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Year 11 Higher Mathematics

Curriculum Overview Autumn 1

Solving Quadratics Proof

Trigonometrical Graphs

Autumn 2 Algebraic Fractions

Functions and Calculus

Spring 1 Revision Cycle 1 Revision Cycle 2 Revision Cycle 3

Spring 2 Revision Cycle 4 Revision Cycle 5 Revision Cycle 6

Summer 1 Practice Exam Papers

Summer 2

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Contents

Quadratic Equations ................................................................................................................................................................................................. 3

Mathematical Proof ................................................................................................................................................................................................... 8

Trigonometry – Graphs ........................................................................................................................................................................................... 12

Working with Algebraic Fractions ............................................................................................................................................................................ 15

Working with Functions ........................................................................................................................................................................................... 18

Higher GCSE Mathematics Exam Revision ............................................................................................................................................................ 22

Cycle 1 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 23

Cycle 1 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 24

Cycle 2 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 25

Cycle 2 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 26

Cycle 3 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 27

Cycle 3 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 28

Cycle 4 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 29

Cycle 4 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 30

Cycle 5 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 31

Cycle 5 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 32

Cycle 6 Aiming for Grades 5 to 6 ............................................................................................................................................................................ 33

Cycle 6 Aiming for Grades 7 to 9 ............................................................................................................................................................................ 34

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Quadratic Equations

Students learn how to solve a range of quadratic equations using factorization, applying the formula, and completing the square. Learning

progresses from solving equations to deriving and sketching quadratics as a graph.

Prerequisite Knowledge

• Simplify and manipulate algebraic expressions by:

• Expanding products of two or more binomials

• Factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares

• Simplifying expressions involving sums, products and powers, including the laws of indices

• Factorising quadratic expressions of the form ax2 + bx + c.

Success Criteria

• Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

• Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c.

• Understand and use standard mathematical formulae; rearrange formulae to change the subject

• Identify and interpret roots, intercepts, turning points of quadratic functions graphically

• Deduce roots algebraically and turning points by completing the square

• Recognise, sketch and interpret graphs of quadratic functions

• Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph

• Solve two simultaneous equations in two variables linear/quadratic algebraically; find approximate solutions using a graph

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Key Concepts

• Check brackets have been factorised correctly by multiplying them back out.

• To solve quadratics by factorising students need to identify two numbers that have a product of c and a sum of b. Roots are found when each bracket is made to equal zero and are solved for x.

• When a quadratic cannot be solved by factorising students should use completing the square or the quadratic formula.

• Students should be able to derive the quadratic formula from the method of completing the square.

• A sketched graph is drawn freehand and includes the roots, turning point and intercept values.

• Quadratic identities in the form (x + a)2 + b ≡ ax2 + bx + c can be solved either through completing the square to RHS = LHS or by expanding the brackets to LHS = RHS and equating the unknowns.

• Quadratic and linear simultaneous equations should be sketched before solved algebraically to ensure students know to find and the x and y values.

Common Misconceptions

• The method of trial and improvement is often incorrectly used to try and solve quadratics.

• When solving quadratic and linear simultaneous equations students often forget to find the y values as well the x.

• When using the formula to solve quadratics students often forget to evaluate the negative solution. Some students also incorrectly apply the division by reducing the terms it covers.

• Students tend to struggle deriving quadratic equations from geometrical facts.

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Lessons

Solving Quadratics by Factorising

Solving Complex Quadratics by Factorising

Completing the Square

Solving Quadratic Identities

Using the Quadratic Formula

Form and Solve Quadratic Equations

Sketching Quadratics

Quadratic and Linear Simultaneous Equations

Quadratic Inequalities

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Problem Solving and Revision Lessons

Solving Quadratics by Factorisation

Solving Complex Quadratics by Factorisation

Completing the Square

Forming and Solving Quadratic Equations

Quadratic and Linear Simultaneous Equations

Setting Up and Solving Quadratic Equations

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Additional Departmental Resources

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Mathematical Proof

Students learn how to construct a mathematical proof using algebraic notation and geometrical reasoning. This higher GCSE topic take place in Term 3 Year 11. Before learning about proof students should be able to manipulate algebraic expressions. Prerequisite Knowledge

• Know the difference between an equation and an identity;

• Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c

• Understand and use standard mathematical formulae; rearrange formulae to change the subject

• Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

Success Criteria

• Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

• Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

• Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

• Use vectors to construct geometric arguments and proofs

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Key Concepts

• Know the difference between an equation and an identity;

• Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c

• Understand and use standard mathematical formulae; rearrange formulae to change the subject

• Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

• Parallel lines have vectors that are multiples of each other.

Common Misconceptions

• A common incorrect approach is to attempt to prove an algebraic and geometrical property through numerical demonstrations.

• Students often struggle generalising the rules of arithmetic to produce a reasoned mathematical argument.

• Some students expand brackets incorrectly when proving a quadratic identity.

• Students often lose marks when attempting to prove geometrical properties due to not connecting the various angle properties.

• Incorrect application of ratio notation leads to difficulty when proving geometrical properties.

• Students often fail to label vector diagrams sufficiently to identify known paths.

• Providing a proof of geometrical facts tends to separate the most able from the majority.

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Lessons

Algebraic Proof

Geometrical Proof

Vector Problems with Ratio

Proof with Vectors

Problem Solving and Revision Lessons

Algebraic Proof

Geometric Proof

Vectors with Geometry

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Additional Departmental Resources

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Trigonometry – Graphs

Students learn how to visualise and sketch the Sine, Cosine and Tangent graphs. They use these graphs to find all the solutions to trigonometric equations. Learning progresses from this to finding the exact solutions to Sine, Cos and Tan of 30, 45, 60 and 90 using an equilateral and right-angled triangle. Prerequisite Knowledge

• Know the trigonometric ratios.

• Apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two- and three-dimensional figures

Success Criteria

• Know the exact values of and for = 0°, 30°, 45°, 60° and 90°; know the exact value of for = 0°, 30°, 45°, 60°

• Recognise, sketch and interpret graphs of trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size

Key Concepts

• Trigonometric graphs have lines of symmetry at that can be used to find additional solutions equations.

• Trigonometric ratios of 30°, 45° and 60° have exact forms that can be calculated using the special triangles.

• The relationship Tanθ = Sinθ/Cosθ can be seen fro the asymptotes in the tan graph.

Common Misconceptions

• Students often forget to rationalise Sin45° and Cos45°.

• When solving trigonometric equations students often forget to use the graphs to include all solutions.

• In examinations students often confuse the coordinates, e.g., (0,180) with (180,0)

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Lessons

Trigonometrical Graphs

Exact Solutions

Problem Solving and Revision Lessons

Trigonometric Graphs and Equations

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Additional Departmental Resources

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Working with Algebraic Fractions Students learn how to simplify and perform addition, subtraction, multiplication and division with algebraic fractions. As learning progresses, they combine these skills to solve equations with algebraic fractions using the quadratic formula. Prerequisite Knowledge

• Solve linear equations in one unknown algebraically including those with the unknown on both sides of the equation)

• Apply the four operations, including formal written methods, simple fractions (proper and improper)

• Calculate exactly with fractions

• Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c

Success Criteria

• Simplify and manipulate algebraic fractions by: o collecting like terms o multiplying a single term over a bracket o taking out common factors o expanding products of two or more binomials o simplifying expressions involving sums, products and

powers, including the laws of indices

• argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula

Key Concepts

• Students need to apply the same numerical techniques with algebraic fractions as they have done with numerical ones.

• Like numerical fractions algebraic fractions need to have a common denominator when performing addition or subtraction.

• Simplifying algebraic fractions involves factorising the expression into either one or more brackets.

• Multiply the fractions through by a common denominator to cancel out the division when solving fractions.

Common Misconceptions

• Students who understand the need for common denominators when adding or subtracting fractions are often let down by their poor algebraic skills. Particularly when multiplying out by a negative.

• When attempting to simplify fractions students tend to cancel down incorrectly thus losing marks for final accuracy.

• Students can forget to use the difference of two squares when finding common denominators.

• Students struggle with factorising quadratics when the coefficient of x2 is greater than one.

• It is common for students to try and solve for the unknown when they have only been asked to simplify.

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Lessons

Simplifying Algebraic Fractions

Addition with Algebraic Fractions

Multiplying and Dividing with Algebraic Fractions

Equations with Algebraic Fractions

Quadratic Equations with Algebraic Fractions

Problem Solving and Revision Lessons

Simplifying Algebraic Fractions

Equations with Algebraic Fractions

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Additional Departmental Resources

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Working with Functions

Students learn how to use function notation to transform graphs by a translation and stretch. As learning progresses, they calculate composite and inverse functions. Finally, students are challenged to solve equations through iteration and calculate a rate of change along a non-linear graph using differentiation. Prerequisite Knowledge

• Plot graphs of equations that correspond to straight-line graphs in the coordinate plane

• Recognise, sketch and interpret graphs of linear and non-linear functions

• Identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement.

Success Criteria

• Where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.

• Sketch translations and reflections of a given function

• Calculate or estimate gradients of graphs (including quadratic and other non-linear graphs),

• Find approximate solutions to equations numerically using iteration

Key Concepts

• A function is any algebraic expression in which x is the only variable. It is denoted as f(x) = x ….

• Understanding the notation for transformation of functions is critical to accessing this topic.

o f(x) ±a = Vertical Translation o f(x ± a) = Horizontal Translation o af(x) = Horizontal stretch o f(ax) = Vertical stretch

• Composite functions combine more than one function to an input.

• Inverse functions perform the opposite operation to a function.

• A gradient function calculates and approximate the instantaneous rate of change for given values of x.

• Iterative solutions can diverge or converge.

Common Misconceptions

• -f(x) is often incorrectly taken as a reflection in the y axis rather than the x.

• f(x + a) is a translation of ‘a’ units to the left rather than to the right.

• Students often struggle with writing the equation of the new function after a transformation.

• Students need to be precise when drawing the transformed function.

• Students can confuse f-1(x) with f’(x).

• The order of a composite function is often confused, fg(x) -> g acts on x first then f acts on the result.

• Students are often able to differentiate functions with little understanding of how to apply the gradient function correctly.

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Lessons

Working with Functions

Translating Functions

Stretching Functions

Composite Functions

Inverse Functions

Solving Equations through Iteration

Instantaneous Rates of Change

Estimating the Area Under a Curve

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Problem Solving and Revision Lessons

Functions and Composite Functions

Area Under a Curve

Inverse Functions

Gradients of Curves

Transforming Graphs

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Additional Departmental Resources

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Higher GCSE Mathematics Exam Revision

The program of revision reminds students of the key concepts, at their potential target grade. It is based around a cycle of formal fortnightly, hour

long assessments which are provided by examination boards.

The revision cycle takes place over 7 lessons per fortnight with the assessment taking place in the 7th lesson. Lesson 1 in the next cycle is used

to go through the assessment with the students. This ensures every student has sufficient exam style practice applying the skills they revised in

the previous five lessons.

All revision lessons begin with a range of traditional questions that remind students of the skill they are revising. As learning progresses, they

apply this skill to solve wider ranging problems that connect to other areas of mathematics.

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Cycle 1 Aiming for Grades 5 to 6

Multiplication with Decimal Numbers

Division with Decimal Numbers

Performing and Describing Transformations

Interior and Exterior Angles of Polygons

Volume of Prisms

Writing Numbers in Standard Form

Assessment Set 1 Paper 1

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Cycle 1 Aiming for Grades 7 to 9

Limits of Accuracy and Error Intervals

Variation

Similarity

Drawing and Interpreting Histograms

Indices with Fractional and Negative Powers

3D Trigonometry

Assessment Set 1 Paper 1

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Cycle 2 Aiming for Grades 5 to 6

Model Solutions to Set 1 Paper 1

Exchange Rates

Pythagoras’ Theorem

Scatter Graphs and Correlation

Setting up and Solving Equations

Setting up and Solving Inequalities

Assessment Set 1 Paper 2

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Cycle 2 Aiming for Grades 7 to 9

Model solutions to Set 1 Paper 1

Conditional Probability

Completing the Square

Functions and Composite Functions

Sine and Area Rules

Expanding Cubic Expressions

Assessment Set 1 Paper 2

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Cycle 3 Aiming for Grades 5 to 6

Model Solutions Set 1 Paper 2

Angles in Parallel Lines

Sharing to a Ratio

Problems with Circles

Compound Percentage Change

Solving Quadratics by Factorisation

Assessment Set 1 Paper 3

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Cycle 3 Aiming for Grades 7 to 9

Model Solutons Set 1 Paper 2

Parallel and Perpendicular Gradients

Sine, Cosine and Area Rules

Rearrange Complex Formulae

Area Under Non-Linear Graphs

Algebraic Proof

Assessment Set 1 Paper 3

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Cycle 4 Aiming for Grades 5 to 6

Model solutions to Set 1 Paper 3

Density and Pressure

Linear Simultaneous Equations

Right-Angled Trigonometry

Sectors and their Formulae

Reverse Percentages

Assessment Set 2 Paper 1

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Cycle 4 Aiming for Grades 7 to 9

Model solutions to Set 1 Paper 3

Trigonometric Graphs and Equations

Solving a2 + bx + c = 0 by Factorisation

Gradients of Curves

Non-Linear Sequences

Inverse Functions

Assessment Set 2 Paper 1

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Cycle 5 Aiming for Grades 5 to 6

Model Solutions to Set 2 Paper 1

Changing the Subject of a Formula

Equations with Fractions

Rules of Indices

Calculations with Standard Form

Kinematics Formulae

Assessment Set 2 Paper 2

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Cycle 5 Aiming for Grades 7 to 9

Model solutions to Set 2 Paper 1

Simplify Algebraic Fractions

Spheres, Cones and Pyramids

Geometrical Proof

Calculations with Surds

Transforming Functions

Assessment Set 2 Paper 2

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Cycle 6 Aiming for Grades 5 to 6

Model Solutions Set 2 Paper 2

Probability Trees and Independent Events

Similar Shapes

Equation of Straight Line Graphs

Surds

Nth Term of Linear Sequences

Assessment Set 2 Paper 3

Model Solutions to Set 2 Paper 3

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Cycle 6 Aiming for Grades 7 to 9

Model Solutions Set 2 Paper 2

Equations with Algebraic Fractions

Circle Theorems

Vectors and Geometry

Setting Up and Solving Quadratics

Linear and Quadratic Simultaneous Equations

Assessment Set 2 Paper 3

Model Solutions to Set 2 Paper 3