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Via Afrika Mathematics

Language: EnglishGrade 12 Teacher s GuideVia Afrika MathematicsVia Afrika understands, values and supports your role as a teacher. You have the most important job in education, and we realise that your responsibilities involve far more than just teaching. We have done our utmost to save you time and make your life easier, and we are very proud to be able to help you teach this subject successfully. Here are just some of the things we have done to assist you in this brand-new series was written to be aligned with CAPS. See page 5 to see how CAPS requirements are possible work schedule has been included. See page 6 to 9 to see how much time this could save topic starts with an overview of what is taught, and the resources you need. See page 31 to find out how this willhelp with your is advice on pace-setting to assist you in completing all the work for the year on time. Page 31shows you how thisis on how to introduce concepts and scaffold learning is given for every topic.

Unit 2 Geometric sequences and series Unit 3 The sum to n terms(S. n): Sigma notation Unit 4 Convergence and sum to infinity : ... Unit 1 Future value annuities . Unit 2 Present value annuities : Unit 3 Calculating the period . Unit 4 Analysing investments and loans :

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Transcription of Via Afrika Mathematics

1 Language: EnglishGrade 12 Teacher s GuideVia Afrika MathematicsVia Afrika understands, values and supports your role as a teacher. You have the most important job in education, and we realise that your responsibilities involve far more than just teaching. We have done our utmost to save you time and make your life easier, and we are very proud to be able to help you teach this subject successfully. Here are just some of the things we have done to assist you in this brand-new series was written to be aligned with CAPS. See page 5 to see how CAPS requirements are possible work schedule has been included. See page 6 to 9 to see how much time this could save topic starts with an overview of what is taught, and the resources you need. See page 31 to find out how this willhelp with your is advice on pace-setting to assist you in completing all the work for the year on time. Page 31shows you how thisis on how to introduce concepts and scaffold learning is given for every topic.

2 See page 31 for an the answers have been given to save you time doing the exercises yourself. See page 32 for an included is a CD filled with resources to assist you in your teaching and assessment. See the inside front accompanying Learner s Book is written in accessible language and contains all the content your learners need to exciting design and layout will keep their interest and make teaching a pleasure for would love to hear your feedback. Why not tell us how it s going by emailing us at Alternatively, visit our teacher forum at Brynn Taylor, TeacherFor me, teaching is about applying myself in the lives of the learners and inspiring flight inspiring them to reach their full Afrika MathematicsGrade 12 Study GuideM. Malan, Schalekamp, Brown, L. Bruce, G. du Toit, Smith, Botsane, J. Bouman, M. PillayM. MalanStudy GuideVia Afrika Mathematics Grade 12 ISBN: 978-1-41546-335-2 Exponents and Surds Via Afrika Mathematics Grade 12 1 Contents Introduction.

3 3 Chapter 1 Number patterns, sequences and series ..4 OVERVIEW ..4 Unit 1 Arithmetic sequences and series Unit 2 geometric sequences and series Unit 3 The sum to n terms(Sn): Sigma notation Unit 4 Convergence and sum to infinity Mixed exercises ..8 Chapter 2 Functions ..10 OVERVIEW ..10 Unit 1 The definitions of a function Unit 2 The inverse of a function Unit 3 The inverse of y = ax + q Unit 4 The inverse of the quadratic function y = ax2 Mixed exercises ..21 Chapter 3 Logarithms ..24 OVERVIEW ..24 Unit 1 The definition of a logarithm Unit 2 Solving exponential equations using logarithms Unit 3 The graph of y = logbx where b > 1 and 0 < b < 1 Mixed exercises ..28 Chapter 4 Finance, growth and decay ..29 OVERVIEW ..29 Unit 1 Future value annuities Unit 2 Present value annuities Unit 3 Calculating the period Unit 4 Analysing investments and loans Mixed exercises ..35 Chapter 5 Compound angles.

4 37 OVERVIEW ..37 Unit 1 Deriving a formula for cos( ) Unit 2 Formula for cos( + ) and ( ) Unit 3 Double angles Unit 4 Identities Unit 5 Equations Unit 6 Trigonometric graphs and compound angles Mixed exercises ..46 Chapter 6 Solving problems in three dimensions ..48 OVERVIEW ..48 Unit 1 Problems in three dimensions Unit 2 Compound angle formulae in three dimensions Mixed exercises ..51 Exponents and Surds Via Afrika Mathematics Grade 12 2 Chapter 7 Polynomials ..53 OVERVIEW ..53 Unit 1 The Remainder Theorem Unit 2 The Factor Theorem Mixed exercises ..57 Chapter 8 Differential calculus ..58 OVERVIEW ..58 Unit 1 Limits Unit 2 The gradient of a graph at a point Unit 3 The derivative of a function Unit 4 The equation of a tangent to a graph Unit 5 The graph of a cubic function Unit 6 The second derivative (concavity) Unit 7 Applications of differential calculus Mixed exercises.

5 70 Chapter 9 Analytical geometry ..72 OVERVIEW ..72 Unit 1 Equation of a circle with centre at the origin Unit 2 Equation of a circle centred off the origin Unit 3 The equation of the tangent to the circle Mixed exercises ..77 Chapter 10 Euclidean geometry ..81 OVERVIEW ..81 Unit 1 Proportionality in triangles Unit 2 Similarity in triangles Unit 2 Theorem of Pythagoras Mixed exercises ..94 Chapter 11 Statistics: regression and correlation ..97 OVERVIEW ..97 Unit 1 Symmetrical and skewed data Unit 2 Scatter plots and correlation Mixed exercises ..106 Chapter 12 Probability ..108 OVERVIEW ..108 Unit 1 Solving probability problems Unit 2 The counting principle Unit 3 The counting principle and probability Mixed exercises ..112 ANSWERS TO MIXED EXERCISES ..113 EXEMPLAR PAPER 1 ..151 EXEMPLAR PAPER 2 ..168 Exponents and Surds Via Afrika Mathematics Grade 12 3 Introduction to Via Afrika Mathematics Grade 12 Study Guide Woohoo!

6 You made it! If you re reading this it means that you made it through Grade 11, and are now in Grade 12. But I guess you are already well aware of It also means that your teacher was brilliant enough to get the Via Afrika Mathematics Grade 12 Learner s Book. This study guide contains summaries of each chapter, and should be used side-by-side with the Learner s Book. It also contains lots of extra questions to help you master the subject matter. Mathematics not for spectators You won t learn anything if you don t involve yourself in the subject-matter actively. Do the maths, feel the maths, and then understand and use the maths. Understanding the principles Listen during class. This study guide is brilliant but it is not enough. Listen to yourteacher in class as you may learn a unique or easy way of doing something. Study the notation, properly. Incorrect use of notation will be penalised in testsand exams.

7 Pay attention to notation in our worked examples. Practise, Practise, Practise, and then Practise some more. You have to practiseas much as possible. The more you practise, the more prepared and confident youwill feel for exams. This guide contains lots of extra practice opportunities. Persevere. We can t all be Einsteins, and even old Albert had difficulties learningsome of the very advanced Mathematics necessary to formulate his theories. If youdon t understand immediately, work at it and practise with as many problems fromthis study guide as possible. You will find that topics that seem baffling at first,suddenly make sense. Have the proper attitude. You can do it!The AMA of Mathematics ABILITY is what you re capable of doing. MOTIVATION determines what you do. ATTITUDE determines how well you do it. Pure Mathematics is, in its way, the poetry of logical ideas. Albert EinsteinNumber patterns, sequences and series Via Afrika Mathematics Grade 12 4 Chapter 1 Overview Chapter 1 Page 8 Number patterns, sequences and series Unit 1 Page 10 Arithmetic sequences and series Formula for an arithmetic sequenceUnit 2 Page 14 geometric sequences and series Formula for the nth termof a sequenceUnit 3 Page 18 The sum to terms ( ): Sigma notation The sum to terms in anarithmetic sequence The sum to terms in ageometric sequenceUnit 4 Page 28 Convergence and sum to infinity Convergence REMEMBER YOUR STUDY APPROACH SHOULD BE: 1 Work through all examples in this chapter of your Learner s Bok.

8 2 Work through the notes in this chapter of this study guide. 3 Do the exercises at the end of the chapter in the Learner s Book. 4 Do the mixed exercises at the end of this chapter in this study guide. Number patterns, sequences and series Via Afrika Mathematics Grade 12 5 Chapter 1 TABLE 1: SUMMARY OF SEQUENCES AND series TYPE GENERAL TERM: SUM OF TERMS: EXAMPLES Arithmetic Sequence (AS) (also named the linear sequence) = +( 1) = 1 = . = 2 1 3 2 etc. = 2[2 +( 1) ] or = 2[ + ] where = the last term of the sequence A)2 ; 5 ; 8 ; 11 ; .. = +3 +3 +3 = 2 +( 1)(3) = 2 + 3 3 = 3 1 B) 1 ; -4 ; -9 ; .. = -5 -5 = 1 +( 1)( 5) =1 5 +5 = 5 +6 geometric Sequence (GS) (also named exponential sequence) = 1 = 1 = = 2 1 3 2 = ( 1) 1Or = (1 )1 Or = 1 Where 1 < < 1 (Converging series ) A)2 ; -4 ; 8 ; -16.

9 = x-2 x-2 x-2 = 2( 2) 1 NOT CONVERGING as < 1 B)3 ; 32; 34; 38 ; .. = x12x12x12 =3 12 1 CONVERGING as 1< <1 Quadratic Sequence (QS) = 2+ + = 1st difference = 2nd difference Determine , and using simultaneous equations (see example) Alternatively: = 2 = 1 3 = 1 where 1= first term of first differences 3 ; 8 ; 16 ; 27 ; .. : 5 8 11 : 3 3 Setup three equations using the first three terms: 1= 3: 3 = + + ..(1) 2= 8: 8 = 4 +2 + ..(2) 3=16: 16=9 + 3 + ..(3) Solving simultaneously leads to: =32 2+12 +1 Constant 2nd differenceConstant ratioConstant 1st difference Number patterns, sequences and series Via Afrika Mathematics Grade 12 6 Chapter 1 TYPES OF QUESTIONS YOU CAN EXPECT STRATEGY TO ANSWER THIS TYPE OF QUESTION EXAMPLE(S) OF THIS TYPE OF QUESTION Identify any of the following three types of sequences: Arithmetic (AS), geometric (GS) and Quadratic (QS) Determine whether sequence has a constant 1st difference (AS) constant ratio (GS) constant 2nd difference (QS)See Table 1 above Determine the formula for the general term, , of AS, GS and QS (from Grade 11) You need to find: and for an AS and for a GS , and for a QSSee Table 1 above Determine any specific term for a sequence 30 Substitute the value of into See Text Book : Example 1, nr.

10 1 d and 2 d, (AS) Example 1, nr. 1 b, 3 b, (AS) Example 1, nr. 1, p. 15 (GS) Determine the number of terms in a sequence, , for an AS, GS and QS or the position, , of a specific given term or when the sum of the series is given Substitute all known variables into the general term to get an equation with as the only unknown. Solve for . OR Substitute all known variables into the -formula to get an equation with as the only unknown. Solve for . Remember: must be a natural number (not negative, not a fraction) See Text Book: Example 1, c, Example 1, c, Example 1, nr. 3, Example 2, , Example 3, nr. 2, When given two sets of information, make use of simultaneous equations to solve: and (for an AS) and (for a GS) For each set of information given, substitute the values of and or and . You then have 2 equations which you can solve simultaneously (by substitution) See Text Book: Example 1, nr.


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