year 11 set 1 and 2 higher

Circle theorems

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Recall the definition of a circle and identify (name) and draw parts of a circle, including sector, tangent, chord, segment;

· Prove and use the facts that:

· the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference;

· the angle in a semicircle is a right angle;

· the perpendicular from the centre of a circle to a chord bisects the chord;

· angles in the same segment are equal;

· alternate segment theorem;

· opposite angles of a cyclic quadrilateral sum to 180°;

· Understand and use the fact that the tangent at any point on a circle is perpendicular to the radius at that point;

· Find and give reasons for missing angles on diagrams using:

· circle theorems;

· isosceles triangles (radius properties) in circles;

· the fact that the angle between a tangent and radius is 90°;

· the fact that tangents from an external point are equal in length.

Practice for mocks exam

· first two weeks of term to be use for revision for mocks

Exam preparation

Probability

OBJECTIVES

By the end of the unit, students should be able to:

· Write probabilities using fractions, percentages or decimals;

· Understand and use experimental and theoretical measures of probability, including relative frequency to include outcomes using dice, spinners, coins, etc;

· Estimate the number of times an event will occur, given the probability and the number of trials;

· Find the probability of successive events, such as several throws of a single dice;

· List all outcomes for single events, and combined events, systematically;

· Draw sample space diagrams and use them for adding simple probabilities;

· Know that the sum of the probabilities of all outcomes is 1;

· Use 1 – p as the probability of an event not occurring where p is the probability of the event occurring;

· Work out probabilities from Venn diagrams to represent real-life situations and also ‘abstract’ sets of numbers/values;

· Use union and intersection notation;

· Find a missing probability from a list or two-way table, including algebraic terms;

· Understand conditional probabilities and decide if two events are independent;

· Draw a probability tree diagram based on given information, and use this to find probability and expected number of outcome;

· Understand selection with or without replacement;

· Calculate the probability of independent and dependent combined events;

· Use a two-way table to calculate conditional probability;

· Use a tree diagram to calculate conditional probability;

· Use a Venn diagram to calculate conditional probability;

· Compare experimental data and theoretical probabilities;

· Compare relative frequencies from samples of different sizes.

Mocks

Graphs of trigonometric functions

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Recognise, sketch and interpret graphs of the trigonometric functions (in degrees)
y = sin x, y = cos x and y = tan x for angles of any size.

· Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45° , 60° and 90° and exact value of tan θ for θ = 0°, 30°, 45° and 60° and find them from graphs.

· Apply to the graph of y = f(x) the transformations y = –f(x), y = f(–x) for sine, cosine and tan functions f(x).

· Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(x + a)
for sine, cosine and tan functions f(x).

Reciprocal and exponential graphs; Gradient and area under graphs

· Estimate area under a quadratic or other graph by dividing it into trapezia;

· Interpret the gradient of linear or non-linear graphs, and estimate the gradient of a quadratic or non-linear graph at a given point by sketching the tangent and finding its gradient;

· Interpret the gradient of non-linear graph in curved distance–time and velocity–time graphs:

· for a non-linear distance–time graph, estimate the speed at one point in time, from the tangent, and the average speed over several seconds by finding the gradient of the chord;

· for a non-linear velocity–time graph, estimate the acceleration at one point in time, from the tangent, and the average acceleration over several seconds by finding the gradient of the chord;

· Interpret the gradient of a linear or non-linear graph in financial contexts;

· Interpret the area under a linear or non-linear graph in real-life contexts;

· Interpret the rate of change of graphs of containers filling and emptying;

· Interpret the rate of change of unit price in price graphs.

Changing the subject of formulae (morecomplex), algebraic fractions, solving equations arising from algebraic fractions, rationalising surds, proof

OBJECTIVES

By the end of the unit, students should be able to:

· Rationalise the denominator involving surds;

· Simplify algebraic fractions;

· Multiply and divide algebraic fractions;

· Solve quadratic equations arising from algebraic fraction equations;

· Change the subject of a formula, including cases where the subject occurs on both sides of the formula, or where a power of the subject appears;

· Change the subject of a formula such as , where all variables are in the denominators;

· Solve ‘Show that’ and proof questions using consecutive integers (n, n + 1), squares a2, b2, even numbers 2n, odd numbers 2n +1;

· Use function notation;

· Find f(x) + g(x) and f(x) – g(x), 2f(x), f(3x) etc algebraically;

· Find the inverse of a linear function;

· Know that f –1(x) refers to the inverse function;

· For two functions f(x) and g(x), find gf(x).

Circle geometry

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Select and apply construction techniques and understanding of loci to draw graphs based on circles and perpendiculars of lines;

· Find the equation of a tangent to a circle at a given point, by:

· finding the gradient of the radius that meets the circle at that point (circles all centre the origin);

· finding the gradient of the tangent perpendicular to it;

· using the given point;

· Recognise and construct the graph of a circle using x2 + y2 = r2 for radius r centred at the origin of coordinates.

Quadratics, expanding more than two brackets, sketching graphs, graphs of circles, cubes and quadratics

OBJECTIVES

By the end of the unit, students should be able to:

· Sketch a graph of a quadratic function, by factorising or by using the formula, identifying roots and y-intercept, turning point;

· Be able to identify from a graph if a quadratic equation has any real roots;

· Find approximate solutions to quadratic equations using a graph;

· Expand the product of more than two linear expressions;

· Sketch a graph of a quadratic function and a linear function, identifying intersection points;

· Sketch graphs of simple cubic functions, given as three linear expressions;

· Solve simultaneous equations graphically:

· find approximate solutions to simultaneous equations formed from one linear function and one quadratic function using a graphical approach;

· find graphically the intersection points of a given straight line with a circle;

· solve simultaneous equations representing a real-life situation graphically, and interpret the solution in the context of the problem;

· Solve quadratic inequalities in one variable, by factorising and sketching the graph to find critical values;

· Represent the solution set for inequalities using set notation, i.e. curly brackets and ‘is an element of’ notation;

· for problems identifying the solutions to two different inequalities, show this as the intersection of the two solution sets, i.e. solution of x² – 3x – 10 < 0 as {x: –3 < x < 5};

· Solve linear inequalities in two variables graphically;

· Show the solution set of several inequalities in two variables on a graph;

· Use iteration with simple converging sequences.

Exam preparation

Students will continue to practice using past papers and low stakes testing to prepare for their real test.

Assessment

Reciprocal and exponential graphs; Gradient and area under graphs

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Recognise, sketch and interpret graphs of the reciprocal function with x ≠ 0

· State the value of x for which the equation is not defined;

· Recognise, sketch and interpret graphs of exponential functions y = kx for positive values of k and integer values of x;

· Use calculators to explore exponential growth and decay;

· Set up, solve and interpret the answers in growth and decay problems;

· Interpret and analyse transformations of graphs of functions and write the functions algebraically, e.g. write the equation of f(x) + a, or f(x – a):

· apply to the graph of y = f(x) the transformations y = –f(x), y = f(–x) for linear, quadratic, cubic functions;

· apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(x + a)
for linear, quadratic, cubic functions;

Vectors and geometric proof

OBJECTIVES

By the end of the unit, students should be able to:

· Understand and use vector notation, including column notation, and understand and interpret vectors as displacement in the plane with an associated direction.

· Understand that 2a is parallel to a and twice its length, and that a is parallel to –a in the opposite direction.

· Represent vectors, combinations of vectors and scalar multiples in the plane pictorially.

· Calculate the sum of two vectors, the difference of two vectors and a scalar multiple of a vector using column vectors (including algebraic terms).

· Find the length of a vector using Pythagoras’ Theorem.

· Calculate the resultant of two vectors.

· Solve geometric problems in 2D where vectors are divided in a given ratio.

· Produce geometrical proofs to prove points are collinear and vectors/lines are parallel.

Multiplicative reasoning: direct and inverse proportion, relating to graph form for direct, compound measures, repeated proportional change

OBJECTIVES

By the end of the unit, students should be able to:

· Express a multiplicative relationship between two quantities as a ratio or a fraction, e.g. when A:B are in the ratio 3:5, A is B. When 4a = 7b, then a = or a:b is 7:4;

· Solve proportion problems using the unitary method;

· Work out which product offers best value and consider rates of pay;

· Work out the multiplier for repeated proportional change as a single decimal number;

· Represent repeated proportional change using a multiplier raised to a power, use this to solve problems involving compound interest and depreciation;

· Understand and use compound measures and:

· convert between metric speed measures;

· convert between density measures;

· convert between pressure measures;

· Use kinematics formulae from the formulae sheet to calculate speed, acceleration, etc (with variables defined in the question);

· Calculate an unknown quantity from quantities that vary in direct or inverse proportion;

· Recognise when values are in direct proportion by reference to the graph form, and use a graph to find the value of k in y = kx;

· Set up and use equations to solve word and other problems involving direct proportion (this is covered in more detail in unit 19);

· Relate algebraic solutions to graphical representation of the equations;

· Recognise when values are in inverse proportion by reference to the graph form;

· Set up and use equations to solve word and other problems involving inverse proportion, and relate algebraic solutions to graphical representation of the equations.

Further trigonometry

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Know and apply Area = ab sin C to calculate the area, sides or angles of any triangle.

· Know the sine and cosine rules, and use to solve 2D problems (including involving bearings).

· Use the sine and cosine rules to solve 3D problems.

· Understand the language of planes, and recognise the diagonals of a cuboid.

· Solve geometrical problems on coordinate axes.

· Understand, recall and use trigonometric relationships and Pythagoras’ Theorem in right-angled triangles, and use these to solve problems in 3D configurations.

· Calculate the length of a diagonal of a cuboid.

• Find the angle between a line and a plane.