Alcester Academy Curriculum Planning: Key Stage 4

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1 Department: Maths Year Group: 10 Higher Term Topic/ subject Assessment Objectives And Knowledge Autumn 1 3D forms & Volume, Find Surface area of prisms using Cylinders, Cones, & areas of simple shapes with or without (7 weeks) Spheres. a diagram. Draw 3D sketches of solids, identify planes of symmetry. Use Volume of cuboids calculations, convert between metric volume measures. Use estimation to 1sf. Find volume and surface area of cylinders, pyramids, spheres and cones. Solve problems involving parts of 3d shapes including segments and frustums. Use equations involving complex shapes. Skills Include detail of any differentiation Form equations involving more complex shapes and solve these equations. Work out the length given the area of the cross-section and volume of a cuboid. Understand that answers in terms of π are more accurate. Given two solids with the same volume and the dimensions of one, write and solve an equation in terms of π to find the dimensions of the other, e.g. a sphere is melted down to make ball bearings of a given radius, how many will it make? Literacy, numeracy and SMSC opportunities Final assessment task and title Accuracy & Bounds. Transformations. Find upper and lower bounds and appropriate degrees of accuracy. Use bounds in calculations involving perimeter, area and volume in 2D and 3D shapes. Use inequalities to identify an error interval. Recognise, describe and draw rotations giving centre of rotation, direction and number of degrees. Use reflections of shapes including simple equations of straight lines. Use translations defined by a column vector. Enlarge shapes through a scale factor enlargement and centre of enlargement. Identify a scale factor, and area scale factors. Use congruence of shapes within transformations, and combinations of transformations. Use front, side and plan elevations, draw 3D representations of shapes. Use scales and maps. Use and construct bearings. Use bearings to solve problems. Use a ruler and compasses to construct angle bisector, a perpendicular line from any point, angles of 90 and 45 degrees. Identify regions bounded by a circle and line, a Work out the upper and lower bounds of a formula where all terms are given to 1 decimal place. Be able to justify that measurements to the nearest whole unit may be inaccurate by up to one half in either direction. Use inequality notation to specify an error interval due to truncation or rounding. Recognise similar shapes because they have equal corresponding angles and/or sides scaled up in same ratio. Understand that translations are specified by a distance and direction (using a vector). Recognise that enlargements preserve angle but not length. Understand that distances and angles are preserved under rotations, reflections and translations so that any shape is congruent to its image. Understand that similar shapes are enlargements of each other and angles are preserved. Describe and transform 2D shapes using combined rotations, reflections, translations, or enlargements. Describe the changes and invariance achieved by combinations of rotations, reflections and translations. Interpret a given plan and side view of a 3D form to be able to produce a sketch of the form. Able to read and construct scale drawings. When given the bearing of a point A from point B, can work out the bearing of B from A. Know that scale diagrams, including bearings and maps, are Use of key words: upper and lower bounds, accuracy SMSC: this unit will provide students with the opportunity to develop counter argument skills. Use of key words: Transformation, rotation, reflection, enlargement, translation, single, combination, scale factor, mirror line, centre of rotation, centre of enlargement, column vector, vector, similarity, congruent, angle, direction, coordinate, describe SMSC: Multi-cultural links to carpet designs using transformations can be made. The use of formal mathematical language is encouraged when describing transformations. Use of key words: Construct, circle, arc, sector, face, edge, vertex, twodimensional, three-dimensional, solid,

2 Constructions, Loci & Bearings. distance from a point, equal distance from two points or lines. Use greater than or less than within drawings. Identify and sketch regions through loci. Know that a perpendicular line is the shortest distance. similar to the real-life examples. Able to sketch the locus of point on a vertex of a rotating shape as it moves along a line, of a point on the circumference and at the centre of a wheel. elevations, congruent, angles, regular, irregular, bearing, degree, bisect, perpendicular, loci, map, scale, plan, region SMSC: Students will continue to develop their problems solving skills, teamwork, creative thinking, discussion, explaining and presenting skills whilst exploring the beauty of mathematics. Autumn 2 (8 weeks) Half term assessment 1. Solving Quadratic & Simultaneous equations. Factorise and solve quadratics using factorisation, completing the square, rearrangement, the quadratic formula. Find exact solutions of two simultaneous equations between straight lines only, between straight line and quadratic, between straight line and circle. Problems that require setting up and solving a pair of simultaneous equations in a real-life context. Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns including where both need multiplying; Linear/linear; linear / quadratic; linear / x 2 + y 2 = r 2. Know that the quadratic formula can be used to solve quadratic equations, and often provides a more efficient method than factorising or completing the square. Use of key words: Quadratic, solution, root, linear, solve, simultaneous, inequality, completing the square, factorise, rearrange, surd, function, solve, circle, sets, union, intersection SMSC: The concept of the inappropriate solution. Half Term Assessment 1 Inequalities. Probability. Use inequalities on a number line, identify whole number values in a range, solve simple inequalities involving 1 or 2 variables. Use systematic listing strategies, use experimental outcomes to complete tables and frequency trees. Identify expected outcomes through randomness and likelihood. Use probability between 0 and 1. Use exhaustive results of outcomes in mutually exclusive events. Compare Use inequality symbols to compare numbers. Given a list of numbers, represent them on a number line using the correct notation. Solve equations involving inequalities. Use the correct notation to show inclusive and exclusive inequalities. If the probability of outcomes are x, 2x, 4x, 3x, calculate x. Find the probability of successive events, such as several throws of a single dice. List all outcomes for single events, and combined events, systematically. Use a two-way, a tree diagram or a Venn diagram to calculate conditional probability. Use of key words: Probability, dependent, independent, conditional, tree diagrams, sample space, outcomes, theoretical, relative frequency, fairness, experimental SMSC: Through exploring probability students will develop an awareness of

3 sample size in experiments to theoretical probability. Use sets and combinations of sets to construct tables, grids, Venn diagrams and tree diagrams. Use tree diagrams and other representations for independent and dependent events. Use conditional probability with two way tables, tree diagrams and venn diagrams. fairness both in a mathematical; context and in real-life scenarios. Multiplicative Reasoning. Use inverse operations to allow cancellation within problems. Use fractions or percentages as operators. Use standard units of mass, length, time, money and others using decimals where appropriate. Change freely between standard types of units. Use ratio and fractions for multiplicative relationships. Solve problems involving direct and inverse proportion, including the use of algebra and graphs. Use compound units including speed, pay, pricing, density and pressure. Know that inverse proportion is 1/y. Use graphs representing direct and inverse proportion. Use iterative processes to interpret answers on growth and decay or compound interest. Set up and use equations to solve word and other problems involving inverse proportion, and relate algebraic solutions to graphical representation of the equations. Change g/cm 3 to kg/m 3, kg/m 2 to g/cm 2, m/s to km/h. Solve word problems involving direct and inverse proportion. Understand direct proportion as: as x increases, y increases. Understand inverse proportion as: as x increases, y decreases. Use of key words: Ratio, proportion, best value, unitary, proportional change, compound measure, density, mass, volume, speed, distance, time, density, mass, volume, pressure, acceleration, velocity, inverse, direct, constant of proportionality SMSC: Multiplicative reasoning involving real-life scenarios develops an awareness of maths use in a wider context. Spring 1 (6 weeks) End of term assessment 2. Similarity & Congruence. Use multiplicative properties between two quantities to give a ratio or fraction. Compare length, area and volume using ratio notation. Use congruence on triangles, quadrilaterals to include angle properties, similarity, areas, volumes. Compare relationships between length, area, and volume. Complete simple proofs using similarity and congruence. Recognise that all corresponding angles in similar shapes are equal in size when the corresponding lengths of sides are not. Understand that enlargement does not have the same effect on area and volume. Understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not. Solve problems involving frustums of cones where you have to find missing lengths first using similar triangles. Use of key words: Congruence, side, angle, compass, construction, shape, volume, length, area, volume, scale factor, enlargement, similar, perimeter, frustum SMSC: Formal proof provides an opportunity to discuss the universal nature of maths. Autumn Term Assessment 2 Graphs of Trigonometric Functions. Recognise and sketch graphs for Trigonometric functions. Know exact values of some trig functions. Use transformations of graphs in f(x) notation. Match the characteristic shape of the graphs to their functions and transformations. Know the exact values of sin θ and cos θ for θ = 0, 30, 45, 60 and 90 and exact value of tan θ for θ = 0, 30, 45 and 60 and find them from graphs. Apply to Use of key words: Axes, coordinates, sine, cosine, tan, angle, graph, transformations, side, angle, inverse, square root, 2D, 3D, diagonal, plane, cuboid

4 the graph of y = f(x) the transformations y = f(x), y = f( x) for sine, cosine and tan functions f(x). Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(x + a) for sine, cosine and tan functions f(x). SMSC: cross-curricular links with Science & music. Further Trigonometry Half term assessment 3 Know how to use the formula for the area of a triangle (1/2absinC) and the sine and cosine rules. Use Trigonometry and Pythagoras in 3D. Find lengths across 3D diagonals, and angles between sides and planes. Find the area of a segment of a circle given the radius and length of the chord. Justify when to use the cosine rule, sine rule, Pythagoras Theorem or normal trigonometric ratios to solve problems. Understand, recall and use trigonometric relationships and Pythagoras Theorem in rightangled triangles, and use these to solve problems in 3D configurations. Half Term Assessment 3 Spring 2 (5 weeks) Collecting data. Know how to specify a problem and set up a plan including what data to collect, what types of primary and secondary data are needed, sample size and population. Identify bias and write questions that avoid or eliminate it. Explain why a sample may not be representative of a whole population. Understand what is meant by a sample and a population. Understand how different sample sizes may affect the reliability of conclusions drawn. Use of key words: Sample, population, fraction, decimal, percentage, bias, stratified sample, random, cumulative frequency, box plot, histogram, frequency density, frequency, mean, median, mode, range, lower quartile, upper quartile, interquartile range, spread, comparison, outlier Cumulative frequency, Box plots, & Histograms. Know when and how to construct a cumulative frequency diagram, and how to interpret the data presented. Compare mean, median range and quartiles of distributions including the use of box plots. Use and construct a histogram including uneven width bars and frequency density. Estimate a mean from a histogram, and identify numbers involved. Interpret two or more data sets from box plots and relate the key measures in the context of the data. Construct cumulative frequency graphs, box plots and histograms from frequency tables. Compare two data sets and justify comparisons based on measures extracted from diagrams. Use and understand frequency density. SMSC: Bias and the implications of media bias can be introduced. The UK census allows students to develop an awareness of the wider culture. Quadratics, expanding more than two brackets, sketching graphs including quads, cubic and circles. Calculate exact answers using surds. Expand two or more brackets to produce polynomials, then identify the turning points by solving or completing the square or sketching. Solve quadratics that requires approximate answers. Use iteration to find approximate solutions. Form equations to solve including simultaneous equations. Use graphical representation of inequalities Interpret a pair of simultaneous equations as a pair of straight lines and their solution as the point of intersection. Be able to state the solution set of x² 3x 10 < 0 as {x: x < -3} {x: x > 5}. Match equations to their graphs and to real-life scenarios. Solve quadratic inequalities in one variable, by factorising and sketching the graph to find critical values. Use of key words: Sketch, estimate, quadratic, cubic, function, factorising, simultaneous equation, graphical, algebraic SMSC; Cross curricular links to PE and ICT.

5 including quadratics. Use set notation and work generally with iterative processes. Summer 1 (6 weeks) End of term assessment 4 Circle Theorems. Know all parts of the circle including sector, tangent, chord and segment. recall a variety of circle properties including the use of proof. Find the value and give reasons for missing angles on diagrams using circle theorems and triangle properties in circles. Justify clearly missing angles on diagrams using the various circle theorems. Problems that involve a clear chain of reasoning and provide counterarguments to statements. Find and give reasons for missing angles on diagrams using: circle theorems; isosceles triangles (radius properties) in circles; the fact that the angle between a tangent and radius is 90 ; the fact that tangents from an external point are equal in length. Use of key words: Radius, centre, tangent, circumference, diameter, gradient, perpendicular, reciprocal, coordinate, equation, substitution, chord, triangle, isosceles, angles, degrees, cyclic quadrilateral, alternate, segment, semicircle, arc, theorem SMSC: Link to Ben Heine, Kandinsky (artists) Spring Term Assessment 4 Circle Geometry. Apply construction techniques and understanding of Loci to draw graphs base on circles and perpendicular lines. Find the equation of a tangent to a point on a circle using gradient of radius, gradient of tangent, a given coordinate. Draw circles using equation of a circle formula. Find the gradient of a radius of a circle drawn on a coordinate grid and relate this to the gradient of the tangent. Justify the relationship between the gradient of a tangent and the radius. Produce an equation of a line given a gradient and a coordinate. Recognise and construct the graph of a circle using x 2 + y 2 = r 2 for radius r centred at the origin of coordinates. Justify if a straight-line graph would pass through a circle drawn on a coordinate grid. Summer 2 (7 weeks) Changing the subject of formulae, Algebraic fractions including solving equations, Rationalising surds, and Proof. Half term assessment 5 Vectors and geometric proof. Simplify and rationalise surds, simplify and manipulate algebraic expressions involving surds and algebraic fractions. Use algebraic fractions up to factorising quadratics within the fraction. Use algebraic proof in a variety of contexts. Solve quadratic equations including those that require rearrangement. Use vector notation including column notation. Understand parallel vectors are multiples of given vectors. Find the sum, difference or multiple of two vectors. Find the length of a vector using Pythagoras theorem. Calculate resultant of vectors and solve 1 1 Rationalise: 3 1,, ( ) Explain the difference between rational and irrational numbers. Given a function, evaluate f(2). When g(x) = 3 2x, find g 1 (x). Solve Show that and proof questions using consecutive integers (n, n + 1), squares a 2, b 2, even numbers 2n, odd numbers 2n +1. Add and subtract vectors algebraically and use column vectors. Solve geometric problems and produce proofs. Find the area of a parallelogram defined by given vectors. Solve geometric problems in 2D where vectors are divided in a given ratio. Produce geometrical proofs to prove points are collinear and vectors/lines are parallel. Use of key words: Rationalise, denominator, surd, rational, irrational, fraction, equation, rearrange, subject, proof, function notation, inverse, evaluate Use of key words: Vector, direction, magnitude, scalar, multiple, parallel, collinear, proof, ratio, column vector SMSC: Use in real life re. air traffic control and link to Physics Half Term Assessment 5

6 geometric problems involving vector diagrams. Reciprocal & exponential graphs, Gradient & area under graphs. Recognise, interpret and sketch reciprocal graphs and power graphs including stating values where x is not defined. Use a calculator to solve problems involving growth and decay. Interpret and analyse transformation of graphs for a variety of functions. Estimate the area under quadratic and other graphs using trapezia. Use gradient with linear and non-linear graphs including speed distance time and velocity time graphs. Use gradient and area of graphs in real life or financial contexts. Use rate of change of graphs. Explain why you cannot find the area under a reciprocal or tan graph. Interpret the gradient of a linear or non-linear graph in financial contexts. Interpret the area under a linear or non-linear graph in real-life contexts. Interpret the rate of change of graphs of containers filling and emptying. Interpret the rate of change of unit price in price graphs. Use of key words: Reciprocal, linear, gradient, quadratic, exponential, functions, direct, indirect, proportion, estimate, area, rate of change, distance, time, velocity, transformations, cubic, transformation, constant of proportionality SMSC: Financial links Direct & inverse proportion. Recognise graphs represented by direct and inverse proportion. Use direct and inverse proportionality with square, cube, or other powers. Set up equations involving direct and inverse proportion to solve. Understand that when two quantities are in direct proportion, the ratio between them remains constant. Know the symbol for is proportional to. Justify and infer relationships in real-life scenarios to direct and inverse proportion such as ice cream sales and sunshine. Use y = kx to solve direct proportion problems, including questions to find k, and then use k to find another value. Solve problems involving inverse proportion using graphs by plotting and reading values from graphs. End of year assessments - assessment 6 End of year assessments - assessment 6