TABLE OF CONTENTS YEAR 12 MATHEMATICS ESSENTIAL. 17 November 2016

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1 17 November 2016 TABLE OF CONTENTS YEAR 12 MATHEMATICS ESSENTIAL CHAPTER 1: GEOMETRY WACE syllabus reference A 2-dimensional shapes 3.2.1, B 3-dimensional shapes C Sketching solids D Views of solids 3.2.9, E Nets of solids This is a fairly straightforward chapter to ease students into the school year. Much of the content of this chapter is revision of the geometry work in Year 11. The emphasis this year is looking at the number of vertices, edges, and faces of the shapes, and drawing three-dimensional solids using nets and perspective diagrams. The material in Section D: Views of solids provides the groundwork for the three-dimensional scale diagrams encountered in Chapter 3. CHAPTER 2: LENGTH AND AREA A Metric length B Imperial length C Perimeter D Area E Area formulae 3.1.2, F Areas of irregular shapes Section B (Imperial length) is not in the WACE syllabus, so Western Australian students may skip this section. In addition to dealing with exact measurements, students should be encouraged to improve their skills in estimating lengths. This chapter contains activities to assist with this. CHAPTER 3: SCALE DIAGRAMS A Scale factor 3.2.4, B Scale diagrams 3.2.5, 3.2.6, 3.2.7, 3.2.8, C Problems involving bearings

2 Students should be familiar with scale diagrams from the Ratio chapter in Year 11. Emphasis has been placed on the types of scale diagrams students are likely to encounter in their adult lives, such as maps and house plans. Students may be unfamiliar with bearings, so some extra time may need to be spent exploring this concept, before using them in a scale diagram context. CHAPTER 4: PYTHAGORAS THEOREM A Solving x k B Pythagoras theorem C The converse of Pythagoras theorem D Problem solving This chapter comprises a fairly standard treatment of Pythagoras theorem. Students who have previously only encountered equations with one solution may need extra 2 help solving equations of the form x k. Section A will help students with this. Of course, in applying Pythagoras theorem to right angled triangles, only the positive solution is considered. Students are given opportunities to solve problems in real-world contexts, including landscaping and design. CHAPTER 5: FURTHER MEASUREMENT A Surface area 3.1.4, B Volume C Capacity 3.1.6, D Mass E Density In this chapter, students extend the measurement work done in Year 11 to consider cones and pyramids. Students will have another opportunity to use Pythagoras theorem here. Many of the problems are grounded in real-world contexts, including nursing, hospitality, and nutrition. Mass and density are not in the WACE syllabus, so Sections D and E can be omitted. CHAPTER 6: RIGHT ANGLED TRIANGLE TRIGONOMETRY A Labelling right angled triangles B The trigonometric ratios C Finding sides and angles , D Problem solving using trigonometry ,

3 It is likely that students will have struggled with trigonometry in previous years, if indeed they have encountered trigonometry at all. The material in this chapter should therefore be covered very carefully. When students are introduced to trigonometric ratios in Section B, students should understand that trigonometric ratios such as sin57 are not variables but have a particular value, and they should understand what that value represents in the context of right angled triangles. Having this understanding early will help greatly for the sections which follow. The problem solving section includes some questions involving bearings, which should reinforce the work done on bearings in Chapter 3. CHAPTER 7: NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY A Trigonometric ratios of obtuse angles B The cosine rule SACE only C The sine rule D Problem solving Non-right angled triangle trigonometry is in the SACE syllabus, but not the WACE syllabus. Therefore, only South Australian students must complete this chapter. CHAPTER 8: BUSINESS APPLICATIONS A Planning business premises B Costing calculations C Depreciation D Insurance SACE only E Cost of goods sold F Profit and loss statements G Break-even analysis H Business structures and taxation Again, this chapter is in the SACE syllabus, but not the WACE syllabus. Therefore, Western Australian students should skip this chapter. CHAPTER 9: LINE GRAPHS A The Cartesian plane B Graphing linear relationships 3.3.2, 3.3.7, 3.3.8, C Graphing lines from equations 3.3.3, D Line graphs 3.3.4, 3.3.5, E The intersection of line graphs In Sections B and C, students are presented with a variety of contexts involving linear relationships, and are asked to interpret the gradient and y-intercept of the linear graph. This will help improve their understanding of the line of best fit in Chapter 12. 3

4 CHAPTER 10: PROBABILITY A Probability 4.1.1, B Experimental probability 4.1.4, 4.1.5, C Sample space D Theoretical probability E Compound events F Tree diagrams 4.1.9, G Simulations 4.1.3, 4.1.4, Mathematics Essential students did not study probability in Year 11, so this chapter will need to be worked through slowly. While the syllabus makes no specific mention of compound events, there seems little point in studying tree diagrams without them. CHAPTER 11: STATISTICS A Sampling 3.4.1, 3.4.2, 3.4.3, 3.4.4, 3.4.5, 3.4.6, 3.4.7, 3.4.8, 3.4.9, , B Displaying data C Measuring the centre of data D Measuring the spread of data E Back-to-back stem plots F Parallel box plots Section A looks at various sampling methods, and the possible errors that can result from sampling. Students following the WACE syllabus need only study this section. CHAPTER 12: LINEAR CORRELATION A Correlation , , , , B Measuring correlation C Line of best fit by eye , , D Least squares regression line In this chapter, students explore the linear correlation between pairs of variables. The WACE syllabus has removed the use of technology in finding the correlation coefficient and the least squares regression line. That being the case, Western Australian students can omit Sections B and D, and focus on drawing a line of best fit by eye. 4

5 CHAPTER 13: INVESTMENTS A Simple interest B Compound interest 4.3.2, 4.3.3, 4.3.4, 4.3.5, C Future value annuities D Tax and inflation E Superannuation Western Australian students need only complete Sections A and B of this chapter. Students may have studied simple interest briefly in Year 11, in the context of percentages. Simple interest will be covered in more detail, and students will encounter compound interest for the first time. CHAPTER 14: LOANS A Reducing balance loans B Home loans C Strategies to minimise interest D Comparing loans This chapter completes the finance work for this course by considering loans. Much of the work is done using technology, so students should assess the reasonableness of the answer the calculator gives. This chapter contains information which should be useful to the students in their adult lives, such as strategies to minimise interest on a home loan, and the dangers of short term loans. CHAPTER 15: EARTH GEOMETRY AND TIME ZONES A Latitude and longitude B Distance on the Earth s surface 4.2.2, C Time zones 4.2.4, 4.2.5, 4.2.6, 4.2.7, 4.2.8, In this final chapter, students learn about latitude and longitude, calculating distances on the Earth s surface, and time zones. In accordance with the syllabus, distances on the Earth s surface are only calculated along lines of longitude. We have included software which students can use to calculate the distance between any two points on the Earth. 5

6 17 November 2016 TABLE OF CONTENTS YEAR 12 MATHEMATICS APPLICATIONS CHAPTER 1: SIMULTANEOUS LINEAR EQUATIONS WACE syllabus reference A Linear functions B Simultaneous linear equations SACE only C Solving simultaneous equations using technology D Problem solving with simultaneous equations Western Australian students should have studied linear simultaneous equations in Year 11. Therefore, this chapter should be omitted. CHAPTER 2: LINEAR PROGRAMMING A Feasible regions B Constructing constraints SACE only C Linear programming D Non-integer vertices This chapter is for South Australian students only, so Western Australian students can omit this chapter. CHAPTER 3: NUMBER SEQUENCES A Number sequences B Arithmetic sequences 3.2.1, 3.2.2, 3.2.3, C Geometric sequences 3.2.5, 3.2.6, 3.2.7, D Recurrence relations 3.2.9, , Students will see how geometric sequences can be used to model growth and decay problems in real-world contexts. The last section uses recurrence relations to model loans and investments. More work is done on this in Chapters 7 and 8.

7 CHAPTER 4: BIVARIATE STATISTICS A Association between categorical variables 3.1.2, 3.1.3, B Association between numerical variables 3.1.5, 3.1.6, 3.1.8, 3.1.9, , C Measuring correlation 3.1.7, D Line of best fit by eye E Least squares regression line , , , F Residual plots G Exponential regression In Section A, two-way tables are used to explore associations between categorical variables. Students will use the correlation coefficient and a residual plot to assess the suitability of the linear model. Line of best fit by eye is not explicitly mentioned in the syllabus, so students may skip this section if they wish. However, we feel that studying the line of best fit by eye first will give students a greater understanding of linear regression when they find the least squares regression line using technology. Section G (Exponential regression) is for South Australian students only, so Western Australian students should omit this section. CHAPTER 5: THE NORMAL DISTRIBUTION A The normal distribution B Probabilities using a calculator SACE only C Quantiles Western Australian students study the normal distribution in Year 11. Therefore, this chapter can be omitted. CHAPTER 6: TIME SERIES A Time series data 4.1.1, B Smoothing data C Deseasonalising data 4.1.4, D Forecasting 4.1.6, In this chapter, students construct and interpret time series plots, and represent time series data in ways that make it easier to identify long term trends. Deseasonalising data involves many stages of calculations with many different data values. If students obtain a slightly different answer to that in the back of the book, it may simple be due to a difference in rounding procedures. 2

8 CHAPTER 7: INVESTMENTS A Simple interest B Compound interest 4.2.1, C Future value annuities 4.2.6, D Effective rates E Tax and inflation F Superannuation In this chapter, students extend the work they did on compound interest in Year 11 to consider annuity models, with regular payments coming in or out of the account. There are Activities in this chapter and Chapter 8 involving the use of recurrence relations in investment and loans problems. CHAPTER 8: LOANS A Reducing balance loans 4.2.4, B Home loans 4.2.4, C Strategies to minimise interest D Comparison rates E Interest-only loans and sinking funds Students consider strategies for minimising the interest paid on loans, and calculate comparison rates to make a more informed choice when deciding between loans. Interest-only loans and sinking funds are in the SACE syllabus but not the WACE syllabus, so Western Australian students should omit Section E. CHAPTER 9: MODELLING WITH MATRICES A Networks 3.3.1, B Connectivity matrices C Dominance matrices D Transition matrices The matrix work done in Year 11 involved defining the matrix structure, and performing operations with matrices. In Year 12, we extend on this work to see how matrices can be applied to networks. Western Australian students should only complete Sections A and B. CHAPTER 10: NETWORKS: CONNECTION PROBLEMS A Terminology B Shortest path problems C Shortest connection problems 4.3.1, 4.3.2, D Maximum flow problems

9 In this chapter, students build on the networks studied in the previous chapter to consider shortest path, shortest connection, and maximum flow problems. Students will be given the opportunity to first use trial and error to solve these problems. This will allow them to see that the algorithms provide a method to solve the problems much faster. CHAPTER 11: NETWORKS: GRAPH THEORY A Routes on graphs B Eulerian graphs C Hamiltonian graphs D Planar graphs 3.3.4, This Unit 3 material is presented after the Unit 4 material in Chapter 10, as we feel that this order presents a more logical progression of concepts. It is important that students become familiar with the terminology associated with routes on graphs early in the chapter. This will assist them in their understanding of Eulerian and Hamiltonian graphs later in the chapter. CHAPTER 12: DISCRETE MODELS A Critical path analysis 4.3.4, 4.3.5, 4.3.6, 4.3.7, B Assignment problems , We complete the course by studying critical path analysis and assignment problems. These topics have been placed in a chapter of their own because they are the network topics that are included in the SACE syllabus. Section B may be unfamiliar to many teachers. An example of an assignment problem is determining which members of a swimming medley relay should swim each leg, if each swimmer s time for swimming each stroke type is known. 4

10 17 November 2016 TABLE OF CONTENTS YEAR 12 MATHEMATICS METHODS CHAPTER 1: FUNCTIONS WACE syllabus reference A Exponential functions B Logarithms 4.1.1, 4.1.2, 4.1.3, 4.1.4, 4.1.5, 4.1.9, C Logarithmic functions 4.1.6, 4.1.7, D Trigonometric functions The main purpose of this chapter is for students to familiarise themselves with exponential, logarithmic, and trigonometric functions, before studying the calculus of these functions in later chapters. Section B provides a brief recap of logarithms, on the basis that some students may have already encountered logarithms in Year 11. However, students following the WACE syllabus may not have studied logarithms in Year 11. For this reason, an introduction to logarithms, based on the work in the Year 11 book, is provided as an online link. Dealing with logarithms at this early stage allows us to include the derivatives and integrals of logarithmic functions in the calculus chapters, rather than having to address logarithmic functions separately later. CHAPTER 2: DIFFERENTIAL CALCULUS A First principles B Simple rules of differentiation C The chain rule D The product rule E The quotient rule F Derivatives of exponential functions 3.1.1, 3.1.2, 3.1.3, G Derivatives of logarithmic functions , H Derivatives of trigonometric functions 3.1.5, 3.1.6, I Second derivatives , In this chapter we recap first principles and simple rules of differentiation from Year 11. Students are then introduced to the chain, product, and quotient rules, before differentiating the exponential, logarithmic, and trigonometric functions studied in Chapter 1.

11 CHAPTER 3: APPLICATIONS OF DIFFERENTIAL CALCULUS A Equations of tangents B Increasing and decreasing functions C Stationary points D Inflection and shape , , E Kinematics F Rates of change G Optimisation , In this chapter we look at the applications of differential calculus. The chapter gives students the opportunity to practise differentiating a wide variety of functions. Many of the concepts explained in this chapter will be familiar to students, as they were studied in Year 11. The focus this year is in applying the concepts to more complicated functions. Students also have the chance to explore the surge function and the logistic function in real-world contexts. CHAPTER 4: INTEGRATION A The area under a curve B Antidifferentiation 3.2.1, C The Fundamental Theorem of Calculus , , , , D Integration 3.2.2, E Rules for integration 3.2.3, 3.2.4, 3.2.5, 3.2.6, 3.2.9, F Integrating f(ax+b) G Definite integrals , , We begin our study of integration by calculating the area under a curve, using the ideas of limits. We feel this approach is consistent with how integral calculus was developed historically. We then move on to consider how the area under a curve relates to antiderivatives of functions. CHAPTER 5: APPLICATIONS OF INTEGRATION A The area under a curve B The area between two functions C Kinematics , D Problem solving by integration We now explore some applications of integration, including the area under and between curves, kinematics, and problem solving. We now formally establish that for functions f (x) 0, we must negate the integral to find the area between f (x) and the x- axis. 2

12 It is at this point that the Methods and Specialist classes will need to be well coordinated. Chapter 7 of the Specialist textbook follows directly on from this chapter, so this chapter must be completed before the Specialist classes reach Chapter 7. CHAPTER 6: A B C STATISTICS Key statistical concepts Measuring the centre of data Variance and standard deviation This chapter gives students the opportunity to revise some important statistical concepts. Students following the WACE syllabus did not do any statistics in Year 11, and it is possible that they have never encountered variance and standard deviation. It is important that these students understand the basic ideas behind variance and standard deviation, before they encounter more advanced treatments of these measures of spread in later chapters. CHAPTER 7: DISCRETE RANDOM VARIABLES A Random variables B Discrete probability distributions 3.3.1, 3.3.2, 3.3.3, C Expected value D Variance and standard deviation E Properties of ax + b 3.3.7, F The Bernoulli distribution 3.3.9, , , G The binomial distribution , , , We now explore the properties of discrete random variables. In Section D, we study the variance and standard deviation of discrete random variables, which is why it is important that students gain some familiarity with these concepts in Chapter 6. Students should recognise that the Bernoulli random variable is a special case of the binomial random variable, where the trial is performed only once. The relationship between these variables will be further developed in Chapter 9. CHAPTER 8: CONTINUOUS RANDOM VARIABLES A Continuous random variables B Probability density functions 4.2.2, 4.2.3, C The normal distribution 4.2.5, 4.2.6, D Probabilities using a calculator E The standard normal distribution (Z-distribution) F Quantiles

13 When defining the mean and standard deviation of a continuous random variable, the syllabus describes an integral over the infinite domain from - to. However, in the overwhelming majority of cases the students will encounter, continuous random variables are defined over a finite domain [a, b], and we have defined the mean and standard deviation accordingly. It seems inappropriate to provide these definitions in a form that the students do not have the tools to evaluate. CHAPTER 9: SAMPLING AND CONFIDENCE INTERVALS A Sampling distributions B Distributions of sample means Mathematics Specialist C The Central Limit Theorem D Confidence intervals for means E Sample proportions 4.3.4, 4.3.5, F Confidence intervals for proportions 4.3.7, 4.3.8, 4.3.9, When presenting the Central Limit Theorem, we state that n = 30 is used as a rule of thumb to decide whether a sample size is large enough for the distribution of sample means to be approximately normal. However, students should understand that this is a guide only, and there is no magical threshold at which distributions change from being non-normal to normal; it is a gradual process through which the approximation to normality improves as the sample size increases. In the SACE syllabus, the work on statistical inference for both sample means and sample proportions is in the Mathematical Methods course, which is why this material all appears in the Methods textbook. For students following the WACE syllabus, the work on statistical inference for sample means is actually part of the Mathematics Specialist course. This should not be a problem, as students studying Mathematics Specialist will also be studying Mathematics Methods, and so they should have both textbooks. If possible, it would be best if these students complete the material on sample means and sample proportions together rather than separately, as the ideas involved are very similar, and the results from both components each stem from the Central Limit Theorem in Section C. Students studying only Mathematics Methods need only complete Sections E and F. That being said, their understanding of the work would be enhanced by studying some relevant aspects of the earlier sections, especially the Central Limit Theorem in Section C. 4

14 17 November 2016 TABLE OF CONTENTS YEAR 12 MATHEMATICS SPECIALIST CHAPTER 1: MATHEMATICAL INDUCTION WACE syllabus reference A The process of induction SACE only B The principle of mathematical induction Induction is in the SACE syllabus, but not the WACE syllabus at Year 12. Thus, Western Australian students should omit this chapter. CHAPTER 2: REAL POLYNOMIALS A Operations with polynomials B Zeros, roots, and factors C Polynomial equality D Polynomial division E The Remainder theorem F The Factor theorem G The Fundamental Theorem of Algebra H Sum and product of roots theorem (Extension) I Graphing real polynomials J Polynomial equations In this chapter we consider the properties, operations, theorems, and graphs associated with real polynomials. Polynomial division by a linear factor is presented using both long division and synthetic division. The long division method is presented first, and students get practice with this method, as they will need it for when they divide by quadratics. We then present synthetic division, so that students can divide by linear factors faster. CHAPTER 3: FUNCTIONS A Composite functions 3.2.1, B Inverse functions 3.2.3, 3.2.4, C Reciprocal functions D The reciprocal of other functions E Rational functions F Absolute value functions 3.2.6, 3.2.7

15 This Topic 2 chapter has been presented before the remainder of Topic 1. This was done so that the absolute value function and properties of modulus are explained in a real number context before we study the modulus of complex numbers. At the end of Section B there is an Investigation on the inverse trigonometric functions. Students should be encouraged to complete this Investigation as these functions appear in the vector and integration chapters later in the book. CHAPTER 4: COMPLEX NUMBERS A The complex plane 3.1.1, 3.1.2, 3.1.3, 3.1.8, B Modulus and argument 3.1.4, C Polar form 3.1.5, D Euler s form E De Moivre s theorem F Roots of complex numbers , Sections A and B are largely revision of what was done in Year 11, so these sections should be worked through swiftly. An Activity on the triangle inequality is included at the end of Section B, however this was also addressed in Year 11, and students who completed the work in Year 11 need not complete the Activity. The remaining sections extend what was done in Year 11 to consider polar and Euler forms of complex numbers. Just as we use the Cartesian form of a complex number to represent addition and subtraction of complex numbers on an Argand plane, students should understand the power of the polar form to represent the multiplication of complex numbers. This will help students appreciate the use of polar form in finding powers and roots of complex numbers. CHAPTER 5: VECTORS A Vectors in space 3.3.1, B Operations with vectors in space C Vector algebra D The vector between two points E Parallelism 3.3.2, F The scalar product of two vectors G The angle between two vectors H Proof using vector geometry I The vector product of two vectors In Year 11, students explored vectors in two dimensions. In Year 12, the focus moves to three dimensional vectors. With the extra dimension, it becomes more difficult to visualise the vectors on the page. This makes it all the more important that students can operate with vectors algebraically. 2

16 CHAPTER 6: VECTOR APPLICATIONS A Area B Lines in 2 and 3 dimensions 3.3.4, C The angle between two lines D Constant velocity problems 3.3.5, E The shortest distance from a point to a line F Intersecting lines G Relationships between lines H Planes 3.3.7, I Angles in space J Solving 3 3 linear systems K Intersecting planes 3.3.9, We now look at some applications of vectors. In Section B we consider the equations of lines in 2 and 3 dimensions. Instead of presenting the 2-dimensional and 3-dimensional cases separately, we simply give the equation for the 3-dimensional case, and instruct students to ignore the z-coordinate for the 2-dimensional case. We begin the study of planes with an Investigation showing that, given two nonparallel vectors in the plane, we can reach any point on the plane by taking a linear combination of these vectors. This helps to motivate the equation of the plane. CHAPTER 7: INTEGRATION A Rules for integration 4.1.1, 4.1.3, B Integrating 1 and a a x a x C Integration by substitution D Integration by parts E The area between two functions F Solids of revolution 4.1.6, This chapter follows directly from the integration studied in the Mathematics Methods course. For this reason, it is important that the Mathematics Methods integration chapters are completed before this chapter is started. In Section B, we differentiate the inverse trigonometric functions. This allows us to integrate expressions of the form 1 and a. It will be beneficial for a x a x students to complete the Investigation on inverse trigonometric functions in Chapter 3, so they are familiar with the functions upon reaching this section. 3

17 CHAPTER 8: RATES OF CHANGE AND DIFFERENTIAL EQUATIONS A Implicit differentiation B Related rates C Differential equations D Differential equations of the form dy f (x) dx E Separable differential equations 4.2.4, F Slope fields G Problem solving 4.2.4, H Logistic growth I Equations of motion J Simple harmonic motion In this chapter, we extend the differentiation work done in Mathematics Methods, and look at implicit differentiation, related rates, and solving differential equations. This chapter is quite long, so teachers should make sure to allocate plenty of time to complete it. CHAPTER 9: VECTOR CALCULUS A Parametric equations , B Pairs of uniformly varying quantities , C Pairs of non-uniformly varying quantities , D Bézier curves E Trigonometric parameterisation , F Arc lengths of parametric curves We complete our study of calculus by considering motion represented as parametric curves. Sections D and F cover material explicitly mentioned in the SACE syllabus only, so Western Australian students can omit these sections. 4