## What will I learn?

The aims of all DP mathematics courses are to enable students to:

- Develop a curiosity and enjoyment of mathematics, and appreciate its elegance and power;
- Develop an understanding of the concepts, principles and nature of mathematics;
- Communicate mathematics clearly, concisely and confidently in a variety of contexts;

- Develop logical and creative thinking, and patience and persistence in problem solving to instil confidence in using mathematics;
- Employ and refine their powers of abstraction and generalisation;
- Take action to apply and transfer skills to alternative situations, to other areas of knowledge and to future developments in their local and global communities.

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 and proof by induction.

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, there is a strong emphasis on the ability to construct, communicate and justify correct mathematical arguments. There will be a recognition that the development of mathematical thinking is important for a student.

Students who choose this subject at HL should be comfortable in the manipulation of algebraic expressions and enjoy the recognition of patterns and understand the mathematical generalisation of these patterns. Students who wish to take Mathematics: Analysis and Approaches at HL 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.

## What is the structure of the course?

The course is structured around these major areas of mathematics:

- Number and Algebra
- Functions
- Geometry and Trigonometry

- Statistics and Probability
- Calculus

## How will I be assessed?

**Assessment**

**Format**

**Mathematical Exploration (Coursework)**

Internal assessment in mathematics is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. Usually 12-20 pages long. 20% weighting.

**Paper One**

No technology allowed. **Section A:** compulsory short-response questions based on the syllabus.**Section B:** compulsory extended-response questions based on the syllabus. 30% weighting.

**Paper Two**

Technology allowed. **Section A:** compulsory short-response questions based on the syllabus. **Section B:** compulsory extended-response questions based on the syllabus. 30% weighting.

**Paper Three**

Technology allowed.

Two compulsory extended-response problem-solving questions. 20% weighting.

## Frequently Asked Questions

*Which CAS opportunities are available?*

There are many CAS projects that require mathematical skills, but in addition to such opportunities that you might explore you may seek to:

- Join the maths club and take part in the UKMT Maths Challenges and support young year groups in their preparation
- Support students with preparation for the GCSE Maths Exam
- Attend university style public lectures on interesting areas of maths and science

*Which opportunities for further study are available? *

The Mathematics IB prepares you for any university course that requires a deep understanding of mathematics, such as courses in Mathematical Sciences, Physics, Engineering and Economics (where there is a focus on mathematical analysis) as well as any other course that requires a higher qualification in mathematics.

*Is there anything else I need to know?*The study of mathematics can be one of the most challenging academic experiences a student can take on, but it is also one of the most rewarding and useful subjects to study due to its applicability in such a wide array of academic disciplines at University. Be prepared to study hard in your own time in order to understand some of the most challenging mathematics you have ever encountered. You will require a graphical calculator for this course.

## Curriculum map

**Topics / Units**

Functions 1 & 2; including introduction to complex numbers

**Core Declarative Knowledge****What should students know?**

- General form of straight lines, their gradients and intercepts
- Parallel and perpendicular lines
- Different methods to solve a system of linear equations up tp 3 unknowns
- Guassian Elimination
- Function notation in all forms
- Domains and ranges of functions
- Inverse and composite functions, and their characteristics
- Self inverse functions
- Transformations of graphs and copositive transformations of graphs
- Graphs of the modulus of functions, the square of a function and the inverse graph of a function
- Polynomial functions and their graphs
- Zeros and factors of polynomial functions; roots of polynomial equations
- Factor and remainer theorems
- Sum and product of roots of a polynomial equation
- Quadratic equations and the different formus in which to express them
- The characteristics of a parabola
- Discriminant of the quadratic formula and interpret its results
- Rational functions, their graphs and their asymptotes
- Partial Fractions
- The fundamenal theory of algebra
- What a complex number and its complex conjugate is
- The real and imaginary parts of a complex number
- From the discriminate the need to find the complex solutions of a quadratic
- The difference between a reducible quadratic and an irreducible quadratic
- The sum of two squares factorisation

**Core Procedural Knowledge****What should students be able to do?**

- Find the equation of straight lines, their gradients and intercepts
- Find the equations of parallel and perpendicular lines
- Solve a system of linear equations up tp 3 unknowns
- Complete a Guassian reduction to echelon form and interpret the result
- Use function notation
- State the domains and ranges of functions including following a transformation
- Find inverse and composite functions, and interpret their characteristics
- Recongnise self inverse functions
- Sketch transformations of graphs and compositive transformations of graphs
- Sketch graphs of the modulus of functions, the square of a function and the inverse graph of a function
- Sketch polynomial functions and their graphs
- Find the zeros and factors of polynomial functions; roots of polynomial equations
- Use the factor and remainer theorems to solve problems
- Use the sum and product of roots of a polynomial equation
- Manipulate quadratic equations and the different formus in which to express them
- Use the characteristics of a parabola to find the line of symmetry and vertex
- Use the discriminant of the quadratic formula and interpret its results
- Sketch rational functions and their asymptotes
- Manipulate partial fractions
- Use the fundamenal theory of algebra to solve problems
- Recognise a complex number and its complex conjugate is
- Find real and imaginary parts of a complex number and equate them
- Find the solutions to quadratics and other polynomials without real solutions
- Carry out the sum of two squares factorisation

**Links to TOK**

- Does studying the graph of a function contain the same level of mathematical rigour as studying the function algebraically?
- What are the advantages and disadvantages of having different forms and symbolic language in mathematics?
- How does language shape knowledge? For example, do the words “imaginary” and “complex” make the concepts more difficult than if they had different names?
- Could we ever reach a point where everything important in a mathematical sense is known?
- Reflect on the creation of complex numbers before their applications were known.

**Links to Assessment**

Functions 1 Assessment

**Topics / Units**

Functions 2; Sequences & Series; Exponentials & Logarithms

**Core Declarative Knowledge****What should students know?**

- Rational functions, their graphs and their asymptotes
- Partial Fractions
- The fundamenal theory of algebra
- Properties of arithmetic and geometric sequences
- Sigma notation
- Sum of arithmetic and geometric sequences, both finite and infinite.
- Binomial theorem
- Counting principals, including permutations and combinations
- How applications of series to compound interest calculations
- Exponential functions and their graphs
- Concepts of exponential growth, decay and their applications
- The nature and significance of the number e
- Logarithmic functions and their graphs
- Properties and laws of logarithms

**Core Procedural Knowledge****What should students be able to do?**

- Sketch rational functions and their asymptotes
- Manipulate partial fractions
- Use the fundamenal theory of algebra to solve problems
- Use sigma notation
- Calculate the sum of arithmetic and geometric sequences, both finite and infinite.
- Employ the binomial theorem and Pascal’s Triangle
- Compute permutations and combinations
- Calculate compound interest and related problems
- Sketch exponential functions
- Calculate exponential growth, decay and solve problems related to their applications
- Work with e
- Solve problems using logarithmic functions and their graphs
- Calculate using logarithms

**Links to TOK**

- What counts as understanding in mathematics? Is it more than just getting the right answer?
- Why might it be said that e^iπ + 1 = 0 is beautiful? What is the place of beauty and elegance in mathematics? What about the place of creativity?

**Topics / Units**

Exponentials and Logarithms; Proof; Trigonometric functions and equations

**Core Declarative Knowledge****What should students know?**

- Exponential functions and their graphs
- Concepts of exponential growth, decay and their applications
- The nature and significance of the number e
- Logarithmic functions and their graphs
- Properties and laws of logarithms
- The language of logic and proof
- How to produce statements, negations and compound statements
- The differences between direct proofs, proofs with contrapositives, contradictions or counter examples.
- Proof by induction
- How to measure in radians
- The unit circle and its links with sine, cosine and tangent.
- Exact trig. values
- Pythagorean identities and double angle identities for sine and cosine.
- Amplitude and periods of trigonometric functions.
- Compositive functions of sine and cosine.
- Reciprocal trigonometric ratios of sec, csc and cot
- Pythagorean identities for tan, sec, csc and cot
- Compound angle identities
- Double angle identity for tan

**Core Procedural Knowledge****What should students be able to do?**

- Sketch exponential functions
- Calculate exponential growth, decay and solve problems related to their applications
- Work with e
- Solve problems using logarithmic functions and their graphs
- Calculate using logarithms
- Recognise tautology and contradictions
- Use the notation of logic
- Understand the modus ponens or the law of detatchment
- Carry out a direct, contrapositive, contradiction or induction proof.
- Convert degrees to radians
- Plot trigonometric functions
- Evaluate all trigonometric functions
- Prove the pythagorean identities
- Find the compound angle indentities
- Use reciprocal identities
- Recognise symmetrical/translation/odd/even identities

**Links to TOK**

- Is mathematics invented or discovered? For instance, consider the number e or logarithms–did they already exist before we defined them?
- How have seminal advances, such as the development of logarithms, changed the way in which mathematicians understand the world and the nature of mathematics?
- What is the role of the mathematical community in determining the validity of a mathematical proof?
- Do proofs provide us with completely certain knowledge?
- What is the difference between the inductive method in science and proof by induction in mathematics?

**Topics / Units**

Differential Calculus 1

**Core Declarative Knowledge****What should students know?**

- The concept of the limit and its notation
- Differentiation by first principles
- The gradient function and how to find it
- The relationship between the gradient function and rate of change
- The derivatives of polynomials and trigonometric functions
- When a function is increasing and decreasing
- Local minimum, maximum and points of inflection
- The applications to displacement, velocity and acceleration
- The relationship between tangents and normals
- Composite functions
- Product and quotients
- Derivatives of exponentials
- Implicit differentiation
- how to optimise
- L’Hôpital’s rule

**Core Procedural Knowledge****What should students be able to do?**

- Differentiate a polynomial
- Find the gradient of a function at a given point
- Find a derivative of a trigonometric function
- Determine whether a function is increasing or decreasing
- Find local minimum, maximum and points of inflection
- Find the equations of tangents and normals
- Differentiate composite functions
- Use the chain rule, product rule and quotient rule
- Find the derivative on an exponential
- Find higher derivatives
- Recognise the need for implicit differentiation
- Apply L’Hôpital’s rule

**Links to TOK**

- What value does the knowledge of limits have?
- Is infinitesimal behaviour applicable to real life?
- Is intuition a valid way of knowing in mathematics?
- The seemingly abstract concept of calculus allows us to create mathematical models that permit human feats such as getting a man on the Moon.
- What does this tell us about the links between mathematical models and reality?
- How can you justify a rise in tax for plastic containers, eg lastic bags plastic bottles, etc using optimisation?

**Topics / Units**

Differential Calculus 1; Integral Calculus 1

**Core Declarative Knowledge****What should students know?**

- Derivatives of exponentials
- Implicit differentiation
- How to optimise
- L’Hôpital’s rule”

**Mocks**

- Integration as antidifferentiation of functions
- The general form for definite integrals
- Applications of integration (area under curves)
- Applications to kinemative problems involving displacement, velocity and acceleration
- How to apply integration to different types of functions
- How when to apply different ways to integrate (inspection, partial fractions, substitution, parts including repeated by parts)
- Applications to volumes of revolution

**Core Procedural Knowledge****What should students be able to do?**

- Find higher derivatives
- Determine whether a function is increasing or decreasing
- Find local minimum, maximum and points of inflection
- Find the equations of tangents and normals
- Recognise the need for implicit differentiation
- Apply L’Hôpital’s rule

**Mocks**

- How to calculate and apply definite integrals
- Find the area under and between curves and the x-axis.
- Use boundary conditions
- Solve problems in kinematics
- Integrate polynomials, trigonometric, inverse trigonometric and exponentials
- Integrate by inspection, with partial fractions, by parts and repeated by parts
- Find volumes of revolution about the x-axis/y-axis

**Links to TOK**

- Does personal experience play a role in the formation of knowledge claims in mathematics?
- Does it play a different role in mathematics compared to other areas of knowledge?

**Topics / Units**

Integral Calculus 1; Introduction to the IA

**Core Declarative Knowledge****What should students know?**

- Integration as antidifferentiation of functions
- the general form for definite integrals
- applications of integration (area under curves)
- applications to kinemative problems involving displacement, velocity and acceleration
- how to apply integration to different types of functions
- how when to apply different ways to integrate (inspection, partial fractions, substitution, parts including repeated by parts)
- applications to volumes of revolution
- Students to explore ideas for their IA and research possible project ideas. The specific purposes of the exploration are to:
- develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics
- provide opportunities for students to complete a piece of mathematical work over an extended period of time
- enable students to experience the satisfaction of applying mathematical processes independently provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics
- encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool
- enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work
- provide opportunities for students to show, with confidence, how they have developed mathematically.

- develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics

**Core Procedural Knowledge****What should students be able to do?**

- how to calculate and apply definite integrals
- find the area under and between curves and the x-axis.
- use boundary conditions
- Solve problems in kinematics
- integrate polynomials, trigonometric, inverse trigonometric and exponentials
- integrate by inspection, with partial fractions, by parts and repeated by parts
- find volumes of revolution about the x-axis/y-axis
- Decide on IA title.

**Links to TOK**

- Can a mathematical statement be true before it has been proven?