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Diploma Programme subject outline—Group 5: mathematics and computer science School name Westlake High School Name of the DP subject Mathematics SL School code 923373 Level Higher (indicate with X) Name of the teacher who completed this outline Gladden, Latricia Date when outline was completed January 2015 Standard completed in two years X Standard completed in one year * Date of IB training Name of workshop (indicate name of subject and workshop category) Summer 2014 Mathematics SL (Category 1) * All Diploma Programme courses are designed as two-year learning experiences. However, up to two standard level subjects, excluding languages ab initio and pilot subjects, can be completed in one year, according to conditions established in the Handbook of procedures for the Diploma Programme. 1. Course outline – Use the following table to organize the topics to be taught in the course. If you need to include topics that cover other requirements you have to teach (for example, national syllabus), make sure that you do so in an integrated way, but also differentiate them using italics. Add as many rows as you need. – This document should not be a day-by-day accounting of each unit. It is an outline showing how you will distribute the topics and the time to ensure that students are prepared to comply with the requirements of the subject. – This outline should show how you will develop the teaching of the subject. It should reflect the individual nature of the course in your classroom and should not just be a “copy and paste” from the subject guide. – If you will teach both higher and standard level, make sure that this is clearly identified in your outline. Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 1 Topic 2: Functions and Equations Topic 1: Algebra 2.4 Quadratic Functions Solve(2.7) Graph (2.2) Characteristics o Domain/Range o Relative extrema o Intercepts o Axis of symmetry transformations(2.3) Applications(2.8) 1.2 Elementary treatment of exponents and logarithms. Laws of exponents Laws of logarithms Change of base Solving equations(2.7) 2.6 Exponential functions Solve(2.7) Graph (2.2) Characteristics o Domain/Range o asymptotes o Intercepts o increasing/decreasing transformations(2.3) Applications(2.8) 2.1 Inverse function Establish the relationship between exponents and logarithms as inverses algebraically and graphically. 2.6 Logarithmic functions Solve(2.7) Graph (2.2) Characteristics o Domain/Range o asymptotes o Intercepts o increasing/decreasing transformations(2.3) Applications(2.8) One class is 90 In one week there are 2/3 classes. 5 class periods 3 class periods 3 class periods 1 class periods 3 class periods minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 1 Topic 1: Algebra Topic 2: Functions and Equations One class is 90 In one week there are 2/3 classes. 1.1 Arithmetic sequences and series Sum of a finite series using sigma notation Geometric sequences and series Sum of a finite and infinite series using sigma notation Applications 3 class periods 1.2 The binomial theorem Expansion of (a+b)n Calculation of binomial coefficients using Pascal’s triangle and () 2 class periods 2.5 The reciprocal functions Solve(2.7) Graph (2.2) Self-inverse nature(2.1) Characteristics o Domain/Range o asymptotes o Intercepts o increasing/decreasing Transformations(2.3) Applications(2.8) 3 class periods 2.1 Composite Functions 1 class periods minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 1 Topic 3: Circular functions and trigonometry One class is 90 In one week there are 2/3 classes. 3.1 The circle: Degrees to radian measure of angles Length of an arc Area of a sector. 2 class periods 3.2 Definition of cosθ and sinθ in terms of the unit circle Definition of tanθ as sin θ /cos θ . Exact values of trigonometric ratios of π/ 6, π/4,π/3, π/2,0 and their multiples. 3 class periods 3.4 The circular functions sin x , cos x and tan x : Solve(2.7) Graph (2.2) Characteristics o Domain/Range o Asymptotes o amplitude o periodic nature o Intercepts Transformations(2.3) Applications(2.8) 3 class periods 3.3 Trigonometric Identities Pythagorean identity Double angle identity (sine and cosine) Relationship between trigonometric ratios 3 class periods 3.5 Solving trigonometric equations on a finite 3 class periods interval including quadratic equations in sine cosine or tangent. Analytically Graphically 3.6 Solutions if triangles Law of Sines (including the ambiguous case) Law of Cosines 1 Area of a triangle 2 Applications(2.8) 4 class periods minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 1 Topic 6: Calculus One class is 90 In one week there are 2/3 classes. 6.1 Informal ideas of limit and convergence 5 class periods Investigate limits graphically, numerically and algebraically Definition of the derivative o f’(x)=lim o f’(x)=lim ℎ→0 → (+ℎ)−() ℎ ()−() − Interpret the derivative as a rate of change Interpret the derivative as the slope function Find and use the equation of tangent lines to approximate values of functions Find equations of normal lines 6.2 Use of Derivatives rules on polynomials, trigonometric functions, ex, lnx, (include ax, rational functions, radical functions, and inverse functions) Sum and difference Product rule Quotient rule Chain rule for composite functions Second derivatives Higher order derivatives Implicit differentiation 3 class periods 10 class periods 6.3 Applications of Derivatives Curve Analysis Find local and global extrema using derivative tests Find points of inflection using the second derivative test for concavity Relate the behavior of the graphs of f f’ and f” Solve optimization problems Solve Related rates problems using implicit differentiation Solve motion problems determining when an object is at rest, moving in a positive/negative direction, speeding up/slowing down minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 1 Topic 6: Calculus One class is 90 In one week there are 2/3 classes. 6.4 Indefinite integration 3 class periods Polynomials Trigonometric functions 1 ex and Composite functions with a linear function Integration using U substitution for ∫ (())′ () minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD 6.5 Definite integration with applications 10 class periods Analytically, graphically and using technology Differential equations Slopefields Find area under a curve Find area between two curves Find volumes of revolution , about horizontal and vertical axis Find volumes using cross sections of different shapes using bases with different shapes. i.e. squares, circles, triangles, and rectangles. 3 class periods Year 2 Topic 4: Vectors 6.6 Integration problems n involving displacement, total distance traveled, velocity and acceleration. 4.1 Vectors as displacements in a plane and in three dimensions Components of a vector and column representation Algebraic and geometric approaches to o Sum and difference of two vectors, o The zero vector, o The negative vector o Multiplication by a scalar o Magnitude of a vector and o Unit vectors i,j and k o Position vectors 5 class periods 5 class periods 4.2 Scalar product of two vectors, Perpendicular vectors Parallel vectors angles between two vectors Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 2 Topic 4: Vectors Topic: 5 Statistics and Probability One class is 90 In one week there are 2/3 classes. 4.3 Representation of a line as r=a+tb and the angle 5 class periods between two lines; 4.4 Distinguish between coincident and parallel lines 5 class periods Determine if lines intersect locate the point of intersection of two lines as the solution to a system 5.1 Comparison of concepts 5 class periods Population Sample Random sample Discrete data Continuous data Graphical Representations of data Frequency table Frequency histograms Box-and-whisker plots o Use of upper and lower boundaries o Interval widths o Mid-interval values 5.2 Calculations of Statistical measures and make interpretations of data using the results. Mean, median, mode, Range, interquartile range, quartiles, percentiles, variance and standard deviation 5.3 Find median, quartiles and percentiles from cumulative frequencies and their graphs 5 class periods 5 class periods 5.8 Binomial distribution Binomial probabilities 5 class periods Mean and Variance of the binomial distribution Determine when random variables are best described using binomial distributions 5 class periods 5.9 Normal distribution and the “Bell” curve Standardization of normal variables Z-values and z-scores Characteristics of a normal distribution minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Students will be assessed Safari Montage, Ti-84 Plus GDC’s, using formative assessments, Promethean Board and Active weekly topic quizzes, monthly expressions, All-in-Learning Student cumulative assessments, response remotes and data analysis. performance assessments, the Fathom software, Calculus in motion internal assessment Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD Allocated time Topic (as identified in the IB subject guide) Contents State the topics in the order you are planning to teach them. Year 2 Topic: 5 Statistics and Probability One class is 90 In one week there are 2/3 classes. Various Math Exploration Activities 10 class periods 5.4 Linear correlation of bivariate data 5 class periods Pearson’s product-moment correlation coefficient r Scatter plots and linear regression equations Use regressions lines to make predictions Use mathematical findings to make interpretations in the context of the problem. 5 class periods 5.5 Probability Concepts trials, outcomes, equally likely outcomes, sample space and event probability of an event and its complement Use of Venn diagrams, tree diagrams and tables to solve problems 5 class periods 5.6 Find probabilities Combined events Mutually exclusive events Conditional probability definition Independent events definition Probabilities with and without replacement 5.7 Concept of discrete random variables and their probability distributions Find expected value for discrete data Investigate real world uses of probability distribution and expected value IB Exam Review 5 class periods 15 class periods minutes. Resources Assessment List the main resources to be instruments to be used used, including information technology if applicable. Safari Montage, Ti-84 Plus GDC’s, Promethean Board and Active expressions, All-in-Learning Student response remotes and data analysis. Fathom software, Calculus in motion Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD 2. IB internal assessment requirement to be completed during the course Briefly explain how and when you will work on it. Include the date when you will first introduce the internal assessment requirement to your students, the different stages and when the internal assessment requirement will be due. The internal assessment (IA) will involve investigations and problem solving assignments. Students will be assessed on their ability to read, interpret and solve a given problem using appropriate mathematical terms, organize and present information and data in tabular, graphical and/or diagrammatic forms, using appropriate notation and terminology, formulate a mathematical argument and communicate it clearly, select and use appropriate mathematical strategies and techniques, demonstrate an understanding of both the significance and the reasonableness of results, recognize patterns and structures, make generalizations, recognize and demonstrate an understanding of the practical applications of mathematics, use appropriate technological devices as mathematical tools and demonstrate an understanding of and the appropriate use of mathematical modeling. The topic will be introduced and students will begin working on the IA when Calculus is introduced. The IA will be due at the beginning of the second semester of the second year. 3. Links to TOK You are expected to explore links between the topics of your subject and TOK. As an example of how you would do this, choose one topic from your course outline that would allow your students to make links with TOK. Describe how you would plan the lesson. Topic Link with TOK (including description of lesson plan) 1.1 Arithmetic sequences and Students in pairs would be challenged to find the sum of the first 100 integers without the use of a calculator. After they work on the task for a few minutes I’ll let them series: Sum of a finite know that an 8 year old was able to find the sum in his head. After pairs arrive at their conclusions we will discuss their methods and any patterns they observed. Students series using sigma notation will be prompted to find an equation that describes their pattern or process and verify that it works for any number of integers. We will then discuss the idea of mathematical intuition and the basis for formal proof. 4. International mindedness Every IB course should contribute to the development of international mindedness in students. As an example of how you would do this, choose one topic from your outline that would allow your students to analyze it from different cultural perspectives. Briefly explain the reason for your choice and what resources you will use to achieve this goal. Topic 3.1 Circular functions and trigonometry Contribution to the development of international mindedness (including resources you will use) Students will investigate the origins of the metric system and countries willingness to adopt the metric system for their units of measure for distance, weight and capacity. Students will then investigate the answer the question. Why haven’t circular functions been converted to the metric system? Students will research the Babylonian origins of the 360 degree circle and its influences on the world. 5. Development of the IB learner profile Through the course it is also expected that students will develop the attributes of the IB learner profile. As an example of how you would do this, choose one topic from your course outline and explain how the contents and related skills would pursue the development of any attribute(s) of the IB learner profile that you will identify. Topic Random sampling from populations; graphical displays of data 6. Contribution to the development of the attribute(s) of the IB learner profile Contribution to the development of the attribute(s) of the IB learner profile: Inquiry, risk-taking, reflection. Sampling from populations and then displaying the results requires a fundamental knowledge of the tools necessary to achieve accurate results. Clear presentation is also necessary. The impact of displaying accurate versus inaccurate results on public opinion and elections throughout the world will be discussed. Suggestions for presenting and displaying clear and meaningful results will be debated. Resources Describe the resources that you and your student will have to support the subject. Indicate whether they are sufficient in terms of quality, quantity and variety. Briefly describe what plans are in place if changes are needed. Students and the teacher will utilize various resources such as Safari Montage, Ti-84 Plus GDC’s, Promethean Board and Active expressions; All-in-Learning Student response remotes and data analysis, Fathom software, Calculus in motion Sketchpad to visually display concepts. Test generator software, USA Test prep where applicable, Text book TBD