Form 2 Teacher s Guide

Transcription

1 Form 2 Teacher s Guide Lorrainne Bowie l Gert van der Westhuizen Reviewed by: Badisa Letlotlo TP.indd /07/13 2:25 PM

2 Pearson Botswana (Longman and Heinemann) PO Box 1083, Gaborone, Botswana Plot 14386, New Lobatse Road, Gaborone, West Industrial Site Botswana Pearson Botswana, Longman and Heinemann are imprints of Longman Publishing Company SA (Pty) LTD Copyright Longman Publishing Company SA (Pty) LTD All rights reserved. No part of this publications may be reproduced, stored in retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or other wise, without the prior permission of the publishers. First published in 2010 ISBN Teacher s Pack (includes a Teacher s Guide and a Test CD) ISBN Teacher s Guide Cover design by Firebrand Typesetting by A1 Graphics Artwork by A1 Graphics Printed by It is illegal to photocopy any page of this book without the written permission of the copyright holder. Maths in Action Form 2 TG.indd 2 6/29/10 3:53:20 PM

3 Contents Introduction iv Topic 1 Work with numbers... 2 Topic 2 Squares, square roots, cubes and cube roots... 9 Topic 3 Indices Topic 4 Matrices Topic 5 Angle properties Topic 6 Geometrical constructions Topic 7 Polygons Topic 8 Coordinate geometry Topic 9 Transformations Topic 10 Perimeter and area Topic 11 Total surface area Topic 12 Volume Topic 13 Algebraic variables Topic 14 Money Topic 15 Data processing and presentation Topic 16 Central tendency Topic 17 Scatter graphs Topic 18 Probability Topic 19 Time, distance and speed Topic 20 Formulae Topic 21 Equations Topic 22 Graphs Topic 23 Problem solving and research Answers for Assessments 1 to Assessment Assessment Assessment Answers for End-of-year Practice Examinations Maths in Action Form 2 TG.indd 3 6/29/10 3:53:21 PM

4 Introduction This Teacher s Guide forms part of a series and will help you to interpret and present the activities in the accompanying Student s Book. Introduction to the revised junior secondary school syllabi The revised junior secondary school syllabi have not been changed drastically. The revised syllabi aim to convey the ideals reflected in the Revised National Policy on Education and those of Vision The syllabi are based on the ten-year basic education philosophy and aim to equip students with knowledge and skills relevant to today s world. The syllabi encourage students to excel within their own capabilities and encourage teachers to use participatory teaching and diverse learning approaches. The aim of these approaches is to recognise individual talents, needs and learning styles. This inclusive approach urges teachers to accommodate all students regardless of their physical, intellectual, social, emotional, linguistic or other conditions. The syllabi also focus on the ongoing instruction of attitudes and values that need to be nurtured. Emerging issues help students understand and cope with challenges and developments that happen around them. The syllabi infuse emerging issues, such as environmental education, HIV/AIDS education, gender equality and the world of work. Assessment is a crucial part of the teaching and learning process, and should take students with special needs into account. Features of the Student s Book The Student s Book contains topics, each consisting of general objectives that comprise specific objectives. The first page of each topic lists the specific objectives covered in that topic and there is a list of specific objectives at the bottom of each page. Activities enable students to explore, experiment, research and discover. Exercises help students reinforce their work. Emerging issues link current social and economic issues to the students environment and make learning more practical. These emerging issues appear in boxes alongside the text. Extension work provides advanced work for students when they need it. Case studies and projects reinforce learning. Summaries help students reflect on their work. exercises test students understanding of their work. Assessments provide exam-style questions for extra practice. End-of-year Practice Examinations give students the opportunity to prepare for examinations. Features of the Teacher s Guide We have designed the Teacher s Guide so that it is easy to use and organised it into topics to match the Student s Book. Each topic includes the following features. 1 Syllabus checklist The course meets all the general and specific objectives. The syllabus checklist is a table at the beginning of each topic that summarises where and how we address the objectives. The table also lists the activities and exercises, along with Student s Book page numbers, in which we address the objectives. 2 Background information After the syllabus checklist, we provide additional information under the headings Rationale and Additional information. These sections include facts, tips, resources and suggested teaching methods. 3 Answers Each topic has an answer section that contains the answers to all the activities and exercises in the Student s Book. iv Maths in Action Form 2 TG.indd 4 6/29/10 3:53:21 PM

5 Teaching methodology Focus on student-centred, activity-based learning. We have designed the activities and exercises to give students a variety of opportunities in which to learn. The importance of revision At the beginning of each topic, revise what students already know about the topic. Use the oral activities to lead a discussion that tests prior knowledge, and create an interest in the work that follows. Before beginning a lesson, spend a few minutes revising the previous lesson. Use the revision section at the end of each topic to give students a chance to revise their work. This allows you to find out whether there are gaps in their knowledge, and where you may need to give additional help. The importance of assessment Assessment measures whether students have achieved the objectives of a teaching and learning process. It also allows teachers to monitor progress, provide feedback and guidance, and diagnose barriers to learning. Simple classroom observation, peer assessment, self-evaluation, tests and projects all play a role in assessment. Continuous assessment is the key principle of assessment. Teachers and students should reflect on the learning processes at appropriate times, and assess their own strengths and weaknesses. Ensure that you record these comments. At various times in each term, perform a formative assessment of relevant objectives. Give students feedback to help them improve their performance. Towards the end of each term, perform a summative assessment. This provides an overall picture of each student s progress during the term. The following is a list of important tips for teachers. 1 Build on what students already know. 2 Use materials that are meaningful, clear and interesting. 3 Allow students to master simple concepts first. Then apply these concepts to more complex ones. 4 Accurately assess students so that you can plan better. 5 Give students positive feedback to motivate them. Let Longman in action help students achieve success in Form 2. v Maths in Action Form 2 TG.indd 5 6/29/10 3:53:23 PM

6 TOPIC 1 Work with numbers Unit Topic General objective 2.1 Numbers and operations Numbers Acquire knowledge on approximation and estimation. Specific objective Identify significant figures in a number. Activity/ Exercise Activity 2 Activity 3 Exercise 1 Exercise 2 SB page number Acquire knowledge on sequences Round numbers to a given number of significant figures. Activity 1 Activity 3 Exercise 1 Exercise 2 Exercise 3 Case study Approximate computations that involve sums, differences, products, and quotients to a specified place value, decimal place or significant figure that shows a reasonable accuracy. Exercise 2 Case study Solve mathematical puzzles. Activity 6 Exercise Complete a sequence that consists of whole numbers, directed numbers, fractions, decimals and percentages. Activity 4 Exercise 4 Activity Rationale Approximation and estimation form part of the basic knowledge and skills that students need to understand and solve real-life problems. This topic also helps students to understand and appreciate patterns in nature, number sequences, relationships between numbers, and models. In addition, they will learn to describe the world around them. Investigating patterns in nature relates to the emerging issue of environmental education. Additional information Oral activity (SB p. 2) Let students work in pairs. Ask different pairs to share their views on the questions with the class. Guide students to revise all the work covered on approximation and estimation in Form 1. Write the names of three places on the map on the board. Ask students to measure the distances between the places in centimetres. They must then use the scale to estimate the actual distances between the places in kilometres. Point out that a small division on the scale represents 5 km. Discuss the advantages of approximations and estimations. Ensure that students realise that they allow us to understand and interpret situations. Approximations and estimations also allow us to communicate 2 Maths in Action Form 2 TG.indd 2 6/29/10 3:53:23 PM

7 Topic 1 Work with numbers information, compare measurements such as distance, length, height, mass, capacity and time and make decisions in cases where we do not need accurate measurements and answers. Approximation (SB p. 3) Ensure that students understand that approximation of a number or measurement gives a value that is close to, but not equal to the actual value. Students must also understand that we use rounding to approximate numbers and measurements. Discuss Example 1 on page 3 of the Student s Book. Use a tape measure and measure the lengths of objects in the classroom, for example the board, a desk, a table and a window. For each object, ask students to round the length to the nearest unit and to the nearest 10. Round to a given number of decimal places (SB p. 4) Introduce rounding to a given number of decimal places by explaining that we do not always have to work with a large number of decimal places. When we work with small units, the markings on measuring instruments are often larger than these small units. Examples of some everyday measuring instruments include rulers, measuring tapes, scales, measuring jugs and thermometers. Ask students to look at their rulers and compare the width of one marking with the width of one millimetre. They should understand that if we cannot measure extremely accurately using everyday measuring instruments, it is unnecessary to calculate answers to a large number of decimal places. Work through Examples 2 and 3 on page 4 of the Student s Book and ensure that students understand how to round a number to a given number of decimal places. Ask students to work in pairs to complete Activity 1. Round to a given number of significant figures (SB p. 5) Introduce significant figures as digits in a measurement that contribute to the accuracy of the number. Ask students to draw three lines that are 3.5 cm, 3.52 cm and cm long. Discuss the accuracy of each line length and the factors that affect the accuracy. Factors include: The thickness of the markings on the ruler How sharp the pencil is The difficulty of estimating tenths of a millimetre. Explain to students that we cannot accurately measure the length of a line to tenths of a millimetre and to thousandths of a millimetre. We need special measuring instruments to make such accurate measurements. The relationship between accuracy and significant figures is that the more accurate the measuring instrument, the more significant figures in the number. Introduce the rules to find the number of significant figures on page 5 of the Student s Book. Work through Example 4 on page 5 of the Student s Book on the board and ask students to work in pairs to complete Activity 2. Introduce rounding to a given number of significant figures and work through Example 5 on page 6 of the Student s Book. Ask students to work in pairs to complete Activity 3. Once students have completed Activity 3, let them move on to Exercise 1. Assess their work and repeat relevant sections if necessary. Estimation (SB p. 7) Ensure that students understand that we estimate distance, mass, capacity, time and the answers of calculations to compare and communicate information. Estimation also helps us to make quick decisions. Another advantage of estimation is that we can check our final calculated answer to the original estimation. Work through Examples 6 and 7 on page 7 of the Student s Book and ask students to do Exercise 2. Assess their work and help them if necessary. Multi-step computations (SB p. 8) Revise the order of operations by explaining the BODMAS rule. Use pen-and-paper methods to work through Example 8 on page 9 of the Student s Book. Ensure that students can interpret, read and write calculator key sequences. 3 Maths in Action Form 2 TG.indd 3 6/29/10 3:53:24 PM

8 Topic 1 Work with numbers The following are the pen-and-paper methods for the two examples: Example 8(1) Step 1 Brackets Step 2 Multiplication Round to two decimal places: The number of decimal places in the answer of equals the total number of decimal places in the two numbers. Example 8(2) Step 1 Brackets = 6.14 Step 2 Division = [ Multiply by a power of 10 to make the divisor a whole number.] Let students begin Exercise 3 in class and complete it for homework. Ask students to read through the case study on pages 9 and 10 of the Student s Book. Discuss the work done by the Youth Health Organisation and ask students to answer the questions in their books. Number sequences (SB p. 10) Ensure that students understand that a number sequence is an ordered set of numbers that follows a certain rule. Explain the difference between finite sequences and infinite sequences. Work through Examples 9 and 10 on page 11 of the Student s Book. Ensure that students can identify the rule for a number sequence and that they can use the rule to find missing numbers in a number sequence. Ask students to work in pairs and complete Activity 4. Check that they can find the rule for each number sequence and that they can use the rule to find the missing numbers in each sequence. Let students begin Exercise 4 in class and complete it for homework. Patterns in nature (SB p. 13) Find examples of patterns and number sequences in nature. Discuss the patterns and look at examples and pictures. Let students work in pairs to complete Activity 5. Mathematical puzzles (SB p. 13) Explain that it is fun to solve mathematical puzzles. Solving mathematical puzzles involves applying mathematical knowledge and skills. These include working with numbers and relationships between numbers, drawing shapes, completing tables and reasoning logically. Ask students to work in groups to complete Activity 6. Observe them and help them if necessary. Let students begin Exercise 5 in class and complete it for homework. 4 Maths in Action Form 2 TG.indd 4 6/29/10 3:53:24 PM

9 Topic 1 Work with numbers Answers Oral activity Discuss approximation and estimation (SB p. 2) Students own responses Activity 1 Find approximate values (SB p. 4) 1 Approximations: a) 3 b) 2 c) 2 d) 5 e) 6 f) 50 2 If the answer differs a lot from the approximation, it may be incorrect. 3 a) 2.67 b) 1.71 c) 2.22 d) 5.23 e) 6.43 f) Activity 2 Find the number of significant figures (SB p. 5) 2 a) 2 b) 4 c) 3 d) 6 e) 2 f) 4 g) 5 h) 2 i) 3 Activity 3 Round to a given number of significant figures (SB p. 6) Exercise 1 Practise approximating (SB p. 6) 1 a) 4 b) 2 c) 4 d) 3 e) 1 2 a) b) c) 73 d) e) 4.1 f) a) b) c) d) e) a) b) c) d) a) b) 4.85 c) d) e) 0.01 f) a) 5.8 b) c) d) a) b) c) d) Exercise 2 Estimate and calculate answers (SB p. 8) 1 a) Estimate: 300 Answer: b) Estimate: 400 Answer: c) Estimate: 20 Answer: d) Estimate: 130 Answer: a) Estimate: 200 Answer: b) Estimate: 10 Answer: 8.62 c) Estimate: 500 Answer: d) Estimate: 200 Answer: a) i) ii) 810 b) i) ii) 88 c) i) ii) d) i) 0.04 ii) Maths in Action Form 2 TG.indd 5 6/29/10 3:53:24 PM

10 Topic 1 Work with numbers Exercise 3 Calculate multi-step computations (SB p. 9) Case study Programme summary of the Youth Health Organisation (SB p. 9) 1 YOHO aims to reduce HIV/AIDS infections, teenage pregnancies and sexually transmitted infections. 2 Peer Education Programme, and Theatre and Arts Programme 3 Estimate: km P3/km = P6 000 Actual cost = km P2.84/km = P Estimate: 20 min P100 = P2 000 Actual cost = 23 min P112.17/min = P Activity 4 Describe number sequences (SB p. 11) 1 a) An ordered set of numbers means that the numbers are arranged from smallest to largest or from largest to smallest. b) Follows a certain rule means that there is a relationship between the numbers. We can use this relationship or rule to find more members of the number sequence. 2 a) to c) Students own responses 3 a) 31; 41; 52; 64 b) Total = = 232 Exercise 4 Work with number sequences (SB p. 12) 1 a) First member = 1; last member = 22; rule: add 3 b) First member = 1; last member = 128; rule: multiply by 2 c) First member = 18; last member = 81 2 ; rule: divide by 3 2 a) i) Add 3 ii) 3 1 ; 4; 4 3 ; b) i) Add 2% ii) 7%; 9%; 11%; 13% c) i) Add 0.7 ii) 5.1; 5.8; 6.5; a) 14; 17; 20; 23 b) 3 1 ; 4 1 ; 5 1 ; 6 1 c) 9.25; 11.25; 13.25; a) 144 b) c) a) 4; 5; 6; 7; 8; 9; 10 The sequence starts at 4, increases by 1 each time and ends at 10. b) 4; 11; 18; 25 The sequence starts at 4, increases by 7 each time and ends at 25. c) 4; 12; 20; 28 The numbers form a sequence. The sequence starts at 4, increases by 8 each time and ends at 28. d) 5; 13; 21; 29 6; 14; 22; 30 e) 1; 7; 13; 19; 25 The sequence starts at 1, decreases by 6 each time and ends at 25. f) Total first column = 58 Total second column = 62 g) Each number in the second column is 1 more than the corresponding number in the first column. h) 66; 70; 75; 80; 85 Activity 5 Investigate patterns in nature (SB p. 13) Students own responses 6 Maths in Action Form 2 TG.indd 6 6/29/10 3:53:25 PM

11 Topic 1 Work with numbers Activity 6 Investigate square numbers (SB p. 13) 2 In each case, the counters form a square: 1 1; 2 2; counters The members of the sequence are square numbers. 7 36; 49 8 We can arrange the counters in a square pattern for each member of the pattern. Exercise 5 Solve mathematical puzzles (SB p. 14) 1 a) 1; 3; 6; 10 b) 15 c) d) 28 counters e) a) b) A = 6; B = 4; C = 2 7 Maths in Action Form 2 TG.indd 7 6/29/10 3:53:27 PM

12 Topic 1 Work with numbers (SB p. 15) 1 a) b) c) a) 2.87 b) c) a) b) c) 33 d) a) 640 cm b) 640 cm 20 = 32 cm c) cm a) 9; 11; 13 b) 81; 243; 729 c) 0.01; 0.001; a) b) c) d) e) a) b) c) d) e) Maths in Action Form 2 TG.indd 8 6/29/10 3:53:28 PM

13 TOPIC 2 Squares, square roots, cubes and cube roots Unit Topic General objective 2.1 Numbers and operations Squares, square roots, cubes and cube roots Acquire knowledge on squares, square roots, cubes and cube roots. Specific objective Find squares, square roots, cubes and cube roots of positive whole numbers and fractions. Activity/ Exercise Activity 1 Exercise 1 Exercise 2 Exercise 3 Exercise 4 SB page number Use a calculator to find squares, square roots, cubes and cube roots of any number. Exercise 1 Exercise 3 Exercise 4 Exercise Indices Acquire knowledge on indices and apply it to solve problems Express a number as a product of its factors and as a product of its prime factors using indices. Exercise 5 22 Rationale This topic helps students to apply computational skills to solve everyday problems that involve squares, square roots, cubes and cube roots. Through doing the exercises, students will learn how to use their calculators and that their calculators have certain limitations. By the end of this topic, students should be able to classify numbers into three groups: square numbers, cube numbers and numbers that are neither squares nor cubes. Additional information Square numbers and cube numbers (SB p. 17) Introduce square numbers and cube numbers and give a few more examples of each. Ensure that students understand that we get a square number when we multiply a number by itself. For example, 25 is a square number because 5 5 = 25. Ask students to work individually to complete Activity 1. Observe them while they work and help them if necessary. Squares and square roots of whole numbers (SB p. 17) Relate the concept of a square root to the concept of a square. For example, in the equation 9 9 = 81 the square root of 81 is 9 and the square of 9 is 81. Explain that 81 has two square roots because 9 9 = 81 and ( 9) ( 9) = 81. Therefore, the square root of 81 is 9 or 9. Highlight the fact that the square root of a negative number does not exist because = and =. Discuss this point with the class and write examples on the board if necessary. Introduce the symbol for square root. Explain that 81 = 9 and 81 = 9. If there is no sign in front of the symbol, we assume a positive sign. Therefore, 4 = 2. Remind students that knowing their multiplication tables will help them to find square roots more easily. Work through Examples 1 and 2 on pages 17 and 18 of the Student s Book and do extra examples on the 9 Maths in Action Form 2 TG.indd 9 6/29/10 3:53:28 PM

14 Topic 2 Squares, square roots, cubes and cube roots board if necessary. Ensure that students can interpret, read and write calculator key sequences. Let students begin Exercise 1 in class and complete it for homework. Squares and square roots of common fractions (SB p. 18) Write a few examples of squares of common fractions on the board, for example 1, 4 and 4 9. Then write the 9 16 square roots of these fractions on the board: 1 4 = 1 2, 4 9 = 2 3 and 9 16 = 3. Remind students that when we 4 multiply two fractions, we multiply the numerators with each other and the denominators with each other. Work through Example 3 on page 18 of the Student s Book. Ensure that students understand how to find the square of a common fraction and the square root of a common fraction. Let students begin Exercise 2 in class and complete it for homework. Squares and square roots of decimals (SB p. 19) Write a few examples of squares of decimal fractions on the board, for example = 0.01 and = Explain that the number of decimal places in the answer equals the sum of the number of decimal places in the two numbers that you multiply. Write a few examples of square roots of decimal fractions on the board, for example 0.09 = 0.3, 0.36 = 0.6 and 0.49 = 0.7. Ensure that students understand how to determine the sign of a square root. Use these examples to demonstrate how to use a calculator to find the square roots of decimals. We use a calculator to find approximate values of decimals that are not perfect squares. Write a few examples on the board and demonstrate how to find approximate values and round them to a given number of decimal places. For example, 0.34 = 0.58, 0.9 = 0.95 and 7.82 = Ask students to work in pairs to discuss the text and Examples 4 and 5 on page 19 of the Student s Book. Follow this up with a class discussion. Ensure that students know how to use a calculator to find the square and square root of decimal numbers. Let students begin Exercise 3 in class and complete it for homework. Cubes and cube roots of whole numbers and decimals (SB p. 20) Relate the concept of a cube root to the concept of a cube. For example, in the equation = 8 the cube of 2 is 8 and the cube root of 8 is 2. Ensure that students understand that 8 has only one cube root because = 8 and ( 2) ( 2) ( 2) = 8. Therefore, the cube root of 8 can only be 2. 3 Introduce the symbol for cube root and write a few examples on the board. For example, 3 27 = 3, 3 64 = 4, = 10, = 0.1 and = 0.5. Remind students that knowing their multiplication tables will help them to find cube roots more easily. Ask students to work through the text and Examples 6, 7 and 8 on pages 20 and 21 of the Student s Book in pairs. Follow this up with a class discussion. Encourage students to give their input in discussing the examples. Ensure that students can use mental and pen-and-paper methods to find cubes and cube roots in simple cases. Students should also know how to use a calculator to find cubes and cube roots. Let students begin Exercise 4 in class and complete it for homework. Cubes and cube roots of common fractions (SB p. 22) Write a few examples of cubes of common fractions on the board, for example 1, 8 and 1. Let students identify the fractions that they need to multiply by twice to get these cubes. 3 Write examples of cube roots of common fractions on the board, for example 27 8 = 3 2, = 1 3 and = 2 5. Work through the text and Example 9 on page 22 of the Student s Book. Ensure that students can find the cube of a common fraction and the cube root of a common fraction. Let students begin Exercise 5 in class and complete it for homework. Assess their work and repeat parts of the topic if necessary. Answers Oral activity Discuss squares, square roots, cubes and cube roots (SB p. 16) Students own responses 10 Maths in Action Form 2 TG.indd 10 6/29/10 3:53:29 PM

15 Topic 2 Squares, square roots, cubes and cube roots Activity 1 Identify square numbers and cube numbers (SB p. 17) 1 Students own responses 3 a) 1; 4; 9; 16; 25; 36; 49; 64; 81; 100 b) 2; 4; 6; 8; 10; 12; 14; 16; 18 Yes, each term is 2 more than the previous term. c) 121; 144 Check: 11 2 = = a) 1; 8; 27; 64 b) 125; 216; 343 Exercise 1 Find squares and square roots of numbers (SB p. 18) 1 a) 1 b) 4 c) 9 d) 16 e) 36 2 a) 49 b) 64 c) 144 d) 100 e) a) 841 b) c) d) e) a) 1 b) 2 c) 3 d) 4 e) 5 5 a) 8 b) 10 c) 7 d) 9 e) 11 f) 5 6 a) 2 b) 5 c) 8 d) 10 e) 12 f) 7 Exercise 2 Find squares and square roots of common fractions (SB p. 19) 1 a) 25 4 b) 9 16 c) 9 25 d) 4 49 e) a) 9 64 b) 1 4 c) d) 1 25 e) 4 49 f) a) 2 3 b) 1 2 c) 3 4 d) 2 5 e) a) 1 2 b) 1 3 c) 4 7 d) 3 8 e) 5 9 f) 7 5 Exercise 3 Find squares and square roots of decimals (SB p. 20) 1 a) 0.64 b) 0.81 c) 1.96 d) 8.41 e) a) 2.43 b) 8.01 c) d) e) a) b) c) d) e) f) a) 3; 3.13 b) 4; 4.20 c) 5; 4.91 d) 7; 7.10 e) 8; a) 2.14 b) 3.21 c) 5.93 d) 8.90 e) f) Exercise 4 Find cubes and cube roots of whole numbers and decimals (SB p. 21) 1 a) 343 b) 729 c) d) e) a) 2.74 b) c) d) e) a) 2.20 b) 9.26 c) d) e) f) a) 27 = 3 3 3; cube root = 3 b) 64 = = 2 6 ; cube root = 2 2 = 4 c) 125 = 5 5 5; cube root = 5 d) 343 = 7 7 7; cube root = 7 e) 729 = = 3 6 ; cube root = 3 2 = 9 5 a) 1.2; 1.12 b) 1.7; 1.96 c) 3.7; 3.80 d) 4.8; 4.93 e) 5.6; 5.86 Exercise 5 Find cubes and cube roots of common fractions (SB p. 22) 1 a) 1 8 b) 1 27 c) 1 64 d) e) a) 8 27 b) c) d) e) a) 1 8 b) 1 27 c) d) e) f) a) 1 2 b) 1 3 c) 2 3 d) 3 4 e) a) 1 4 b) 2 5 c) 1 6 d) 4 3 e) 4 5 f) Maths in Action Form 2 TG.indd 11 6/29/10 3:53:30 PM

16 Topic 2 Squares, square roots, cubes and cube roots (SB p. 23) 1 a) 25 b) 2.25 c) 0.09 d) 0.06 e) 9 4 f) 27 2 a) b) 25 9 c) 2.07 d) e) f) a) b) c) d) e) f) a) b) c) d) e) f) a) b) 0.68 c) 6.47 d) 4.10 e) 5.18 f) a) cm b) cm 7 a) cm 3 b) cm 2 12 Maths in Action Form 2 TG.indd 12 6/29/10 3:53:30 PM

17 TOPIC 3 Indices Unit Topic General objective 2.1 Numbers and operations Indices Acquire knowledge on indices and apply it to solve problems. Specific objective Write a repeated multiplication as a power Express a number as a product of its factors and as a product of its prime factors using indices. Activity/ Exercise Exercise 1 Exercise 2 Exercise 3 SB page number Derive the laws of indices by investigation. Activity 1 Activity 2 Activity 3 Activity Solve problems that involve indices by applying the laws of integral indices. Exercise 4 Exercise Express a number in standard form. Exercise Solve problems that involve the practical use of standard form. Exercise Rationale In this topic, students will use the laws of indices in calculations that involve multiplication, division and powers of numbers in index form. They will also use critical thinking and problem-solving skills to solve real-life problems. In addition, they will explore and develop basic mathematical concepts for further study in mathematics. Additional information Index notation (SB p. 25) Discuss repeated multiplication expressions and write on the board. Explain that this is difficult to read, especially if the expression contains a large number of 3s. This is why we use index notation to write repeated multiplication expressions. Index notation is a short way of writing a repeated multiplication, for example = 3 5. Refer students to the text on page 25 of the Student s Book. Introduce the terms index notation, power, base and index. Work through Questions 1 and 2 of the first example on page 25 of the Student s Book and write more examples on the board if necessary. Discuss how to use a calculator to find the value of a power. Work through Question 3 of Example 1 on page 25 of the Student s Book and ensure that students can use a calculator to find the value of a power. Work through Example 2 on page 25 of the Student s Book. Let students work individually to complete Exercise 1. Observe students and help them if necessary. 13 Maths in Action Form 2 TG.indd 13 6/29/10 3:53:31 PM

18 Topic 3 Indices Factors and prime factors (SB p. 26) Introduce the terms factor and prime factor. Write a few examples of each on the board. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. The prime factors of 12 are 2 and 3, because they are factors of 12 and they are also prime numbers. Work through the text and Example 3 on page 26 of the Student s Book. Ensure that students can find the prime factors of a given number. Let students begin Exercise 2 in class and complete it for homework. Laws of indices (SB p. 27) Explain that we follow certain rules when we work with indices. These laws of indices help us simplify calculations such as , and (4 2 ) 3. In this section, students will learn about the laws of indices through investigation. Let students work in pairs to complete Activity 1. Observe them and help them if necessary. Ensure that students understand that when we multiply two powers that have the same base, we add the indices to simplify the expression. Write more examples on the board if necessary. Then let students work individually to complete Exercise 3. Let students work in pairs to complete Activity 2. Observe them and help them if necessary. Ensure that students understand that when we divide a power by another power that has the same base, we subtract the indices to simplify the expression. Write more examples on the board if necessary. Let students work in pairs to complete Activity 3. Observe them and help them if necessary. Ensure that students understand that when we raise a power to another power, we multiply the indices to simplify the expression. Write more examples on the board if necessary. Ask students to look at the summary of the laws of indices on page 29 of the Student s Book. It is the students responsibility to ensure that they understand the laws of indices and that they know how to apply them. Work through Example 4 on page 29 of the Student s Book. Ask students to complete Exercise 4. Assess their work and repeat aspects of the topic if necessary. The meaning of a 0 and a 1 (SB p. 30) Let students work in pairs to complete questions 1 and 2 of Activity 4. Observe their progress and ensure that they find that a 0 = 1 and a 1 = 1 a. They must be able to apply this knowledge in new situations, for example in Question 9 of Activity 4. Revise the terms reciprocal and additive inverse. For example, 1 is the reciprocal of 2. 2 Let students complete Activity 4 in pairs and tell them that they can ask for help if they need it. It is important to create opportunities for students to take responsibility for their own learning. Guide them in this and ask questions to ensure that they can work with negative indices. Work through Example 5 on page 31 of the Student s Book. Let students complete Exercise 5 in class. Observe them and repeat parts of the section if necessary. Standard form (SB p. 31) Introduce the term standard form and write a few examples on the board, for example and Explain that the first part is a number between 1 and 10, and the second part is a power of 10. Demonstrate how to convert numbers in standard form back to decimals, for example = = and = = Let students work through Example 6 on page 31 of the Student s Book in pairs. Observe them and help them if necessary. Explain that we can add, subtract, multiply and divide numbers in standard form. Work through Example 7 on page 32 of the Student s Book and ensure that they can perform the calculations. Let students begin Exercise 6 in class and complete it for homework. Assess their work, and support and guide them to correct their mistakes. Discuss the work covered in this topic. Ask questions and let students write examples on the board to show that they can apply knowledge and skills related to indices in new situations. 14 Maths in Action Form 2 TG.indd 14 6/29/10 3:53:31 PM

19 Topic 3 Indices Answers Oral activity Discuss indices (SB p. 24) Group discussion Exercise 1 Work with index notation and expanded form (SB p. 25) 1 a) 4 b) 3 2 a) 2 4 = 16 b) 8 3 = a) 343 b) a) b) 32 c) d) a) 5 8 b) Exercise 2 Find factors and prime factors (SB p. 26) 1 a) and b) A composite number is a number that has more than two factors. For example, 6 is a composite number because it has 1, 2, 3 and 6 as factors. A prime number is a number that has only two different factors. For example, 3 is a prime number because it has only 1 and 3 as factors. 2 a) 1; 5 b) 1; 2; 3; 6 c) 1; 2; 4; 8 d) 1; 2; 4; 5; 10; 20 3 a) 1; 2; 3; 5; 6; 10; 15; 30 b) 2; 3; 5 4 a) 2; 3 b) 36 = a) 2; 3 b) 54 = Activity 1 Investigate the product of two powers with the same base (SB p. 27) 2 Expression Factors in expanded notation Answer in Result index notation (3) (3 3) = = (3 3) (3 3 3) = = (3 3 3) ( ) = = ( ) ( ) = = ( ) ( ) = = m 3 n (3 3 m factors) (3 3 n factors) 3 m n 3 m 3 n = 3 m n 3 To multiply two powers that have the same base, add the indices. 4 add 5 3; 5; 8 Exercise 3 Apply the first law of indices (SB p. 27) 1 a) 5 8 b) 7 2 c) a 3 2 a) 2 4 b) c) a) b) c) a) b) c) Maths in Action Form 2 TG.indd 15 6/29/10 3:53:32 PM

20 Topic 3 Indices Activity 2 Investigate the quotient of two powers with the same base (SB p. 28) 2 Expression Factors in expanded notation Cancel factors to simplify Answer in index notation Result = = = = 3 4 3m 3 n m factors n m factors factors n factors 3m n 3 m 3 n = 3 m n 3 To divide two powers that have the same base, subtract the indices. 4 subtract Activity 3 Investigate a power raised to a power (SB p. 28) 2 Expression Factors in expanded notation Answer in index notation Result (3 2 ) (3 2 ) 2 = = 3 4 (3 2 ) (3 2 ) 3 = = 3 6 (3 2 ) (3 2 ) 5 = = 3 10 (3 m ) n 3 m 3 m n factors 3 3 (m n) factors 3 m n (3 m ) n = 3 m n = 3 mn 3 To raise a power to another power, multiply the indices. 4 multiply Exercise 4 Apply the laws of indices (SB p. 29) 1 a) 2 7 b) 4 5 c) 3 5 d) 5 6 e) 7 6 f) 6 4 g) 8 17 h) 2 14 i) a) 3 3 b) 2 3 c) 6 2 d) 4 3 e) 5 3 f) 1 5 g) 8 1 h) 7 1 i) a) 5 10 b) 3 8 c) 2 6 d) 7 8 e) 4 15 f) 6 2 g) 1 30 h) 8 10 i) a) 512 b) 512 c) d) 81 e) f) 16 g) 1 h) i) 243 Activity 4 Investigate zero indices and negative indices (SB p. 30) 2 Expression Factors in expanded notation Cancel factors to simplify Answer in index notation Result = = = = = = = = = = = = = = = Value 16 Maths in Action Form 2 TG.indd 16 6/29/10 3:53:34 PM

21 Topic 3 Indices = = = = = 1 Therefore, 3 0 = 1. 4 The value of a number raised to the power of 0 equals 1, for example 5 0 = = = 2 0 = = = = = 1 9 Therefore, 3 2 = The value of 5 3 = ( 1 5 ) 3 = sign 9 a) ( 1 4 ) 2 = 4 2 b) 7 3 = ( 1 7 ) 3 c) = 5 4 d) 3 2 = e) ( 1 2 ) 4 = 2 4 f) 3 7 = ( 1 3 ) 7 g) ( 3 6 ) 4 = ( 6 3 ) 4 h) 8 2 = ( 1 8 ) 2 Exercise 5 Use zero indices and negative indices (SB p. 31) 1 a) = 5 0 b) = 1 c) 8 0 = 1 d) = 2 0 = 1 2 a) 1 b) 3 c) 3 d) a) ( 1 5 ) 2 b) 7 3 c) ( 1 3 ) 2 d) ( 10 1 ) 3 4 a) 3 1 b) 4 4 c) 2 1 d) a) 2 3 = ( 1 2 ) 3 b) = 7 2 c) 3 3 = ( 1 3 ) 3 d) = 1 0 = 1 Exercise 6 Work with numbers in standard form (SB p. 32) 1 a) b) c) d) e) f) a) b) c) d) e) f) a) b) c) d) e) f) a) b) c) d) e) f) a) b) c) d) amperes 7 Time = = 79 s (SB p. 33) a) 24 = b) 18 = a) 3 2 b) 5 4 c) 2 12 d) a) 125 b) 0.01 c) 4 d) 1 5 a) b) a) b) c) a) b) c) a) 2.71 b) 0.67 c) ( ) = N/m m 11 a) i) kg ii) kg iii) kg b) kg = km 17 Maths in Action Form 2 TG.indd 17 6/29/10 3:53:35 PM

22 TOPIC 4 Matrices Unit Topic General objective 2.1 Numbers and operations Matrices Acquire knowledge on matrices. Specific objective Represent information in matrix form Determine the order of matrices. Activity/ Exercise Exercise 1 Exercise 1 SB page number Add matrices. Exercise 2 Exercise Subtract matrices. Exercise 2 Exercise Multiply a matrix by a scalar. Exercise 3 Exercise Multiply a matrix by another matrix up to a 2 2 matrix. Exercise Rationale This topic will increase students ability to use mathematics to solve real-life problems. Learning about matrices and their applications in problem solving will help students grow intellectually. Their capacity to think and reason logically and critically will increase. Additional information Introduce matrices as rectangular arrangements of numbers that we use to solve problems. Ask students to look at the examples of matrices on page 35 of the Student s Book. Then, write a few more examples of matrices on the board. Ensure that students can distinguish between rows and columns, and that they know how to determine the order of a matrix. In this topic, students will learn how to represent information in matrix form. They will also learn how to perform basic operations addition, subtraction and multiplication on matrices. Work through Example 1 on page 35 of the Student s Book. Let students work in pairs to complete Exercise 1. Add and subtract matrices (SB p. 36) Explain that we can add and subtract matrices that have the same number of rows and columns. Demonstrate how to add two matrices. Work through Example 2 on page 36 of the Student s Book and ensure that students can add and subtract matrices. Let students work in pairs to complete Exercise 2. Multiply a matrix by a scalar (SB p. 37) Explain that in this topic, a scalar is an integer. A scalar in front of a matrix means that we must multiply the matrix by the scalar. When we multiply a matrix by a scalar, we multiply each element of the matrix by the scalar. 18 Maths in Action Form 2 TG.indd 18 6/29/10 3:53:35 PM

23 Topic 4 Matrices Work through Examples 3 and 4 on page 37 of the Student s Book and ensure that they can multiply a matrix by a scalar. Demonstrate one more example on the board. Let students work individually to complete Exercise 3 and let them complete Exercise 4 for homework. Multiply a matrix by another matrix (SB p. 38) Explain that we can multiply two matrices if the number of rows in the first matrix equals the number of columns in the second matrix. Write Example 5 on page 38 of the Student s Book on the board. Demonstrate that when we multiply two matrices, we start in the upper left-hand corner of the first matrix. We multiply each element in the first row of the matrix by each element in the first column of the second matrix, and then add the products. Repeat multiplying the elements of the first row with the remaining columns of the second matrix. Write Example 6 on page 38 of the Student s Book on the board. Show how to perform multi-step computations that involve multiplying two matrices. Work through the solution step by step. Ask students to look at these examples again and to ensure that they can multiply two matrices. Let students work individually to complete Exercise 5. Answers Oral activity Discuss matrices (SB p. 34) Group discussion Exercise 1 Work with matrices (SB p. 35) 1 a) 2 b) 3 c) 6 2 a) 2 2 b) 3 2 c) Individual responses 4 a) 3 b) 2 c) 3 d) 2 Exercise 2 Add and subtract matrices (SB p. 36) 1 a) a) b) b) c) c) Exercise 3 Multiply a matrix by a scalar (SB p. 37) Exercise 4 Perform multi-step computations (SB p. 37) Maths in Action Form 2 TG.indd 19 6/29/10 3:53:36 PM

24 Topic 4 Matrices Exercise 5 Multiply a matrix by another matrix (SB p. 38) = = = = = = = = (SB p. 39) 1 a) 2 b) 3 c) a) b) = m 2 = = = = Maths in Action Form 2 TG.indd 20 6/29/10 3:53:37 PM

25 TOPIC 5 Angle properties Unit Topic General objective 2.2 Geometry Angle properties Understand angle properties of triangles and quadrilaterals. Specific objective Calculate unknown angles of triangles using properties of triangles that include the sum of the interior angles of a triangle, base angles of an isosceles triangle, angles in an equilateral triangle and the sum of the exterior angles of a triangle. Activity/ Exercise Activity 1 Exercise 1 Exercise 2 Activity 2 Activity 3 Extension SB page number Calculate unknown angles of quadrilaterals using angle properties of squares, rectangles, parallelograms, rhombuses, kites and trapeziums. Activity 4 Exercise 3 Activity 5 Exercise Rationale In this topic, students generalise the angle properties of triangles and quadrilaterals. This knowledge forms the basis of geometry and it is important that students are comfortable with this work. Students need to understand 2D shapes so that they can understand 3D objects. We live in a 3D world, but students often struggle with 3D concepts. Prepare students for later topics by ensuring that they understand all the concepts in this topic. Additional information In geometry, we give reasons for statements where necessary. You may find that students can see the solution, but they do not know what the reason is. Students are used to simply finding an answer and not thinking about how they reasoned. Reasoning is an important part of geometry. Encourage students to give good geometrical reasons for their answers. Answers Oral activity Discuss two-dimensional shapes in a photograph (SB p. 40) 1 Zimbabwe and Zambia 2 Mainly triangles; also rectangles and a curve (possibly an arc of a circle) 3 Triangles 4 Triangles are very strong structures if they are designed and constructed accurately. Activity 1 Discuss what you remember about triangles (SB p. 41) 1 Right-angled triangle 2 a) Scalene triangle b) The angles of a scalene triangle are all different sizes. 3 a) Isosceles triangle b) The two base angles of an isosceles triangle are equal. 21 Maths in Action Form 2 TG.indd 21 6/29/10 3:53:38 PM

26 Topic 5 Angle properties 4 a) Equilateral triangle b) All three angles of an equilateral triangle are equal. Each angle is a) i) Yes ii) If the two angles that are not right angles are different sizes, the sides opposite them will be different lengths. b) i) Yes ii) If the two angles that are not right angles are the same size, the sides opposite them will be the same length. c) i) No ii) If all three sides are the same length, the angles will all be the same size. This means that the triangle will not have a right angle. Exercise 1 Investigate the properties of triangles (SB p. 42) 2 z = x y 3 e = f 4 p = q = r = 60 Exercise 2 Calculate unknown angles of triangles (SB p. 43) 1 a) 2x 3x x = 180 [Sum of angles of a triangle = 180.] \ x = 30 B = 3 30 = 90 \ ABC is a right-angled scalene triangle. b) x x 5 x 5 = 180 [Sum of angles of a triangle = 180.] \ x = 60 D = 60, E = 65 and F = 55 \ DEF is an acute-angled scalene triangle. c) x = 60 2 = 120 [Exterior angle of an equilateral triangle.] \ JKL is an equilateral triangle. d) R = P = x [Isosceles PQR; QP = QR] \ 2x = 130 [Exterior angle of a triangle.] \ x = 65 \ PQR is an acute-angled isosceles triangle. e) 2x = [Angles on a straight line = 180.] \ x = 55 U = [Sum of angles of a triangle = 180.] \ U = 55 \ UVW is an acute-angled isosceles triangle. f) M = x [Isosceles KLM; KL = KM] 2x = [Sum of angles of a triangle = 180.] \ x = 60 \ KLM is an equilateral triangle. 2 a) According to the markers, PQR is equilateral, but the angle sizes are all different. b) The two equal angles are not opposite the two equal sides. c) The sum of the angles is 240, not 180. d) B C = 190. This is impossible, because the sum of two angles of a triangle must be less than 180. e) X Y = 120, but exterior Z = 110. f) Exterior W should be = 120. Activity 2 Investigate different triangles (SB p. 44) Let students discover the basic rule that applies to any triangle: in any triangle, the longest side is always opposite the largest angle, the shortest side is always opposite the smallest angle and the remaining side is 22 Maths in Action Form 2 TG.indd 22 6/29/10 3:53:38 PM