Birkdale High School - Higher Scheme of Work

Birkdale High School - Higher Scheme of Work Module 1 - Integers and Decimals Understand and order integers (assumed) Use brackets and hierarchy of operations (BODMAS) Add, subtract, multiply and divide integers, negative numbers and decimals. Round whole numbers to the nearest, 10, 100, 1000, (assumed) Round and approximate decimals to a given number of decimal places or significant figures Multiply and divide whole numbers by a given multiple of 10 (assumed) Check their calculations by rounding, e.g. 29 31 30 30 Write decimals in ascending order of size (assumed) Multiply and divide decimal numbers by whole numbers and decimal numbers (up to 2 d.p.), e.g. 266.22 0.34 Know that, e.g. 13.5 0.5 = 135 5 Module 2 Coordinates Use axes and coordinates to specify points in all 4 quadrants in 2D Add a point to a coordinate grid to complete a given shape (parallelogram; rhombus; trapezium; square) Use the formula to calculate the midpoint of a line segment Module 3 Fractions Recognise and write fractions in everyday situations (assumed) Write a fraction in its simplest form and recognise equivalent fractions Find fractions of an amount Compare the sizes of fractions using a common denominator (assumed) Convert between mixed numbers and improper fractions Add and subtract fractions by using a common denominator Multiply and divide using fractions. Use a calculator to add, subtract, multiply and divide fractions. Manipulation of algebraic fractions Module 4 Algebra Distinguish the different roles played by letter symbols in algebra Understand the meaning between the words equation, formula, identity and expression Write an expression Simplify expressions with like terms, e.g. x 2 + 3x 2 ; 3ab + 5ab +2c 2 Expand and factorise expressions with one pair of brackets, e.g. expand x(2x + 3y); factorise 3xy 2 6x 2 y Expand and simplify expressions involving more than one pair of brackets, e.g. 3(x + 4) 2(x 3); (2x + 3)(3x 4) Factorise quadratic expression (including the difference of two squares) Module 5 Shape and Angle Identify triangles by their properties (scalene, isosceles, equilateral, rightangled, obtuse, and acute) (assumed) Recall and use properties of - Angles at a point - Angles on a straight line - Perpendicular lines - Vertically opposite angles (all assumed) Use the angle properties of a triangle to find missing angles (assumed) Use alternate and corresponding angles in parallel lines to find missing angles (assumed) Prove the angle sum in a triangle is 180 (assumed) Explain why the angle sum of a quadrilateral is 360 (assumed) Identify quadrilaterals by their properties (trapezium, parallelogram, rhombus, rectangle, square, kite and arrowhead) Name a polygon with 3, 4,..., 10 sides Prove the exterior angle of a triangle is equal to the sum of the two opposite interior angles Calculate and use the sums of the interior angles of convex polygons of sides Know, or work out, the relationship between the number of sides of a polygon and the sum of its interior angles Know that the sum of the exterior angles of any polygon is 360 Find the size of each exterior/interior angle of a regular polygon Understand tessellations and explain why some shapes tessellate and why other shapes do not (assumed) Understand, draw and measure bearings (assumed) Calculate bearings and solve bearing problems (assumed) Module 6 Collecting data Module 7 Displaying data

Module 8 Constructions and Loci Construct triangles inc an equilateral triangle with a given side The mid point and perpendicular bisector of a line segment The perpendicular from a point to and on a line. The bisector of an angle. The angles 60, 30 and 45, 90. Construct diagrams from given information involving rectangles, triangles, parallel lines and circles. A regular hexagon inside a circle etc. A region bounded by a circle and an intersecting line. A path equidistant from two points or two line segments. Module 9 Types of Numbers Find: squares, cubes, square roots, cube roots of numbers, with and without a calculator (square and roots up to 15x15 and cubes of 2,3,4,5,10) Understand odd and even numbers, factors, multiples and prime numbers (assumed knowledge) Find the HCF and the LCM of numbers Write a number as a product of its prime factors, e.g. 108 = 2 2 3 3 Understand the standard form convention Convert numbers to, and from, standard form Use of calculator to calculate standard form Use index rules to simplify and calculate numerical expressions involving powers, eg (2 3 2 5 ) 2 4, 4 0 for integer powers only in this module. Module 10 Patterns and Sequences Find the missing numbers in a number pattern or sequence or diagram Find the nth term of a number sequence as an algebraic expression Explain why a number is, or is not, a member of a given sequence Produce a sequence of numbers from a given nth term formula Module 12 Perimeter and Area Find perimeters of shapes Use the area formulae for triangles, rectangles, parallelograms and trapeziums Calculate the area and perimeter of compound shapes Find the surface area of simple prisms using formulae for rectangles and triangles. Learn the formula for circumference and area of a circle Solve problems involving the circumference and area of a circle, possibly leaving answers in terms of π Calculate the lengths of arcs and sectors. Find the area of a segment given the radius and the length of the chord. Module 13 Fractions, Decimals and Percentages Understand that a percentage is a fraction in hundredths Write a percentage as a decimal; or as a fraction in its simplest terms Calculate the percentage (or fraction) of a given amount Find a percentage increase/decrease of an amount Use a multiplier to increase by a given percent, e.g. 1.10 64 increases 64 by 10% Convert a recurring decimal into a fraction Find a percentage increase/decrease of an amount Find a reverse percentage, e.g. find the original cost of an item given the cost after a 10% deduction Use a multiplier to increase by a given percent, e.g. 1.1 64 increases 64 by 10% Calculate simple and compound interest for two, or more, periods of time Solve Functional Elements questions involving percentages Write a number as a fraction or percentage of another (assumed) Percentage profit or loss Module 11 2D and 3D shapes Use 2D representations o 3D shapes on isometric grids Draw nets and show how they fold to make 3D shapes Understand and draw plan, side and front elevations from simple solids Draw a sketch of the 3D solid given plan, side and front elevations

Module 14 Formulae and Linear Equations Substitute positive and negative numbers into simple algebraic formulae Substitute positive and negative numbers into algebraic formulae involving powers Generate a formula from given information, e.g. find the formula for the perimeter of a rectangle given its area A and the length of one side Solve linear equations with one, or more, operations (including fractional coefficients) Solve linear equations involving brackets and/or variables on both sides Form linear equations from word problems in a variety of contexts and relating the answer back to the original problem Solve linear inequalities in one variable and present the solution set on a number line Simple change of subject of a formula, e.g. convert the formula for converting Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade Change the subject of the formula when the variable appears more than once (questions could involve powers, roots, fractions or reciprocals) Module 15 Linear Graphs Substitute values of x into linear functions to find corresponding values of y Plot points for linear functions on a coordinate grid and draw the corresponding straight lines Find the gradient and intercept of a straight-line graph (assumed) Interpret m and c as gradient and y-intercept in linear functions Understand that the graphs of linear functions are parallel if they have the same value of m Know that the line perpendicular to y = mx + c has gradient Understand linear functions in practical problems, e.g. distance-time graph Draw linear graphs from tabulated data, including real-world examples (assumed) Interpret linear graphs (assumed) Draw the graphs of linear inequalities in two variables and interpret the solution sets given by regions in the coordinate plane, or to identify all the integer coordinates with crosses 1 m Module 16 Simultaneous equations Solve algebraically two simultaneous equations Interpret the solution of two simultaneous equations as the point of intersection the corresponding lines Model worded problems as a pair of linear simultaneous equations and interpret the answer Module 17 Probability Module 18 Ratio and Scale Write a ratio in its simplest form Appreciate that, e.g. the ratio 1:2 represents 1/3 and 2/3 of a quantity Divide quantities in a given ratio, e.g. divide 20 in the ratio 2:3 Solve word problems involving ratios and proportion, e.g. find the cost of 8 pencils given that 6 pencils cost 78p Use and interpret maps and scale drawings (assumed) Solve simple direct & inverse proportion problems using the Unitary method or by proportional change (from a table of values) Interpret direct and inverse proportions as algebraic equations, e.g. y x 2 as y = kx 2 Use given information to find the value of the constant of proportionality Use algebraic functions for direct and inverse proportionality, with their value of k, to find unknown values Recognise and sketch the graphs for direct and inverse variation (y x, y x 2, y x 3, y 1/x, y 1/x 2 ) Module 19 Averages and range

Module 20 Pythagoras Theorem and Trigonometry Find missing sides of right-angle triangles by using Pythagoras Find the distance between two points using Pythagoras Giving answers as decimals or surds for Pythagoras problems Use trigonometric ratios (sin, cos and tan) to calculate angles in right-angled triangles Use the trigonometric ratios to calculate lengths in right-angled triangles (2- D) Understand bearings and solve problems involving bearings using Pythagoras/trigonometry Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras and/or trigonometry Calculate the length of a diagonal of a rectangle given the lengths of the sides of the rectangle Calculate the length of the diagonal through a, or across the face of a Find the angle between the diagonal through a and the base of the Find the angle between a sloping edge of a pyramid and the base of the pyramid Module 21 Trial and Improvement Solve cubic functions by successive substitution of x Use systematic trial and improvement to find approximate solutions of equations Module 22 Surface Area and Volume Use V = l w h to solve problems involving the volume and dimensions of a Work out how many small boxes fit into a large box Convert between volume measures inc cubic cms and cubic ms. Use volume = cross-section length to find the volume of a regular prism, e.g. with trapezium cross-section Count the vertices, faces and edges of 3-D shapes Find the surface area and volume of a cylinder Find the surface area and the volume of more complex shapes, e.g. find the volume of an equilateral triangular prism Solve more complex problems, e.g. given the surface area of a sphere find the volume Find the surface area and volume of more complex shapes, e.g. find the curved surface area of a cone, volume of frustums of cones, shapes constructed from cubes, cones, pyramids, spheres, etc. Module 23 Compound Measures Convert between units of measure in same system Know rough metric and imperial equivalents and be able to convert between them Use time, distance and speed to solve problems inc fractions of an hour Use density, mass and volume to solve problems. Understand that measurements can not be precise, and write down the maximum and minimum possible values Work out the maximum/minimum possible error in a calculation involving measures Find when numbers are given to a specific degree of accuracy, the upper and lower bounds of perimeters, area and volume Apply upper and lower bounds to compound units, eg speed Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation Module 24 Transformations Understand translation as a combination of a horizontal and vertical shift including signs for directions, written as a vector Understand rotation as a turn about a given origin Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = x Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater and less than one (and understand the effects that negative and fractional scale factors have on the image) Find the centre of enlargement Understand that image produced by translation, rotation and reflection are congruent to the original shape Describe a transformation using a combination of reflections, rotations, translations or enlargements

Module 25 Similarity and Congruence Use integer and non-integer scale factors to find the length of a missing side in each of two similar shapes, given the lengths of a pair of corresponding sides Understand and use SSS, SAS, ASA, and RHS conditions to prove the congruence of triangles Prove formally geometric properties of triangles, e.g. that the base angles of an isosceles triangle are equal (assumed) Know the relationship between linear, area and volume scale factors of similar shapes Module 26 Quadratic Functions, equations and graphs Plot the graphs of quadratic functions for positive and negative values of x Find graphically the solutions of quadratic equations by considering the intercept on the x-axis Find graphically the approximate solutions of linear and quadratic simultaneous equations Solve simple quadratic equations by factorisation and completing the square Solve simple quadratic equations by using the quadratic formula Solve equations involving algebraic fractions, which lead to the quadratic equation Derive the quadratic equation by completing the square Module 27 Index Notation and surds Simplify surds Use surds and π in exact calculations, without a calculator Write (3 3)2 in the form a + b 3 Rationalise the denominator of fractions, e.g. 1 3 = and 3 3 e.g. write ( 18 + 10) 2 in the form p + q 2 Use index rules to simplify and calculate numerical expressions involving powers, e.g. (2 3 2 5 ) 2 4, 4 0. 8-2/3 1 Know that, e.g. 3 3 x = 8 x = 8 Use calculators to explore exponential growth and decay Module 28 Circle theorems Understand, prove and use circle theorems Use circle theorems to find unknown angles and explain their method - quoting the appropriate theorem(s) Module 29 Trigonometry and sin and cosine rules Find the unknown lengths, or angles, in non right-angled triangles (in 2-D and 3-D) using the sine and cosine rules Find the area of triangles given two lengths and the included angle Module 30 Vectors Understand that 2a is parallel to a and twice its length Understand that a is parallel to a and in the opposite direction Use and interpret vectors as displacements in the plane (with an associated direction) Use standard vector notation to combine vectors by addition Represent vectors, and combinations of vectors, in the plane Solve geometrical problems in 2-D, e.g. show that joining the mid points of the sides of any quadrilateral forms a parallelogram Module 31 Further graphs and functions Plot and recognise quadratic, cubic, reciprocal, exponential and circular (trig) functions (see above) within the range 360 to +360 Use the graphs of these functions to find approximate solutions to equations, e.g. given x find y (and vice versa) Match equations with their graphs Sketch graphs of given functions Draw a circle of radius r centred at the origin Find the approximate solutions of linear equations and the equation of a circle graphically Find the exact solutions of linear equations and the equation of a circle Module 32 Transformation of functions Understand the notation y = f(x) Represent translations in the x and y direction, reflections in the x-axis and the y-axis, and stretches parallel to the x-axis and the y-axis Apply to general graphs or specific curves such as trigonometric functions (i.e. curved graphs) Sketch the graph of y = 3sin(2x), given the graph of y = sinx Sketch the graph of y = f(x + 2), y = f(x) + 2, y=2f(x), y = f(2x) given the shape of the graph y = f(x) Find the coordinates of the minimum of y = f(x + 3), y = f(x) + 3 given the coordinates of the minimum of y = x2 2x