KS3 GCSE Year 9 Transition Scheme of Work

KS3–GCSE Year 9 Transition Scheme of Work

Welcome!

We’ve created this Scheme of Work (SoW) to help you to start preparing your Year 9s for the new Edexcel GCSE from September 2014. You can use this between now and the end of the Spring term 2015, after which you will be able to use the new GCSE materials.

This Scheme of Work covers two terms of new National Curriculum content, which your Year 9s need to get the best start on the new GCSE and which they won’t have covered if they followed the old National Curriculum in Years 7 and 8. The stated teaching hours act as a guide to how much time should be spent on the topic. Please note, this Scheme of Work acts as a bridge between the old and the new KS3 curriculum in preparation for the new GCSE and therefore isn’t a comprehensive guide as to what should be taught in Year 9. 2

Overview

Key for (draft*) Edexcel GCSE Specification Objectives  All students will develop confidence and competence with the content identified by standard type.  All students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content.  Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.  N = Number A = Algebra R = Ratio, proportion and rates of change G = Geometry and measures P = Probability S = Statistics

* Please note, at time of writing (August 2014) the new Edexcel GCSE specification is subject to accreditation by Ofqual. We will revise this Scheme of Work if necessary once the new Edexcel GCSE specification has been accredited.

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels  enumerate sets and unions/intersections of sets P6 enumerate sets and combinations of sets systematically, using tables, grids and Venn systematically, using tables, grids, Venn diagrams and diagrams tree diagrams 6  generate theoretical sample spaces for single and 5a–8b Probability P7 construct theoretical possibility spaces for single and combined events with equally likely, mutually combined experiments with equally likely outcomes and exclusive outcomes and use these to calculate 1 use these to calculate theoretical probabilities T theoretical probabilities H

m  use integer powers and associated real roots u t (square, cube and higher), recognise powers of 2, 3, N6 use positive integer powers and associated real u

A 4, 5 and distinguish between exact representations roots (square, cube and higher) recognise powers of 2, Indices and of roots and their decimal approximations 3, 4, 5; estimate powers and roots of any given number 8  appreciate the infinite nature of the sets of 6b–8c N7 calculate with roots, and with integer and fractional Standard Form integers, real and rational numbers indices  interpret and compare numbers in standard form N9 calculate with and interpret standard form A × 10n, A × 10n 1 ≤ A < 10, where n is a positive or negative where 1 ≤ A < 10 and n is an integer integer or zero

KS3–GCSE Year 9 Transition Scheme of Work © Pearson Education Limited 2014 3

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels  substitute numerical values into formulae and expressions, including scientific formulae  understand and use standard mathematical A2 substitute numerical values into formulae and formulae; rearrange formulae to change the expressions, including scientific formulae Formulae 6 5a–7a subject A5 understand and use standard mathematical  model situations or procedures by translating them formulae; rearrange formulae to change the subject into algebraic expressions or formulae and by using 1

T graphs H

m A24 recognise and use sequences of triangular, square u t and cube numbers, simple arithmetic progressions, u

A  recognise arithmetic sequences and find the nth Fibonacci type sequences, quadratic sequences, and term simple geometric progressions ( r n where n is an integer Sequences 4 5b–7c  recognise geometric sequences and appreciate and r is a rational number > 0 or a surd) and other other sequences that arise sequences A25 deduce expressions to calculate the nth term of linear and quadratic sequences

Half Term

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels

 interpret mathematical relationships both A9 plot graphs of equations that correspond to straight- algebraically and graphically line graphs in the coordinate plane; use the form  model situations or procedures by translating them y = mx + c to identify parallel and perpendicular lines; into algebraic expressions or formulae and by using find the equation of the line through two given points graphs or through one point with a given gradient  reduce a given linear equation in two variables to A10 identify and interpret gradients and intercepts of the standard form y = mx + c; calculate and linear functions graphically and algebraically interpret gradients and intercepts of graphs of such A12 recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions,

2 linear equations numerically, graphically and T algebraically 1 H the reciprocal function y  with x  0, exponential

n  recognise, sketch and produce graphs of linear and 10 6b–7a x

m Graphs quadratic functions of one variable with x u functions y = k for positive values of k, and the t

u appropriate scaling, using equations in x and y and trigonometric functions (with arguments in degrees) A the Cartesian plane y = sin x, y = cos x and y = tan x for angles of any size  use linear and quadratic graphs to estimate values A14 plot and interpret graphs (including reciprocal of y for given values of x and vice versa and to find graphs and exponential graphs) and graphs of non- approximate solutions of simultaneous linear standard functions in real contexts to find approximate equations solutions to problems such as simple kinematic  find approximate solutions to contextual problems problems involving distance, speed and acceleration from given graphs of a variety of functions, A19 solve two simultaneous equations in two variables including piece-wise linear, exponential and (linear/linear or linear/quadratic) algebraically; find reciprocal graphs approximate solutions using a graph

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels G12 identify properties of the faces, surfaces, edges and  derive and apply formulae to calculate and solve vertices of: cubes, cuboids, prisms, cylinders, pyramids, problems involving: perimeter and area of cones and spheres triangles, parallelograms, trapezia, volume of G16 know and apply formulae to calculate: area of cuboids (including cubes) and other prisms triangles, parallelograms, trapezia; volume of cuboids (including cylinders) and other right prisms (including cylinders)  use the properties of faces, surfaces, edges and 2 G17 know the formula: circumference of a circle T vertices of cubes, cuboids, prisms, cylinders, 2 H

 2r  d , area of a circle  r ; perimeters of 2D

n Perimeter, Area pyramids, cones and spheres to solve problems in 10 6c–8b shapes, including circles; areas of circles and composite m and Volume 3D u

t shapes; surface area and volume of spheres, pyramids,  use approximation through rounding to estimate u cones and composite solids A answers and calculate possible resulting errors N15 round numbers and measures to an appropriate expressed using inequality notation a < x ≤ b degree of accuracy (eg to a specified DP or SF); use *2D has already been studied in Y7/8. However, the inequality notation to specify simple error intervals due emphasis is now on knowing the formulae and to truncation or rounding therefore 2D shapes should be revised here before 3D N16 apply and interpret limits of accuracy, including shapes upper and lower bounds End of Term 1

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels G3 apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)  interpret mathematical relationships both G6 apply angle facts, triangle congruence, similarity and algebraically and geometrically properties of quadrilaterals to conjecture and derive  identify and construct congruent triangles, and results about angles and sides, including Pythagoras’ construct similar shapes by enlargement, with and Theorem and the fact that the base angles of an without co-ordinate grids Angles, isosceles triangle are equal, and use known results to  apply angle facts, triangle congruence, similarity obtain proofs Pythagoras and 15 and properties of quadrilaterals to derive results 5c–8b G19 apply the concepts of congruence and similarity, Trigonometry about angles and sides, including Pythagoras’ including the relationships between lengths, areas and Theorem and use known results to obtain simple volumes in similar figures proofs G20 know the formula for: Pythagoras’ Theorem 1  use Pythagoras’ Theorem and trigonometric ratios 2 2 2 T a + b = c , and trigonometric ratios, H in similar triangles to solve problems involving opposite adjacent g sin  cos  n right-angled triangles , and i

r hypotenuse hypotenuse p

S opposite tan  ; apply them to find angles and adjacent lengths in right-angled triangles and where possible, general triangles in two and three dimensional figures R9 define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these  solve problems involving percentage change, multiplicatively; express one quantity as a percentage of including: percentage increase, decrease and another; compare two quantities using percentages; 4 original value problems and simple interest in 6b–8c Percentages work with percentages greater than 100%; solve financial mathematics problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics Half Term KS3–GCSE Year 9 Transition Scheme of Work © Pearson Education Limited 2014 7

(Old NC) Term SoW Unit Teaching hours National Curriculum 2014 Programme of Study Refs Leading into (draft) GCSE Specification Objectives Levels

 relate the language of ratios and the associated R8 relate ratios to fractions and to linear functions calculations to the arithmetic of fractions and to R10 solve problems involving direct and inverse linear functions proportion, including graphical and algebraic Ratio and  solve problems involving direct and inverse 8 6a–8c representations proportion, including graphical and algebraic Proportion R11 use compound units such as speed, rates of pay, representations unit pricing, density and pressure  use compound units such as speed, unit pricing and density to solve problems 2 T H

g S4 interpret, analyse and compare the distributions of n i

r data sets from univariate empirical distributions

p  describe, interpret and compare observed

S through: distributions of a single variable through:  appropriate graphical representation involving appropriate graphical representation involving discrete, continuous and grouped data, 4 discrete, continuous and grouped data; and 5b–7a Statistics including box plots appropriate measures of central tendency (mean,  appropriate measures of central tendency mode, median) and spread (range, consideration of (median, mean, mode and modal class) and outliers) spread (range including consideration of outliers, quartiles and inter-quartile range)

End of Term 2

There will be an end of term 1 assessment and a GCSE baseline assessment (to be taken at the end of term 2 before pupils begin their GCSE course) available free from Edexcel.

Autumn Term: Unit Plans

SoW Unit 1 – Probability

Teaching Time Guidance - 6 hours

2014 Programme of Study Refs  enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams  generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

Leading into (draft) GCSE Specification Objectives P6 enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams P7 construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities

Fluency Before starting this unit, students will need to be fluent in:  the basics of fractions, decimals and percentages for use within probability  the foundations of probability, its scale and calculating theoretical and experimental probabilities

Foundation Route: To familiarise students with the learning objectives Higher Route: To master the learning objectives

Learning Objectives: You will learn to:  calculate probabilities from tables, diagrams and experiments  compare experimental and theoretical probabilities  use of Venn diagrams, tables and sample space diagrams to list all possible outcomes of 1 and 2 events  generate Venn diagrams, tables and sample space diagrams to calculate probabilities of outcomes of 1 and 2 events  use tree diagrams to calculate probabilities of combining 2 independent events

Focus on Reasoning 1. Create a set of objects (e.g. pens, counters, socks) to match given probabilities for picking items from the set. 2. Consider the difference between theoretical and experimental outcomes for a game with certain outcomes (e.g. spinners, dice), and how you decide whether the game is fair or not. 3. Use a pack of playing cards to consider how you calculate the probability of picking cards that meet more than one criteria (e.g. black and a king, hearts and a 3). 4. Consider the relative likelihood of different total scores when you roll two dice and add their scores together. Repeat with dice with different numbers of sides. 5. Use a tree diagram and 1 – p to explore different ways of finding the probabilities of different types of events happening (e.g. throwing a dice twice and getting at least one six).

Focus on Problem-solving 1. Find the probabilities of picking different coloured counters from a bag. Work out how many counters have been added so that a given new probability is correct. 2. Spin two numbered spinners and add their scores together. Find out whether the total scores are more likely to be above or below a given total. 3. Use given probabilities and relative frequencies to complete a two-way table for outcomes to an event. 4. Use a probability table for counters in a bag to find the minimum number of counters that must be in the bag. 5. Use the outcomes from an experiment repeated different numbers of times, and use the best estimate to solve a fairness or relative frequency problem.

SoW Unit 2 – Indices and Standard Form

2014 Programme of Study Refs  use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations  appreciate the infinite nature of the sets of integers, real and rational numbers  interpret and compare numbers in standard form A × 10n 1 ≤ A < 10, where n is a positive or negative integer or zero

Leading into (draft) GCSE Specification Objectives N6 use positive integer powers and associated real roots (square, cube and higher) recognise powers of 2, 3, 4, 5; estimate powers and roots of any given number N7 calculate with roots, and with integer and fractional indices N9 calculate with and interpret standard form A × 10n, where 1 ≤ A < 10 and n is an integer

Fluency Before starting this unit, students will need to be fluent in:  understanding how to read/write powers, and use BIDMAS/BODMAS.  estimating answers to calculations  using the 4 operations with positive and negative numbers to apply with powers

Learning Objectives: You will learn to:  use the index laws to simplify and evaluate expressions  calculate combinations of powers, roots, fractions and brackets and give estimates to such calculations  understand negative and 0 indices  use powers of 10 and their prefixes  read and write both large and small numbers in standard form  use numbers written in standard form in context with and without a calculator

Focus on Reasoning 1. Establish a rule for how to raise a power of a product, using examples such as (2 × 4)2 and 22 × 42. Then do the same for a power of a division, using, for example (4 ÷ 2)2 and 42 ÷ 22. Try similar exercises with fractions, and with negative powers. 2. Explore the difference between a negative number raised to a power and a bracketed negative number raised to a power, e.g. –32 and (–3)2. Create more complex expressions to explore BIDMAS further. Try a similar exercise with negative powers. 3. Create calculations that use BIDMAS, remove the brackets, and ask students to add brackets to the calculations to reach the given answers. 4. Use the index law for multiplying two powers to find a rule for how to raise a power to another power, e.g. 52 × 52 × 52, and (52)3.

Focus on Problem-solving 1. Calculate the thickness of the paper used in different types of books, newspapers and magazines. Discuss what this tells you about them. 2. For models of objects of different sizes (e.g. famous landmarks, countries, the solar system), express their scales in standard form. 3. Use the average speed formula to estimate different calculations involving space rockets travelling to other planets, e.g. distance to planet, time taken to reach planet, average speed of rocket.

SoW Unit 3 – Formulae

2014 Programme of Study Refs  substitute numerical values into formulae and expressions, including scientific formulae  understand and use standard mathematical formulae; rearrange formulae to change the subject  model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

Leading into (draft) GCSE Specification Objectives A2 substitute numerical values into formulae and expressions, including scientific formulae A5 understand and use standard mathematical formulae; rearrange formulae to change the subject

Fluency Before starting this unit, students will need to be fluent in:  understanding and using algebraic notation  substitution into simple algebraic expressions  working with the 4 operations with integers (BIDMAS/BODMAS)

Learning Objectives: You will learn to:  substitute integers into simple and more complex formulae including scientific formulae  write expressions and formulae from a variety of situations and contexts (including simple graphs)  solve equations by substituting into formulae  change the subject of a formula

Focus on Reasoning 1. Create various expressions that include indices and more than one variable (e.g. 2xy2 or 3ab3 + c2). If the values used for the various terms are always negative, decide whether the expression will always be positive, negative, or either positive or negative.

Focus on Problem-solving 1. With given values for a set of variables (e.g. x = 2, y = –3, etc), create expressions that will produce given answers. 2. Use a set of six or seven formula cards to work out the values of a set of variables. Some of the formulae should include brackets and indices, e.g. a = 5, b = a + c, d = c2 – a, d – 1 = 2a, e = (b – a)2. 3. Substitute given values into scientific formulae to find the missing value, which need not be the subject of the formula. 4. Using a table of given values for various letters (e.g. a = –4, b = 5), match up a set of expressions with their respective numerical values. The expressions should include brackets, fractions, indices and roots. 5. Substitute given values into two interrelated formulae to find a missing value that appears in one of the formula only.

SoW Unit 4 – Sequences This can be combined with Unit 5.

2014 Programme of Study Refs  recognise arithmetic sequences and find the nth term  recognise geometric sequences and appreciate other sequences that arise

Leading into (draft) GCSE Specification Objectives A24 recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( r n where n is an integer and r is a rational number > 0 or a surd) and other sequences A25 deduce expressions to calculate the nth term of linear and quadratic sequences

Fluency Before starting this unit, students will need to be fluent in:  finding the next term and missing terms in a given sequence  recognising sequences of odds, evens, primes, triangular, squares, cubes and Fibonacci

Learning Objectives: You will learn to:  use the nth term to generate arithmetic sequences  find the nth term of an arithmetic sequence  recognise and continue geometric sequences  recognise and continue quadratic sequences

Focus on Reasoning 1. Find the values of the first five terms in an arithmetic sequence (e.g. 3n – 1), and plot these values on a graph with the term number on the x-axis and the values of the terms on the y-axis. Consider why arithmetic sequences are sometimes called linear sequences. 2. For quadratic sequences with the nth term n2 + a, find the second differences of the first five terms. Repeat for sequences with the nth term an2 + b, and consider how to use second differences to tell if a sequence is quadratic or not. 3. Use examples of term-to-term rules to determine which of the four operations (+, –, ×, ÷) lead to ascending sequences, and which lead to descending sequences.

Focus on Problem-solving 1. For a range of geometric sequences that involve division (e.g. 200, 100, 50…), find the first term that is less than 1, and determine whether any of the sequences have any negative terms or not. 2. Create the first few terms of a real-life geometric sequence and a statement that a particular term will be greater or less than a given number. Determine if the statement is true or false. 3. For a knockout tournament with a given number of players, work out the number of matches in each round, the number of rounds, and the total number of matches. 4. Solve an equation in terms of n to find which number term in a sequence gives the right-hand side of the equation (e.g. 3n + 4 = 19); solve other equations with different numbers to determine whether or not these numbers are in the sequence or not. 5. Create a pattern sequence involving two different colours of tiles, find the nth term for each colour, and work out the largest complete tiled pattern that can be made from given numbers of each colour of tile.

SoW Unit 5 – Graphs

2014 Programme of Study Refs  interpret mathematical relationships both algebraically and graphically  model situations or procedures by translating them into algebraic expressions or formulae and by using graphs  reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically  recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane  use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations  find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

Leading into (draft) GCSE Specification Objectives A9 plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points or through one point with a given gradient A10 identify and interpret gradients and intercepts of linear functions graphically and algebraically A12 recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal 1 function y  with x  0, exponential functions y = kx for positive values of k, and the trigonometric functions (with x arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size A14 plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration A19 solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph

Fluency Before starting this unit, students will need to be fluent in:  sequences  substitution  interpreting simple real life graphs  constructing axes with 4 quadrants and plotting co-ordinates accurately

Learning Objectives: You will learn to:  use distance time graphs to solve problems and interpret rates of change  draw a graph from its linear equation and without working out co-ordinates  understand, use and interpret y = mx + c  solve simultaneous equations by drawing graphs  draw quadratic and other non-linear graphs

Focus on Reasoning 1. Draw a set of parallel lines on a coordinate grid. Explore the similarities and differences between the equations of the lines, and use this information to write the equations of some more parallel lines. 2. Use examples of distance–time graphs to determine the meaning of the gradient of the line. 3. Water runs at a constant rate into containers of different shapes. Consider the relationships between the cross-sections of the containers and the graphs showing the depth of water in each container over time. 4. Draw the graph of a quadratic equation of the form y = ax2 + b and draw the line y = c (where c > b) to find solutions to the equation ax2 + b = c. Solve the equation algebraically and compare with the graphical solutions. Consider a second line y = d (where d < b) and whether the equation ax2 + b = d has any solutions or not. 1 5. Create a table of values for y = , and draw the graph on a software package. Examine whether the graph ever reaches the x x-axis and y-axis and the reason for this. 6. Use a software package to plot the graphs of cubic equations of the form y = x3 + c, and use these graphs to predict other cubic equations of the same form. 7. Match simple cubic and quadratic equations to sketches of their graphs.

Focus on Problem-solving 1. Provide information on a journey that someone makes, along with a sketch of the distance–time graph that includes a deliberate error, and identify and correct the error. Use the corrected graph to find out further information about the journey. 2. Create a situation with three objects moving along given equations of lines, and determine whether these objects can meet or not, and if so where. Try a similar exercise to see whether one moving object makes contact with other objects at fixed coordinates. 3. Without drawing the graphs, find the coordinates of the points where given equations of lines intersect with lines in the form y = c or x = c. 4. Create a linear equation for a real-life situation (e.g. salary plus commission, or taxi hire rates), and draw and interpret the graph. 5. Draw graphs to show the regions satisfied by given inequalities, and find pairs of values that satisfy both inequalities. 6. Draw half of a shape that is reflected in a given equation of a line (e.g. x = 2, y = –x). Find the equations of the lines needed to complete the symmetrical shape.

SoW Unit 6 – Perimeter, Area and Volume

2014 Programme of Study Refs  derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)  use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D  use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b

Leading into (draft) GCSE Specification Objectives G12 identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres G16 know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders) G17 know the formula: circumference of a circle  2r  d , area of a circle  r 2 ; perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids N15 round numbers and measures to an appropriate degree of accuracy (eg to a specified DP or SF); use inequality notation to specify simple error intervals due to truncation or rounding N16 apply and interpret limits of accuracy, including upper and lower bounds

Fluency Before starting this unit, students will need to be fluent in:  understanding the definitions of perimeter, area and volume  finding the perimeter, area and volume of shapes by counting  using given formula to work out the area of 2D shapes and volume of 3D shapes (including compound shapes)

Learning Objectives: You will learn to:  Find the perimeter, area and volume of shapes and ‘learn’ such formulas (rectilinear based shapes)  Solve problems involving perimeter, area and volume of rectilinear shapes  Find the area and circumference of a circle, volume and surface of a cylinder  Solve problems involving circles in 2D and 3D with and without a calculator  Find errors, upper and lower bounds for a measurement

Focus on Reasoning 1. Round the corners of a square and find the area of the rounded shape. Consider the limits of the rounding. Repeat with a rectangle. 2. Given the diameter of a circle, find its circumference and area. Double the diameter and find out what happens to the circumference and area. Consider what happens in general to the circumference and area as the diameter increases.

Focus on Problem-solving 1. For a wheel of given diameter, find the number of rotations for each wheel as it completes a circuit of a circular course. Make the calculations in terms of π. 2. Given its dimensions, work out the volume of water needed to fill a triangular water tank. If the tank is part-filled with a given volume of water, find the height of the water in the tank.

3. Find the difference in circumference between the inner and outer lanes of a circular running track. 4. Find the cross-section of a hexagonal prism given its height and the size of one side. Given the prism’s volume, find its length. 5. Give the dimensions of a dartboard and its bull, each with upper and lower bounds, find the bounds of the area of each. Find the upper and lower bounds of the probability that a randomly thrown dart that hits the dartboard also hits the bull. 6. Given the area of the roof of a semicircular hut and the length of the hut, make a scale drawing of the end of the hut. 7. Given the perimeter of a semicircle or quarter-circle, find its radius. 8. A hospital patient receives fluid at a given rate from a cylindrical container of given diameter and height. Find how long it takes to empty the container and represent the height of the fluid in the container over time.

SoW Unit 7 – Angles, Pythagoras and Trigonometry

2014 Programme of Study Refs  interpret mathematical relationships both algebraically and geometrically  identify and construct congruent triangles, and construct similar shapes by enlargement, with and without co-ordinate grids  apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs  use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

Leading into (draft) GCSE Specification Objectives G3 apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons) G6 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 proofs G19 apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures opposite G20 know the formula for: Pythagoras’ Theorem Theorem a 2 + b 2 = c 2, and trigonometric ratios, sin  , hypotenuse adjacent opposite cos  and tan  ; apply them to find angles and lengths in right-angled triangles and where hypotenuse adjacent possible, general triangles in two and three dimensional figures

Fluency Before starting this unit, students will need to be fluent in:  solving geometrical problems using angle facts, properties and reasoning  knowledge of 2D shapes, their names and properties

Learning Objectives: You will learn to:  identify if shapes are congruent, similar or neither  solve problems involving similar triangles using ratio  calculate missing sides in right angled triangles using Pythagoras or Trigonometry

Focus on Reasoning 1. For a set of shapes on a grid, determine which shapes are congruent, similar or neither to each other. 2. Use examples to explore the SSS, SAS, ASA or AAS criteria for congruent triangles. 3. Consider which sides and angles in a right-angled triangle need to be given to meet the SSS, SAS, ASA or AAS criteria for congruence. 4. Create a diagram with two or more similar triangles and find missing lengths and angles. 5. For a set of similar right-angled triangles, compare their trigonometric ratios for one of the angles. 6. Explain why two triangles, one of which is an enlargement of the other, are similar. Explore how to use trigonometry to find missing sides and angles in similar right-angled triangles. 7. Explore similar pairs of different types of quadrilaterals, using their specific angle and line properties.

Focus on Problem-solving 1. Create a set of pairs of congruent triangles, with either SSS, SAS, ASA or AAS given. Then trace the shapes and match up the congruent pairs, and consider why the given sides or angles mean the triangles must be congruent. (You could also add pairs of non-congruent triangles with other sides and angles given.) 2. Two similar right-angled triangles are lined up at one matching point and along one side, one smaller triangle within the other, and with some lengths given. Use their similar properties to find the lengths of missing sides. 3. Create real-life problems that require deciding which of the trigonometric ratios are needed to find missing angles or lengths. For missing lengths, consider what would be an appropriate degree of accuracy for the answers. 4. Create further real-life problems where the trigonometric ratios are required, but where also there are fixed parameters, additional lengths to be added, or similar triangles to be found. 5. For a ladder of given length, determine whether it will reach a window at a given height at different angles to the ground. 6. For a tent in the shape of a triangular isosceles prism with some given information, find missing lengths or angles.

SoW Unit 8 – Percentages

2014 Programme of Study Refs  solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics

Leading into (draft) GCSE Specification Objectives R9 define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics

Fluency Before starting this unit, students will need to be fluent in:  finding the percentage of amounts and increasing/decreasing amounts by percentages  comparing (by equivalence) fractions, decimals & percentages  writing percentages changes as multipliers  solving problems involving percentages

Learning Objectives: You will learn to:  find a percentage change  find an original value using inverse operations

Focus on Reasoning 1. Create a table with annual data such as population or revenue. Find percentage change between years – include percentage increases and decreases.

Focus on Problem-solving 1. Two friends buy similar items in a sale. Given the sale prices and the percentage reductions from the original prices, find which of the friends saved more money on their item. 2. The number of visitors at an event varies over a given number of years. Given the number of visitors in the last year, and the percentage changes year-by-year (both increases and decreases), find the number of visitors in the first year.

SoW Unit 9 – Ratio & Proportion

2014 Programme of Study Refs  relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions  solve problems involving direct and inverse proportion, including graphical and algebraic representations  use compound units such as speed, unit pricing and density to solve problems

Leading into (draft) GCSE Specification Objectives R8 relate ratios to fractions and to linear functions R10 solve problems involving direct and inverse proportion, including graphical and algebraic representations R11 use compound units such as speed, rates of pay, unit pricing, density and pressure

Fluency Before starting this unit, students will need to be fluent in:  using ratios (involving decimals)  solving simple proportion problems.

Learning Objectives: You will learn to:  solve problems using compound measures, constant rates and related formulae  solve problems using percentage change, ratios and scale factors  set up equations that show direct proportion  recognise data sets that are proportional  use algebra to solve problems involving proportion

Focus on Reasoning 1. An object grows at a given daily rate. Consider how reliable estimates of its length/size will be after varying periods of time (e.g. after a week, month, year, 10 years). 2. Compare the densities of different objects, given volume and mass. 3. Find best value for an object sold in different sizes at different prices. Discuss what is meant by best value.

Focus on Problem-solving 1. A volume of sand contains a given number of grains. Determine how many grains there will be in a larger volume of sand presented in a larger unit of measure. A similar problem can be set with molecules of water and units of capacity. 2. Measure the sides of an A4 piece of paper. Cut the longer side in half to create an A5 piece of paper and measure the sides. Cut in half again to create A6 and measure the sides. Compare the ratios between the longer and shorter sides of A4, A5 and A6 paper. 3. Calculate the average speed of a journey given departure and arrival times and distance travelled.

SoW Unit 10 – Statistics

2014 Programme of Study Refs  describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)  relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

Leading into (draft) GCSE Specification Objectives S4 interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:  appropriate graphical representation involving discrete, continuous and grouped data, including box plots  appropriate measures of central tendency (median, mean, mode and modal class) and spread (range including consideration of outliers, quartiles and inter-quartile range)

Fluency Before starting this unit, students will need to be fluent in:  calculating the mean, mode and median of a set of data or from a frequency diagram and make some statements  understanding the advantages and disadvantages of using each average in different situations  interpreting data displayed in a variety of statistical diagrams

Learning Objectives: You will learn to:  find the mean, mode, median from a (grouped) frequency table  display and analyse data  identify and explain outliers

Focus on Reasoning 1. Find the mean of the ages of a group of people whose ages are given in months and years. 2. Find the mean, median, mode and range of a small set of numbers. Consider the minimum information you would need to be able to identify the numbers exactly given the averages and range.

Focus on Problem-solving 1. A set of numbers have a given mode, mean and median. Work out what the numbers are. Try the same exercise with negative numbers. 2. For different sets of data, consider the advantages and disadvantages of presenting the data in a variety of graphs and charts. Consider what needs to be displayed, and how easy it will be to find the measures of average and range from the different graphs and charts.