Mathematics: applications and interpretation SL
Syllabus Outline
Chapter 1: Approximations and errorSyllabus Outline
A Rounding numbers
B Approximations
C Errors in measurement
D Absolute and percentage error
Description:
- The first two sections of this chapter are likely to be quite straightforward for most students: indeed, much of the work students would have done in the Core Topics SL book will have assumed a familiarity with rounding numbers. Teachers should feel free to skip these early sections if the students are already comfortable with this material.
Chapter 2: Loans and annuities
A Loans
B Annuities
Description:
- This chapter follows on from the work done in Chapter 5 of the Core Topics SL book, where technology was used to solve problems involving compound interest investments. In this chapter we consider the more complex cases of loans and annuities, where regular payments or withdrawals are made from the account. It would be useful for students to see loans and annuities as the “reverse models” of each other, in which the bank and the individual are essentially swapping roles.
Chapter 3: Functions
A Relations and functions
B Function notation
C Domain and range
D Graphs of functions
E Sign diagrams
F Transformations of graphs
G Inverse functions
Description:
- Although much of the content of this chapter is shared with the Analysis and approaches course, this chapter has been put in the Applications and interpretation book because this course requires a more informal treatment of functions. In particular, inverse functions are explained geometrically (as a reflection in the line y=x) rather than algebraically. There is also a more informal treatment of asymptotes, although students are expected to identify asymptotes from a graph.
- Although transformations of graphs are not explicitly included in the syllabus, in this chapter we will consider vertical translations, and vertical and horizontal stretches of functions, as these will help students in the study of quadratic, exponential, and trigonometric functions in later chapters. Reflections of functions are also presented in preparation for reflecting functions in the line y=x to give inverse functions.
- Sign diagrams are presented in this chapter, as they will be useful in calculus.
Chapter 4: Modelling
A The modelling cycle
B Linear models
C Piecewise linear models
D Systems of equations
Description:
- In this chapter, students are introduced to the concept of a mathematical model. We see that we can take real-life problems, and then make assumptions about the situation in order to simplify it to a form which can be represented mathematically. However, it is important to assess the reasonableness of the assumptions, and to understand that the assumptions affect the accuracy of the final answer. We must therefore strike a balance between simplicity and accuracy.
- In the final section, we use technology to find unknown coefficients in a model. This is done by using given information to create a 2x2 or 3x3 system of simultaneous equations. This method will be used in future chapters to determine quadratic, exponential, and trigonometric models.
Chapter 5: Bivariate statistics
A Association between numerical variables
B Pearson’s product-moment correlation coefficient
C The coefficient of determination
D Line of best fit by eye
E The least squares regression line
F Spearman’s rank correlation coefficient
Description:
- Most of this chapter is similar in style and presentation as in previous books and the MYP series.
- Section F (Spearman’s rank correlation coefficient) is new. We motivate the need for the rank correlation by focussing on the direction of a trend that may not necessarily be linear.
- We would expect classes to reach the end of this chapter by the end of the first year.
Chapter 6: Quadratic functions
A Quadratic functions
B Graphs from tables of values
C Axes intercepts
D Graphs of the form y = ax^2
E Graphs of quadratic functions
F Axis of symmetry
G Vertex
H Finding a quadratic from its graph
I Intersection of graphs
J Quadratic models
Description:
- Since solving quadratic equations is not part of the Applications and interpretation SL syllabus, this chapter has much more of a technology focus than the equivalent chapter in the Analysis and approaches SL book. Technology is used to find the x-intercepts of a quadratic function, and more generally to find the value(s) of x for a given value of y.
- When finding a quadratic given information about the graph, students will sometimes need to solve a system of linear equations, which was studied in the Modelling chapter. Although some quadratic models were considered in the modelling chapter, here we extend that work now that students have the tools to consider properties of the quadratic model such as the axis of symmetry and vertex.
Chapter 7: Direct and inverse variation
A Direct variation
B Powers in direct variation
C Inverse variation
D Powers in inverse variation
E Determining the variation model
F Using technology to find variation models
Description:
- The material in Sections A to D is unlikely to be assessed directly, but is important so that students understand the concept of direct and inverse proportion. That being said, this material can be progressed through quickly if the students are familiar with it.
- In Section E, students are generally given the type of variation relationship that exists between two variables (for example, y is proportional to x^3), and must use given data to find the proportionality constant. The models in this section are exact.
- In Section F, students must use technology to find the best variation model connecting two variables, given a set of data. Students should be familiar with this process from their work on linear regression in Chapter 5. As is the case in Chapter 5, students should use the correlation coefficient r to determine how well the model fits the data. Students should also think about the context the data is given in, and use the context to sensibly round the parameters of the model. For example, we may suspect a shape’s area will be proportional to the square of its perimeter as it is enlarged. If the calculator returns a model with power 2.0001, this power should simply be rounded to 2.
Chapter 8: Exponentials and logarithms
A Exponential functions
B Graphing exponential functions from a table of values
C Graphs of exponential functions
D Exponential equations
E Growth and decay
F The natural exponential
G Logarithms in base 10
H Natural logarithms
Description:
- In this chapter students are introduced to exponential functions. After briefly encountering asymptotes in Chapter 3, students will investigate them in more detail here.
- Since solving exponential equations algebraically is not part of the Applications and interpretation SL course, students will use graphical methods to find the value of x for a given value of y.
- Students will once more use technology to find unknown coefficients of exponential functions, by using known information to construct systems of equations.
- The work done in transforming functions in Chapter 3 will come in useful here, as students should be able to recognise that y=2-x is a reflection of y=2x in the y-axis.
- In Section F, students are introduced to the natural exponential e. Since calculus involving e is not included in this course, it is difficult to motivate the use and importance of e. We present students with an Investigation to see how e appears in continuously compounding interest. Transformation of graphs again comes into play here, as students should recognise that y=erx is a horizontal stretch of y=ex with scale factor 1/r.
- The chapter ends with a brief introduction to logarithms. It is perhaps unfortunate that laws of logarithms are not included in this syllabus, as there is not much of use that students can do with the logarithms. The final question of the chapter guides students to the idea that y=lnx is the inverse function of y=ex.
Chapter 9: Trigonometric models
A The unit circle
B Periodic behaviour
C The sine and cosine functions
D General sine and cosine functions
E Modelling periodic behaviour
Description:
- This chapter begins by using the unit circle, to extend the definition of trigonometric ratios to all angles.
- We then use the transformations studied in Chapter 3 to construct graphs of trigonometric functions. Radians are not part of this course, so all trigonometric functions will be given in terms of angles in degrees. We use the modulus symbol to talk about the amplitude of a trigonometric function. If needed, some material on modulus is provided in the Background Knowledge.
- When using information to construct a trigonometric model, students should be careful to consider all the information given. For example, when modelling the height of a rotating windmill blade, students will need to consider the starting position of the blade, and whether the blade is rotating clockwise or anticlockwise.
Chapter 10: Differentiation
A Rates of change
B Instantaneous rates of change
C Limits
D The gradient of a tangent
E The derivative function
F Differentiation
G Rules for differentiation
Description:
- This chapter provides students with their first look at differential calculus. Although much of the calculus content is common between the SL courses, we expect the classes will be separated by the time they encounter calculus. Having the calculus chapters in the separate books allows a more targeted approach to calculus for each course.
- This chapter begins with rates of change, which is used to motivate an informal study of limits. A blended learning investigation, which combines limits, previous results, and technology, is used to explore the instantaneous rate of change for a curve.
- Since rules such as the product and quotient rules are not part of this course, we have included a section entitled “Rules of differentiation” at the end of this chapter, rather than it being a chapter of its own as in the Analysis and approaches SL book. This Section covers the rule about the derivative of xn, as well as the linearity of differentiation. These rules are sufficient to differentiate polynomial functions.
Chapter 11: Properties of curves
A Tangents
B Normals
C Increasing and decreasing
D Stationary points
Description:
- This chapter allows students to apply the calculus they have learnt to discover the properties of curves.
- This chapter is considerably shorter than the corresponding chapter in the Mathematics: analysis and approaches SL book. This is because, in this course, students can only differentiate polynomial functions, and inflection points are not part of this course. Only stationary inflections are mentioned, in the context of stationary points.
- Students should be encouraged to think of the concepts of increasing and decreasing in terms of intervals, rather than at a particular point. This will help students understand why, for example, the graph of y=x^2 is increasing for x>=0, and decreasing for x<=0.
Chapter 12: Applications of differentiation
A Rates of change
B Optimisation
C Modelling with calculus
Description:
- In this final chapter of differential calculus we explore some of its real world applications.
- Students should be encouraged to keep in mind the constraints imposed by the context of the problem, and to make sure their solution makes sense in this context.
- In Section C, we use known information to find unknown coefficients in models, like we have done in previous chapters. Now, however, some of the information is given in the context of the derivative, rather than of the model itself.
Chapter 13: Integration
A Approximating the area under a curve
B The Riemann integral
C The Fundamental Theorem of Calculus
D Antidifferentiation and indefinite integrals
E Rules for integration
F Particular values
G Definite integrals
H The area under a curve
Description:
- This material is may be challenging for some students, as integral calculus was not in the previous Mathematics Studies SL course. However, we believe the focus here will be on positive functions and the area under a curve, and for this application we believe integration is actually more intuitive to weaker students than differentiation.
- We begin with some numerical methods for finding the area under a curve, which is consistent with the history of integration, and so ties in well with IB thinking. Section B is titled “The Riemann Integral”, which may sound complicated, but all we are really doing is giving some integral notation to describe the area under a curve. Only then do we discuss the Fundamental Theorem of Calculus, which links the area under a curve with the differentiation we have studied previously. The Fundamental Theorem of Calculus therefore gives us motivation to antidifferentiate. We generalise from examples to give basic rules for integration, then tie it all together by returning to definite integrals and the area under a curve.
Chapter 14: Discrete random variables
A Random variables
B Discrete probability distributions
C Expectation
D The binomial distribution
E Using technology to find binomial probabilities
F The mean and standard deviation of a binomial distribution
Description:
- We start the chapter with an introduction to the concept of (discrete) random variables and their probability distributions. If you are going through the Core book and this book in chapter order, it will have been a long time since the students have seen probability. Before starting this chapter, it may be beneficial to briefly revise key probability concepts as they are assumed throughout this chapter, Chapter 15: “The normal distribution”, and Chapter 16: “Hypothesis testing”.
- Section C: “Expectation” continues on directly from Section J: “Making predictions with probability” from Chapter 10 of the Core book.
- Since the binomial theorem is not included in the Applications syllabus, we motivate and define the binomial coefficient using Pascal’s triangle instead. In calculating binomial probabilities without technology, we encourage students to use Pascal’s triangle to help find binomial coefficients.
Chapter 15: The normal distribution
A Introduction to the normal distribution
B Calculating probabilities
C Quantiles
Description:
- In the chapter’s introduction, we briefly mention probability density functions as the continuous analogue of probability mass functions. Here, we give the definition that the probability is the area under the curve without further comment or investigation. We are simply using the probability density function as a tool to justify the notion that area under the normal curve = probability later in the chapter, and nothing more.
- The first section introduces the normal distribution by focussing on how the normal distribution arises and exploring the shape of the distribution. Probability calculations are treated separately in the following section.
Chapter 16: Hypothesis testing
A Statistical hypotheses
B Student’s t-test
C The 2-sample t-test for comparing population means
D The goodness of fit test
E The test for independence
Description:
- We start the chapter by introducing the student to the terminology used in hypothesis testing. In particular, we first focus on statistical hypotheses: their definition, formulation, and role in a hypothesis test.
- We then introduce the hypothesis testing procedure via the t-test. At this point, it is important that students clearly understand what each component and step of the testing procedure means, as these concepts will reappear in every section following.
- Although the t-distribution is not in the SL Applications course, we have included a brief definition of it for the sake of conceptual understanding, as the concept of a test statistic’s distribution is fundamental to one’s understanding of the p-value as a probability. We are aware that students and teachers may skip the relevant Investigation and theory due to its non-examinable nature. Thus, we have provided an Example and calculator instructions for conducting the t-test using technology.
- Section C (Comparing population means) looks at the 2-sample version of the t-test. The calculations in this section are done completely with technology, unlike in Section B. We felt that with the two different cases for pooling data, giving the formulae for the test statistics of each case was counter-productive to conceptual understanding.
- In Section D (The goodness of fit test), we introduce the goodness of fit test and the concepts of observed and expected frequencies. Both p-values and critical values are covered for the decision rule. The Activity at the end of the Section shows how the goodness of fit test can be used in the context of modelling -- the primary theme of this course. Although subtle, it serves as a reminder that probability, statistics, and inference are just other facets of modelling.
- Section E (The test for independence) should be familiar to teachers who have previously taught the Studies SL course. However, this section has seen a major overhaul from its incarnation in the Studies book. It has been rewritten to be more like the preceding sections and follows on directly from Section D. Since the test for independence is just a special case of the goodness of fit test, it should be noted that the distribution and hence critical values are exactly the same as those used in the previous section. Only how the df (degrees of freedom) is calculated has changed.
- Although we provide procedures for each test type, it should be emphasised that the student shouldn’t be remembering them as completely separate procedures. Being able to identify the common elements, and how and why the procedures differ are paramount in successfully understanding the concepts in this chapter.
- It should also be noted that not all calculators will have the relevant hypothesis test functions. For example, the HP Prime Graphing calculator lacks functions to perform the goodness of fit test and the test for independence.
Chapter 17: Voronoi diagrams
A Voronoi diagrams
B Constructing Voronoi diagrams
C Adding a site to a Voronoi diagram
D Nearest neighbour interpolation
E The Largest Empty Circle problem
Description:
- We conclude this book with a study of Voronoi diagrams, a topic which is likely to be unfamiliar to many teachers.
- We begin with presenting students with Voronoi diagrams, and asking students to interpret the diagrams. We hope this will allow students to get used to the concept and the terminology, as well as to develop an intuition as to how the diagrams should look, before the students are asked to construct a Voronoi diagram of their own.
- We then guide students through the construction of Voronoi diagrams, including adding an extra site to a diagram, finding a missing site or edge, and finding the largest empty circle within a diagram. We hope that the contextual nature of these problems will make this chapter engaging for students.