# Maths P3 Grade 13 A2

## Overview

The Math Syllabus at GEMS Wesgreen International Secondary School aims to support students to develop their ability to calculate fluently, to reason and solve problems through application of knowledge and transferable skills. Throughout the year we recover and extend objectives as the focus is on securing an understanding in the subject by developing a greater depth.

## Learning Outcomes

The aims of all subjects state what a teacher may expect to teach and what a student may expect to experience and learn. These aims suggest how the student may be changed by the learning experience.

The aims of the Math Syllabus are to encourage and enable students to:

- understand mathematics and mathematical processes in a way that promotes confidence, fosters enjoyment and provides a strong foundation for progress to further study
- extend their range of mathematical skills and techniques.
- understand coherence and progression in mathematics and how different areas of mathematics are connected.
- apply mathematics in other fields of study and be aware of the relevance of mathematics to the world of work and to situations in society in general
- use their mathematical knowledge to make logical and reasoned decisions in solving problems both within pure mathematics and in a variety of contexts, and communicate the mathematical rationale for these decisions clearly

## Unit Overviews

**Term 1 **

**Unit 1 – 3.1 Algebra **

**Approximate length: 2 weeks **

In this unit the students will:

- understand the meaning of |
*x*|, sketch the graph of y = |ax + b| and use relations such as |a| = |b| ⇔ a 2 = b2 and |x – a| < b ⇔ a – b < x < a + b when solving equations and inequalities. - divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero) .
- use the factor theorem and the remainder theorem e.g. to find factors and remainders, solve polynomial equations or evaluate unknown coefficients. Including factors of the form (ax + b) in which the coefficient of x is not unity, and including calculation of remainders.
- recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than – (ax + b)(cx + d)(ex + f) – (ax + b)(cx + d) 2 – (ax + b)(cx 2 + d) Excluding cases where the degree of the numerator exceeds that of the denominator.
- use the expansion of (1 + x) n , where n is a rational number.

Specific CIE Objectives Covered:

- Understand and use the laws of indices for all rational exponents
- Use and manipulate surds, including rationalising the denominator
- Work with quadratic functions and their graphs;
- the discriminant of a quadratic function, including the conditions for real and repeated roots;
- Completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.

**Unit 2 3.2 Logarithmic and exponential functions**

**Approximate length: 2 weeks **

In this unit the students will understand and use graphs of functions; sketch curves defined by simple equations including polynomials, find the modulus of a linear function including their vertical and horizontal asymptotes; interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations.

Specific CIE Objectives Covered:

- Understand and use proportional relationships and their graphs
- Understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
- Understand the definition and properties of e
*x*and ln*x*, including their relationship as inverse functions and their graphs - Use logarithms to solve equations and inequalities in which the unknown appears in indices
- Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

**Unit 3 3.3 Trigonometry **

**Approximate length: 2 weeks **

In this unit the students will understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle and work with radian measure, including use for arc length and area of sector.

Specific CIE Objectives Covered:

- understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude
- use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of sec2
*θ**≡*1 + tan2*θ*and cosec2*θ**≡*1 + cot2*θ* - the expansions of sin(
*A*±*B*), cos(*A*±*B*) and tan(*A*±*B*) - the formulae for sin 2
*A*, cos 2*A*and tan 2*A* - the expression of
*a sin θ*+*b cos θ*in the forms*R*sin(*θ*± α) and*R*cos(*θ*± α), - Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity,

**Unit 4 –3.4 Differentiation**

**Approximate length: 4 weeks **

In this unit the students will understand and use the derivative of as the gradient of the tangent to the graph of at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x

Specific CIE Objectives Covered:

- use the derivatives of e x , ln x, sinx, cos x, tanx, tan–1 x, together with constant multiples, sums, differences and composites.
- Differentiate products and quotients.
- find and use the first derivative of a function which is defined parametrically or implicitly
- Understand and use the second derivative as the rate of change of gradient.

**Unit 5 – 3.5 Integration**

**Approximate length: 3 weeks **

In this unit the students will know and use the Fundamental Theorem of Calculus

Specific CIE Objectives Covered:

- Extend the idea of ‘reverse differentiation’ to include the integration of e ax + b, sin(ax + b), cos(ax + b), sec2 (ax + b) and 1/(x2 +a2).
- use trigonometrical relationships in carrying out integration
- E.g. use of double-angle formulae to integrate sin2 x or cos2 (2x).
- Integrate rational functions by means of decomposition into partial fractions.
- Recognize when an integrand can usefully be regarded as a product, and use integration by parts e.g..
- Use a given substitution to simplify and evaluate either a definite or an indefinite integral.
- Integrate xn (excluding
*n*= -1), and related sums, differences and constant multiples . - Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.
- Understand and use integration as the limit of a sum.

**Unit 6 – 3.6 Numerical solution of equations**

**Approximate length: 2 weeks **

In this unit the students will apply the knowledge of this topic by using a graph plotter to demonstrate both sign changes and graphical considerations e.g. Change of sign

Specific CIE Objectives Covered:

- Locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change.
- Notes and examples e.g. finding a pair of consecutive integers between which a root lies.
- Understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation.
- Understand how a given simple iterative formula relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

**Term 2 **

**Unit 7 – 3.7 Vectors**

**Approximate length: 3 weeks **

In this unit the students will:

- Use standard notations for vector.
- Carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms.
- Calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors In 2 or 3 dimensions.
- Understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb, and find the equation of a line, given sufficient information.
- Determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists.
- Calculation of the shortest distance between two skew lines is not required.
- Finding the equation of the common perpendicular to two skew lines is also not required. • use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points.

Specific CIE Objectives Covered:

- Use vectors in two dimensions.
- Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form
- Understand and use position vectors
- Calculate the distance between two points represented by position vector.

**Unit 8 – 3.8 Differential equations**

**Approximate length: 2 weeks**

In this unit the students will differentiate using the product rule, the quotient rule and the chain rule including problems involving connected rates of change and inverse functions.

Specific CIE Objectives Covered:

- Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
- Formulate a simple statement involving a rate of change as a differential equation.
- find by integration a general form of solution for a first order differential equation in which the variables are separable Including any of the integration techniques from topic 3.5
- use an initial condition to find a particular solution
- interpret the solution of a differential equation in the context of a problem being modelled by the equation

**Unit 9 – 3.9 Complex numbers**

**Approximate length: 3 weeks**

In this unit the students will know the Fundamental Theorem of Algebra and show that it is true for quadratic polynomials.

Specific CIE Objectives Covered:

- Find the Formula for the solution of the quadratic equation ax2+bx+c=0
- Understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal.
- carry out operations of addition, subtraction, multiplication and division of two complex numbers express the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs E.g. in solving a cubic or quartic equation where one complex root is given.
- Represent complex numbers geometrically by means of an Argand diagram in Cartesian form x + iy.
- Find the two square roots of a complex number e.g. the square roots of 5 + 12i in exact Cartesian form. Full details of the working should be shown. • understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers
- illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram

## Blended Learning

- Throughout this year there will be blended learning and we are using multiple teaching methods in order to help our students learn more effectively. All students whether face to face or learning from home will have the opportunity to access all the lessons and resources. Students will use Phoenix, Teams and myimaths. Each lesson begins with a set of clearly stated objectives and an explanation of its place in the overall CIE AS/A level syllabus. Effective learning is encouraged through frequent activities and self-assessment questions.
- Links for solving past papers and links related to concepts covered to reinforce classroom learning followed will be available to students. Students will have access to worksheets with progression of difficulty, online assessments, tasks/open ended questions and fun activities through online platforms.

## Assessment

**Formative: **Throughout the units, the students will complete graded work, quizzes and problem solving activities which allows the teacher to assess the students’ attainment and inform their planning.

For each unit the students complete written quizzes, online quizzes as well as Chapter- wise tests (Topic Tests). Quizzes are taken based on 1-chapter assessment, where Tests are combined as per the requirement i.e. 2 to 3 chapters/topics – sections. This allows us to see progress across the units and align our planning.

**Summative**: At the end of each term we complete internal and standardized tests. This allows us to measure the students’ progress throughout the term and year. End of term 2 they have CIE format exam for P3. This is practice / preparation for their final CIE examinations. At the end of one year course, students appear for their final CIE examination for Syllabus - Cambridge CIE® Mathematics 9709.