Reading various documents on the new GCSE specifications I thought it would be useful to create a simple summary of new content. (This will be updated with additional information and resources in the coming weeks). Note that I have very recently updated (May 2015) the post on Venn Diagrams which includes several resources for teaching this new topic; currently I have added a link to Brilliant for combinatorics problems, Trigonometry demonstrations, Nrich and TES resources for Frequency Trees and Desmos graph pages for inequalities including quadratic, circles and tangents to curves. See each section below.
Ratio proportion and rates of change
Geometry and Measures
Structure and Calculation
5. apply systematic listing strategies including use of the product rule for counting
Resources: Combinatorics on Brilliant (create a free account to view)
Measures and Accuracy
15. round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding
Note that this is introduced in the new Key Stage 3 programme of study. (From La Salle education a very useful document which is annotated clearly with changes at Key Stage 3 can be found here. Note that you can also find information on changes to the Primary curriculum in this collection of resources).
use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b
Ratio, proportion and rates of change
1. change freely between related standard units (e.g. time, length, area, volume/capacity, mass) and compound units (e.g. speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
9. 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
11. use compound units such as speed, rates of pay, unit pricing, density and pressure
15. interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts
16. set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes.
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Notation, vocabulary and manipulation
7. 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’.
11. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square
12. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function, with x ≠ 0, exponential functions and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size
15. calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
16. 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.
Resources: Trigonomery demonstrations
Use Desmos to explore tangents to a curve
Use Desmos to explore circles and tangents
BBC Bitesize – Finding the equation of a tangent to a circle
Solving equations and inequalities
17. solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph
18. 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
20. find approximate solutions to equations numerically using iteration
22. solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph
Resources: Desmos for inequalities – including quadratic
24. recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd) and other sequences
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Geometry and measures
Properties and Constructions
2. use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line
8. describe the changes and invariance achieved by combinations of rotations, reflections and translations
Mensuration and calculation
21 know the exact values of sinθ and cosθ for θ = 00, 300, 450, 600 and 900; know the exact value of tanθ for θ = 00, 300, 450 and 600
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1. record describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
6. enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams
9. calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.
Frequency Tree TES resource – Alison Gilroy
Resources: Venn Diagrams
From Nrich Prize Giving and note the Frequency Tree representation
From TES: Frequency Trees