MTG Math Gr 12 Web.pdf

Grade Mat h e m a t i c s Study Guide 12 © Department of Basic Education 2015 This content may not be sold or used for commercial purposes. Curriculum and Assessment Policy Statement (CAPS) Grade 12 Mind the Gap study guide for Mathematics ISBN 978-1-4315-1935-4 Thi...

Grade Mat h e m a t i c s Study Guide 12 © Department of Basic Education 2015 This content may not be sold or used for commercial purposes. Curriculum and Assessment Policy Statement (CAPS) Grade 12 Mind the Gap study guide for Mathematics ISBN 978-1-4315-1935-4 This publication has a Creative Commons Attribution NonCommercial Sharealike license. You can use, modify, upload, download, and share content, but you must acknowledge the Department of Basic Education, the authors and contributors. If you make any changes to the content you must send the changes to the Department of Basic Education. This content may not be sold or used for commercial purposes. For more information about the terms of the license please see: http://creativecommons.org/licenses/by-nc-sa/3.0/. Copyright © Department of Basic Education 2015 222 Struben Street, Pretoria, South Africa Contact person: Dr Patricia Watson Email: [email protected] Tel: (012) 357 4502 http://www.education.gov.za Call Centre: 0800202933 The first edition, published in 2012, for the Revised National Curriculum Statement (RNCS) Grade 12 Mind the Gap study guides for Accounting, Economics, Geography and Life Sciences; the second edition, published in 2014 , aligned these titles to the Curriculum and Assessment Policy Statement (CAPS) and added more titles to the series in 2015, including the CAPS Grade 12 Mind the Gap study guide for Mathematics. ISBN 78-1-4315-1935-4 Mind the Gap team Series managing editor: Dr Patricia Watson Production co-ordinators: Lisa Treffry-Goatley and Radha Pillay Production assistants: Nomathamsanqa Hlatshwayo and Motshabi Mondlane Authors: Lynn Bowie, Ronald Peter Jacobs, Sue Jobson, Terrence Mongameli Mbusi, Sello Gert Motsoane, Nonhlanhla Rachel Mthembu, Ntshengedzeni Steven Muthige, Mpho Francis Phatlane, Josephina Mamaroke Phatlane, Peter Ketshepile Raadt, Percy Stephen Tebeila, Anita van Heerden and Carol Wyeth. Expert readers: Prof Bruce Watson, Leonard Mudau and Karen van Niekerk Proofreaders: John Ostrowick and Angela Thomas Designers: Sonja McGonigle and Erika van Rooyen Illustrators: Michele Dean, Vusi Malindi, Khosi Pholosa, John Ostrowick, Kenneth Tekane Cover illustration: Alastair Findlay Afrikaans Translation: Marga Vos Onsite writers’ workshop IT support: Wayne Cussons Special thanks to Winning Teams, CEO Denzil Hollis, for the organisation’s subject expertise and workshop support Winning Teams board game facilitators: Mantse Khoza and Sue Jobson  © Department of Basic Education 2015 Ministerial foreword The Department of Basic Education (DBE) has pleasure in releasing the second edition of the Mind the Gap study guides for Grade 12 learners. These study guides continue the innovative and committed attempt by the DBE to improve the academic performance of Grade 12 candidates in the National Senior Certificate (NSC) examination. The study guides have been written by teams of exerts comprising teachers, examiners, moderators, subject advisors and coordinators. Research, which began in 2012, has shown that the Mind the Gap series has, without doubt, had a positive impact on grades. It is my fervent wish that the Mind the Gap study guides take us all closer to ensuring that no learner is left behind, especially as we celebrate 20 years of democracy. Matsie Angelina Motshekga, MP The second edition of Mind the Gap is aligned to the 2014 Curriculum and Minister of Basic Education Assessment Policy Statement (CAPS). This means that the writers have considered the National Policy pertaining to the programme, promotion requirements and protocols for assessment of the National Curriculum Statement for Grade 12 in 2014. The CAPS aligned Mind the Gap study guides take their brief in part from the 2013 National Diagnostic report on learner performance and draw on the Grade 12 Examination Guidelines. Each of the Mind the Gap study guides defines key terminology and offers simple explanations and examples of the types of questions learners can expect to be asked in an exam. Marking memoranda are included to assist learners to build their understanding. Learners are also referred to specific questions from past national exam papers and examination memos that are available on the Department’s website – www.education.gov.za. The CAPS editions include Accounting, Economics, Geography, Life Sciences, Mathematics, Mathematical Literacy and Physical Sciences Part 1: Physics and Part 2: Chemistry. The series is produced in both English and Afrikaans. There are also nine English First Additional Language (EFAL) study guides. These include EFAL Paper 1 (Language in Context); EFAL Paper 3 (Writing) and a guide for each of the Grade 12 prescribed literature set works included in Paper 2. These are Short Stories, Poetry, To Kill a Mockingbird, A Grain of Wheat, Lord of the Flies, Nothing but the Truth and Romeo and Juliet. (Please remember when preparing for EFAL Paper 2 that you need only study the set works you did in your EFAL class at school.) The study guides have been designed to assist those learners who have been underperforming due to a lack of exposure to the content requirements of the curriculum and aim to mind-the-gap between failing and passing, by bridging the gap in learners’ understanding of commonly tested concepts, thus helping candidates to pass. All that is now required is for our Grade 12 learners to put in the hours required to prepare for the examinations. Learners, make us proud – study hard. We wish each and every one of you good luck for your Grade 12 examinations. __________________________________ Matsie Angelina Motshekga, MP Minister of Basic Education 2015 © Department of Basic Education 2015  © Department of Basic Education 2015 Table of contents Dear Grade 12 learner.................................................................ix How to use this study guide.........................................................xi Top 10 study tips..........................................................................xii Mnemonics...................................................................................xiii Mind maps.................................................................................... xiv On the day of the exam............................................................... xv Question words to help you answer questions........................ xvi Vocabulary................................................................................... xvii General terms............................................................................. xvii Technical terms........................................................................... xix The maths you need................................................................ xxviii Unit 1: Exponents and surds........................................................ 1 1.1 The number system...................................................................... 1 1.2 Working with irrational numbers................................................. 3 1.3 Exponents..................................................................................... 6 1.4 Exponential equations................................................................ 12 1.5 Equations with rational exponents............................................ 14 1.6 Exam type examples................................................................... 17 Unit 2: Algebra.............................................................................19 2.1 Algebraic expressions................................................................. 19 2.2 Addition and subtraction............................................................ 19 2.3 Multiplication and division......................................................... 20 2.4 Factorising................................................................................... 21 2.5 Notes on factorising a trinomial................................................ 22 2.6 Quadratic equations................................................................... 24 2.7 Quadratic inequalities................................................................ 30 2.8 Simultaneous equations............................................................ 34 2.9 The nature of the roots............................................................... 37 Unit 3: Number patterns, sequences and series.................... 42 3.1 Number patterns........................................................................ 42 3.2 Arithmetic sequences................................................................. 43 3.3 Quadratic sequences................................................................. 45 3.4 Geometric sequences................................................................ 48 3.5 Arithmetic and geometric series................................................ 50 © Department of Basic Education 2015 CONTENTS v Mind the Gap Mathematics © Department of Basic Education 2015 Unit 4: Funtions............................................................................60 4.1 What is a function?..................................................................... 60 4.2 Function notation........................................................................ 62 4.3 The basic functions, formulas and graphs............................... 63 4.4 Inverse functions........................................................................ 81 4.5 The logarithmic function............................................................ 84 Unit 5: Trig functions...................................................................88 5.1 Graphs of trigonometric functions............................................ 88 5.2 The effect of a on the shape of the graph: change in amplitude.................................................................................... 91 5.3 The effect of q on the shape of the graph: vertical shift.......... 93 5.4 The effect of b on the shape of the graph: change in period.......................................................................................... 94 5.5 The effect of p on the shape of the graph: horizontal shift..... 95 Unit 6: Finance growth and decay...........................................101 6.1 Revision: Simple and compound interest............................... 101 6.2 Calculating the value of P, i and n...........................................104 6.3 Simple and compound decay formulae.................................. 107 6.4 Nominal and effective interest rates.......................................109 6.5 Investments with time and interest rate changes..................111 6.6 Annuities...................................................................................113 Unit 7: Calculus..........................................................................123 7.1 Average gradient.......................................................................123 7.2 Average rate of change............................................................125 7.3 The derivative of a function at a point....................................126 7.4 Uses of the derivative............................................................... 131 7.5 Drawing the graph of a cubic polynomial................................132 Unit 8: Probability......................................................................145 8.1 Revision.....................................................................................145 8.2 Theoretical probability and relative frequency.......................146 8.3 Venn diagrams.......................................................................... 147 8.4 Mutually exclusive events........................................................149 8.5 Complementary events............................................................150 8.6 Events which are not mutually exclusive................................152 8.7 Summary of symbols and sets used in probability................154 8.8 Tree diagrams and contingency tables...................................158 8.9 Contingency tables................................................................... 161 8.10 Counting principles...................................................................164 8.11 Use of counting principles in probability................................. 170 vi CONTENTS  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 Unit 9: Analytical Geometry.....................................................172 9.1 Revise: Analytical Geometry.................................................... 172 9.2 The equation of a line.............................................................. 177 9.3 The inclination of a line............................................................ 179 9.4 Circles in analytical geometry..................................................184 Unit 10: Trigonometry...............................................................191 10.1 Revise: Trig ratios...................................................................... 191 10.2 Trig ratios in all the quadrants of the Cartesian plane..........194 10.3 Solving triangles with trig.........................................................196 10.4 Using a calculator to find trig ratios......................................... 197 10.5 The trig ratios of special angles...............................................198 10.6 Using reduction formulae.........................................................201 10.7 Trigonometric identities............................................................205 10.8 More trig identities................................................................... 207 10.9 Solving trigonometric equations..............................................209 10.10 More solving trig equations using identities........................... 213 10.11 Compound and double angle identities.................................. 215 10.12 Determining x for which the identity is undefined.................220 Unit 11: Trigonometry: Sine, cosine and area rules.............222 11.1 Right-angled triangles..............................................................222 11.2 Area rule.................................................................................... 224 11.3 Sine rule....................................................................................226 11.4 Cosine rule................................................................................228 11.5 Problems in two and three dimensions..................................230 Unit 12: Euclidean Geometry...................................................235 12.1 Revise: Proportion and area of trianges.................................235 12.2 Proportion theorems................................................................ 237 12.3 Similar polygons....................................................................... 240 Unit 13: Statistics......................................................................248 13.1 Bar graphs and frequency tables............................................ 249 13.2 Measures of central tendency.................................................250 13.3 Measures of dispersion (or spread)........................................254 13.4 Five number summary and box and whisker plot..................256 13.5 Histograms and frequency polygons.......................................260 13.6 Cumulative frequency tables and graphs (ogives).................263 13.7 Variance and standard deviation.............................................267 13.8 Bivariate data and the scatter plot (scatter graph)................ 271 13.9 The linear regression line (or the least squares regression line)......................................................................... 274 © Department of Basic Education 2015 CONTENTS vii Mind the Gap Mathematics © Department of Basic Education 2015  © Department of Basic Education 2015 Dear Grade 12 learner We are confident that this This Mind the Gap study guide helps you to prepare for the end-of-year Mind the Gap study CAPS Grade 12 exam. guide can help you to prepare well so that you The study guide does NOT cover the entire curriculum, but it does focus on pass the end-of-year core content of each knowledge area and points out where you can earn exams. easy marks. You must work your way through this study guide to improve your understanding, identify your areas of weakness and correct your own mistakes. To ensure a good pass, you should also cover the remaining sections of the curriculum using other textbooks and your class notes. Overview of the Grade 12 exam The following topics make up each of the TWO exam papers that you write at the end of the year: Paper Topics Duration Total Date Marking Patterns and sequences 3 hours 150 October/ Externally Finance, growth and decay November Functions and graphs 1 Algebra, equations and inequalities Differential Calculus Probability Euclidean Geometry 3 hours 150 October/ Externally Analytical Geometry November 2 Statistics and regression Trigonometry © Department of Basic Education 2015 introduct ion ix Mind the Gap Mathematics © Department of Basic Education 2015 Approximate Cognitive level Description of skills to be demonstrated Weighting number of marks in a 150-mark paper Recall Knowledge Identification of correct formula on the information 20% 30 marks sheet (no changing of the subject) Use of mathematical facts Appropriate use of mathematical vocabulary Algorithms Estimation and appropriate rounding of numbers Proofs of prescribed theorems and derivation of Routine formulae 35% 52–53 marks Procedures Perform well-known procedures Simple applications and calculations which might involve few steps Derivation from given information may be involved Identification and use (after changing the subject) of correct formula Generally similar to those encountered in class Problems involve complex calculations and/or Complex higher order reasoning 30% 45 marks Procedures There is often not an obvious route to the solution Problems need not be based on a real world context Could involve making significant connections between different representations Require conceptual understanding Learners are expected to solve problems by integrating different topics. Non-routine problems (which are not necessarily Problem Solving difficult) 15% 22–23 marks Problems are mainly unfamiliar Higher order reasoning and processes are involved Might require the ability to break the problem down into its constituent parts Interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts. x introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 How to use this study guide Look out for This study guide covers selected parts of the different topics of these icons in the the CAPS Grade 12 curriculum in the order they are usually taught study guide. during the year. The selected parts of each topic are presented in the following way: PAY SPECIAL ATTENTION An explanation of terms and concepts; NB NB PAY SPECIAL ATTENTION NB Worked examples to explain and demonstrate; NB Activities with questions for you to answer; and PAY SPECIAL ATTENTION Answers for you to use to check your own NB work. NB HINT! HINT! hint hint Hints to help you remember a concept NB PAY SPECIAL ATTENTION Pay special attention NBhint HINT! orEGguide you in e.g. EG - worked examples Worked examples e.g. - worked examples solving problems exams HINT! Step-by-step e.g. exams EG - worked examples Refers you to the Activities with hint questions for you instructions exemplar paper to answer exams e.g. EG - worked examples activity The activities are based on exam-type questions. activity Cover the answers ACTIVITIES boy and girl-left and provided and do each activity on your own. Then check your answers. ACTIVITIES boy right of page exams and girl-left and Reward yourself for things you get right. If you get any incorrect right of page activity answers, make sure you understand where activity you went wrong before activity ACTIVITIES boy moving on to the next section. and girl-left and right of page In these introduction pages, we will go through the mathematics that you activity need to know, in particular, algebra activity and graphs. These are Step by step comment comment Step by step comment comment crucial skills that you willACTIVITIES need boy for any subject that makes use of mathematics. Make surerightyou of pageunderstand these pages before you go and girl-left and any further. activity Step by step comment comment Go to www.education.gov.za to download past exam papers for you to practice. Step by step comment comment Use this study guide as a workbook. Make notes, draw pictures and highlight important concepts. © Department of Basic Education 2015 introduct ion xi Mind the Gap Mathematics © Department of Basic Education 2015 Top 10 study tips 1. Have all your materials ready before you begin studying – pencils, pens, highlighters, paper, etc. Try these study tips to make learning easier. 2. Be positive. Make sure your brain holds on to the information you are learning by reminding yourself how important it is to remember the work and get the marks. 3. Take a walk outside. A change of scenery will stimulate your learning. You’ll be surprised at how much more you take in after being outside in the fresh air. 4. Break up your learning sections into manageable parts. Trying to learn too much at one time will only result in a tired, unfocused and anxious brain. 5. Keep your study sessions short but effective and reward yourself with short, constructive breaks. 6. Teach your concepts to anyone who will listen. It might feel strange at first, but it is definitely worth reading your revision notes aloud. 7. Your brain learns well with colours and pictures. Try to use them whenever you can. 8. Be confident with the learning areas you know well and focus your brain energy on the sections that you find more difficult to take in. 9. Repetition is the key to retaining information you have to learn. Keep going – don’t give up! 10. Sleeping at least 8 hours every night, eating properly and drinking plenty of water are all important things you need to do for your brain. Studying for exams is like strenuous exercise, so you must be physically prepared. If you can’t explain it simply, you don’t understand it well enough. Albert Einstein xii introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 Mnemonics A mnemonic code is a useful technique for learning information that is difficult to remember. Here’s the most useful mnemonic for Mathematics, Mathematical Literacy and Physical Science: BODMAS: Mnemonics encode information and make it easier to remember B – Brackets O – Of or Orders: powers, roots, etc. D – Division M – Multiplication A – Addition S – Subtraction Throughout the book you will be given other mnemonics to help you remember information. The more creative you are and the more you link your ‘codes’ to familiar things, the more helpful your mnemonics will be. Education helps one cease being intimidated by strange situations. Maya Angelou © Department of Basic Education 2015 introduction xiii Mind the Gap Mathematics © Department of Basic Education 2015 Mind maps There are several mind maps included in the Mind the Gaps guides, summarising some of the sections. Mind maps work because they show information that we have to learn in the same way that our brains ‘see’ information. As you study the mind maps in the guide, add pictures to each of the branches to help you remember the content. You can make your own mind maps as you finish each section. How to make your own mind maps: 1. Turn your paper sideways so your brain has space to spread out in all directions. 2. Decide on a name for your mind map that summarises the information you are going to put on it. 3. Write the name in the middle and draw a circle, bubble or picture Mind around it. mapping 4. Write only key words on your branches, not whole sentences. Keep it your notes makes short and simple. them more interesting and easier to 5. Each branch should show a different idea. Use a different colour for remember. each idea. Connect the information that belongs together. This will help build your understanding of the learning areas. 6. Have fun adding pictures wherever you can. It does not matter if you can’t draw well. xiv introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 On the day of the exam 1. Make sure you have all the necessary stationery for your exam, i.e. pens, pencils, eraser, protractor, compass, calculator (with new batteries). Make sure you bring your ID document and examination admission letter. 2. Arrive on time, at least one hour before the start of the exam. 3. Go to the toilet before entering the exam room. You don’t want to waste valuable time going to the toilet during the exam. 4. Use the 10 minutes reading time to read the instructions carefully. This helps to ‘open’ the information in your brain. Start with the question you think is the easiest to get the flow going. 5. Break the questions down to make sure you understand what is being asked. If you don’t answer the question properly you won’t get any marks for it. Look for the key words in the question to know how to answer it. Lists of difficult words (vocabulary) is given a bit later on in this introduction. 6. Try all the questions. Each question has some easy marks in it so make sure that you do all the questions in the exam. 7. Never panic, even if the question seems difficult at first. It will be linked with something you have covered. Find the connection. 8. Manage your time properly. Don’t waste time on questions you are unsure of. Move on and come back if time allows. Do the questions that you know the answers for, first. GOOD LUCK! 9. Write big and bold and clearly. You will get more marks if the marker can read your answer clearly. 10. Check weighting – how many marks have been allocated for your answer? Take note of the ticks in this study guide as examples of marks allocated. Do not give more or less information than is required. © Department of Basic Education 2015 introduct ion xv Mind the Gap Mathematics © Department of Basic Education 2015 Question words to help you answer questions It is important to look for the question words (the words that tell you what to do) to correctly understand what the examiner is asking. Use the words in the table below as a guide when answering questions. Question word/phrase What is required of you Analyse Separate, examine and interpret Calculate This means a numerical answer is required – in general, you should show your working, especially where two or more steps are involved Classify Group things based on common characteristics Compare Point out or show both similarities and differences between things, concepts or phenomena Define Give a clear meaning Describe State in words (using diagrams where appropriate) the main points of a structure/process/ phenomenon/investigation Determine To calculate something, or to discover the answer by examining evidence Differentiate Use differences to qualify categories Discuss Consider all information and reach a conclusion Explain Make clear; interpret and spell out Identify Name the essential characteristics PAY SPECIAL ATTENTION Label Identify on a diagram or drawing List Write a list of items, with no additional detail Mention Refer to relevant points Name Give the name (proper noun) of something State Write down information without discussion Suggest Offer an explanation or a solution Tabulate Draw a table and indicate the answers as direct In every exam pairs question, put a CIRCLE around the question word and underline any other important key words. These words tell you exactly what is being asked. xvi introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 Vocabulary C The following vocabulary consists of all the category (n). Class or group of things. difficult words used in Mind the Gap Mathematics, complex (adj). Consisting of many different Mathematical Literacy, and Physical Science. We parts; not easy to understand suggest that you read over the list below a few (n). a group or system of things times and make sure that you understand each connected in a complicated way. term. Tick next to each term once you understand component (n). A part. it so you can see easily where the gaps are in your compose (v). To make up from parts. knowledge. composite (n). Something made up of parts; KEY (adj). made up of several parts. Abbreviation Meaning conjunction (n). When two or more things come together at the same point; (v) verb: doing-word or action word, in grammar, a part of speech such as “walk” that connects words, sentences, (n) noun: naming word, such as phrases or clauses, e.g.: “and” “person” consecutive (adj). One after another without any (adj) adjective: describing word such as gaps or breaks. “big” consider (v). think about. (adv) adverb: describing word for verbs, contrast (v). Show the difference between; such as “fast” (n). something that is very different (prep) preposition: a word describing a from what it is being compared position, such as “on”, “at” with. (sing) singular: one of conversely (adv). The opposite of. (pl) plural: more than one of (abbr) abbreviation D General terms data (pl), (n). Information given or found. datum (sing) Term Meaning deduce (v). To work something out by A reasoning. deduction (n). Conclusion or idea that abbreviate (v). Make shorter. someone has worked out. account for (v). Explain why. define (v). Give the meaning of a word or adjacent (adj). Next to something. words. analyse (v). Examine something in detail. definition (n). The meaning of a word or words. annotated (adj). Something that has comments or explanations, usually denote (v). To refer to or mean something. written, added to it. descending (adj). Going down. apply (v). Make a formal application; be determine (v). Work out, usually by experiment relevant to; work hard; place on. or calculation. approximate (v. & adj.). Come close to (v); discreet (adj). Careful, polite. roughly, almost, not perfectly discrete (adj). Single, separate, distinct, a accurate, close but not exact. The part. verb is pronounced “approxi-mayt” and the adjective is pronounced “approxi-mitt”. E ascending (adj). Going up. arbitrary (adj). Based on random choice; establish (v). Show or prove, set up or create. unrestrained and autocratic. exceed (v). Go beyond. excess (n). More than necessary. © Department of Basic Education 2015 introduction xvii Mind the Gap Mathematics © Department of Basic Education 2015 excluding (prep). Not including. inter- (adj). Can be swapped or exclusive (adj). Excluding or not admitting changeable exchanged for each other. other things; reserved for one investigate (v). Carry out research or a study. particular group or person. exemplar (n). A good or typical example. M exempt (v). To free from a duty. exempt (adj). Be freed from a duty. magnitude (adj). Size. exemption (n). Being freed from an obligation. manipulate (v). Handle or control (a thing or a exhibit (v). To show or display. person). exhibit (n). A part of an exhibition. motivate (v). Give someone a reason for extent (n). The area covered by something. doing something. Limit. multiple (adj). Many. F N factor (n). A circumstance, fact or negligible (adj). Small and insignificant; can influence that contributes to be ignored. From “neglect” (ignore). a result; a component or part. numerical (adj). Relating to or expressed as a A number that is divisible into number or numbers. another number without a numerous (adj). Many. remainder. factory (n). A place where goods are made or put together from parts. O find (v). Discover or locate. find (n). Results of a search or discovery. obtain (v). Get. finding (n). Information discovered as the optimal (adj). Best; most favourable. result of an inquiry. optimum (adj). Best; (n) the most favourable format (n). Layout or pattern; the way situation for growth or success. something is laid out. P H provide (v). Make available for use; supply. horizontal (adj). Across, from left to right or right to left. (From “horizon”, the R line dividing the earth and the sky). hypothesis (n). A theory or proposed explanation. reciprocal (adj). Given or done in return. hypothetical (adj). Theoretical or tentative; record (v). Make a note of something in waiting for further evidence. order to refer to it later (pronounced ree-cord). record (n). A note made in order to refer I to it later; evidence of something; a copy of something (pronounced identify (v). Recognise or point out. rec-cord. illustrate (v). Give an example to show what relative (adj). Considered in relation to is meant; draw. something else; compared to. imply (v). Suggest without directly saying relative (n). A family member. what is meant. represent (v). Be appointed to act or speak for indicate (v). Point out or show. someone; amount to. initial (n). First. resolve (v). Finalise something or make it clear; bring something to a insufficient (adj). Not enough. conclusion. xviii introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 respect (v). Admire something or someone; angel (n). In Abrahamic religions, a consider the needs or feelings of messenger from God. Note the another person. spelling. respectively (adj). In regards to each other, in angle (n). The difference in position relation to items listed in the same between two straight lines which order. meet at a point, measured in degrees. Note the spelling. S annual (adj). Once every year. (E.g. “Christmas is an annual holiday”). annum, per (adv). For the entire year. (E.g. “You simultane- (adv). At the same time. should pay R 100 per annum”). ously annuity (n). A fixed sum of money paid suffice (v). Be enough. to someone each year, typically surplus (adj). More than is needed. for the rest of their life, as an survey (n). A general view, examination, insurance policy. See policy. or description of someone or apex (n). The tip of a triangle or two lines something. meeting. survey (v). Look closely at or examine; approach (v). To approximate or come close consider a wide range of opinions to in value. or options. area (n). Length x breadth (width). In common usage: a place. T asymptote (n). A line that continually approaches a given curve but does not meet it at any finite distance. tendency (n). An inclination to do something in a particular way; a habit. average (n). Mathematics: The sum of parts divided by the quantity of parts. In transverse (adj). Extending across something. common use: neither very good, strong, etc., but also neither very V weak, bad, etc; the middle. If you are asked to find the average, you always have to calculate it using verify (v). Show to be true; check for truth; the information you have. For confirm. example, the average of (1;2;3) is vice versa (adv). The other way round. 2, because (1+2+3)/3 = 2. See versus (prep). Against. Abbreviated “vs” also mean, median and mode. and sometimes “v”. axiom A basic truth of mathematics. vertical (adj). Upright; straight up; standing. axis (sing), (n). A line along which points axes (pl, can be plotted (placed), showing Technical terms pronounced “akseez”) how far they are from a central point, called the origin. See origin. “Vertical axis” or “y-axis” refers A to how high up a point is above the origin (or how far below). abscissa (n). The distance from a point to “Horizontal axis” or “x-axis” refers the vertical or y-axis, measured to how far left or right a point is parallel to the horizontal or x-axis; away from the origin. the x-coordinate. See ordinate. acute (adj). Having an angle less than B 90˚. algebra (n). A mathematical system base (n). The horizontal lowest line where unknown quantities are on a diagram of a geometrical represented by letters, which shape, usually of a triangle. Or: can be used to perform complex a number used as the basis of a calculations through certain rules. numeration scale. Or: a number in altitude (n). Height. terms of which other numbers are expressed as logarithms. © Department of Basic Education 2015 introduction xix Mind the Gap Mathematics © Department of Basic Education 2015 bias (n). To be inclined against consecutive (adj). Following on from one something or usually unfairly another. opposed to something; to not continuous (adj). Mathematics: having no accurately report on something; to breaks between mathematical favour something excessively. points; an unbroken graph or curve binomial (n). An algebraic expression of the represents a continuous function. sum or the difference of two terms. See function. bisect (v). To cut into two. control (n. and v.). To ensure something bivariate (adj). Depending on two variables. does not change without being allowed to do so (v); breadth (n). How wide something is. From an experimental situation to the word “broad”. which nothing is done, in order to compare to a separate C experimental situation, called the ‘experiment’, in which a change is attempted. The control is then calculus (n). A branch of mathematics compared to the experiment to see that deals with the finding and if a change happened. properties of derivatives and control (n). A variable that is held constant integrals of functions, by methods variable in order to discover the relationship originally based on the summation between two other variables. of infinitesimal (infinitely small) “Control variable” must not be differences. The two main types confused with “controlled variable” are differential calculus and (see independent variable). integral calculus. coordinate (n). The x or y location of a point on cancel (v). To remove a factor by dividing a Cartesian graph, given as an x or by the factor. y value. Coordinates (pl) are given chance (n). The same as possibility or as an ordered pair (x, y). likelihood; that something might correlate (v). To see or observe a happen but that it is hard to relationship between two things, predict whether it will. without showing that one causes chart (v). To draw a diagram comparing the other. values on Cartesian axes. correlation (n). That there is a relationship chord (n). A line cutting across a circle between two things, without or arc at a position other than the showing that one causes the other. diameter. Note spelling. correspond (v). To pair things off in a circum- (n). The distance around the outer correlational relationship. For two ference rim of a circle. things to agree or match. E.g. A coefficient (n). A constant value placed corresponds to 1, B corresponds to next to an algebraic symbol as a 2, C corresponds to 3, etc. multiplier. Same as constant (see cubed (adj). The power of three; below). Or: a multiplier or factor multiplied by itself three times. that measures a property, e.g. cubic (adj). Shaped like a cube; having coefficient of friction. been multiplied by itself three complement (n).Geometry: the amount in times. degrees by which a given angle is cut (n). A subdivision of a line or less than 90°. point where one line crosses over Mathematics: the members of a another. set or class that are not members of a given subset. Do not confuse cyclic (adj). Pertaining to a circle. with compliment (praise). cylinder (n). A tall shape with parallel sides composite (adj). Made of parts. and a circular cross-section – think of a log of wood, for example, or a compound (n). Interest charged on an amount tube. See parallel. The formula for interest due, but including interest charges the volume of a cylinder is πr2h. to date. Compare to simple interest. constant (n). See coefficient. Means “unchanging”. xx introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 D divisor (n). The number below the line in a fraction; the number that is dividing the other number above dependent (adj/n). A variable whose value the fraction line. See numerator, (variable) depends on another; the thing denominator. that comes out of an experiment, the effect; the results. See also domain (n). The possible range of x-values independent variable and control for a graph of a function. See variable. The dependent variable range. has values that depend on the independent variable, and we plot E it on the vertical axis. derivation (n). Mathematics: to show the working of your arithmetic or element (n). Mathematics: part of a set of answer or solution; the process of numbers. Popular use: part of. finding a derivative. eliminate (v). To remove from an equation. derivative (n). Mathematics: The rate of See cancel. change of a function with respect equiangular (adj). Having the same angle. to an independent variable. See equidistant (adj). Having the same distance or independent variable. In common length. use: something that comes from equilateral (adj). Having sides of the same length. something else. estimate (n., v.). To give an approximate determine(s) (v). To cause; to ensure that; to set value close to an actual value; an (causation) up so that; to find out the cause of. imprecise calculation. deviation (n). A variation from a statistical Euclidean (adj). Pertaining to geometry of norm; not as far out as an outlier. straight lines on flat planes. An amount by which a single measurement differs from a fixed even (adj). Divisible by two without a value such as the mean. A significant remainder. deviation from the average value. exponent (n). When a number is raised to di- (prefix). Two. a power, i.e. multiplied by itself as many times as shown in the diagonal (adj. & n.). A line joining two power (the small number up opposite corners of an angular above the base number). So, 23 shape. means 2 x 2 x 2. See also cubed. diameter (n). The line passing through the exponential (adj). To multiply something many centre of a shape from one side times; a curve representing an of the shape to the other, esp. a exponent. circle. Formula: d = 2r. See radius, radii, circumference. expression (n). A formula or equation. difference (n). Mathematics: subtraction. extrapolation (n). To extend the line of a graph Informally: a dissimilarity. How further, into values not empirically things are not the same. documented, to project a future event or result. In plain language: digit (n). A number represented in to say what is going to happen writing. based on past results which were dimension (n). The measurable size or extent obtained (gotten) by experiment of usually a geometric shape and and measurement. If you have often on a Cartesian Coordinate a graph and have documented system, e.g. the x-dimension certain results (e.g. change vs (breadth). time), and you draw the line discriminant (n). A function of the coefficients of further in the same curve, to say a polynomial equation whose value what future results you will get, gives information about the roots that is called ‘extrapolation’. See of the polynomial. predict. Mathematics: to project distribution (n). How something is spread another iteration, value or solution, out. Mathematics: the range and based on a formula that covers or variety of numbers as shown on a formulates a previous solution. graph. © Department of Basic Education 2015 introduction xxi Mind the Gap Mathematics © Department of Basic Education 2015 F homologous (n). Belonging to the same group of things; analogous. factorial (n). The product of an integer hyperbola (n). Mathematics: a graph of a and all the integers below it; e.g. section of a cone with ends going factorial four (4!) is equal to 24. off the graph; a symmetrical (both sides the same) open curve. factorise (v). To break up into factors. hypotenuse (n). The longest side of a right- formula (n). See expression. angled triangle. fraction (n). Mathematics: Not a whole number; a representation of a division. A part. E.g. the third I 2 fraction of two is 0,666 or ​ __ 3​ meaning two divided into three illuminate (v). To explain or light up. parts. incline (n. & v.). Slope. See gradient (n); to frequency (n). How often. Usually represented lean (v). 12 1 as a fraction, e.g. ___ ​ 48 ​ = ​ __ 4 ​or 0,25. independent (n). The things that act as input function (n). Mathematics: when two (variable) to the experiment, the potential attributes or quantities correlate. causes. Also called the controlled If y changes as x changes, then variable. The independent variable y = f(x). See correlate, graph, is not changed by other factors, Cartesian, axis, coordinate. Also: and we plot it on the horizontal a relation with more than one axis. See control, dependent variable (mathematics). variable. inequality (n). A relation between two G expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than’, or < geometric (adj). Progressing or growing in a ‘less than’. regular ratio. inflation (n). That prices increase over time; geometry (n). The mathematics of shape. that the value of money decreases gradient (n). A slope. An increase or over time. General use: the action decrease in a property or of getting bigger. measurement. Also the rate of such insufficient (adj). Not enough. a change. In the formula for a line integer (n.) a whole number not a fraction, graph, y=mx+c, m is the gradient. can be negative. gradually (adv). To change or move slowly. intercept (n.) Where a line cuts an axis on a graph (n). A diagram representing graph. See cut. experimental or mathematical interest (n). Finance: money paid regularly values or results. Cartesian at a particular rate for the use or Coordinates. loan of money. It can be paid by graphic (n., adj.). A diagram or graph a finance organisation or bank (n). Popular use: vivid or clear or to you (in the case of savings), remarkable (adj.). or it may be payable by you to a graphically (adv). Using a diagram or graph. finance organisation on money you Popular use: to explain very clearly. borrowed from the organisation. See compound interest and simple interest, see also borrow. H interquartile (adj). Between quartiles. See quartile. histogram (n). A bar graph that represents intersection (n). Where two groups overlap on a continuous (unbroken) data (i.e. Venn Diagram. data with no gaps). There are no spaces between the bars. A interval (n). Gap. A difference between two histogram shows the frequency, or measurements. the number of times, something inverse (n). The opposite of. Mathematics: 1 happens within a specific interval or one divided by. E.g. ​ __ 2 ​is the “group” or “batch” of information. inverse of 2. xxii introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 imaginary (n). I; a number which is a multiple minimise (v). To make as small as possible. numbers of the square root of (–1). The minimum (n). The smallest amount possible. opposite of real numbers. Not examinable/advanced. modal (adj). Pertaining to the mode, or method. Can mean: about the irrational (n). Fractions which recur, or which mathematical mode or about the numbers cannot be expressed as a ratio of method used. See mode. whole numbers. Decimals. mode (n). the most common number in a isosceles (n., adj.). A triangle with two sides series of numbers. See also mean, (triangle) of equal length. median. model (n). A general or simplified way L to describe an ideal situation, in science, a mathematical description that covers all cases of law (n). A formula or statement, the type of thing being observed. A deduced (discovered) from prior representation. axioms (truths), used to predict a result. mutually (adj). In respect to each other, affecting each other. likely (adj). To be probable; something that might well happen. linear (adj). In a line. Mathematics: in a N direct relationship, which, when graphed with Cartesian coordinates, natural (n). Any number which is not a turns out to be a straight line. numbers fraction and greater than –1 logarithm (n). A quantity representing the (includes zero). Positive whole power by which a fixed number (the numbers. base) must be raised to produce negative (adj). Below zero. a given number. The base of a common logarithm is 10, and normal (n., adj.). Mathematics and that of a natural logarithm is the Physics: a force, vector or line that number e (2,7183…). A log graph acts at right angles to another can turn a geometric or exponential force, vector or line or object. (n). relationship, which is normally Common use: Regular or standard curved, into a straight line. (adj). numerator (n). The opposite of a denominator; the number on top in a fraction. M O magnitude (n). Size. manipulate (v). To change, or rearrange something. Usually in Mathematics obtuse (adj). Having an angle greater than it means to rearrange a formula to 90˚ but less than 180˚. solve for (to get) an answer. odd (adj). Not divisible by two without a mean (n). See average. remainder. median (n). Mathematics: the number in ogive (adj). A pointed arch shape; a the middle of a range of numbers cumulative frequency graph. written out in a line or sequence. optimal (adj). Best, most. metric (adj). A measurement system, ordinate (n). A straight line from any point using a base of 10 (i.e. all the drawn parallel to one coordinate units are divisible by 10). The USA axis and meeting the other, uses something known as the especially a coordinate measured Imperial system, which is not used parallel to the vertical. See in science. The Imperial system is abscissa. based on 12. Examples: 2,54 cm origin (n). Mathematics: the centre of (metric) = 1 inch (imperial). 1 foot a Cartesian coordinate system. = 12 inches = approx. 30 cm; General use: the source of 1 metre = 100 cm. 1 Fl.Oz (fluid anything, where it comes from. ounce) = approx 30 mℓ. © Department of Basic Education 2015 introduction xxiii Mind the Gap Mathematics © Department of Basic Education 2015 outlier (n). Statistics: a data point which pi (n). π, the Greek letter p, the lies well outside the range of ratio of the circumference of a related or nearby data points. circle to its diameter. A constant without units, value approximately 3,14159. P plan (n). Architecture: a diagram representing the layout and parallel (adj). Keeping an equal distance structure of a building, specifically along a length to another item as viewed from above. More (line, object, figure). Mathematics: general use: any design or two lines running alongside each diagram, or any intended sequence other which always keep an equal of actions, intended to achieve a distance between them. goal. parallelogram (n). Any four-sided figure with two plane (n). A flat surface. sides parallel. Abbr.: parm. plot (v). To place points on a Cartesian parameter (n). A value or algebraic symbol in coordinate system; to draw a a formula. Statistics: a numerical graph. characteristic of a population, poly- (prefix). Many. as distinct from a statistic of a sample. polygon (n). Any shape with many (at least three) equal sides and angles. A quantity whose value is selected for the particular circumstances polyhedron (n). A three-dimensional shape and in relation to which other with many usually identical flat variable quantities may be sides. expressed. polynomial (n). An expression of more than particular (adj). A specific thing being pointed two algebraic terms, especially the out or discussed; to single out or sum of several terms that contain point out a member of a group. different powers of the same variable(s). pent- (prefix). Five. population (n). Statistics: the larger body from pentagon (n). A five sided figure with all sides which the statistical sample is equal in length. taken. per (prep). For every, in accordance positive (adj) Above zero. with. predict (v). General use: to foresee. per annum (adv). Once per year; for each year. Physical Science: to state what will percent (adv). For every part in 100. The happen, based on a law. See law. rate per hundred. prime number (n). Any number divisible only by percentile (n). A division of percentages one and itself. into subsections, e.g. if the scale probability (n). How likely something is. See is divided into four, the fourth likely. Probability is generally a percentile is anything between 75 mathematical measure given and 100%. as a decimal, e.g. means perimeter (n). The length of the outer edge; unlikely, but [1,0] means certain, the outer edge of a shape. and [0,5] means just as likely period (n). The time gap between events; versus unlikely. [0,3] is unlikely, a section of time. and [0,7] is quite likely. The most common way to express periodic (adj). Regular; happening regularly. probability is as a frequency, or permutation (n). The action of changing the how often something comes up. 1 arrangement, especially the linear E.g. an Ace is ___ ​ 13 ​or 0,077 likely, order, of a set of items. in a deck of cards, because perpendicular (adj). Normal; at right angles to there are 4 of them in a set of (90˚). 52 cards. xxiv introduction  © Department of Basic Education 2015  Mind the Gap Mathematics  © Department of Basic Education 2015 product (n). Mathematics: the result of quartile (n). A quarter of a body of data multiplying two numbers. represented as a percentage. This is the division of data into project (n. & v.). A project (n., pronounced 4 equal parts of 25% each. To PRODJ-ekt) is a plan of action determine the quartiles, first divide or long-term activity intended the information into two equal to produce something or parts to determine the median reach a goal. To project (v., (Q2), then divide the first half into pronounced prodj-EKT), is to two equal parts, the median of throw something, or to guess or the first half is the lower quartile predict (a projection). To project (Q1), then divide the second half a result means to predict a into two equal parts, and the result. See extrapolate. median of the second half is the proportion (n). To relate to something else upper quartile (Q3). Data can be in a regular way, to be a part of summarised using five values, something in relation to its volume, called the five number summary, size, etc; to change as something i.e. the minimum value, lower else changes. See correlate and quartile, median, upper quartile, respectively. and maximum value. pyramid (n). A polyhedron of which one quotient (n). A ratio. face is a polygon of any number of sides, and the other faces are triangles with a common vertex. R Pythagoras’s (n). The square on the hypotenuse Theorem is equal to the sum of the squares radical (n). A root. on the other two sides of a right- radius (sing), (n). The distance between the angled triangle. Where h is the radii (plur) centre of an object, usually a hypotenuse, a is the side adjacent circle, and its circumference or to the right angle, and b is the outer edge. Plural is pronounced other side: h2 = a2 + b2. “ray-dee-eye.” random (n). Unpredictable, having no cause Q or no known cause. Done without planning. quadrilateral (n). A shape with four sides. range (n). The set of values that can be supplied to a function. The set of qualitative (adj). Relating to the quality possible y-values in a graph. See or properties of something. domain. A qualitative analysis looks at changes in properties like rate (n). How often per second (or per colour, that can’t be put into any other time period). Physics: numbers. Often contrasted with number of events per second; see quantitative. frequency. Finance: the exchange rate or value of one currency when quantitative (adj). Relating to, or by comparison exchanged for another currency; to, quantities. Often contrasted how many units of one currency with qualitative. A quantitative it takes to buy a unit of another analysis is one in which you currency. Also “interest rate”, or