Chapter

Content

Time

Resources

Number and Algebra

Writing numbers in Standard Form

Laws of Indices for Integer exponents

Ratio and Proportion

Factorising, Algebra in fractions, Simultaneous equations,

Changing the subject of the formula

Straight line graphs, gradient and y-intercept.

Finding the distance between two points

3 hours

3 hours

2 hours

Number and algebra 1

Number and algebra 2

Straight line graphs

Shape and Space

Basic Trigonometry

Sine Rule, Cosine Rule, Area of a Triangle

Volume and Surface Area of a Cone and Sphere

Circle theorems:

(a)     angle in a semicircle is a right angle.

(b) the perpendicular drawn from the centre of a circle to a chord bisects the chord.

(c) the radius of a circle drawn to the point of contact of a tangent to the circle is at right angles to the tangent.

2 hours

4 hours

1 hour

2 hours

Trigonometry

Sine rule, Cosine rule, areas of triangles

Surface Area and Volume

Circles

Chapter

Content

Time

Resources

P1.1. Proof

Proof by direct methods. Proof by a direct method including a sequence of logical steps may be required. For example, prove that the equation x² + px + q = 0 has distinct real roots if, and only if, p² > 4q.

2 hours

Proof

P1.2. Algebra

Laws of indices for all rational exponents. The equivalence of a^m/n and ⁿ√a^m.

Use and manipulate surds including rationalising denominators.

Solution of quadratic equations. The discriminant of a quadratic function. Proof of existence of roots. Solution by factorisation, formula and completing the square. Proof of the quadratic formula.

Algebraic manipulation of polynomials, including expanding brackets and collecting like terms.

Factorising and use of the Factor Theorem.  (If f(x) = 0 when x = a, then (x – a) is a factor of f(x).) Factorisation of polynomials of degree n, n ≤ 3, eg x³ - 4x² + 3. The notation f(x). Evaluate f(x) for specific values of x. Identities. Algebraic division. Equating coefficients.

Quadratic functions and their graphs.

Solution of simultaneous equations in particular where one equation is linear and one equation is quadratic.

Solution of linear and quadratic inequalities. For example,  px² + qx + r ≤ 0.

2 hour

1 hour

3 hours

1 hour

3 hours

2 hours

2 hours

2 hours

P1 – 5A,5B,5C

P1 – 1E

P1 – 2B

The Discriminant

P1 – 1A,  Identities

P1 – 1B,1C,1D

Long Division

Factor Theorem

Quadratic Functions & Graphs

P1 – 2C

P1 – 2D

P1.3. Trigonometry

Radians. Use of the formulas s = rq and A = ½r²q for arc length and area of sector

Trigonometric functions for any angle in radians and degrees. Their graphs, symmetries and periodicity. Knowledge of graphs of curves with equations such as y = 3 sin x, y = sin(x + π/6), y = sin 2x will be required.

Knowledge and use of                             and   sin²q + cos²q = 1 to solve  trigonometric equations in a given interval.

                                                                   For example, 2sin (x - π/2) = ¾ for 0 < x < 2π

      cos (2x + 30) = -½ for –180 ≤ x ≤ 180,

      6cos²x + sin x – 5 = 0 for 0 ≤ x ≤ 360,

2 hours

3 hours

3 hours

P1 – 7A

P1 –7B,7C,7D (Questions 1 & 2) 7E

P1 – 7D

Trigonometric equations

Chapter

Content

Time

Resources

P1.4. Co-ordinate geometry in the (x,y) plane

Forming the equation of a straight line in forms y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0 from either being given a) 2 points or b) a parallel / perpendicular line and 1 point For example the line perpendicular to the line 3x + 4y = 18 through the point (2,3) has equation y – 3 = ⁴/₃(x – 2)

Conditions for two straight lines to be parallel or perpendicular to each other.

The co-ordinates of the mid-point of a line segment joining two given points.

3 hours

1 hour

P1 – 4B,4C

Perpendicular /  Parallel Lines

Equations of Lines

P1.5. Sequences and series

Sequences, including those given by a formula for the nth term.

Arithmetic series, including the formula for the sum of the first n natural numbers. The general term and the sum to n terms of the series are required. The proof of the sum formula should be known.

Geometric series. The general term and the sum to n terms are required. The proof of the sum formula should be known.  The sum to infinity of a convergent geometric series.

Use of S notation for both Arithmetic and Geometric series

1 hour

2 hours

2 hours

1 hour

P1 – 6A

P1 – 6D

P1 – 6E

P1 – 6C

Chapter

Content

Time

Resourses

P1.6. Differentiation

The derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change, i.e.     is the rate of change of y with respect to x. Knowledge of the chain rule is not required.

Algebraic differentiation of x^p , where p is rational. For example, the ability to differentiate

expressions such as (2x + 5)(x - 1) and                                   is expected.

Second order derivatives. Increasing and decreasing functions.

Notation such as                                                                                     will be needed.

Applications to gradients, maxima, minima and stationary points. Problems involving finding equations of normals and tangents. Problems involving curve sketching. Practical problems, for example find maximum volume of a box given its surface area.

1 hour

2 hours

4 hours

Notes from P1 Ch8

P1 – 8A

Differentiation

P1 – 8D

Tangents / Normals

P1.7. Integration

Indefinite integration as the reverse of differentiation.  Integration of x^p, where p is rational, p ≠ -1.

The general and particular solution of                     

Evaluation of definite integrals. Interpretation of the definite integral as the area under a curve. Evaluation of  the area of a region bounded by a curve, given lines parallel to the y-axis and the x-axis. Familiarity with the idea that the area under a curve may be obtained as the limit of a sum of the areas of rectangles. Both ∫ y dx and ∫ x dy are required.

2 hours

3 hours

P1 – 9A,9B(question 1)

P1 – 9B,9C

Integration and Area

Chapter

Content

Time

Resources

M1.1. Mathematical Modelling in Mechanics

The basic ideas of mathematical modelling as applied in mechanics. This should not be taught as a singular topic, but the ideas should be constantly emphasised during the course wherever appropriate. Familiarity with terms : particle, lamina, rigid body, rod (light, uniform, non-uniform), inextensible string, smooth and rough surface, light smooth pulley, bead, wire, peg. Students should be familiar with the assumptions made in these models.

M1.2. Vectors

Magnitude and direction of a vector. Properties of vectors. Resultant of vectors.

Use i and j notation.

Application of vectors to displacements, velocities, accelerations and forces in a plane.

Use of velocity = change of displacement / time  in the case of constant velocity, and of

acceleration = change of velocity / time in the case of constant acceleration.

2 hours

2 hours

2 hours

Notes from M1 p16

M1 – 2B

M1 – 2C

M1 – 2D

M1.3. Kinematics

Motion in a straight line with constant acceleration. Knowledge and use of formulae for constant acceleration will be required.

Graphical solutions may be required, including displacement-time, velocity-time, speed-time and acceleration-time graphs.  Greater emphasis should be placed on speed-time graphs.

4 hours

3 hours

M1 – 3A, 3B

M1 – 3C

Vertical Motion

M1.4. Dynamics

Newton’s laws of motion.

Simple applications including the motion of two connected particles. Problems with smooth fixed pulleys and motion on a rough or smooth inclined plane. Coefficient of friction. An understanding of F = mR when a particle is moving.

Momentum and impulse. The impulse-momentum principle. The principle of conservation of momentum applied to two particles colliding directly. Knowledge of Newton’s law of restitution is not required. Problems will be confined to those of a one-dimensional nature.

2 hours

6 hours

2 hours

M1 – 5A

M1 – 5B, 5C, 5D

M1 – 5I, 5J(ignore questions relating to energy)

Chapter

Content

Time

Resources

M1.5 Statics

Forces treated as vectors. Resolution of forces. Resolving forces into components using sin and cos.

Equilibrium of a particle under coplanar forces. Weight, normal reaction, tension and thrust, friction.

Coefficient of friction. An understanding of F £ mR in a situation of equilibrium.

4 hours

3 hours

2 hours

M1 – 4A, 4B

M1 – 4C, 4D

M1 – 4E

M1.6. Moments

Moment of a force. Simple problems involving coplanar parallel forces acting on a body and conditions for

equilibrium in such situations.

3 hours

M1 – 6A, 6B

Chapter

Content

Time

Resources

1. Mathematical models in probability and statistics

The basic ideas of mathematical modelling as applied in probability and statistics.

1 hour

Information sheet

“Mathematical modelling in Probability and Statistics”

2. Representation and summary of data

Histograms

Stem and leaf diagrams

Box plots. Use to compare distributions.

Frequency distribution, cumulative frequency step polygon.

Grouped frequency distributions, cumulative frequency polygons

Back-to-back stem and leaf diagrams may be required.

Measures of location - mean, median, mode. Data may be discrete, continuous, grouped or ungrouped. Understanding and use of coding.

Measures of dispersion - variance, standard deviation, range and interpercentile ranges. Simple interpolation may be required.Interpretation of measures of location and dispersion.

Skewness. Concepts of outliers. Any rule to identify outliers will be specified in the question.

5 hours

4 hours

3 hours

1 hour

T1 – 3B

T1 – 3A(p15 – 19)

T1 – p51 –52

T1 – 3A(p13 – 15)

T1 – 3B(p23 – 28)

T1 – 3A(p18 – 19)

T1 – 4A,

T1 – p67 –76

T1 – 4B(p76 – 78)

3. Probability

Elementary probability. Sample space. Exclusive and complementary events. Conditional probability. Understanding and use of

P(A¢) = 1 - P(A),

P(A È B) = P(A) + P(B) - P(A Ç B),

P(A Ç B) = P(A) P(B½A).

Independence of two events. P(B½A) = P(B), P(A½B) = P(A), P(A Ç B) = P(A) P(B).

Sum and product laws. Use of tree diagrams and Venn diagrams. Sampling with and without replacement.

2 hours

2 hours

2 hours

T1 – p95 – 115

T1 – 5A, 5B, 5C

Chapter

Content

Time

Resources

4. Correlation and regression

Scatter diagrams. The product moment correlation coefficient, its use, interpretation and limitations. Derivations and tests of significance will not be required.

Linear regression. Explanatory (independent) and response (dependent) variables. Applications and interpretations. Use to make predictions within the range of values of the explanatory variable and the dangers of extrapolation. Derivations will not be required. Variables other than x and y may be used. Linear change of variable may be required.

3 hours

3 hours

Scatter diagrams

Correlation

Product Moment Correlation Coefficient

Regression

Linear regression

5. Discrete random variables

The concept of a discrete random variable. The probability function and the cumulative distribution function for a discrete random variable. Simple uses of the probability function p(x) where p(x) = P(X = x). Use of the cumulative

distribution function

F(x₀ ) = P(X £ x₀) =            p(x)

Mean and variance of a discrete random variable. Use of E(X), E(X² ) for calculating the variance of X.

Knowledge and use of E(aX + b) = aE(X) + b, Var (aX + b) = a² Var (X).

The discrete uniform distribution. The mean and variance of this distribution.

2 hours

3 hours

1 hour

T1 – 6A

T1 – p124 – 135

T1 – p143 – 159(discrete cases only)

T1 – 6B

T1 – 7A(p161 – 165)

6. The Normal distribution

The Normal distribution including the mean, variance and use of tables of the cumulative distribution function.

Knowledge of the shape and the symmetry of the distribution is required. Knowledge of the probability density function is not required. Derivation of the mean, variance and cumulative distribution function is not required. Interpolation is not necessary. Questions may involve the solution of simultaneous equations.

3 hours

T1 – 8A, 8B, 8C, 8D

T1 – 198 – 212