Dyffryn School Ysgol Y Dyffryn Mathematics Faculty. Higher GCSE Course Content

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1 Dyffryn School Ysgol Y Dyffryn Mathematics Faculty Higher GCSE Course Content

2 NOTE: Every topic will need to be revised for the Mathematics GCSE. Green parts do NOT need to be revised for Numeracy GCSE. 1. Number. 4 rules Multiplying/dividing by multiples of 10 Factors, multiples, primes 4 rules of negative numbers Order directed numbers BIDMAS/BODMAS Use of inverse operations 4. Round to an appropriate degree of accuracy. Round to any given number of decimal places Round to any given number of significant figures Estimate answers Know when to round up or down as appropriate Round an answer to a reasonable degree of accuracy in light of the context Recognise limitations on the accuracy of data and measurements Knowledge of rounding to an appropriate degree of accuracy 7. Percentages. Percentages of quantities with & without a calculator One number as a percentage of another with & without a calculator Percentage increase/decrease Use of multipliers for increase & decrease Profit/loss as a percentage of the original Simple/compound interest including depreciation Repeated proportional changes (use of formula P (1 ± r/100) n ) Finding the original quantity Number 2. Types of numbers & use of index notation. Odds, evens, multiples, factors, primes, squares, cubes, reciprocals Indices writing in index form Find values of indices eg. 6 3 Multiplication/division of index numbers Zero index Negative & fractional indices Powers & roots Use of calculator for powers & roots Prime factors in index form Use the above to make perfect square & find square roots, HCF & LCM 5. Fractions. Equivalent fractions, top heavy to mixed & vice versa Ordering fractions Fractions of quantities One number as a fraction of another 4 rules of fractions Calculate fractional changes (increase & decrease) Problems involving fractions 8. Fractions, decimals, ratios & percentages. Interchange between fractions, decimals, percentages & ratios Use equivalences between decimals, fractions, ratios & percentages Order & compare sizes of fractions, decimals, ratios & percentages Recognise that recurring decimals are exact fractions Recognise that some exact fractions are recurring decimals 3. Decimals. Place value & order decimals Add/Subtract whole numbers & decimals Multiply/divide whole numbers & decimals by a decimal number 6. Ratio & proportion. Simplify ratios Use ratios to find unknown quantities eg as in scale diagrams or maps Division in a given ratio Number based direct/inverse proportion Recognise and interpret graphs that illustrate direct and inverse proportion 9. Everyday Maths. Exchange rates & commissions TV schedules Bus and rail timetables Holiday bookings Distance charts Best buys Personal and household finance including fuel and other bills Hire purchase Discount & VAT Tax Wages & salaries Loan repayments Mortgages

3 11. Standard Form. Interpret numbers written in standard form Change numbers into standard form Change from standard form to normal numbers Non calculator methods for standard form Use the calculator (EXP button) for standard form problems Problems involving standard form 13. Rational & Irrational numbers. Distinguish between rational & irrational numbers Recap fractions as recurring decimals Recurring decimals as fractions Manipulate surds Use surds and π in exact calculations Simplify numerical expressions involving surds 11. Venn Diagrams. Understand and use venn diagrams to solve problems Understand and interpret set notation Budgeting Enterprise, saving & borrowing Investing & use of AER & APR 12. Error Approximation/Limits of Accuracy. Upper & lower bounds Upper & lower bounds used in calculations (+ - ) Use of min,min & max,max for +/- Use of min,max & max,min for / Percentage errors 14. Simplify in Algebra. Collect like terms (add & subtract) Multiply & divide, including rules of indices Remove/expand brackets Remove/expand touching brackets Remove/expand brackets such as (x + 3) 2 Function machines Write algebraic expressions for worded problems Distinguish in meaning between equations, formulae, identities and expressions Form, simplify expressions involving sums, differences, products & powers 17. Simultaneous equations. Solve simultaneous equations algebraically by the method of elimination Problems form & then solve simultaneous equations Algebra 15. Substitution in Algebra. Substitution of positive & negative whole numbers, fractions & decimals into simple formulae expressed in words or symbols 18. Straight line graphs. Plot coordinates & set up X & Y axes Draw, interpret, recognise & sketch the graphs of x=a, y=b, y=ax+b Tables of values & drawing linear graphs of type y=ax+b Gradients of parallel lines The gradient(m) & y-intercept(c) 16. Solve linear equations with whole number & fractional coefficients. Solve linear equations including brackets Solve equations with fractions that have only numbers as denominators Eg. x 2-2x 1 = Form & solve equations Form & solve linear equations in solving problems set in real-life contexts 19. Construct & interpret graphs in everyday life Construct and interpret conversion graphs Construct and interpret graphs that describe real-life situations Interpret graphical representation used in the media & recognise that some graphs may be misleading

4 20. Distance/Velocity-time Graphs. Construct & interpret travel graphs Use speed = distance time based on graph Velocity/time, distance/time graphs Find velocities & time from graph Find acceleration by drawing tangents Construct and use tangents to curves to estimate rates of change for non-linear functions, and use appropriate compound measures to express results, including finding velocity in distance-time graphs and acceleration in velocity-time graphs. Distance travelled (area under graph) Interpret the meaning of the area under a graph, including the area under velocity-time graphs and graphs in other practical and financial contexts 23. Trial & Improvement. Solve a range of quadratic & cubic equations by trial & improvement methods. Answers correct to 1dp (& 2dp) Find the equation of the line, using y=mx + c, when given either the points on the line or given the line Identify equations of lines parallel or perpendicular to a given line to satisfy given conditions Solve simultaneous equations graphically Find the coordinates of the mid-point of a line 21. Algebra - Factorising. Factorise common factors Factorise the difference of 2 squares Factorise simple quadratics eg x 2 + 7x + 12 Factorise harder quadratics eg 2x 2-7x Changing the subject of a formula Change the subject of a formula when the subject appears in one term Change the subject of a formula when the subject appears in more than one term 22. Solve Quadratic Equations Solve quadratic equations by factorising first using 1. common factors 2. difference of 2 squares 3. quadratics (including harder ones which have a coefficient of x 2 2) Solve quadratic equations using the formula Form & solve quadratic equations 25. Sequences. Recognise & continue sequences (to include the difference method) Generate a sequence from a given nth term (linear & non-linear) Find the nth term of a linear or quadratic sequence from numbers or diagrams 26. Inequalities & regions. Concept of an inequality Symbols used The number line & representation of inequalities Solve linear inequality equations, including fractional ones Solve linear double inequality equations Regions graphical solution to inequalities State the coordinates within a region 27. Variation. Direct variation Inverse variation Link with graphical representation & find the value of k (& other constants where necessary) from a table of information & graph Construct and use equations that describe direct and inverse proportion Recognise & interpret graphs that illustrate direct & indirect proportion 28. Algebraic Fractions. Simplify algebraic fractions (may have to factorise too) Recap solving equations with numbers only in the denominator Solve equations with letters in denominator leading to solving a quadratic equation either by factorising or by using the formula

5 29. Curved algebraic graphs. Tables of values & plotting points Draw, interpret, recognise & sketch graphs of y=ax 2 + b, y=a/x, y=ax 3 Draw & interpret quadratic & cubic graphs (y = ax 2 + bx + c, y = ax 3 + b) Draw and interpret graphs of y = ax + b + c/x with x not equal to 0, y = ax 3 + bx 2 + cx + d, y = k x for integer values of x and simple positive values of k Draw and interpret graphs when y is given implicitly in terms of x. Solve equations using graphs (by drawing lines such as y=3) Find the gradient at a point on algebraic graph Area under graph using trapezium rule 30. Graph Sketching. Basic quadratic, cubic, reciprocal & exponential graphs Understand & use function notation Interpret & apply the transformation of functions in the context of their graphical representation, including y=f(x + a) y=f(kx) y=kf(x) y=f(x) + a y= -f(x) to y=f(x) Sketch the graph of functions from other functions such as f(x) Eg. y=f(x + a) y=f(kx) y=kf(x) y=f(x) + a y= -f(x)

6 31. Angle Facts. Basic angle facts Properties of triangles & quadrilaterals Angles in triangles & quadrilaterals Angles in parallel lines Interior & exterior angles of polygons Use angle facts for tessellating shapes 34. Pythagoras Theorem. Find the hypotenuse Find a shorter side Test for right angled triangles Problems involving Pythagoras Theorem 37. Perimeter, area, volume & density. Find perimeters of shapes Find areas of shapes Parts of a circle Find the circumference & area of circles Problems involving the above to include inverse problems & semicircles etc. Surface area of cubes/cuboids/prisms/ pyramids/ cylinders/ cones/spheres/tetrahedron Volume of cubes/cuboids/prisms/ pyramids/ cylinders/ cones/spheres Problems involving density, mass & volume 40. Trigonometry. Label the sides of a right angled triangle Sine, cosine & tangent formulae Use of SOHCAHTOA Calculate the length of sides Calculate the size of angles Angles of elevation & depression Problems involving Trigonometry & Pythagoras Theorem, including the use of bearings Use Trigonometry and Pythagoras Theorem in Geometry and Measure 32. Constructions. Construct triangles & 2-D shapes accurately Draw plans & elevations of any 3D solid Use a ruler & pair of compasses to do constructions: Bisect a given line Bisect a given angle Construct angles 35. Congruent Shapes. Identify congruent shapes Understand & use SSS, SAS, ASA & RHS conditions to prove the congruence of triangles using formal arguments 38. Dimensions of formulae. Considerations of dimensions in order to determine between perimeter (1D), area (2D) & volume (3D) 41. Transformations. Revision of coordinates Reflection of 2D shapes in x axis y axis y = a x = a y = +/- x Rotational symmetry & order Rotate about a given point 33. Locus/loci. Find the path of a point moving under given criteria 4 main loci 1. Fixed distance from a point 2. Fixed distance from a line 3. Equidistant from 2 points 4. Equidistant from 2 lines which meet at a point Regions satisfied by the criteria of the locus Loci questions involving inequalities 36. Bearings & Scale Drawings. Compass directions Understand & use (draw & measure) 3-figure bearings Understand scales written in various forms eg 1cm represents 500m or 1:500 Interpret & construct scale drawings Use & interpret maps Problems involving bearings & scale diagrams 39. Compound measures. Use of speed = distance time Use of miles per gallon Use of density Use of population density 42. Non-right angled triangles. Introduction to cosine rule Introduction to sine rule Areas of non-right angled triangles using 1 /2absinC Problems involving the above, including the use of bearings Use in 2D & 3D shapes

7 3D shapes. clockwise/anticlockwise through a given angle Translation (under a given column vector) Enlarge a shape from a given scale factor Enlarge a shape from a given centre of enlargement Use of positive, negative & fractional scale factors Find the centre of enlargement Transform shapes using 2 successive transformations Describe the transformation(s) that a shape has gone through 43. Arcs, sectors & segments. Parts of a circle Length of arcs Area of sectors Area of segments Problems involving the above including reverse problems 46. Trigonometric Graphs. Draw, sketch & know the behaviour of the graphs Y = sin Y = cos Y = tan Solution of trigonometric graphs, eg of the form 5 sin = Similar shapes. Understand & use mathematical similarity, knowing that angles remain unchanged & that sides are in the same ratio Use the knowledge that for 2 similar 2D or 3D shapes one is an enlargement of the other Use the knowledge that in similar shapes corresponding dimensions are in the same ratio Prove that shapes are similar by looking at corresponding sides Find the lengths of missing sides using scale factors and the ratio of corresponding sides Understand & use the relationship between surface areas of similar shapes & volumes of similar 3D solids. Recap volume of 3D shapes where similar shapes need to be used in order to find volumes or areas 45. Circle Theorems. Angle subtended at centre is twice one at circumference Angle subtended at the circumference in a semicircle is a right angle Angles in the same segment are equal (angles that are at the circumference and are subtended from the same arc) Opposite angles in a cyclic quadrilateral add to 180 Tangent at any point to the circle is perpendicular to the radius Tangents from one point outside the circle are equal in length A line drawn from the radius to a chord at 90 bisects the chord Alternate segment theorem Understand and construct geometrical proofs using circle theorems Use of equal radii and tangents to create isosceles triangles

8 47. Averages/Representing data. Find mean, median & mode from a frequency table Estimate the mean, find the median & modal class from a grouped frequency table Draw grouped frequency diagrams Draw frequency polygons Comparison of 2 distributions Statistics 48. Cumulative frequency. Complete a cumulative frequency table Draw a cumulative frequency diagram Find the median, upper/lower quartiles & interquartile range from the diagram Answer questions based on graph (less/more than) Comparison of 2 or more distributions 49. Box-And-Whisker plots. Produce box & whisker plots. Use box & whisker plots to compare distributions. 50. Draw accurately & interpret Pie charts. Draw pie charts by calculating angle size Calculate angles from percentages on chart Extract information from pie charts Find frequencies from given angles on pie charts 53. Scatter diagrams. Set up axes for scatter graphs Plot points Types of correlation Draw the line of best fit by eye Draw the line of best fit through the mean point if it is given Obtain information from scatter graphs 51. Questionnaires. Design, criticise questions on questionnaires to include fairness & bias Test hypotheses, taking into account the limitations of the data available Specify the data needed and consider potential sampling methods. 54. Probability. Definition of probability Calculate theoretical probabilities using scale 0-1 Use of P(event not occurring) = 1 P(event occurs) Listing all possible outcomes Possibility space diagrams & calculate probabilities Estimate the probability of an event as the proportion of times it has occurred Relative frequency The AND & OR rules Probability trees Conditional Probability(not replaced) Solve problems with or without tree diagrams 52. Sampling. Systematic sampling Random sampling Stratified sampling Consider the effect of sample size and other factors that affect the reliability of conclusions drawn 55. Venn Diagrams. Venn Diagrams. Understand and use venn diagrams to solve problems Understand and interpret set notation 56. Histograms. Construction of histograms with unequal class widths Frequency density Find the frequency from a histogram Problems involving histograms Interpret histograms with unequal class widths Interpret histograms representing distributions

9 with reference to mean & dispersion