page 23) 1. In words, (D ∪ E )' represents the complement of the union of sets D and E. It includes all the skilled workers in the electronic factory (ξ) who do not possess either a diploma or a degree, or both. In other words, it represents the group of workers who possess neither a diploma nor a degree. The value of n (D ∪ E ) = 100. 2. n (D ∪ E ) = 100, n (ξ) = 120, n [(D ∪ E )’ ] = 20. Therefore, the equation is correct. Guide students to see Example 30 – Example 33 on pages 24 – 25. Students will learn and understand the concept of union of sets and also how to find the complement of union of sets. The teacher is encouraged to provide more examples to the class to ensure better understanding. D Solving problems involving the intersection and union of sets Once the students mastered intersection and union of sets, go through Example 34 – Example 37 with them. These are problem solving examples on intersection and union of sets. E Difference of sets (I) Difference of two sets Begin the lesson by introducing the concept of the difference of sets. Explain that the difference of two sets, A and B, denoted as A – B, is a new set that contains all the elements from set A that are not present in set B. Define the difference of sets A and B as the set that includes all elements from A, excluding those that are also in B. Symbolically, the difference of A and B is represented as A – B. For the case A ∪ B ≠ φ For the case A ∪ B = φ ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 17 The teacher can use Venn diagrams to visually represent the difference of sets. Draw two circles representing sets A and B, and show the overlapping region, which represents the elements that are common to both sets. Then, show the region specific to set A, which represents the elements that are in A but not in B. Provide examples to illustrate the concept of the difference of sets. Use Venn diagrams to demonstrate how to identify the elements unique to set A and form the difference set. Discuss the relationship between the complement of the difference of sets and the union of sets. Emphasise that (A – B)' is equivalent to A' ∪ B, meaning it includes all elements not in A – B. (II) Find the difference of sets using Venn diagram Review the concept of sets and the notation for finding the difference of sets. Remind students that the difference of two sets, A and B, denoted as A - B, is the set that includes all elements from set A that are not present in set B. Revisit the basics of Venn diagrams, explaining that they are graphical representations of sets. Draw a simple Venn diagram with two circles representing sets A and B and their overlapping region, which represents the intersection of A and B. Explain the process of finding the difference of sets using Venn diagrams. To find A – B , students should focus on the region that belongs to set A but not to the overlapping region. Similarly, the difference B – A is the set of all those elements of B that do not belong to A. Provide more examples of finding the difference of sets using Venn diagrams. Use simple sets and gradually progress to more complex examples. Reinforce the relationship between the intersection of sets and the difference of sets. Emphasise that (A ∩ B) represents the common elements in both sets, while A – B includes only the elements in A that are not in B. ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 18 Focus-on Mathematics Teacher’s Guide Grade 8 Proceed to Example 38 and Example 39 on page 29, which is about the application fo the concept of difference of sets. Let students attempt Practice 1.3 (Textbook Grade 8: Page 30 – 31) and discuss about the answers with them. Practice 1.3 Answers: 1. (a) {7, 8, 9} (b) {9} (c) {9} 2. (a) {a, u} (b) 3. (a) {4, 6, 7, 10, 11} (b) {4, 6, 7, 9, 11} 4. (a) A ∩ B' (b) (A ∩ B )' ∩ C 5. ξ = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} X = {11, 13, 15, 17, 19} (a) Y = {11, 13, 17, 19} X ∪Y = {11, 13, 15, 17, 19} ∪ {11, 13, 17, 19} = {11, 13, 15, 17, 19} (b) 6. (a) {2, 6, 9} (b) {2, 3, 6} 7. {16, 17} 8. (a) 5 (b) 16 (c) 2 9. 225 students 10. (a) (b) A’ ∩B A’ ∩ B ’ (c) (d) A ∪B A ∪B 11. (a) Papaya, orange and apple (b) Grape, watermelon, kiwi, mango, guava and banana 12. (a) (b) (i) 13 (ii) 27 13. (a) Nasi lemak, Fried chicken, Curry noodles, Fried egg, Noodle soup, Tom yam, Fried rice (b) Nasi lemak, Fried chicken ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 19 Independent Practice Assign students to complete Exercises 1.3 – 1.5 (Workbook Grade 8: Page 13 – 21). Closing Guide the whole class to conclude the concept of sets. Go to page 32, 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 (P32 – 35). Then, discuss about the answers with them. 14. (a) (b) (i) 43 (ii) 7 (iii) 10 15. P = {11, 12, 13, 14, 15} Q = {10, 12, 14, 16, 18} R = {7, 9, 11, 14, 18, 20} (a) P – Q = {Elements of set P which are not in set Q } = {11, 13, 15} (b) Q – R = {Elements of set Q not belonging to set R } (c) R – P = {Elements of set R which are not in set P } = {7, 9, 18, 20} (d) Q – P = {Elements of set Q not belonging to set P } = {10, 16, 18} ©Praxis Publishing_Focus On Maths

Chapter 1 SETS 20 Focus-on Mathematics Teacher’s Guide Grade 8 Mastery Practice 1 Answers: Section A 1. C 2. B 3. D 4. A 5. A 6. B 7. C 8. B 9. B 10. C 11. B 12. C Section B 1. (a) (i) The set of the first five prime numbers. (ii) {2, 3, 5, 7, 11} (b) (i) False (ii) False 2. (a) True (b) False (c) True (d) True 3. (a) 7 (b) (i) False (ii) False 4. 5. (a) (b) { }, {2}, {4}, {2, 4} (c) {2, 4, 5, 7, 8} (d) 6 6. (a) { }, {4} (b) {2, 5, 7, 8, 10} (c) 16 (d) 2 7. (a) P = {20, 24, 28} Q = {20, 21, 22, 23, 24, 30} (b) {20, 21, 22, 23, 24, 25, 26, 27} (c) 1 8. (a) 5 (b) 26 (c) 11 (d) 16 (e) 15 9. (a) M ' = {17, 19, 20, 22, 24, 26, 28} (b) { }, {3}, {7}, (3, 7} 10. (a) 124 (b) 8 11. (a) K ∪ L = {1, 2, 3, 5, 6, 7, 9, 18} (b) P ∩Q = {12, 15} 12. (a) x = 12 (b) 24 13. (a) {1, 4} (b) {5, 7} 14. (a) {13, 15} (b) {10, 16} (c) {7, 9, 20} ©Praxis Publishing_Focus On Maths

Chapter 1 SETS Focus-on Mathematics Teacher’s Guide Grade 8 21 Independent Practice Assign students to complete Enrichment Exercises (Workbook Grade 8: Page 22 – 27). Assessment At the end of this chapter, teacher needs to make sure that students should be able to • explain what a set is. • define a set using set notation and description. • represent sets using Venn diagrams. • recognise universal sets and empty sets. • identify the cardinality of sets. • list and state the number of elements in a set. • understand and use the concept of subset, universal set and complement of a set. • understand intersection between sets. • understand the union of sets. • perform operations on sets. Materials • GeoGebra App • Focus-on Mathematics Textbook Grade 8 • Focus-on Mathematics Workbook Grade 8 • Focus-on Mathematics Grade ©Praxis 8 — PowerPoint Publishing_Focus On Maths

22 Focus-on Mathematics Teacher’s Guide Grade 8 Objective To introduce students to the concept of sets and their application in categorising everyday objects. Materials • Various everyday objects (e.g., pencils, books, fruits, toys, etc.) • Index cards or sticky notes • Large poster paper or whiteboard • Writing materials Instructions 1. Begin by briefly explaining the concept of sets to the students. Use simple language to convey that sets are groups of objects or things that share common characteristics. 2. Present a collection of everyday objects to the students. These could be items gathered from around the classroom or items brought in from home. 3. Instruct students to work individually or in small groups. Each group should select a set of objects based on a specific characteristic. For example, they could create a set of ‘red objects,’ ‘things that can be eaten,’ ‘items found in a classroom,’ etc. 4. Provide index cards or sticky notes to each group. Ask them to write the name of their selected set on the card or note. Group Presentation 5. Have each group present their set to the class. As they present, they should explain the common characteristic that defines their set. 6. As groups present their sets, the teacher can arrange the index cards or sticky notes on a large poster paper or whiteboard. This creates a visual representation of different sets. Discussion 7. After all groups have presented, engage the class in a discussion. Ask students if they can find any commonalities between the different sets. Discuss how items can belong to multiple sets (e.g., a red apple belongs to both the ‘red objects’ set and the ‘things that can be eaten’ set). ©Praxis Publishing_Focus On Maths

STEM ACTIVITY: Set Sorting Challenge Focus-on Mathematics Teacher’s Guide Grade 8 23 8. Distribute blank index cards or sticky notes to each student. Instruct them to create their own sets based on personal criteria. For instance, they could make a set of ‘favourite things,’ ‘items in my backpack,’ etc. Sharing and Display 9. Allow students to share their self-created sets with the class. Add the new sets to the classroom display, emphasising that sets can vary based on individual perspectives. Conclusion 1. Summarise the activity by reiterating the concept of sets and how they are used to categorise objects based on common characteristics. 2. Encourage students to observe sets in their surroundings and identify different ways in which objects can be grouped. Extensions The teacher can extend the activity by discussing the intersection of sets (common elements between sets) and the concept of the universal set. ©Praxis Publishing_Focus On Maths

24 Focus-on Mathematics Teacher’s Guide Grade 8 Time 14 hours Elemen Capaian Pembelajaran Aljabar Di akhir fase D peserta didik dapat mengenali, memprediksi dan menggeneralisasi pola dalam bentuk susunan benda dan bilangan. Mereka dapat menyatakan suatu situasi ke dalam bentuk aljabar. Mereka dapat menggunakan sifat-sifat operasi (komutatif, asosiatif, dan distributif) untuk menghasilkan bentuk aljabar yang ekuivalen. Peserta didik dapat memahami relasi dan fungsi (domain, kodomain, range) dan menyajikannya dalam bentuk diagram panah, tabel, himpunan pasangan berurutan, dan grafik. Mereka dapat membedakan beberapa fungsi nonlinear dari fungsi linear secara grafik. Mereka dapat menyelesaikan persamaan dan pertidaksamaan linear satu variabel. Mereka dapat menyajikan, menganalisis, dan menyelesaikan masalah dengan menggunakan relasi, fungsi dan persamaan linear. Mereka dapat menyelesaikan sistem persaman linear dua variabel melalui beberapa cara untuk penyelesaian masalah. Learning Objectives Students will be taught to: • distinguish between monomials and polynomials. • determine the degree of a polynomial. • model polynomials using algebra tiles. • perform addition and subtraction involving polynomials. • perform multiplication and division involving polynomials. • determine the values of algebraic expressions. • apply the knowledge of algebraic expressions to solve mathematical problems and real-life problems. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 25 Bilingual Key Terms Algebraic expression Ekspresi aljabar Term Suku Monomial Monomial Polynomial Polinomial Coefficient Koefisien Variable Variabel Constant Konstanta Exponent Pangkat Degree of a term Tingkat suku Degree of a polynomial Tingkat polinomial Pattern Pola Formula Hubungan antar peubah START UP Background Information Assess the students’ prior knowledge about algebra by asking what the students know and what they want to know more about algebra. Encourage them to share what they already know about algebra and inquire about their curiosity, prompting them to express what they would like to learn more about the subject. Introduce algebra as a branch of mathematics that deals with representing problems through mathematical expressions. These expressions encompass both numbers with fixed values, such as 2, –8, or 0.0068, and variables denoted by letters like x, y, or z, which can take on various values. Make algebra relatable to the students’ lives by discussing its real-world applications. Explore the different types of algebra that manifest in various aspects of daily life. Examples may include cooking, where algebra is employed to adjust or modify recipes, interior and landscape design, where measurements and calculations play a crucial role, and business and financial management, which rely heavily on algebraic analysis for decision-making processes. Additionally, emphasise how algebra finds relevance in sports, tax calculations, computer programming, and astrological predictions, among others. By connecting algebra to practical situations, students are more likely to grasp its significance and develop a deeper appreciation for its utility in diverse fields. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 26 Focus-on Mathematics Teacher’s Guide Grade 8 Real World Connection Refer the chapter opener (Textbook Grade 8: Page 36). Let students observe the picture. Encourage them to share their observations and select one or two students to provide a brief presentation to the class about the picture. Afterward, ask if they can identify other ways in which algebra is expressed in everyday life. To cultivate students’ interest, introduce real-life applications of algebra. Discuss practical examples that demonstrate how algebra is applied in various situations. Relate these examples back to the topic being discussed to highlight the relevance of algebra in our daily lives. Next, engage students’ imaginations by prompting them to come up with their own algebraic examples. For instance, they can imagine scenarios related to cooking, such as baking cakes or preparing vegetable soups, where getting the right ingredient combinations is essential. In these cases, algebra becomes a valuable tool to determine the correct quantity of ingredients for varying serving sizes. (Textbook Grade 8: Page 36) Suggested answer(s): There are many daily-life that applying the concept of algebra. • Shopping and Budgeting: When planning a shopping trip, we often need to find the total cost of items, taking into account discounts, sales tax, or other factors. Using algebra, we can express the relationship between the prices of items, discounts, and the final cost to calculate the unknown total. • Time and Distance: Whether it’s calculating travel time, average speed, or distance covered, algebra can help us model these situations. For instance, if we know the speed of a vehicle and the time it traveled, we can use algebra to find the distance covered. • Mixing and Dilution: In cooking or chemistry, we might need to mix different ingredients or solutions to achieve a desired concentration. Algebra enables us to determine the quantities needed to obtain the desired outcome. • Financial Planning: When planning investments or loan payments, algebra helps us determine the future value of an investment, the interest earned, or the monthly payments required to pay off a loan. • Scaling and Proportions: When dealing with maps, blueprints, or resizing images, algebra allows us to maintain the proportions and calculate the corresponding dimensions. • Temperature Conversion: Converting temperatures from Celsius to Fahrenheit or vice versa can be done using algebraic equations. • Sports and Performance: In sports, coaches and athletes use algebra to analyse statistics, calculate averages, and assess performance improvements. • Geometry and Measurements: Problems involving areas, volumes, and angles can be solved using algebraic expressions and equations. • Planning Events: Organising events often involves determining budgets, attendance, and resource allocation, which can all be handled using algebraic representations. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 27 (Textbook Grade 8: Page 38) Ask students to complete the section and discuss about the answers with them. Answers: 1. (a) Multiples of 2: 2, 4, 6, 8, 10, 12, ... Multiples of 3: 3, 6, 9, 12, 15, 18, ... (b) Common multiples of 2 and 3: 6, 12, ... 2. (a) 7 9 (b) 6 17 (c) 11 12 (d) 14 15 3. (a) 5 2 (b) 2 7 (c) 18 15 (d) 1 8 (Textbook Grade 8: Page 38) Begin this section by explaining the definition of rice seedling. This section introduces the concept of a rice seedling. Initially, rice seeds are planted densely in a single flooded paddy, where they germinate and grow into seedlings. These seedlings are later transplanted into multiple paddy fields. Historically, farmers used to transplant rice manually when the seedlings reached a height of approximately 8 to 10 inches. It's important to understand that transplanting rice serves two significant purposes. Firstly, it ensures a uniform distribution of plants across the field, promoting better growth and yield. Secondly, by transplanting the seedlings, the rice crop gains an advantage over emerging weeds, giving it a head start and reducing competition for nutrients and resources. Answer(s): (a) Adam. (b) 2 m more. (Textbook Grade 8: Page 38) Suggested answer(s): There are more than 40,000 varieties of cultivated rice (the grass species Oryza sativa) said to exist. But the exact figure is uncertain. Over 90,000 samples of cultivated rice and wild species are stored at the International Rice Gene Bank and these are used by researchers all over the world. Rice varieties can differ in size, shape, texture, aroma, and taste. Some popular types of rice include: • Long-grain rice: Examples include Basmati and Jasmine rice, which are well-known for their distinct aroma and fluffy texture, they are commonly used in Indian and Middle Eastern cuisines. • Medium-grain rice: Often used in dishes like paella and risotto, examples include Arborio and Calrose rice. • Short-grain rice: Sticky and moist when cooked, commonly used in sushi and other Asian dishes. Examples include Japanese sushi rice and sticky rice. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 28 Focus-on Mathematics Teacher’s Guide Grade 8 • Brown rice: Whole-grain rice with the bran and germ intact, providing more nutrients than white rice. • White rice: The most common type, with the bran and germ removed, resulting in a milder flavor and softer texture. • Black rice: Also known as forbidden rice, it has a dark color and a nutty flavor, packed with antioxidants. • Red rice: Slightly nutty with a chewy texture, rich in anthocyanins. • Wild rice: Not a true rice but a semi-aquatic grass seed, with a distinctive chewy texture and earthy flavor. Teaching/Learning Activities 2.1 Simplifying Algebraic Expressions A Monomials and polynomials Begin by explaining the definition of a monomial. A monomial is an algebraic expression that contains only one non-zero term. Help students understand that a monomial can consist of numbers, variables, or numbers multiplied with variables. Examples of monomials include: 2, ab, 4pq, 6x2, and 6p3q. Once students grasp the concept of monomials, introduce polynomials. A polynomial is an algebraic expression that represents the sum of one or more monomials. Explain that polynomials consist of terms of the form kxn, where k is any number and n is a positive integer. Show an example of a polynomial like 3x + 2x – 5. Use the table (page 39) to illustrate the differences between monomials and polynomials. Highlight the key distinctions, such as the number of terms and the complexity of the expression. Guide students to Example 1 on page 39 of the textbook, which provides a practical application of monomials and polynomials. (Textbook Grade 8: Page 39) Suggested answer(s): In summary, the names monomial, binomial, and trinomial are used to describe algebraic expressions based on the number of terms they contain. Monomials have one term, binomials have two terms, and trinomials have three terms. Understanding these terms helps to classify and categorise algebraic expressions more efficiently. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 29 B Degree of a polynomial Teach students about the meaning of degree of a polynomial. The degree of a polynomial is determined by the highest or greatest power of the variable in a polynomial equation. It indicates the highest exponential power present in the polynomial expression. Give example to the class how to find degree of a polynomial in a equation. Take the polynomial 6x4 + 2x3 + 3. In this expression, we have three terms: 6x4, 2x3, and 3. The term 6x4 is called the leading term as it has the highest degree. The constant term is 3. The coefficients of the polynomial are 6 and 2. To further reinforce the concept of polynomial degrees, use the table provided on textbook page 40. The table helps to classify polynomials based on their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on. To solidify understanding, guide students to look at Example 2 on page 40 of the textbook. (Textbook Grade 8: Page 40) Suggested answer(s): (a) No, this algebraic expression is not a polynomial. For an expression to be a polynomial term, it must contain no square root of variables, and no fractional or negative powers on the variables. (b) No, this algebraic expression is not a polynomial. For an expression to be a polynomial term, it must contain no square root of variables, and no fractional or negative powers on the variables. C Modelling polynomials Continuing from the previous topic on algebra, let’s now introduce the concept of algebra tiles to the students. Algebra tiles are a valuable visual aid used to represent integers (or constants) and variables in algebraic expressions. These tiles come in square and rectangular shapes, and their dimensions are related to the unit square. They provide a hands-on, area-based model that helps students better understand and manipulate algebraic concepts. To delve deeper into algebra tiles, engage students in Let’s Explore found on page 41 of the textbook. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 30 Focus-on Mathematics Teacher’s Guide Grade 8 (Textbook Grade 8: Page 41) Suggested answer(s): (a) Use a table. Model Description of Tiles Polynomial A Two x2-tiles, eight –x-tiles, and two 1-tiles 2x2 – 8x + 2 B Eight –x-tiles, two x2-tiles, and two 1-tiles –8x + 2x2 + 2 C Four –x-tiles and six 1-tiles –4x + 6 (b) Both models A and B contain the same tiles. The polynomials represented by these tiles have the same degree, and the same terms: 2x2, –8x, and 2. Both polynomials can be written as: 2x2 – 8x + 2 So, 2x2 – 8x + 2 and –8x + 2x2 + 2 are equivalent polynomials. Model C has no x2-tiles, so its degree is different from that of models A and B. Guide students to see Example 3 on pages 41 – 42 to learn about algebra tiles. D Adding and subtracting polynomials Start by reviewing the concept of polynomials and their terms. Introduce the topic of adding and subtracting polynomials as an extension of basic operations with algebraic expressions. Engage students in Let’s Explore 3 (Textbook P13). In this activity, students will use algebra tiles to solve questions involving adding and subtracting polynomials. Provide guidance as needed and encourage students to share their solutions. (Textbook Grade 8: Page 42) Suggested answer(s): (a) (i) The dimensions of the rectangle are 3x and x. So, the perimeter of the rectangle is: 3x + x + 3x + x = 8x (ii) The dimensions of the rectangle are 3x and 2. So, the perimeter of the rectangle is: 3x + 2 + 3x + 2 = 6x + 4 ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 31 (b) (i) The perimeter is 4a + 2. Work backward. Write the polynomial as the sum of equal pairs of terms. 4a + 2 = 2a + 2a + 1 + 1 The dimensions of the rectangle could be 2a and 1. Another solution is: 4a + 2 = a + (a + 1) + a + (a + 1) The dimensions of the rectangle could be a + 1. (ii) The perimeter is 10b. Write the polynomial as the sum of equal pairs of terms. 10b = 4b + 4b + b + b The dimensions of the rectangle could be 4b and b. Another solution is: 10b = 3b + 3b + 2b + 2b The dimensions of the rectangle could be 3b and 2b. Emphasise that adding and subtracting polynomials is similar to combining integers with algebra tiles. Reinforce the use of the same symbols for addition and subtraction of polynomials. Carefully explain the steps involved in adding or subtracting polynomials: Arrange each polynomial with the term of the highest degree first, followed by decreasing order of degree. Group like terms together. Like terms have the same variables and exponents. Simplify the equation by combining like terms. Present Example 4 and Example 5 on page 42 to demonstrate how to add and subtract polynomials. Discuss each step in detail while solving the examples to clarify the process. Introduce the Alternative Method for simplifying polynomials using algebra tiles. Discuss how this method provides a visual representation of the operations and aids in better understanding. Continue with Examples 6 to 10 on pages 43 to 44 to reinforce the concepts of adding and subtracting polynomials. Encourage students to attempt similar problems on their own. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 32 Focus-on Mathematics Teacher’s Guide Grade 8 E Multiplying and Dividing a Polynomial by a number Begin by reviewing the concepts of addition and subtraction of polynomials, which students should have mastered. Explain that multiplying and dividing are fundamental operations in algebra that extend our understanding of polynomials. Emphasise that when multiplying or dividing a polynomial by a number, we apply the operation to every term in the expression. Show that multiplication distributes over addition and subtraction in polynomials, making it essential to understand this operation thoroughly. Explain the steps involved in multiplying and dividing a polynomial by a number: (i) For multiplication: Multiply each term in the polynomial by the given number. (ii) For division: Divide each term in the polynomial by the given number. Work through Example 11 on page 44 to 45 of the textbook with the students. Show step-by-step solutions for both multiplication and division of the given polynomial by a number. Discuss the process and reasoning behind each step to deepen their understanding. The teacher can provide additional problems for students to practice multiplying and dividing polynomials by numbers. Encourage them to attempt various examples independently and seek help as needed. (Textbook Grade 8: Page 45) Suggested answer(s): (a) The product is 21a – 15. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 33 (b) The quotient is –2m2 + 3. Continue to see Example 12 on page 45 to learn on how to multiply and divide polynomials by a number in problem solving questions. F Multiplying and dividing a polynomial by a monomial Remind students that they have already learned about multiplying and dividing polynomials by numbers. Emphasise that to multiply a polynomial by a monomial, we use the distributive property. Remind students that the distributive property states that we multiply each term of the polynomial by the monomial, combining like terms afterward. Explain the steps involved in multiplying a polynomial by a monomial: (i) Distribute the monomial to each term in the polynomial. (ii) Multiply the coefficients of each term with the coefficient of the monomial. (iii) Add the exponents of the variables in each term with the exponents of the monomial. (iv) Explain the steps involved in dividing a polynomial by a monomial: (v) Divide each term in the polynomial by the coefficient of the monomial. (vi) Subtract the exponents of the variables in each term from the exponents of the monomial. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 34 Focus-on Mathematics Teacher’s Guide Grade 8 Work through Example 13 on page 46 of the textbook with the students. Demonstrate how to multiply and divide the given polynomial by a monomial using the distributive property. Discuss the process and reasoning behind each step to deepen their understanding. (Textbook Grade 8: Page 47) Suggested answer(s): (a) (b) 2x + 3x = 5x 4 + 3x Encourage students to explore additional online resources by scanning the provided QR codes of (P39) for further understanding and practice. Let students attempt Practice 2.1 (Textbook Grade 8: Page 47 – 48) and discuss about the answers with them. Practice 2.1 Answers: 1.. Monomials Polynomials 5 7x + 3 6pq d + e 7x3 9mn + m y2 + y + 1 2a4 + a3 – 6a2 – a 2. (a) 3 – coefficient; a – variable (b) 2 – coefficient; b – variable; 2 – exponent (c) –5 – coefficient; hk – variable (d) –m – variable 3. (a) Quadratic (b) Cubic (c) Linear (d) Quartic 4. (a) (b) (c) 5. (a) –2a + 2 (b) 2b2 – 3b + 1 (c) 2hk + 3h2 + 4k2 + 6h – 8k + 10 (d) 4m – 5 (e) 4n2 + n + 5 (f) –pq + 12p2 – 10q2 ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 35 Independent Practice Assign students to complete Exercise 2.1 – 2.4 (Workbook Grade 8: Page 30 – 38). 2.2 Determining the Values of Algebraic Expressions Teacher may start this subtopic by reviewing the concept of expressions and variables. Remind students that variables are represented by letters and can take on different values. Explain to students that to evaluate an algebraic expression, they need to substitute a specific number for each variable and perform the arithmetic operations. Use simple examples like 6 + x = 12 to demonstrate how to find the value of the variable (x = 6) by solving the equation. Emphasise that knowing the value of variables allows us to replace the variables with their respective values, simplifying the expression. This process is crucial for problem-solving and real-life applications of algebra. Explain the steps involved in determining the values of algebraic expressions: (i) Substitute the given values for the variables in the expression. 6. (a) 6a + 27 (b) –6b2 – 3b + 12 (c) 30cd + 18d2 – 24c2 (d) 2h + 3 (e) –m2 + 4 (f) 7p2 + 3p + 1 7. (a) 6e2 – 4e (b) –20h3 + 12h2 – 4h (c) –s – 3 (d) 3t2 – 2t + 4 8. Perimeter = 5n + 9 9. PQ = 1 2 p + 1 10. (a) 28r + 8 (b) 9x2 11. (a) (i) 28x + 24 (ii) 84x + 72 (b) 56x + 96 12. 8πx2 13. 99x2 14. It is incorrect. (– 3x )( –x 2 – 7x + 3) = (– 3x )( –x 2) + (– 3x )(– 7x ) + (– 3x )(3) = 3x2 + 21x2 –9x 15. 7+5 2 ; 7+9 2 ; 5+9 2 ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 36 Focus-on Mathematics Teacher’s Guide Grade 8 (ii) Perform the arithmetic operations (addition, subtraction, multiplication, division, etc.) to simplify the expression and find the result. Work through Examples 14 and 15 on page 48 and 49 of the textbook with the students. Demonstrate how to determine the values of the given algebraic expression by substituting the provided values and evaluating the result. Encourage students to explore additional online resources by scanning the provided QR codes of (P48) for further understanding and practice. Let students attempt Practice 2.2 (Textbook Grade 8: Page 49) and discuss about the answers with them. Independent Practice Assign students to complete Exercise 2.2 (Workbook Grade 8: Page 6 – 9). 2.3 Applying Algebraic Expressions A Making generalisation about patterns Divide the class in groups of four. Guide students through this activity by following the instructions in the textbook. As they work in their groups, ask them questions to assess their understanding and facilitate their discussion. The teacher may monitor while students so this activity in group. Give students some time to explore the activity and learn the patterns and sequences. Instruct each group to select a representative who will present their group's fin’ings to the class. This will give every student an opportunity to participate and share their group's con’lusions. During the presentations, encourage active engagement from the class by asking follow-up questions, encouraging discussion, and facilitating a deeper understanding of the topic. Help students make connections between different groups' find’ngs and draw meaningful conclusions from the collective responses. Practice 2.2 Answers: 1. (a) –16 (b) 80 (c) –45 2. (a) 9x2 + 7x + 6 (b) 56 cm 3. (a) 2x + 4y + 4400 (b) $12 400 4. (a) 4π4r2 + 2π2rh (b) 104π 5. (a) (i) 4x + 6 (ii) 4x2 + 6x (b) (i) 12 cm (ii) 18 cm2 ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 37 Guide students through Examples 16 and 17 on page 51 to 52 to give them better understanding about patterns and sequences. (Textbook Grade 8: Page 52) Suggested answer(s): Diagram Generalisation 3 × 4 = 12 4 × 4 = 16 5 + 5 = 10 4 + 4 = 8 5 × (3 + 4) = 5 × 7 = 35 The number is 35. Guide students to see Example 18 on page 52 – 53 to give them better understanding about patterns and sequences in problem solving question. B Forming algebraic formulae The teacher may Begin by reviewing the concept of algebraic formulae. Emphasise that algebraic formulae are expressions that represent patterns, relationships, and rules in mathematics. Encourage students to identify patterns and relationships in various numerical sequences or real-life scenarios. Demonstrate how these patterns can be expressed using algebraic formulae. Introduce the concept of variables and constants in algebraic formulae. Explain that variables represent unknown quantities that can change, while constants are fixed values. Explain the steps involved in forming algebraic formulae: • Identify the variables and constants involved in the problem. ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS 38 Focus-on Mathematics Teacher’s Guide Grade 8 • Analyse the pattern or relationship to determine how the variables and constants are related. • Write the expression that represents the relationship, using appropriate mathematical operations. The teacher can provide real-life examples where algebraic formulae are used, such as calculating areas, volumes, interest rates, or growth rates. Discuss how algebraic formulae can be applied in different fields, such as science, finance, engineering, etc. (Textbook Grade 8: Page 53) Suggested answer(s): Guide students through Examples 19 and 20 pages 53 – 54 to give students better understanding about forming algebraic formulae. (Textbook Grade 8: Page 52) Answer(s): (a) Q = 2k – 5 (b) 16 Guide students to Example 21 pages 54 – 55 to give students better understanding about forming algebraic formulae. 4x – 3y 2x + y ? Given that perimeter of the triangle is 9x + 2. Third side = Perimeter – (2 sides) = 9x + 2 – [(4x – 3y) + (2x + y)] = 9x + 2 – (4x – 3y + 2x + y) = 9x + 2 – (4x + 2x – 3y + y) = 9x + 2 – (6x – 2y) = 9x + 2 – 6x + 2y = 9x – 6x + 2 + 2y = 3x + 2y + 2 ©Praxis Publishing_Focus On Maths

Chapter 2 ALGEBRAIC EXPRESSIONS Focus-on Mathematics Teacher’s Guide Grade 8 39 Encourage students to explore additional online resources by scanning the provided QR codes of (P50) for further understanding and practice. Let students attempt Practice 2.3 (Textbook Grade 8: Page 55 – 56) and discuss about the answers with them. Independent Practice Assign students to complete Exercise 2.5 (Workbook Grade 8: Page 38 – 41). Practice 2.3 Answers: 1. (a) 4n – 3 (b) 5n + 1 (c) 38 – 3n (d) 45 – 5n 2. (a) V = pqr (b) A = 1 2 st (c) P = 4a 3. (a) y = 9 + x2 (b) P = (c) E = mc2 4. (a) e = h – 4 (b) c = e – 2d (c) b = l + d (d) p = r 2 – h (e) m = 4nr (f) I = (g) r = 2 (h) n = 3 (i) w = 2 – 7 5. t = − 6. 157.1 cm3 7. Using numbers: 2, 3, 4, 5, … Using words: The straight line in Diagram 1 passes through two points and each subsequent straight line passes through 1 point more than the previous straight line. Using algebraic expressions: n + 1, where n represents the number of nth straight line. 8. Using numbers: 2, 4, 6, 8, … Using words: The top row has two boxes and each row below it has two boxes more than the previous row. Using algebraic expressions: 2n, where n represents the number of cereal boxes in n rows. 9. (a) 11 (b) p = 2n + 1 10. (a) Diagram (n ) 1 2 3 Number of circles 1 2 3 Number of dots 3 5 7 (b) Un = 2n + 1 (c) 33 11. (a) h = −22 4 (b) 5 12. k = + 10 4 13. 3n + 3 ©Praxis Publishing_Focus On Maths