Mathematics Grade 12 - INTEC College

1 *I10050441-E1* I10050441-E1 Mathematics Grade 12 Numbers and number Relationships I10050441 Edition 1 _____ Numbers and number Relationships / ICG Mathematics Grade 12 Numbers and number Relationships CONTENTS PAGE How to work through this study unit 1 Learning Outcomes and Assessment Standards 3 Introduction to this study unit 5 Lesson 1: Indices (revision) 7 Lesson 2: Solving exponential equations (revision) 29 Lesson 3: Surds (revision) 46 Lesson 4: Logarithms: Basic principles and applications 60 Lesson 5: Arithmetic sequences and series 75 Lesson 6: Geometric sequences and series 104 Lesson 7: Sigma notation 142 Lesson 8: Compound increase and decrease 163 Lesson 9: Simple and compound interest 189 Conclusion 203 References 204 Please read the Addendum at the end of this study unit before you start Lesson 1.

2 PRINTING HISTORY 1st Edition January 2010 INTERNATIONAL COLLEGES GROUP (ICG) PTY. LTD. 2010 130 Strand Street, Cape Town, 8001, South Africa _____ This document contains proprietary information that is protected by copyright. All rights are reserved. No part of this document may be photocopied, reproduced, electronically stored or transmitted, or translated without the written permission of ICG. _____ Numbers and number Relationships / ICG / (ii) Written by: Allen Taylor _____ Numbers and number Relationships / ICG / Page 1 HOW TO WORK THROUGH THIS STUDY UNIT Welcome to Mathematics Grade 12: Numbers and number Relationships.

3 We have designed the lessons in this unit in such a way that each builds on the knowledge gained from the previous lessons. Therefore, it's essential that you ensure that you have a good understanding of each lesson. It's also important that you work through the lessons in the order in which we present them in the unit. The icons used in this study unit Read the descriptions of icons below. Look out for these icons as you work through the study unit. They will show you at a glance where you need to work through activities, definitions, self-assessment questions, and so on. Icon Description Learning Outcomes: This signals the Learning Outcomes of the unit.

4 You must be able to show competence in the Outcomes after you have worked through the study unit. Competence means that you must be able to demonstrate that you can meet an Outcome with skill and knowledge. Definition: This signals an important definition that you should understand and remember. Mathematical formula: This is a formula that you will use in calculations. It is important that you know what values to substitute into the formula. Important statement: This signals an important point that you must grasp before you continue with the rest of the lesson. It could also signal an interesting snippet of information. Activity: This signals an activity that you should complete as part of your learning.

5 We have interspersed activities throughout your lessons. Self-assessment questions: This signals the questions that will help you to analyse your understanding of the theory that was covered in the lesson. We conclude each lesson with a set of self-assessment questions. Answers to self-assessment questions: This signals the suggested answers to the self-assessment questions. Please do not look at the answers before you have tried to answer the questions yourself. Competence checklist: This signals a checklist to help you discover whether you can meet all the Assessment Standards in the lesson. _____ Numbers and number Relationships / ICG / Page 2 The best way to study To ensure that you get the full benefit of this study unit, we recommend that you do the following: Carefully read the next section.

6 It provides you with the Learning Outcomes and Assessment Standards for each of the lessons in the study unit. Carefully and diligently work through each lesson, keeping in mind the Outcomes that you have to achieve. Ensure that you complete all the activities in the lessons. Ensure that you answer all the self-assessment questions in the lessons. Compare your answers to the answers that we provide in this study unit. If you encounter any words that you do not understand, make a list of them, and then either look them up in a dictionary or ask your tutor for their meanings. _____ Numbers and number Relationships / ICG / Page 3 LEARNING OUTCOMES AND ASSESSMENT STANDARDS Numbers and number Relationships has been developed according to the National Curriculum Statement subject guidelines for Mathematics Grade 12.

7 We also included the National Curriculum Statement subject guidelines for Mathematics Grade 11 as revision as this will also be assessed in the final examination. Read through the lists below. Your aim is to complete the Learning Outcome and Assessment Standards for this unit successfully. In this unit, you will complete one Learning Outcome and six Assessment Standards for Grade 11 (Revision) and five Assessment Standards for Grade 12. You need to have this information because you must keep checking your learning goals. Learning Outcomes and Assessment Standards After you have worked through the content covered in this study unit, you should be able to do all the tasks we have listed below: Learning Outcome 1: Numbers and number Relationships When solving problems, the learner is able to recognise, describe, represent and work confidently with numbers and their relationships to estimate, calculate and check solutions.

8 Assessment Standards ( Grade 11 Revision) Understand that not all numbers are real. (This requires the recognition, but not the study of non-real numbers) (a) Simplify expressions using the laws of exponents for rational exponents. (b) Add, subtract, multiply and divide simple surds. (c) Demonstrate an understanding of error margins. Investigate number patterns (including, but not limited to those where there is a constant second difference between consecutive terms in a number pattern, and the general term is therefore quadratic) and hence: (a) make conjectures and generalisations (b) provide explanations and justifications, and attempt to prove conjectures.

9 Use simple and compound decay formulae ()(1niPA = and niPA)(1 =) to solve problems (including, straight line depreciation and depreciation on a reducing balance). Demonstrate an understanding of different periods of compounding growth and decay (including, effective compounding growth and decay and including effective and nominal interest rates). Solve non-routine, unseen problems. _____ Numbers and number Relationships / ICG / Page 4 Assessment Standards ( Grade 12) Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real-life problems (for example, growth and decay). (a) Identify and solve problems involving number patterns (including, but not limited to) arithmetic and geometric sequences and series.

10 (b) Correctly interpret sigma notation. (c) Prove and correctly select the formula for, and calculate the sum of series, including: ==nin11 =+=ninni121)( [] = += +nidnandia11)(221)( 1;11)(11 = = rrrar annii = > < = 1111;1iirrara (d) Correctly interpret recursive formulae (for example, 11 ++=nnnTTT). (a) Calculate the value of n in the formula niPA)(1 =. (b) Apply knowledge of geometric series to solving annuity, bond repayment and sinking fund problems, with or without the use of the formulae: ()iixFn1)(1 += and ()iixPn + =)(11 Critically analyse investment and loan options, and make informed decisions as to the best option(s), including, pyramid and micro-lenders schemes.