TEACHER NOTES FOR YEAR 11 MATHEMATICAL METHODS AND SPECIALIST MATHEMATICS YEAR 11 MATHEMATICAL METHODS CHAPTER 0: BACKGROUND KNOWLEDGE (ONLINE) 15 June 2015 A Coordinate geometry Unit 1 Unit 1 B The equation of a line Topic 1 Topic 1 C The intersection of lines Sub-topic 1.1 We put this material in a separate online chapter because the students have seen it several times and should know it thoroughly. If classes require this material, they need to move through it swiftly. Otherwise, they will struggle to get through the course. We have left out Solve linear equations completely, because a student unable to do this in Year 11 should not be attempting this course. CHAPTER 1: RELATIONSHIPS BETWEEN VARIABLES A Linear relationships Unit 1 Unit 1 Topic 1 Topic 1 Sub-topic 1.1 B Inverse proportion Unit 1 Unit 1 C Reciprocal functions Topic 1 Topic 1 D Rational functions Sub-topic 1.3 In this chapter we provide brief introductions into direct and inverse proportionality. It is important to not linger on this chapter, since proportionality is not returned to in the course. The rational functions are written to reflect the syllabus. However, it 1 seems strange to not consider a general translation of the hyperbola y. x It is important that students understand the difference between inverse proportionality and a general rational function. Just as y x or y mx is the special case of 1 k a y mx c, so is y or y the special case of y. x x x b
CHAPTER 2: QUADRATICS A Solving quadratic equations B The discriminant of a quadratic C The sum and product of the roots Unit 1 Unit 1 D Quadratic functions Topic 1 Topic 1 E Finding a quadratic from its graph Sub-topic 1.2 F Where functions meet G Problem solving with quadratics H Quadratic optimisation When considering the order for presenting material in this book, we are mindful of how things will actually work in the classroom. Usually, the first few weeks of term are difficult because it takes students time to settle down into a regular work pattern. Knowing that Chapter 1 presents some new material, we want to help students settle by the revision and extension of quadratics from previous years before we get to the rigours of formal function notation, domain and range, and function notation. We are content to use the word function, however, since students had an introduction to functions in Year 10. The other purpose is that when we introduce domain and range in the functions chapter, we can immediately use quadratics in addition to linear functions and graphs. CHAPTER 3: RELATIONS AND FUNCTIONS A Relations and functions B Function notation C Domain and range Unit 1 Unit 1 D Composite functions Topic 1 Topic 1 E Sign diagrams Sub-topic 1.5 F Transforming y f (x) G Graphs of circles Having studied quadratics in detail, we now consider relations and functions more formally. The chapter is somewhat fragmented, which tends to happen in any introduction to functions. This highlights the case for dealing with quadratics first, enabling students to settle in to work. Having done domain and range, and composite functions, we move to sign diagrams which will have later virtue in polynomials and 2 calculus. We highlight y x as an example of a relation which is not a function, since it has been given specific mention in the syllabus. When considering the transformations listed in the syllabus, reflections are not mentioned. There is clearly an assumption, however, that the parameters c and k in y cf (x) and y f (kx) can be negative, amounting as it were to a dilation and reflection combined. We feel this is unnatural and foolish for many reasons, including: 2
it leads to non-unique specification of transformations, which is mathematically incorrect it leads to other errors such as defining polar coordinates in which r can be negative it leads away from the Specialist Mathematics work on linear transformations using matrices, in which reflections are given a specific mention. Therefore, we are including reflections in their own right, and considering the combination of transformations of functions as an example of the use of composition of functions. We close the chapter with circles because they provide a further example of the use of transformations, especially translations to locate the centre. They also belong under the banner of relations, which are then largely put aside for the remainder of the course. Having studied quadratics already, we reinforce the method of completing the square by applying it to circles. CHAPTER 4: POLYNOMIALS A Polynomials B Zeros, roots, and factors Unit 1 Unit 1 C Factorising cubic polynomials Topic 1 Topic 1 D Solving cubic equations Sub-topic 1.4 E Graphing cubic functions In the ACARA syllabus, powers and polynomials are placed together. There seems no 1 sense in this, however, since we have already dealt with y and y x, and x everything else to be covered are polynomials. Hence, we present here polynomials as a discrete unit. It would be advantageous for students to have completed the 10A Polynomials chapter in the previous year. CHAPTER 5: THE UNIT CIRCLE AND RADIAN MEASURE SACE A Radian measure B Arc length and sector area C The unit circle and the trigonometric Unit 1 ratios Topic 2 D Applications of the unit circle Sub-topic 2.2 E Multiples of 6 or 4 F The gradient of a straight line ACARA Unit 1 Topic 2 Having studied right angled triangle trigonometry since Year 9, we feel this material should be well known. A short review section will be placed online, but otherwise we want to get into more advanced trigonometry quickly. 3
It would be advantageous for students to have completed the 10A Advanced trigonometry chapter in the previous year. Feedback over many years and from multiple curricula suggest that radian measure is something students find challenging. For me, this presents a case for introducing it earlier and applying it more often, rather than the other way around. We therefore present radians immediately, and continue working with both angle measures throughout. By the time we consider the sine and cosine of multiples of 6 and 4, we want students to have an ingrained radian mindset. Formulae are now stated in terms of radians only, and the equivalent ratios for degree measure are secondary. In terms of the SACE curriculum, we present sub-topic 2.2 before sub-topic 2.1 so that students have calculated the cosine and sine of obtuse angles before meeting them in the context of the cosine and sine rules. CHAPTER 6: NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY A Areas of triangles Unit 1 Unit 1 B The cosine rule Topic 2 Topic 2 C The sine rule Sub-topic 2.1 D Problem solving This chapter is a fairly traditional presentation of these trigonometric applications. We include proofs for the rules where appropriate, and investigate the ambiguous case for the sine rule. The extensive problem solving section requires students to select the appropriate rule. CHAPTER 7: TRIGONOMETRIC FUNCTIONS A Periodic behaviour B The sine function C The cosine function Unit 1 Unit 1 D The tangent function Topic 2 Topic 2 E Trigonometric equations Sub-topic 2.3 F Using trigonometric functions G Angle sum and difference identities The general introduction to periodic functions provides a range of contextual examples. We develop each component of the general sine function using the transformations we did in Chapter 3. However, for the purpose of modelling, we think it is really important to stick the whole lot together. Otherwise you end up with a silly contextual model of a situation which is not real world at all. We have included questions involving more than one transformation, since this will prove useful for real world modelling. This is where the students learn their craft of identifying and utilising what 4
parameters actually do in modelling. It is unclear whether the syllabus requires this, however. We feel the SACE and ACARA syllabuses are a bit silly in using sin(x+c), cos(x+c), and tan(x+c) rather than (the equivalent) sin(x c), cos(x c), and tan(x c), since this is directly counter to the work on transformations done previously. For this reason we stick to the traditional sin(x c), etc. Once we have moved on to trigonometric functions, it is more important than ever that students are entrenched in the radian mindset. The whole notion of period can get very confusing if they are not thinking in radians. For students who are also studying Specialist Mathematics, it is important that this chapter is completed in semester 1. Otherwise, they will not have the necessary background for the chapter in SM which flows on from this one. CHAPTER 8: COUNTING AND THE BINOMIAL EXPANSION A The product and sum principles B Permutations Unit 1 Unit 1 C Factorial notation Topic 3 Topic 3 D Combinations Sub-topic 3.1 E Binomial expansions F The binomial theorem This chapter provides a general introduction to the principles of permutations and combinations, and explores the link between combinations and the binomial expansion. Students following the ACARA syllabus, who are studying both Methods and Specialist Mathematics, may already have encountered permutations and combinations in their Specialist course. These students may skip through Sections A to D, and focus their study on the binomial expansion in Sections E and F. It is important to lay a solid foundation of understanding for the binomial expansion in Year 11, as this will hold students in good stead for their study of the binomial distribution in Year 12. CHAPTER 9: PROBABILITY A Experimental probability B Sample space Unit 1 C Theoretical probability Topic 3 Unit 1 D Compound events Sub-topic 3.2 Topic 3 E Tree diagrams F Sets and Venn diagrams G The addition law of probability H Conditional probability Unit 1 I Independent events Topic 3 Sub-topic 3.3 5
This chapter is a fairly traditional treatment of probability. Students are given the opportunity to investigate and explore concepts involving experimental probability and compound events. Students should be very familiar with sets from their work in the middle school years. As a result, sets are not studied in their own right in this book, but are considered in this chapter in the context of probability. The notions of Venn diagrams, intersection, and union are revised in the sets Section F. Students have been finding probabilities from Venn diagrams since Year 8, so they should not linger on this section. These concepts are then applied to the study of laws of probability, conditional probability, and independent events. Students who completed Year 10A in the previous year should have already encountered these ideas. CHAPTER 10: NUMBER SEQUENCES A Number sequences Specialist B Arithmetic sequences Mathematics Unit 2 C Geometric sequences Unit 1 Topic 2 D Series Topic 1 E Arithmetic series Sub-topic 1.1, F Geometric series 1.2 In the ACARA syllabus, number sequences are covered in Mathematical Methods, but in the SACE syllabus they are covered in Specialist Mathematics. South Australian students taking both Methods and Specialist will study this chapter as part of their Specialist course. CHAPTER 11: STATISTICS SACE A Key statistical concepts Unit 2 Topic 4 Sub-topic 4.1 B Measuring the centre of data Unit 2 C Measuring the spread of data Topic 4 D Variance and standard deviation Sub-topic 4.2 E The normal distribution Unit 2 Topic 4 Sub-topic 4.3 ACARA Statistics is in the SACE syllabus at Year 11, but not the ACARA syllabus. Therefore, only South Australian students need to study this chapter. Sections A and B are largely revision, looking at graphing discrete and continuous numerical data, and finding the centre of a data set. Students who are comfortable with these concepts should skip quickly through these sections. 6
Students who completed the 10A course should have briefly encountered standard deviation, however those who did not will be seeing it for the first time. Students are asked to calculate the sample standard deviation, first by hand, then using technology, as outlined in the syllabus. However, we also mention population standard deviation, as we feel it is important for students to be familiar with both concepts, and to understand the difference between them. The chapter also contains an introduction to the normal distribution. It includes a discussion of how the normal distribution arises, and uses the 68-95 - 99.7% rule to find the proportion of values within certain intervals. It provides a valuable grounding for the more advanced study of the normal distribution in Year 12. CHAPTER 12: SURDS, INDICES, AND EXPONENTIALS A Surds B Indices Unit 2 C Index laws Topic 5 D Scientific notation Sub-topic 5.1 Unit 2 E Rational indices Topic 1 F Algebraic expansion and factorisation G Exponential equations Unit 2 H Exponential functions Topic 5 I Growth and decay Sub-topic 5.2 J x The natural exponential e Sections A-D should be familiar to many of the students, and should be worked through quickly. Rational indices may or may not be familiar to students, depending on whether they covered the 10A material in the previous year. The ability to write expressions involving surds in terms of rational indices is important in the lead-up to calculus, where students must differentiate functions like f ( x) x x. x The natural exponential e is not explicitly mentioned in the syllabus, but we have included it so that students meet both it and natural logarithms in Year 11. This is important because they need to be comfortable both graphing and performing differentiation with them in Year 12. 7
CHAPTER 13: LOGARITHMS SACE A Logarithms in base 10 B Logarithms in base a C Laws of logarithms Unit 2 D Natural logarithms Topic 5 E Solving exponential equations using Sub-topic 5.3 logarithms F Growth and decay ACARA As with statistics, logarithms are in the SACE syllabus at Year 11, but not the ACARA syllabus. Therefore, only South Australian students are required to study this chapter. Students who completed 10A in the previous year would have encountered logarithms in base 10. However, students who did not take 10A will be seeing logarithms for the first time. Natural logarithms are not in the syllabus, however we have included them because in Year 12, students will be dealing with natural logarithms more so than logarithms in other bases. We feel it is only sensible to introduce them in Year 11, so that students are familiar with them before needing to graph and differentiate with them in Year 12. CHAPTER 14: INTRODUCTION TO DIFFERENTIAL CALCULUS A Rates of change Unit 2 Unit 2 B Instantaneous rates of change Topic 6 Topic 3 C Finding the gradient of the tangent Sub-topic 6.1, D The derivative function 6.2 This chapter provides students with their first look at differential calculus. We had originally planned to present limits first, so it did not interrupt the flow of the chapter as we moved from rates of change to the derivative function. However, seeing as only differential calculus is studied in Year 11, only a very brief study of limits is required anyway. Therefore, we decided to lead with rates of change, and introducing the δy/δx notation. We use this to motivate a quick study of limits, where we only deal with limits where x is finite. Rates of change feature heavily in the syllabus, however we were keen to not get too laborious with them. Spending too much time on rates of change detracts from the power of calculus, where the whole point is that you can generate a function which allows us to quickly find the rate of change at any value of x. We use Leibniz notation as in the syllabus as we introduce the derivative function, but are careful not to overdo it, thus avoiding unnecessary complexity. 8
CHAPTER 15: PROPERTIES AND APPLICATIONS OF DERIVATIVES A Simple rules of differentiation Unit 2 Topic 6 Sub-topic 6.3, 6.4 B Tangents and normals Unit 2 C Increasing and decreasing functions Unit 2 Topic 3 D Stationary points Topic 6 E Kinematics Sub-topic 6.5 F Optimisation We have included the study of normal in with the study of tangents, as they link well with the students previous work on perpendicular gradients. The ACARA syllabus includes anti-derivatives in this topic, however we have omitted it as we believe it is an inappropriate afterthought to stick anti-derivatives at the end of Year 11 without immediate extension to integral calculus. It is our view that spare time would be better spent reinforcing the differential calculus and its applications. This brings our chapter in line with the SACE syllabus, which has also excluded anti-derivatives from Year 11. 9
YEAR 11 SPECIALIST MATHEMATICS In the Mathematical Methods book: NUMBER SEQUENCES SACE A Number sequences B Arithmetic sequences Unit 1 C Geometric sequences Topic 1 D Series Sub-topic 1.1, E Arithmetic series 1.2 F Geometric series Number sequences are in the SACE Specialist syllabus and in the ACARA Methods syllabus. The chapter that includes this material is in the Methods textbook, since SA Specialist students will also have this. The chapter provides a fairly standard treatment of sequences and series, including both the explicit and recursive rules for generating arithmetic and geometric sequences. We explore applications of sequences, such as growth and decay, and compound interest. CHAPTER 1: COUNTING A The product principle B Counting paths C Factorial notation Unit 1 D Permutations Topic 1 E Combinations F The Inclusion-Exclusion principle G The Pigeonhole principle Counting is in the ACARA Specialist Mathematics syllabus, but not the SACE syllabus. South Australian students may therefore skip this chapter, although they will do some work on combinations as part of their Methods course. SACE students may like to do sections F and G as an extension of the Methods course. In the ACARA syllabus, combinations are listed last in Topic 1. We have instead placed them directly after permutations, since they are a logical progression. Permutations introduces the idea of dividing by a factorial. For example, suppose we have 10 letters, and take 4 of them to form a word. The order of the 4 selected letters 10! is important, so the number of ways to do this is 10 9 8 7= 6!. In other words, there are 10! ways to order all of the letters, but we divide by 6! because the order of the 6 letters not in the word is not important. This leads to permutations where certain objects are indistinguishable. Here we have the same discussion: since the objects are indistinguishable, the order in which they are selected is not important. 10
This leads naturally to our work on combinations: if a team of 4 people is selected from a group of 10, the order of the 4 that are selected is not important, and the order of the 6 that are not selected is not important, so the total number of ways to do this is 10! 4!6!. The inclusion-exclusion principle is then presented. Here we assume knowledge of sets and Venn diagrams from previous years, including Venn diagrams with 3 sets. This section is quite short, as there are not many sensible questions which can be written. The chapter finishes with a basic treatment of the pigeonhole principle. This only considers the case where there must be at least one box with more than one pigeon. CHAPTER 2: CIRCLE GEOMETRY A The angle in a semi-circle theorem B The chord of a circle theorem C The radius-tangent theorem Unit 1 Unit 1 D The angle at the centre theorem Topic 2 Topic 3 E Angles subtended by the same arc Sub-topic 2.1, F Tangents from an external point 2.2 G Angle between tangent and chord H Cyclic quadrilaterals I Intersecting chords and secants theorems In this chapter we present a variety of theorems associated with circle geometry. This chapter contains a lot of formal proofs, both as proofs of theorems, and also asking students to provide proofs for other theorems and geometric properties. There is a useful appendix on proof at the end of the book, which gives a more general treatment of the nature of proof, and outlines what a formal proof entails. The 10A chapter Geometry of circles is a very good preparation for this chapter. It introduces the theorems and their application, allowing students in Year 11 to concentrate more on the rigour and proof. The more of this work the students can complete at the 10A level, the faster they will be able to advance. Knowing that this material is not directly examinable in Year 12, students should not get bogged down in this chapter. Cyclic quadrilaterals and the intersecting chords and secant theorems were not covered in 10A. We have included the chord of a circle theorem, even though it was not listed in the syllabus, because we needed it in a proof. It is important that these theorems are presented in the correct order, otherwise you may find that, in trying to prove one theorem, you require another theorem which you have not yet proved. In proof we must avoid circular references. A lot of what is in this chapter will push the students in terms of rigorous proof from what they have done before. It is unfortunate that we were not forewarned that words such as contrapositive, implication, converse, equivalence, and negation would be in 11
this course, otherwise we would have included a chapter on Mathematical Logic in previous years. CHAPTER 3: VECTORS A Vectors and scalars Unit 1, Topic 3 Unit 1 B Geometric operations with vectors Sub-topic 3.1 Topic 2 C Vectors in the plane D The magnitude of a vector E Operations with plane vectors Unit 1 F The vector between two points Topic 3 G Parallelism Sub-topic 3.2 H Problems involving vector operations I The scalar product of two vectors Unit 1 J The angle between two vectors Topic 3 K Vector projection Sub-topic 3.3 L Proof using vector geometry Unit 1, Topic 3, Sub-topic 3.4 This chapter gives students their first look at vectors. In accordance with the syllabus, this chapter deals with vectors in 2 dimensions, leaving 3-dimensional vectors for Year 12. In the SACE syllabus, there are several statements in the right hand column that we feel are inappropriate or misleading: 1) Calculating the projection algebraically leads to the definition of the scalar product and the formula for the cosine of the angle between two vectors. This is like saying that the vector cross product should lead to the definition of a 3 3 matrix determinant. In practice we define 3 3 determinants first, then find that they have an application in the vector cross product. In the same way, we should define the scalar product first, then explore its application in projection. Otherwise we need to introduce the concept of projection, backtrack to a definition of dot product, and then return to the original context. This way, when the angle between two vectors is investigated, we find the scalar product embedded within it, and the implications of a positive, negative, or zero scalar product are clear. 2) The syllabus also cites calculating the assistance a plane receives from wind in a particular direction as an example of an application of projections. However, care must be taken here, because we cannot simply take the projection of the wind vector onto the direction of travel to analyse the change in course. For example, if I want to fly due north, and there is a wind from the south-west, the projection of the wind onto my flight does NOT give the assistance of the wind. This is because the wind is actually blowing the plane off course. Therefore, we need to turn the plane to face slightly north-west, and let the wind blow it back on course. We have included a discussion to clarify this issue. As an application of scalar multiplication, we have included an investigation on linear combinations. 12
The chapter concludes with a section on proof using vector geometry. Teachers should be aware that students may be encountering these types of proofs for the first time. CHAPTER 4: TRIGONOMETRIC FUNCTIONS A The general tangent function B General trigonometric functions C Reciprocal trigonometric functions Unit 2 Unit 2 D Solving trigonometric equations Topic 4 Topic 1 E Modelling using trigonometric functions Sub-topic 4.1, 4.2 F Trigonometric relationships G Double angle identities H Angle sum and difference identities I Trigonometric equations in quadratic form (Extension) This chapter follows directly from the chapter Trigonometric functions in the Methods course. It should not be started until students have completed the Methods chapter. If necessary, Specialist students may move ahead to Chapter 5 Matrices while the Methods trigonometry content is completed, and then return to this chapter after. In this chapter we revise and extend the work done on general trigonometric functions and their graphs. We introduce reciprocal trigonometric functions, and solve more complicated trigonometric equations. We are then in a position to form detailed trigonometric models of real-world situations. Finally, we move on to study and apply important trigonometric identities. CHAPTER 5: MATRICES A Matrix structure B Matrix operations and definitions Unit 2 Unit 2 C Matrix multiplication Topic 5 Topic 2 D The inverse of a 2 2 matrix Sub-topic 5.1 E Simultaneous linear equations F Translations and lines in 2-D G Linear transformations H Rotations about the origin Unit 2 I Reflections Topic 5 J Dilations Sub-topic 5.2 K Compositions of transformations L The inverse of a linear transformation 13
The chapter starts with a fairly traditional treatment of matrices. We look at matrix structure and operations, and we use inverse matrices to solve simultaneous linear equations. We note that only 2 2 inverses are covered, which means that the intersection of planes in Year 12 will need to be done by row operations (as per IB HL Mathematics). The remainder of the chapter is more advanced, looking at transformations in the plane. We know that this material will be unfamiliar to many teachers as well as students, and we are looking to run some PD workshops to help you. It is unclear exactly what is required for translations and their representation as column vectors. We took the opportunity to present the vector work on the parametric equation of a line and its real-world applications. The solution to these problems corresponds to the translation of a point through a scalar multiple of a vector matrix. We finish with the composition of transformations and inverse of transformations, which provides interesting contrast with what we did with functions. When teaching this topic, it is important to keep in mind that matrices are not returned to in Year 12, so students should not spend more time than necessary completing this chapter. CHAPTER 6: A B MATHEMATICAL INDUCTION The process of induction The principle of mathematical induction SACE ACARA C Proof of divisibility Unit 2 Unit 2 D Proofs for sequences and series Topic 6 Topic 3 E Proofs for products Sub-topic 6.2 F G Induction with matrices Proof of inequalities The Principle of Mathematical Induction is a sub-topic of Real and complex numbers, but it can be applied to many other areas, such as sequences and series, matrices, and trigonometry. We therefore took the opportunity to present a more complete picture of what students can do with induction. It ties together material from earlier in the course, providing the opportunity for revision. Teachers should feel free to skip through the induction applied to these other areas if they feel it is not appropriate for their students. 14
CHAPTER 7: REAL AND COMPLEX NUMBERS A Interval notation Unit 2 Unit 2 B Rational numbers Topic 6 Topic 3 C Division by surds Sub-topic 6.1 D Irrational numbers E The invention of imaginary numbers F Complex numbers Unit 2 G Complex conjugates Topic 6 H Complex numbers as 2-D vectors Sub-topic 6.3, I Modulus 6.4, 6.5 Sections A to D focus on the real number line, looking at interval notation, and rational and irrational numbers. We have skipped the basic revision of surds, as this is covered in the Methods course. However, division by surds is included, as this will be important later in the chapter. The remainder of the chapter provides an introduction to complex numbers which will be taken up and extended in Year 12. We find it inconsistent that the right hand column of the SACE syllabus presents division of complex numbers as a rearrangement of the product of complex numbers, for example (2 i)(1 + i) = (3 + i), when we are already asked to draw parallels with the arithmetic surds. In fact, the division of surds and the use of radical conjugates is included solely for the purpose of comparing with complex conjugates. 15