REVISED GCSE Scheme of Work Mathematics Higher Unit 4 For First Teaching September 2010 For First Examination Summer 2011
Version 1: 28 April 10
Version 1: 28 April 10 Unit T4
Version 1: 28 April 10
Unit T4 This is a working document for teachers to adapt for their own needs. Knowledge of the content of Units T1, T2 and T3 is assumed. Topic No. Topic Subject Content 1 Number Indices and index notation, including roots 2 Algebra Direct and indirect proportion 3 Algebra Linear and simultaneous equations 4 Algebra Solving quadratic equations 5 Algebra Graphs of linear functions, including parallel and perpendicular lines 6 Geometry and measures Using Pythagoras Theorem in 3D 7 Geometry and measures Trigonometry in 2D and 3D 8 Geometry and measures Circle Theorems with proofs 9 Handling Data Sampling 10 Handling Data Histograms and frequency densities; use of statistical functions on a calculator or spreadsheet Version 1: 28 April 10 2
TOPIC 1: ALGEBRA - Indices and index notation, including roots Pupils should be able to: calculate using integer indices, positive, zero or negative; express and evaluate all powers and roots in index notation. Revise over the meaning of the terms index, power, and base. Look at examples of the zero index and the reciprocal. Move onto the rules of indices. When the bases are the same a m a n = a m+n, a m a n = a m n, (a m ) n = a mn Progress to examples with coefficients in front of the power terms. Simplify expressions involving positive indices only, such as: Specimen T4 (2010) Qn. 8 6x 6 3x 4, 2x 2 3x 3 and (3x 2 ) 3 Introduce negative powers as being the reciprocal of the same expression with a positive power. a m = 1/a m A fractional power is a root of the base number a ⅓ = 3 a Version 1: 28 April 10 3
TOPIC 1: ALGEBRA - Indices and index notation, including roots (contd.) Use x o = 1, y 3 x ½ x 1½ = x 2 1 2, x / x 3 3 y 1 = x 1 x Evaluate 27 2/3, 8 4/3 without using a calculator. Use index button on a calculator to calculate powers including negative and fractional types. Introduce exponential as another word for power. An exponential equation is any equation in which the unknown is the power. Version 1: 28 April 10 4
TOPIC 2: ALGEBRA Proportion Pupils should be able to: understand and use direct proportion, inverse proportion and inverse square law. Introduce the concept of direct proportion by discussing some everyday examples. Progress to examples being solved using the equation of proportionality. Show some non-linear questions also. Look at inversely proportional situations. Again use examples that will have some meaning to the pupils. The power P varies as the square of the current I. When I = 2, P = 1000. Find P when I = 5. The current, I amps, in a circuit is inversely proportional to the resistance, R ohms. The current is 2 amps when the resistance is 250 ohms. Find I when R = 200. Version 1: 28 April 10 5
TOPIC 3: ALGEBRA Linear and Simultaneous Equations Pupils should be able to: formulate, use and solve linear equations. Revise solving equations with more than one x term. Must get rid of smaller x term first this reduces the question to solving linear equations seen before eg 4x 3 = x + 12 Specimen T4 (2010) Qn. 2(b), (c) Solve equations involving expansion of brackets. Solve equations such as 1 1 ( x 3) ( x 2) 6 4 3 Practice examples where an equation is to be formed from information provided e.g. perimeter of a rectangle with sides of (2x + 2) and (3x 5). Revise standard methods for solving simultaneous equations. Show examples where an equation needs to be rearranged into the required format before solving. Consider questions where the equations are produced from given information. Always highlight the notion of checking that the solutions fit into the two original simultaneous equations. Version 1: 28 April 10 6
TOPIC 4: ALGEBRA Quadratic Equations Pupils should be able to: use factors or the quadratic formula to solve quadratic equations, selecting the most appropriate method for the problem concerned. Revise multiplying out pairs of brackets. Move onto factorising expressions where the coefficient of x 2 is 1. Relate this to solving quadratic equations into two linear brackets. Look at the difference of two squares. Show more complicated quadratic equations where the coefficient of x 2 is other than 1. Introduce pupils to the idea that not all quadratic equations will factorise. Show them how to use the quadratic formula with considerable practice of setting out the working. Specimen T4 (2010) Qn. 2(a) (i) and (ii), Qn. 11, Qn. 16 Examine situations where the quadratic has to be formed from given information and the necessity to check solutions for realism e.g. the variable x represents the side of a triangle and therefore a solution such as 2 is not realistic and must be discarded. Give pupils the chance to see questions like the example below and how they are produced from information given. 2 3 Solve 1 emphasising the need to use a common denominator x 2 2x 1 and how it should be used. Version 1: 28 April 10 7
TOPIC 5: ALGEBRA Graphs of Linear Functions Pupils should be able to: Interpret and use m and c in y = mx + c Find the gradient of lines with equations of the form y = mx + c where m is the gradient and c is the y intercept Understand parallel lines have the same gradients Understand the properties of perpendicular lines. Revise all previous work on straight lines and the form y = mx + c. Highlight what happens when two lines have the same gradient parallel lines eg y = 5x and y = 5x + 3 represent parallel lines with gradient 5. Look at gradients when lines are perpendicular to each other, eg y = 5x and 5y = x + 4 are perpendicular. Emphasise the relationship between the gradients of two perpendicular lines. Consider cases of finding the equation of a line which is parallel to (or perpendicular to) a given line and passes through a given point Specimen T4 (2010) Qn. 10 Version 1: 28 April 10 8
TOPIC 6: GEOMETRY AND MEASURES Using Pythagoras Theorem in 3D Pupils should be able to: Understand and use coordinates in 3D In the case of three dimensions pupils may find it useful to initially refer to the origin as a vertex of a cuboid in order to help visualise movement along the three axes. Specimen T4 (2010) Qn. 13 (a) Apply Pythagoras Theorem to finding the lengths of lines in 3D Define the position of a point on a set three dimensional axes. Use Pythagoras Theorem to find the distance between two points given in the form (a, b, c) and to find the length of a space diagonal of a 3D object. Version 1: 28 April 10 9
TOPIC 7: GEOMETRY AND MEASURES Trigonometry in 2D and 3D Pupils should be able to: understand and apply the sine, cosine and tangent to right-angled triangles in 2-D or 3-D; recall and use the formulae for the area of a triangle using ½ ab sin C; extend their understanding of trigonometry to angles of any size, the graphs of trigonometric functions and the application of trigonometry to the solution of problems in 2-D or 3-D, sine and cosine rules. Introduce the pupil to the concepts of sine, cosine and tangent of an angle with reference to their associated graphical function. Familiarise the pupils with any relevant new calculator methods and the buttons required. Pupils can analyse a problem and identify the appropriate ratio to find an unknown angle/side of a right angled triangle. Revise using sine, cosine and tangent to then calculate the side or angle of a triangle. Calculate the area of a triangle when you know 2 sides and one angle. Given the sine/cosine/tan of an angle, pupils should be able to recall and use trigonometric functions to find the possible solutions within a given range. Set questions using trigonometry to find unknown angles (including the angle between a line and a plane) and distances in 2 and 3 dimensional settings. Pupils can use the sine and cosine rules to solve problems in 2 and 3 dimensions. Pupils understand what is meant by the terms angle of elevation and angle of depression and identify them in a given situation. Both these types of questions must be attempted: 1. questions where the relevant diagram is given and 2. questions where the pupils have to sketch the problem before attempting to solve it. Specimen T4 (2010) Qn. 3, Qn. 12, Qn. 13(b), Qn. 17 Version 1: 28 April 10 10
TOPIC 7: GEOMETRY AND MEASURES Trigonometry in 2D and 3D Questions could involve the use of bearings. Questions could involve bodies moving at a steady speed. For example a plane flying at a constant speed at a height of 7000m is vertically above a given point and 40 seconds later the angle of elevation is 76; find the speed in m/s. Practice the use of memory buttons on a calculator to store early answers to improve accuracy when they are used in a multi-step question. Version 1: 28 April 10 11
TOPIC 8: GEOMETRY AND MEASURES Circle Theorems Pupils should be able to: know and use angle and tangent properties of circles to include: angle in a semicircle, angle at centre and at circumference, angles in same segment, cyclic quadrilaterals, angle between tangent and radius, tangent kite. Alternate segment theorem.. Proof may be required. Specimen T4 (2010) Qn. 5 Version 1: 28 April 10 12
TOPIC 9: HANDLING DATA - Sampling Pupils should be able to: understand and use sampling schemes (including random and stratified sampling). Introduce the notion that if it is not practical to survey a whole population then a sample can be used instead. Discuss the advantages and disadvantages of using a sample: ie. it is practical to collect, cheaper and quicker. Conversely you don t have all the information, so it is really important that you make sure the sample is representative of the population. Progress to a discussion on random and stratified sampling: Random sampling Every item in the population has an equal chance of being chosen. A random sample should be completely unbiased. To choose a random sample the following procedure is useful: assign every item of data a number, use random number tables, a calculator or computer to obtain a sample (ignore duplicate numbers). This method is more suited to a relatively small population where the sampling frame is complete. Stratified sampling This method is useful when the population is divided into categories or groups (age, gender, etc.). A random sample can be chosen from each category with the size of each sample being proportional to the size of each category within the population. This sampling method is useful when the categories or groups within the population are easily defined. Other sampling methods such as systematic, cluster, quota and convenience may also be mentioned with their strengths and weaknesses being identified. Specimen T4 (2010) Qn. 19 Version 1: 28 April 10 13
TOPIC 10: HANDLING DATA Histograms and frequency densities Pupils should be able to: Understand and use: histograms for grouped continuous data; frequency density and Review the material introduced in module N3 that data may be obtained from tables, pictorial representations, charts, graphs or diagrams and that care must be exhibited as these may lead to misleading statements. Introduce a histogram as a form of chart with the area of the bars being the important feature. Discuss that the areas of each of the bars represents the frequencies of the groups. Note also that a histogram may have equal or unequal intervals. Specimen T4 (2010) Qn. 18 use relevant statistical functions on a calculator or spreadsheet A histogram with equal intervals is the same as a frequency diagram. The width of each bar is exactly the same and only the heights vary. A histogram is a frequency diagram where frequency is represented by area so that the frequency is directly proportional to the area. In symbols: f = ka = kih where f is the frequency, A is the area, k (= f A) is the frequency density i.e. number per unit area, i is the interval width and h is the column height. Rearranging the equation gives h = f (ki) = (1/k) (f/i) so that the height of a column is proportional to the frequency per unit interval. When calculating the height of a column the unit interval may be whatever is convenient, e.g. for exam marks 10 may do, for money 10 or 50 or 1000 may suit. The vertical axis of the graph would then be labelled 'Frequency per 10 mark interval' etc. (Some textbooks use 'Frequency Density' for the vertical axis. This is not correct. As explained above, frequency density is constant not variable.) Version 1: 28 April 10 14
TOPIC 10: HANDLING DATA Histograms and frequency densities Introduce the statistics mode on the scientific calculator (one variable statistics at present). Illustrate the benefits of using the available functions on a spreadsheet programme such as MS Excel (C2k network) for statistical analysis. Version 1: 28 April 10 15