# Mathematics Applications and Interpretations

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.

Please click on the arrow for a detailed breakdown:

 THE LEARNING JOURNEY FOR MATHS HL APPLICATIONS Unit / Block of work Key Episodes / Questions Colour Code Length of time (weeks) Learner Attribute(s) Functions Range and domain; assymptotes; inverse functions; composite functions; piecewise functions Blue 3 Thinker Sequences and series Arithmetic series: term rule and summation; geometric series: term rule, summation and infinite series; sigma notation. Red 3 Inquirer Statistical measures Measures of central tendancy; measures of dispersion; box plots; cumulative frequency graphs; histograms; sampling techniques; weighted means and variances. Yellow 4 Open minded Linear functions and regression Scatter graphs; Pearson’s product moment corellation coefficient; Spearman’s rank; linear regression; non-linear regression; corellation t-test. Yellow 6 Communicator Quadratics Interpreting a quadratic: minimum, maximum, roots, y-intercept, vertex; finding the equation of a quadratic. Red 3 Knowledgeable Probability Venn diagrams; probability trees; sample spaces; conditional probability; binomial distribution; Poission distribution; Normal distribution; combined distribution; central limit theorem. Yellow 8 Balanced Testing Distribution testing; t-tests: 1 sample, 2 sample and paired; Chi-squared tests: independance; goodness of fit; estimation of parameters; confidence intervals: t and Z. Yellow 6 Pricipled Matrices Multiplication; determinants; inverses; solving simultaneous equations; linear transformations; affine transformations; eigen values and eigen vectors; diagonalising. Green 8 Knowledgeable

 THE LEARNING JOURNEY FOR MATHS SL APPLICATIONS Unit / Block of work Key Episodes / Questions Colour Code Length of time (weeks) Learner Attribute(s) Functions Function notation; inverse functions; sketching; linear models; asyptotes; modelling using functions; domain and range. Blue 2 Communicator Arithmetic sequences Arithmetic sequences and series; sigma notation; position to term rule; sum of a series. Red 2 Inquirer Statistical measures Sampling techniques; outliers; histograms; cumulative frequency; box and whisker diagrams; averages; measures of spread. Yellow 4 Knowlegable Linear functions and regression Linear correlation; Pearson’s product moment correlation coefficient; scatter diagrams; equation of regression; predictions. Yellow 2 Probabililty Complementary events; expected outcomes; Venn diagrams; tree diagrams; sample spaces; combined events; conditional probability; independent events; mutually exclusive events; discrete random variables Yellow 3 Distributions Binomial distributions; mean and variance; normal distribution; bell curves; normal cumulative distribution; inverse normal. Yellow 3 Testing Spearman’s rank; hypothesis; significance levels; chi-squared independance; chi-squared goodness of fit; t-test. Yellow 4 Thinker Quadratics Sketching; minimum, maximum and intercepts; axis of symmetry; direct and inverse variation; cubi models; domain and range. Red 4 Exponential and logarithmic fuctions Geometric sequences and series; position to termrule; sigma notation; sum of a series; financial applications; laws of exponents; introduction for logarithms; amortization and annuities. Red 4
 THE LEARNING JOURNEY FOR MATHS HL APPLICATIONS Unit / Block of work Key Episodes / Questions Colour Code Length of time (weeks) Learner Attribute(s) Voronoi diagrams Interpretation of cells and boundaries; toxic waste problems; perpendicular bisectors. Green 2 Knowledgable Radians and sinusodial functions Amplitude, period and priciple axis; modelling sinusodial functions; phase shift; radians; sectors and arcs. Green 2 Open minded Exponetials and logarithms Exponential models; logarithms laws; logarithmic models; logistics models. Blue 3 Thinker Complex numbers Quadratic equations; conjugates; real and imaginary numbers; argand diagrams; polar and cartesian form; arguments and modulus; de moivre’s theorem. Red 4 Open minded Networks and graphs Graph terminology; adjacency matrices; transition matrices; Prim’s and Kruskals algorithms; Chinese postman problem; travelling salesperson problem: nearest neighbour algorithm and deleted vertex algorithm. Green 6 Thinker Vectors Scalars and vectors; position vectors; unit vectors; scalar product; angle between two vectors; vector product; area of a parallogram and triangle. Green 2 Thinker Finance Compound interest; loans; investments; inflation. Red 2 Knowledgable Calculus Differentiation; stationary points and gradients; tangents and normals; chain, product and quotient rules; integration; ingration by inspection; area under a curve; volumes of revolution; kinematics (including vectors); seperable differential equations; slope fields; Euler’s method for first order differential equations; coupled differential equations; predator prey models; phase portraits; Euler’s method for coupled differential equations; Euler’s method for second order differential equations. Purple 6 Risk taker

 THE LEARNING JOURNEY FOR MATHS SL APPLICATIONS Unit / Block of work Key Episodes / Questions Colour Code Length of time. Trigonometry and volume Distance between two points; volume and surface area of: right pyramid, cone, sphere and hemisphere; finding angles in 3D; right angled trigonometry; sine rule; cosine rule; area of a triangle; pythagoras; angles of elevation and depression. Green 3 Trigonometric fuctions Sinusoidal models; domain and range. Blue 2 Coordinate geometry, lines, Voronoi diagrams Equation of a straight line; gradient; parallel lines; perpendicular lines; equations of perpendicular bisectors; voronoi diagrams: sites, vertices, edges and cells; toxic waste problems. Green 4 Differential calculus Gradient functions; increasing and decreasing functions; derivative of polynomials; tangents and normals; stationary points; optimisation. Purple 7 Integration Integrate polynomials; definite intergration; area of a region enclosed by a curve; trapezoidal rule. Purple 3 Accuracy and 2D geometry Standard form; upper and lower bounds; percentage error. Red 1