# Maths Grade 12 P1 AS Level

## 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:

- become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- reason mathematically by following a line of enquiry, conjecturing relationships and generalizations, and developing an argument, justification or proof using mathematical language
- can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

## Unit Overviews

### Term 1

**Unit 1 –Co-ordinate geometry (Chapter 3)**

**Approximate length: 2 weeks **

In this unit the children will build on their understanding of Co-ordinates, plotting, sketching and drawing, the gradient of a line, the distance between two points, the midpoint of a line joining two points, finding the equation of a straight line, the intersection of two lines

Specific CIE Objectives Covered:

- Find the equation of a straight line given sufficient information.
- Interpret and use any of the forms y = mx + c, y – y1 = m(x – x1), ax + by + c = 0 in solving problems.
- Understand that the equation (x – a)2 + (y – b)2 = r 2 represents the circle with center (a, b) and radius r.
- Use algebraic methods to solve problems involving lines and circles.
- Understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations.

**Unit 2 – Functions**

**Approximate length: 2 weeks **

In this unit the children will build on their understanding of the language of functions, composite functions, inverse functions.

Specific CIE Objectives Covered:

- Understand the terms function, domain, range, one-one function, inverse function and composition of functions.
- Identify the range of a given function in simple cases, and find the composition of two given functions.
- Determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases.
- Illustrate in graphical terms the relation between a one-one function and its inverse.
- Understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these.

**Unit 3 – Quadratics**

**Approximate length: 2 weeks **

In this unit the children will build on their understanding of quadratic equations, solving quadratic equations, equations that cannot be factorized, completing square, the graphs of quadratic functions, the quadratic formula, simultaneous equations, and inequalities.

Specific CIE Objectives Covered:

- Carry out the process of completing the square for a quadratic polynomial ax2 + bx + c and use a completed square form.
- Find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant.
- Solve quadratic equations, and quadratic inequalities, in one unknown.
- Solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic.
- Recognize and solve equations in x which are quadratic in some function of x.

**Unit 4 – Differentiation**

**Approximate length: 1 week **

In this unit the children will build on their understanding of the gradient of a curve,finding the gradient of a curve , first principle of calculus , differentiating by using standard results.

Specific CIE Objectives Covered:

- understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations
- Only an informal understanding of the idea of a limit is expected. e.g. includes consideration of the gradient of the chord joining the points with x coordinates 2 and (2 + h) on the curve y = x3. Formal use of the general method of differentiation from first principles.

**Unit 5 – Further differentiation **

**Approximate length: 1 week **

In this unit the children will build on their understanding of using differentiation, tangents and normal, maximum and minimum points, increasing and decreasing functions, point of inflection, the second derivative, applications and the chain rule.

Specific CIE Objectives Covered:

- Use the derivative of xn (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule.
- Apply differentiation to gradients, tangents and normal, increasing and decreasing functions and rates of change.
- Locate stationary points and determine their nature, and use information about stationary points in sketching graphs.
**Unit 6 – Integration Approximate length: 2 weeks**Unit 6 – Integration Approximate length: 2 weeks

**Unit 6 – Integration Approximate length: 2 weeks **

In this unit the children will build on their understanding of reversing differentiation, finding area under a curve, area as the limit of a sum, area below the x-axis, the area between two curves, the area between a curve and the y-axis, the reverse chain rule, improper integrals, finding volumes by integration.

Specific CIE Objectives Covered:

- Understand integration as the reverse process of differentiation, and integrate (ax + b)n (for any rational n except –1), together with constant multiples, sums and differences.
- Solve problems involving the evaluation of a constant of integration.
- evaluate definite integral
- Use definite integration to find – the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves – a volume of revolution about one of the axes.

### Term 2

**Unit 7 – Circular measure Approximate length: 2 weeks **

In this unit the children will build on their understanding of circular measure, the length of an arc of a circle, the area of a sector of a circle.

Specific CIE Objectives Covered:

- Understand the definition of a radian, and use the relationship between radians and degrees.
- Use the formulae in solving problems concerning the arc length and sector area of a circle.

**Unit 8 - Trigonometry Approximate length: 2 weeks**

In this unit the children will build on their understanding of trigonometry background, trigonometrical functions, trigonometrical functions for angles of any size, the sine and cosine graphs, the tangent graph, solving equations using graphs of trigonometrical functions.

Specific CIE Objectives Covered:

- Sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians).
- Use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles.
- Use the notations sin–1x, cos –1x, tan–1x to denote the principal values of the inverse trigonometric relations.
- Find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).

**Unit 9 – Sequences and series Approximate length: 2 weeks**

In this unit the children will build on their understanding of definitions and notation, Arithmetic progressions, geometric progressions and binomial expansions.

Specific CIE Objectives Covered:

- Use the expansion of (a + b)n, where n is a positive integer.
- Recognize arithmetic and geometric progressions.
- Use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions.
- Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

## 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 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 children 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 P1. 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.