 # Mathematics Analysis and Approaches

Mathematics has been described as the study of structure, order and relation that has evolved from the practices of counting, measuring and describing objects. Mathematics provides a unique language to describe, explore and communicate the nature of the world we live in as well as being a constantly building body of knowledge and truth in itself that is distinctive in its certainty. These two aspects of mathematics, a discipline that is studied for its intrinsic pleasure and a means to explore and understand the world we live in, are both separate yet closely linked.

Mathematics is driven by abstract concepts and generalization. This mathematics is drawn out of ideas, and develops through linking these ideas and developing new ones. These mathematical ideas may have no immediate practical application. Doing such mathematics is about digging deeper to increase mathematical knowledge and truth. The new knowledge is presented in the form of theorems that have been built from axioms and logical mathematical arguments and a theorem is only accepted as true when it has been proven. The body of knowledge that makes up mathematics is not fixed; it has grown during human history and is growing at an increasing rate.

The side of mathematics that is based on describing our world and solving practical problems is often carried out in the context of another area of study. Mathematics is used in a diverse range of disciplines as both a language and a tool to explore the universe; alongside this its applications include analyzing trends, making predictions, quantifying risk, exploring relationships and interdependence.

While these two different facets of mathematics may seem separate, they are often deeply connected. When mathematics is developed, history has taught us that a seemingly obscure, abstract mathematical theorem or fact may in time be highly significant. On the other hand, much mathematics is developed in response to the needs of other disciplines.

The two mathematics courses available to Diploma Programme (DP) students express both the differences that exist in mathematics described above and the connections between them. These two courses might approach mathematics from different perspectives, but they are connected by the same mathematical body of knowledge, ways of thinking and approaches to problems. The differences in the courses may also be related to the types of tools, for instance technology, that are used to solve abstract or practical problems. The next section will describe in more detail the two available courses.

Mathematics: analysis and approaches

This course recognizes the need for analytical expertise in a world where innovation is increasingly dependent on a deep understanding of mathematics. This course includes topics that are both traditionally part of a pre-university mathematics course (for example, functions, trigonometry, calculus) as well as topics that are amenable to investigation, conjecture and proof, for instance the study of sequences and series at both SL and HL, and proof by induction at HL.

The course allows the use of technology, as fluency in relevant mathematical software and hand-held technology is important regardless of choice of course. However, Mathematics: analysis and approaches has a strong emphasis on the ability to construct, communicate and justify correct mathematical arguments.

Mathematics: analysis and approaches: Distinction between SL and HL

Students who choose Mathematics: analysis and approaches at SL or HL should be comfortable in the manipulation of algebraic expressions and enjoy the recognition of patterns and understand the mathematical generalization of these patterns. Students who wish to take Mathematics: analysis and approaches at higher level will have strong algebraic skills and the ability to understand simple proof. They will be students who enjoy spending time with problems and get pleasure and satisfaction from solving challenging problems.

Mathematics: applications and interpretation

This course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. As such, it emphasizes the meaning of mathematics in context by focusing on topics that are often used as applications or in mathematical modelling. To give this understanding a firm base, this course also includes topics that are traditionally part of a pre-university mathematics course such as calculus and statistics.

The course makes extensive use of technology to allow students to explore and construct mathematical models. Mathematics: applications and interpretation will develop mathematical thinking, often in the context of a practical problem and using technology to justify conjectures.

Mathematics: applications and interpretation: Distinction between SL and HL

Students who choose Mathematics: applications and interpretation at SL or HL should enjoy seeing mathematics used in real-world contexts and to solve real-world problems. Students who wish to take Mathematics: applications and interpretation at higher level will have good algebraic skills and experience of solving real-world problems. They will be students who get pleasure and satisfaction when exploring challenging problems and who are comfortable to undertake this exploration using technology.

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 THE LEARNING JOURNEY FOR MATHS HL ANALYSIS Unit / Block of work Key Episodes / Questions Colour Code Length of time. Learner Attribute(s) Algebra and function basics Equations and graphs; Systems of linear equations; domain and range; function notation; composite functions; inverse functions; transformation of functions; reciprocal functions Blue Communicator Functions Polynomials; Factor and remainder theorem; quadratic functions; rational functions; solving inequalities; partial fractions Blue Thinker Sequences and series Arithmetic and geometric sequences; Arithmetic and geometric series; sum of infinite geometric series; sigma notation; binomial theorem; combinations and permutations Red Inquirer Exponential and logarithmic functions Exponential and logarithmic graphs; Exponential growth and decay; Natural logarithms; solving equations involving exponentials and logarithms Red Reflective Proofs Logic basics; Direct proof; Proof by contrapositive; Proof by contradiction; proof by counter example; proof by mathematial induction Red Open minded Trigonometric functions and equations Arc length and setor area; unit circle; exact values; trig identities; graphs of trig functions; transformation of trig graphs; solving equations; reciprocal trig functions; inverse trig functions; compound angles; double angles Green Knowledgable Geometry and trigonometry 3D distance and midpoint; volume and surface area; sine and cosine rule; area of a triangle; solving problems in 2D and 3D; bearings Green Reflective Complex numbers Cartesian form; polar form; sum, product and quotients of complex numbers; roots of complex quadratic functions; De Moivre’s theorem Red Open minded Differential calculus 1 Limits; first principles; derivatives of functions; applying derivatives to graphs; second derivatives; increasing and decreasing functions; turning points; kinematics; tangents and normals Purple Communicator Integral calculus 1 Calculating definite integrals; Area under curves; area between curves; kinematics; integration by inspection; integration by partial fractions; integration by substitution; integration by parts; volumes of revolution Purple Inquirer

 THE LEARNING JOURNEY FOR MATHS SL ANALYSIS Unit / Block of work Key Episodes / Questions Colour Code Length of time. Learner Attribute(s) Understanding Functions Domain and range Inverse functionsSketching functionsTechnology to graph functionsComposite functions Blue 2 Caring Linear and quadratic functions Equation of a straight lineParallel and perpendicular lines Determine key features of graphs                                               The discriminantSolution of quadratic equations and inequalitiesUse of technology to solveTransformations of graphsComposite transformations Blue 3 Principled Rational Functions The reciprocal function and its graph Rational functions and their graphs.Equations of vertical and horizontal asymptotes. Blue 3 Balanced Geometry and Trigonometry The distance between two points in 3D, and their midpointVolume and surface area of 3D solidsSOHCAHTOAThe sine ruleThe cosine ruleArea of a triangleAngles of elevation and depression. Green 2 Reflective Trigonometric functions length of an arc, area of a sector.Exact values of trigonometric ratiosTrigonometric IdentitiesTrigonometric functions Composite functions of the form Transformations.Solving trigonometric equations Green 3 Thinker Differentiation Derivative interpreted as gradient functionIncreasing and decreasing functions.Tangents and normals and their equationsThe chain rule, product rule and quotient rules.Second derivativeLocal maximum and minimum points.Optimization.Points of inflexion with zero and non-zero gradients.Kinematics Purple 5 Knowledgeable Logs and Exponential Laws of exponentsNumerical evaluation of logarithms using technology.Laws of exponents with rational exponents.Laws of logarithms.Change of base of a logarithm.Solving exponential equations, including using logarithms. Red 3 Open minded Basic Statistics OutliersSampling techniquesPresentation of data                                                      Histograms.Cumulative frequencyBox and whisker diagrams.Mean, median and modeEstimation of mean from grouped data.Modal class.Interquartile range, standard deviation and variance Yellow 2 Inquiror Correlation and Regression Linear correlationPearson’s product-moment correlation coefficientScatter diagramsEquation of the regression line Yellow 1 Communicator Probability Sample spacesThe complementary events                                                                                           Expected number of occurrencesVenn diagrams, tree diagrams, sample space diagrams and tablesCombined events Mutually exclusive eventsConditional probabilityIndependent events Yellow 2 Risk taker
 THE LEARNING JOURNEY FOR MATHS HL ANALYSIS Unit / Block of work Key Episodes / Questions Colour Code Length of time (weeks) Learner Attribute(s) Statistics Disrete and continuous data; reliability; outliers; presenting data; grouped data; measures of central tendancy; measures of spread; linear correlation; regression models Yellow Reflective Probability Sample spaces; combined events; mutually exclusive events; conditional probability; independent events; venn diagrams; Bayes theorem Yellow Open minded Probability distributions Discrete random variables; continuous random variables; linear transformations of X; normal distribution; binomial distribution Yellow Risk taker Vectors, lines, and planes Vectors as dispacements; unit vectors; sum of vectors; zero vector; multiplication by a scalar; magnitude of a vector; position vector; scalar product; angle between vectors; vector equation of a line; angle between two lines; coincident, parallel, intersecting and skew lines; points of intersection; vector product; vector equation of a plane; normal vectors; cartesian equation of a plane; intersection of a line and plane(s); angle between a line and plane(s) Green Thinker Differential calculus 2 Derivative of composite function; derivative of product/quotient; derivative of trigonometric functions and exponentials; rates of change; optimisation; L’Hopital’s rule Purple Inquirer Integral calculus 2 Euler’s method; variable seperable equations; homogeneous equations; solving using integrating factors; Maclaurin series Purple Knowledgable

 THE LEARNING JOURNEY FOR MATHS SL ANALYSIS Unit / Block of work Key Episodes / Questions Colour Code Length of time (weeks) Learner Attribute(s) Probability Distribution Discrete random variables and their probability distributionsExpected value for discrete dataBinomial distributionThe normal distribution Yellow 3 Inquirer Integration Indefinite and definite integrals                                              Definite integrals using technologyArea of a region enclosed by a curve and axisAreas between curves. Purple 4 Thinker Further Calculus Derivative and integrals of trig functions, ln x and exponentialsKinematic problems         Integration by inspection or by substitution Purple 4 Open minded Sequences and series Arithmetic sequences and seriesGeometric sequences and series                                                Sum of infinite geometric sequences.Sigma notation                                                                                         Financial applications                                                                            Deductive proofThe binomial theorem Red 3 Caring