# Maths Grade 12 Mechanics

## 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 – ****Forces and equilibrium**

**Approximate length: 3 weeks **

In this unit the children will develop their idea of forces and different types of forces as possible and types of situations in which they might be present.

Specific CIE Objectives Covered:

- identify the forces acting in a given situation, e.g. by drawing a force diagram
- understand the vector nature of force, and find and use components and resultants; calculations are always required, not approximate solutions by scale drawing
- use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero; solutions by resolving are usually expected, but equivalent methods (e.g. triangle of forces, Lami’s Theorem, where suitable) are also acceptable; these other methods are not required knowledge, and will not be referred to in questions • understand that a contact force between two surfaces can be represented by two components, the normal component and the frictional component
- use the model of a ‘smooth’ contact, and understand the limitations of this model
- understand the concepts of limiting friction and limiting equilibrium, recall the definition of coefficient of friction, and use the relationship F = μR or F ≤ μR, as appropriate; terminology such as ‘about to slip’ may be used to mean ‘in limiting equilibrium’ in questions
- use Newton’s third law, e.g. the force exerted by a particle on the ground is equal and opposite to the force exerted by the ground on the particle

**Unit 2 – Kinematics of motion in a straight line**

**Approximate length: 4 weeks **

In this unit the children will reiterate the difference between scalar and vector quantities and that in one dimension, the direction of a vector will determine whether you use a positive or negative sign. They will develop te understanding and application that negative speed or distance is incorrect and a negative acceleration is equivalent to a positive deceleration.

Specific CIE Objectives Covered:

- understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities; restricted to motion in one dimension only; the term ‘deceleration’ may sometimes be used in the context of decreasing speed
- sketch and interpret displacement–time graphs and velocity–time graphs, and in particular appreciate that: the area under a velocity–time graph represents displacement, the gradient of a displacement–time graph represents velocity, the gradient of a velocity-time graph represents acceleration
- use differentiation and integration with respect to time to solve simple problems concerning displacement, velocity and acceleration; calculus required is restricted to techniques from the content for Paper 1: Pure Mathematics 1
- use appropriate formulae for motion with constant acceleration in a straight line; questions may involve setting up more than one equation, using information about the motion of different particles

**Unit 3 –Momentum**

**Approximate length: 3 weeks **

In this unit the children will be introduced to the concept of linear momentum (p) as the product of the mass (m) and the velocity (v) of a particle and that momentum is a vector quantity that acts in the direction of motion of the particle.

Specific CIE Objectives Covered:

- use the definition of linear momentum and show understanding of its vector nature; for motion in one dimension only
- use conservation of linear momentum to solve problems that may be modelled as the direct impact of two bodies; including direct impact of two bodies where the bodies coalesce on impact; knowledge of impulse and the coefficient of restitution is not required

**Term 2**

**Unit 4 – Newton’s laws of motion**

**Approximate length: 2 weeks **

In this unit the children will identify the direction of motion and understand that forces resolved perpendicular to the motion must be balanced, while the resultant force parallel to the motion is what will cause the acceleration.

Specific CIE Objectives Covered:

- apply Newton’s laws of motion to the linear motion of a particle of constant mass moving under the action of constant forces, which may include friction, tension in an extensible string and thrust in a connecting rod; if any other forces resisting motion are to be considered (e.g. air resistance) this will be indicated in the question
- use the relationship between mass and weight; W = mg; in this component, questions are mainly numerical, and use of the approximate numerical value 10 (ms-2) for g is expected
- solve simple problems which may be modelled as the motion of a particle moving vertically or on an inclined plane with constant acceleration; including, for example, motion of a particle on a rough plane where the acceleration while moving up the plane is different from the acceleration while moving down the plane
- solve simple problems which may be modelled as the motion of connected particles, e.g. particles connected by a light inextensible string passing over a smooth pulley, or a car towing a trailer by means of either a light rope or a light rigid tow bar.

**Unit 5 – Work, Energy and Power**

**Approximate length: 3 weeks **

In this unit the children will learn to define work done by a force and how to use this definition to solve problems when the motion is horizontal and for vertical motion. They will also be introduced to gravitational potential energy and motion on an inclined plane.

Specific CIE Objectives Covered:

- understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force; W = Fd cos θ ; use of the scalar product is not required
- understand the concepts of gravitational potential energy and kinetic energy, and use appropriate formulae
- understand and use the relationship between the change in energy of a system and the work done by the external forces, and use in appropriate cases the principle of conservation of energy; including cases where the motion may not be linear (e.g. a child on a smooth curved ‘slide’), where only overall energy changes need to be considered
- use the definition of power as the rate at which a force does work, and use the relationship between power, force and velocity for a force acting in the direction of motion; including calculation of (average) power as ;P = Fv
- solve problems involving, for example, the instantaneous acceleration of a car moving on a hill against a resistance

## Blended Learning

Throughout this year there will be blended learning. 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 Mechanics. 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.