Year 11 Scheme of work Autumn Term Half term 1 Higher level

Year 11 Scheme of work Autumn Term Half term 1 Higher level

Topic Title: Algebra: further quadratics, rearranging formulae and identities 6 lessons Routemap: 3 Year Higher Pre requisite knowledge: Basic algebraic substitution and equation solving skills. 4 rules with fractions and decimals. Students should be able to: Higher Content: Keywords: Function, inverse, expression, input, output Number of lessons required: 6 lessons. A4 Simplify and manipulate algebraic expressions (including those involving surds) by: expanding products of two or more binomials factorising quadratic expressions of the form including the difference of two squares factorising quadratic expressions of the form simplifying expressions involving sums, products and powers, including the laws of indices A5 Understand and use standard mathematical formulae Rearrange formulae to change the subject A6 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 A7 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 Topic commentary: Quadratics have been covered in year 10. This is an opportunity to consolidate and extend. This unit covers factorising only. Solving is covered in the next unit although you may choose to teach them together. Functions (A7) is covered in detail in the AQA resources. The PowerPoint slides take students through simple functions, inverse, graphing and composite functions. The notes section suggests which activities from the resource fit in where. The worksheet gives some basic practise with simple functions and includes solving equations, also including composite functions. The Jigsaw activity requires students to evaluate composite functions. NB Students will need to be given the functions required. The dominoes activity requires students to match functions with their inverse to complete a puzzle. Each activity can be used alone or alongside the PowerPoint as a complete lesson. Lesson Plans available AQA Pearsons Notes 1 powerpoint,2 activities and 1 worksheet available 2.7 Expanding and factorising quadratics 2.4,17.1 Rearranging formulae 17.7 Functions 17.8 Proofs Lesson Assessment Problem Solving activity

Topic Title: Further equations and graphs 6 lessons Routemap: 3 Year Higher Pre requisite knowledge: Key words Students should be able to: Number of lessons required: 6 lessons. Higher Content: A17 Solve linear equations in one unknown algebraically including those with the unknown on both sides of the equation Find approximate solutions using a graph including use of brackets A18 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 A12 Recognise, sketch and interpret graphs of linear and quadratic functions A11 Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square including the symmetrical property of a quadratic

A21 Translate simple situations or procedures into algebraic expressions or formulae derive an equation, solve the equation and interpret the solution including solution of geometrical problems and problems set in context Topic commentary: Lesson Plans available AQA Pearsons Notes 2.3 Solving linear equations 9.1 9.3 Solving quadratics including completing the square 6.6, 6.8 Quadratic graphs 15.3 Graphs of quadratic functions 15.4 Solving quadratic functions graphically Lesson Assessment Problem Solving activity

Topic Title: Sketching graphs 3 lessons Routemap: 3 Year Higher Pre requisite knowledge: Trigonometrical ratios and how to find on a calculator. Plotting points joining with a curve. Understanding of 2 < x > -2 notation for values of x Key words Cosine, tangent, exponential, period. Students should be able to: draw, sketch, recognise and interpret graphs of the form y = k x for positive values of k know the shapes of the graphs of functions y = sin x, y = cos x and y = tan x Number of lessons required: 3 lessons. Sketch y = sin x, y = cos x and y = tan x, know that the maximum and minimum values for Sin and Cos are 1 and -1. Know that the graphs of sin, cos and tan are periodic. Higher Content: Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function, y = x 1 with x 0, exponential functions y = k x for positive values of k, and the trigonometrical functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size. (A12h) Topic commentary: Trig graphs are introduced in this unit as well as other graphs. The following units recap on basic trig, followed by the sine rule and cosine rule. You may wish to delay trig graphs until you have recapped trigonometry

AQA: The resource is designed to be used in a variety of ways. The PowerPoint covers all the knowledge of graphs which students will need to know. The Trigonometrical Functions worksheet leads students through an investigation looking at graphs of sin, cos and tan. The PowerPoint contains the correct graphs and properties. The True False activity provides a way of checking knowledge of what the graphs should look like. Answers are provided. A set of exam paper questions are provided. Full papers can be found on AQA s website. The resource lends itself to a variety of lesson formats. From a short revision PowerPoint with students sketching graphs on mini whiteboards to a full 1 hour lesson. E.g. Starter discussion of slide 1. Time spent doing Trigonometric Functions use PowerPoint to check and discuss. True/ False in groups or whole class to consolidate. Exam questions to finish off or for homework. In order to do the activities students will need to be able to use their calculators to find trig ratios, draw accurate graphs, and sketch graphs. The resource could be used as a way to revise all of these skills. Lesson Plans available AQA Pearsons Notes Powerpoints and resources. See notes in topic commentary. 6.7 Cubic and reciprocal graphs 15.5 Cubic graphs 19.4 Exponential functions 13.2 13.4 Trig graphs Lesson Assessment Problem Solving activity

Topic Title: Trigonometry recap and extension 3 lessons Routemap: 3 Year Higher (Trigonometry was previously covered in Year 9) Pre requisite knowledge: Convert fractions to decimals. Identify the hypotenuse. Students should be able to: Higher Content: Keywords: Number of lessons required: 3 lessons. G20 Know the formula for Pythagoras' Theorem Apply it to find length in right angled triangles and, where possible, general triangles in two and three dimensional figures Know and use 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 G21 Know the exact values of Know the exact value of 0, 30 45, 60 and 90 0, 30, 45 and 60 G6 Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides including Pythagoras Theorem, use

known results to obtain simple proofs R12 Compare lengths using ratio notation; Make links to trigonometric ratios Topic commentary: Students may not rearrange the equation correctly when the unknown is the denominator of the fraction. Review solving simple equations such as 3 =. Lesson Plans available Notes AQA Pearsons 2 lessons available (5.4,5.5) Lesson Assessment Problem Solving activity

Year 11 Scheme of work Autumn Term Half term 2 Higher level Topic Title: Sine and cosine rules 6 lessons Routemap: 3 Year Higher Pre requisite knowledge: Key words Students should be able to: Number of lessons required: 6 lessons. Higher Content: G22 Know and apply the Sine rule and Cosine rule to find unknown lengths and angles G23 Know and apply to calculate the area, sides or angles of any triangle

Students are expected to know the exact values of sin θ and cos θ for θ = 0, 30, 45, 60 and 90, and the exact values of tan θ for θ = 0, 30, 45 and 60. This is new to the Higher tier GCSE 2015. Students need to know the formula for the cosine rule and be able to apply it to work out unknown lengths and angles. This is new to the Higher tier GCSE 2015. Students need to know the formula for area of a triangle and be able to apply it to work out sides, angles and areas of given triangles. This is new to the Higher tier GCSE 2015. Topic commentary: The only complication is the ambiguous case, where there are two possible angles. One is found using sin 1, and the other is found by subtracting the first value found from 180. Refer back to the graph of the sine function to help you. Students may not understand that practice is needed to quickly decide whether to use the sine rule or the cosine rule to solve a problem. Lesson Plans available Notes AQA Pearsons 13.5,13.6 13.7 3D trig Lesson Assessment Problem Solving activity

Topic Title: Transforming functions 3 lessons Routemap: 3 Year Higher Pre requisite knowledge: Applying simple transformations to coordinates Key words asymptote Students should be able to: Number of lessons required: 3 lessons. Higher Content: A13 Sketch translations and reflections of a given function Topic commentary: Lesson Plans available AQA Pearsons Notes 13.8,13.9 Transforming trig graphs 19.6, 19.7 Transforming graphs Lesson Assessment Problem Solving activity

Topic Title: Algebraic Fractions 3 lessons Routemap: 3 Year Higher Pre requisite knowledge: Students should be confident with basic algebra although there will be revision, and should know how to solve linear and quadratic equations, both by factorising and using the formula. They should be able to apply Pythagoras theorem and be confident with a range of angle, perimeter and area facts. Students should be able to: Higher Content: Keywords: Number of lessons required: 3 lessons. Simplify and manipulate algebraic expressions involving algebraic fractions (A4) Topic commentary: The topic is introduced by revising the four operations with fractions, and simple algebra. This leads through to manipulating algebraic fractions and solving problems. Solving quadratics is also covered as a revision topic. Some of the plenaries are word problems which can be read out, encouraging higher level students to make notes and process information. Some extension ideas take students into work above GCSE level these are clearly indicated and are a good way to support students in the transition to advanced level study. Lesson Plans available Notes AQA 4 lessons Pearsons 4 lessons (17.2 17.4, 17.6) Lesson Assessment Problem Solving activity

Topic Title: Numerical Methods 1 lesson Routemap: 3 Year Higher Pre requisite knowledge: Key words Students should be able to: Solve quadratic equations using an iterative process. Number of lessons required: 1 lesson Higher Content: A20 Find approximate solutions to equations numerically using iteration including the use of suffix notation in recursive formulae Topic commentary: The iterative process (and the notation used in it) is new to the Higher tier GCSE 2015, although the process is not dissimilar to the trial and improvement process. Lesson Plans available Notes AQA Pearsons 15.4 Q12,13 Lesson Assessment Problem Solving activity

MOCK EXAMS AND REVISION Year 11 Scheme of work Spring Term Half term 1 Higher level Topic Title: Vectors 6 lessons Pre requisite knowledge: Students need to know how to add and subtract negative numbers and work with co-ordinates in four quadrants. They should also know properties of plane shapes such as parallelograms. Students may already be familiar with column vectors for using a vector to represent a translation. Routemap: 3 Year Higher Key words vector, column, horizontal, vertical, scalar, negative, sum, difference, resultant, commutative, associative, collinear, parallel, midpoint Higher Content: Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors (G25). Use vectors to construct geometrical arguments and proof (G25). Number of lessons required: 6 lessons. Topic commentary: This topic builds on students possible prior knowledge of column vectors from their use to represent translations in the transformations topic. Students may confuse column vectors with co-ordinate pairs. They may also initially find the construction of a geometrical proof difficult so a lot of opportunities to consolidate this skill are provided. Throughout the topic, the correct use of notation should be emphasized, such as vectors being expressed in bold font in printed documents and with an arrow of the form. Extension materials are provided which offer additional demand for more able students. Lesson Plans available Notes

AQA 8 lessons available Pearsons 18.1 18.5 Lesson Assessment Problem Solving activity

Topic Title: Circle theorems 6 lessons Routemap: 3 Year Higher Pre requisite knowledge: Recall the sum of angles of a quadrilateral. Use correct mathematical vocabulary for parts of a circle Students should be able to: Higher Content: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords and use them to prove related results (G10) Number of lessons required: 6 lessons. Topic commentary: Lesson Plans available Notes AQA Pearsons 4 lessons available (16.1 16.4) Lesson Assessment Problem Solving activity

Topic Title: Equation of a circle 2 lessons Routemap: 3 Year Higher Pre requisite knowledge: Students should know how to calculate the area and perimeter of a circle and be able to use Pythagoras theorem to solve problems. They should also be able to use coordinate geometry to find gradients and equations of straight lines and know that the product of perpendicular lines are -1. Students need to be able to solve both quadratic equations and simultaneous equations where one equation is linear and one is quadratic. It is beneficial if students have some knowledge of circle theorems and rules. Students should be able to: recognise the equation of a circle, centre (0, 0), radius r write down the equation of a circle, centre (0, 0) and radius r work out coordinates of points of intersection of a given circle and a given straight line use the fact that the angle between the tangent and radius is 90 to work out the gradient of a tangent and hence the equation of a tangent at a given point. Number of lessons required: 2 lessons. Higher Content: Recognise and use the equation of a circle with centre at the origin Find the equation of a tangent to a circle at a given point. (A16) Topic commentary: The PowerPoint for this topic demonstrates how the equation of a circle can be derived. There are some simple questions to find the radius from the equation of a circle and vice versa. There is also a revision / introductory activity for relevant circle theorems, leading on to using the fact that gradients of perpendicular lines multiply to -1 to find the equation of a tangent to a circle. The worksheet and homework provide reinforcement of these ideas. The topic requires the candidates to be able to use Pythagoras theorem, circle theorems, and finding equations of lines, but could also be could be used as a revision opportunity for these topics.

Lesson Plans available Notes AQA Powerpoint and resources available. See commentary above. Pearsons 16.5 Lesson Assessment Problem Solving activity

Topic Title: Direct and Inverse Proportion 3 lessons Routemap: 3 Year Higher Pre requisite knowledge: Recognise direct proportion Key words Constant of proportionality Students should be able to: Number of lessons required: 3 lessons. Higher Content: R10 Solve problems involving direct and inverse proportion, including graphical and algebraic representations R13 Understand that is inversely proportional to is equivalent to is proportional to Construct and interpret equations that describe direct and inverse proportion R14 Recognise and interpret graphs that illustrate direct and inverse proportion

Topic commentary: Ratio and proportion was covered in Year 10 Students may write a formula with the two variables the wrong way round (finding the inverse of the constant). Encourage students to check their formula by substitution. Students may substitute into the formula, or process incorrectly. Encourage students to write down each step in their working and check their answers. Students may fail to identify the correct type of relationship. Encourage them to read the question carefully, writing down each piece of information as they go. Some students may carry out operations in the wrong order (such as multiplying before squaring). Remind students of BIDMAS. Lesson Plans available Notes AQA Pearsons 19.1 19.3 Lesson Assessment Problem Solving activity

Topic Title: Gradients and Rate of Change 4 lessons Routemap: 3 Year Higher Pre requisite knowledge: Students need to be familiar with finding the gradient of a line by using a right-angled triangle and calculating the change in y divided by the change in x. They should also have good basic algebra and geometry skills, including plotting coordinates and selecting axis units. Students should be able to: Higher Content: Interpret the gradient of a straight-line graph as a rate of change (R14) Key words Parallel, perpendicular, gradient, positive, negative, rate of change, instantaneous rate of change, average rate of change, chord, tangent, Number of lessons required: 4 lessons Interpret the gradient at a point on a curve as the instantaneous rate of change (R15) Apply the concepts of average and instantaneous rates of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts (R15) Topic commentary: AQA: There are 6 lessons in this topic. The topic has been approached in a systematic way working from revision of gradients of straight lines to finding and using the gradient of a curve. There are opportunities to bring in some basic pre-calculus ideas with students if desired; indeed, there are overlaps with this topic and the later pre-calculus and area under a curve topic, however it is suggested that this topic is completed first. Not all lessons may be needed and several could stand alone if preferred. Lesson Plans available AQA Pearsons Notes 6 lessons available 6.3 Graphing rates of change 11.2,11.3??

Lesson Assessment Problem Solving activity

Year 11 Scheme of work Spring term Term Half term 2 Higher level Topic Title: Pre Calculus and area under a curve 6 lessons Routemap: 3 Year Higher Pre requisite knowledge: Students should already be able to plot straight line and curved graphs. They should be able to read and interpret distance-time and velocity-time graphs. Students are expected to be able to find the gradient of straight line graphs. Students should be able to: Higher Content: Key words curve, gradient, tangent, trapezium, distance-time graph, velocity-time graph, velocity, speed, trapezia, vertical axis, horizontal axis, estimate, rate of change, positive / negative gradient Number of lessons required: 6 Lessons Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs) (A15) Interpret the results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts (A15) Topic commentary: AQA: This topic overlaps with gradients and rate of change. The idea of finding a gradient of a curve is revisited in lesson 1. Although in the examination any method of finding the area is allowed, in this topic we are introducing using the trapezium rule as the best method Lesson Plans available AQA Pearsons Lesson Assessment Notes 6 lessons available

Problem Solving activity

Year 9 and 10 Overview

Year 11 Overview