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 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 functions Sketching functions Technology to graph functions Composite functions | Blue | 2 | Caring |

Linear and quadratic functions | Equation of a straight line Parallel and perpendicular lines Determine key features of graphs The discriminant Solution of quadratic equations and inequalities Use of technology to solve Transformations of graphs Composite 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 midpoint Volume and surface area of 3D solids SOHCAHTOA The sine rule The cosine rule Area of a triangle Angles of elevation and depression. | Green | 2 | Reflective |

Trigonometric functions | length of an arc, area of a sector. Exact values of trigonometric ratios Trigonometric Identities Trigonometric functions Composite functions of the form Transformations. Solving trigonometric equations | Green | 3 | Thinker |

Differentiation | Derivative interpreted as gradient function Increasing and decreasing functions. Tangents and normals and their equations The chain rule, product rule and quotient rules. Second derivative Local maximum and minimum points. Optimization. Points of inflexion with zero and non-zero gradients. Kinematics | Purple | 5 | Knowledgeable |

Logs and Exponential | Laws of exponents Numerical 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 | Outliers Sampling techniques Presentation of data Histograms. Cumulative frequency Box and whisker diagrams. Mean, median and mode Estimation of mean from grouped data. Modal class. Interquartile range, standard deviation and variance | Yellow | 2 | Inquiror |

Correlation and Regression | Linear correlation Pearson’s product-moment correlation coefficient Scatter diagrams Equation of the regression line | Yellow | 1 | Communicator |

Probability | Sample spaces The complementary events Expected number of occurrences Venn diagrams, tree diagrams, sample space diagrams and tables Combined events Mutually exclusive events Conditional probability Independent 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 distributions Expected value for discrete data Binomial distribution The normal distribution | Yellow | 3 | Inquirer |

Integration | Indefinite and definite integrals Definite integrals using technology Area of a region enclosed by a curve and axis Areas between curves. | Purple | 4 | Thinker |

Further Calculus | Derivative and integrals of trig functions, ln x and exponentials Kinematic problems Integration by inspection or by substitution | Purple | 4 | Open minded |

Sequences and series | Arithmetic sequences and series Geometric sequences and series Sum of infinite geometric sequences. Sigma notation Financial applications Deductive proof The binomial theorem | Red | 3 | Caring |