REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
Version 1: 28 April 10
Version 1: 28 April 10 Unit T6
Unit T6 This is a working document for teachers to adapt for their own needs. Knowledge of the content of Unit T5 is assumed. Topic No. Topic Subject Content 1 Number Standard Form 2 Algebra Change of Subject 3 Number Recurring Decimals 4 Algebra Linear Inequalities 5 Algebra Simultaneous equations 6 Geometry and Measures Transformation of Shapes 7 Geometry and Measures Similarity 8 Algebra Real-life Graphs 9 Geometry and Measures Loci Constructions 10 Algebra Indices 11 Number Surds 12 Geometry and Measures Volume 13 Geometry and Measures Perimeters using arcs and Volumes 14 Geometry and Measures Units and Dimensions 15 Algebra Graphs of Reciprocal Functions 16 Algebra Graphs of Cubic, Trigonometric and Exponential Functions 17 Probability Mutually Exclusive and Exhaustive Events 18 Probability Independent Events and Tree Diagrams Version 1: 28 April 10 2
TOPIC 1: NUMBER Standard Index Form interpret, order and calculate with numbers written in standard index form; Be able to write very large and very small numbers presented in context in standard form; Convert between ordinary and standard index form numbers; SAM T62 Q. 7 Interpret a calculator display using standard form; Calculate with numbers in standard form using a calculator. Version 1: 28 April 10 3
TOPIC 2: ALGEBRA Changing The Subject change the subject of formulae including cases where the subject appears in more than one term, or where a power of the subject appears. Show examples of changing the subject with linear equations before progressing on to quadratic equations and equations in which the subject appears more than once. Transform formulae such as v = u + at, A = r 2 100(s c), p c SAM T61 Q. 4(b) understand, construct and evaluate formulae related to mathematics or other subjects or real-life situations. e.g. Work out s = ut + ½at 2 where u or a may have negative values. GMm Use the formula F = 2 to calculate one variable given the others. r SAM T61 Q. 4(a) SAM T62 Q. 8 Version 1: 28 April 10 4
TOPIC 3: NUMBER Recurring decimals change a recurring decimal to a fraction. Know the significance of recurring and non-recurring decimals. Recurring decimals are rational numbers and can be changed into fractions. Non-recurring decimals that are not finite are irrational, e.g.. SAM T61 Q. 13(a) In recurring decimal notation dots are placed over the first and last of the set of 124 recurring digits, e.g. 0.124124124 = 0.12 4 = 999 To change a recurring decimal to a fraction the following method can be used: Find the length of the recurring sequence. If the recurring sequence is one digit then multiply the recurring decimal by 10, if the recurring sequence is two digits then multiply the recurring decimal by 100, etc. Subtract the original recurring decimal from the one that has just been calculated. Divide through to obtain the fractional form of the recurring decimal. e.g. n = 0.234234234 = 0.234, multiply by 1000 1000n = 234.234234234 n = 0.234234234 subtract 999n = 234 Therefore n = 234 999 26 111 Version 1: 28 April 10 5
TOPIC 4: ALGEBRA Linear Inequalities Students should be able to: solve a range of linear inequalities; Introduce inequalities by how they are represented on a number line open circle means the end point is not included, closed (shaded) circle means end point is included, e.g. x < 10, y 6. Include examples of double inequalities. (Progress to examples where an inequality is multiplied or divided through by a negative number, meaning the direction of the inequality sign must be reversed.) Computer package such as Omnigraph Graph paper use straight line graphs to locate regions representing linear inequalities; Introduce locating regions of a graph by first looking at examples of inequalities parallel to one axis. e.g. x < 10, draw the x = 10 line (dotted), shade appropriate region. e.g. y 6, draw the y = 6 line (solid) and shade appropriate region. e.g. 3 < x 8 draw appropriate lines and shade region. Move on to show inequalities involving both x and y on the same graph, e.g. x < 10, y 6. Progress to several inequalities are to be drawn on the same graph, e.g. x < 10, y 6, y < 2x + 3. The solution space should be left unshaded. Show examples of how inequalities can solve problems. Omnigraph can be used to give practice at solving inequalities. SAM T62 Q. 4 Version 1: 28 April 10 6
TOPIC 5: ALGEBRA Simultaneous Equations Students should be able to: use algebraic and graphical methods to solve simultaneous linear equations in two unknowns; use algebraic and graphical methods to solve two simultaneous equations, one linear and the other quadratic, in two unknowns. Introduce the idea of two linear equations having only one point in common by drawing the graph of the equations. The point of intersection is the solution to the equations. Check that the coordinates work in both equations before moving on. Draw the straight line graphs and find the point of intersection. May be helpful to use Omnigraph to draw the two simultaneous equations and read off the point of intersection. Introduce the algebraic method of solving simultaneous equations by first using examples where the coefficients of x or y are equal in both equations subtract the equations to find one unknown. Then use substitution to find the second unknown. Move on to examples where the coefficients of x and y are the same but of opposite signs add the equations to find one unknown. Then use substitution to find the second unknown. Look at examples of simultaneous equations where one equation needs to be multiplied to make the coefficients equal in size. Show examples of simultaneous equations where the two equations need to be multiplied to make the coefficients equal in size. Show examples where an equation needs to be rearranged into the required format before solving. Always highlight the notion of checking that the solutions fit into the two original simultaneous equations. Progress to equations where one is linear and one is quadratic, e.g. solve y = 5x 6 and y = x 2. Computer package such as Omnigraph Version 1: 28 April 10 7
TOPIC 6: GEOMETRY and MEASURES Transformations of Shapes understand transformation of shapes using reflection, rotation, translation and enlargement. enlarge a shape through a given centre of enlargement, by a positive fractional or negative scale factor; recognise that enlargements preserve angle but not length; distinguish properties that are preserved under particular transformations; Ensure that the pupils understand that the term transformation encompasses reflection, rotation, translation, and enlargement. Understand that reflection, rotation and translation all preserve the size and shape of the object being transformed while enlargement just preserves shape. Consider reflections in both axes and y = x. Stress that you need the centre of rotation, the direction and the angle to fully describe a rotation. Be able to complete and describe given rotations Consider simple enlargements with positive integer scale factors. The image is larger and on the same side of the centre of enlargement as the original. Using examples and exemplars allow pupils to decide if an image is an enlargement of the original or not. All enlargements are described using a scale factor and the centre of enlargement. Stress that the enlargement is similar to the original; has the same angles and all corresponding side have the same ratio. Move on to consider positive fractional scale factors. Highlighting that the image is always smaller and on the same side of the centre of enlargement as the original. SAM T61 Q. 6 Version 1: 28 April 10 8
TOPIC 6: GEOMETRY and MEASURES Transformations of Shapes(cont.) use transformations to create and analyse spatial patterns; understand how transformations are related by combinations and inverses. Negative scale factors can now be clarified stressing that the image is now on the other side of the centre of enlargement from the original. Be able to complete and describe given enlargement questions Understand that the image in a translation is congruent to the original. And be able to describe a translation in using vector notation. Pupils should do set question both in class discussion and individual work involving choosing and describing which transformation or combination of transformations maps an original shape to its image. SAM T61 Q. 11(a) Pupils should then understand how to map an image back to its original ie find inverses of transformations. Version 1: 28 April 10 9
TOPIC 7: GEOMETRY and MEASURES - Similarity Students should be able to: understand the effect of enlargement for perimeter, area and volume of shapes and solids; understand and use the relationship between the surface areas of similar 3-D shapes and between the volumes of similar 3-D shapes. Understand how to make correct use of the scale factor. Pupils should consider similar 3-D shapes. An investigation could allow them to establish the relationship between the areas of corresponding surfaces of two similar 3-D shapes. Given 2 sides of one 2-D shape and the length of one corresponding side on a similar shape, pupils should be able to determine the area of the second shape. Given the area and one side of one 2-D shape a pupils should be able to determine the area of a similar 2-D shape with the corresponding side known. Given 2 sides of one 3-D shape and the length of one corresponding side on a similar shape, pupils should be able to determine the area and volume of the second 3-D shape. Given the area and one side of one 3-D shape a pupil should be able to determine the area and volume of a similar 3-D shape with the corresponding side known. SAM T61 Q. 8 SAM T61 Q. 11(b) Version 1: 28 April 10 10
TOPIC 8: ALGEBRA Real-life Graphs interpret and display information on graphs that describe real-life situations; For example: distance-time graphs including intersecting travel graphs. Remind pupils about the 12 and 24 hour clocks and measuring lengths of time. Do some calculations involving time. Spreadsheet package Graph paper Investigate the connection between distance, speed and time and highlight the use of compound units as a measure of rate. Progress to travel graphs and notice the connection between the steepness of the slope and the speed. Work out average speed (distance/time). Practical work could include collection of timings of races over certain distances and could link in with PE. Show examples of real life graphs and discuss the relevance of joining the points up do the values between points have any meaning? Do the lines show any trends? Progress to interpreting conversion graphs. Pupils may use a spreadsheet package to draw conversion graphs. Notice that they are always straight lines, but do not always start at the origin. Give pupils a chance to read information from conversion graphs, in both directions. Progress on to examples that use non-linear graphs. Version 1: 28 April 10 11
TOPIC 9: GEOMETRY and MEASURES Loci and Construction use ruler and compasses to do standard constructions including an equilateral triangle with a given side, the mid-point and perpendicular bisector of a line segment, the perpendicular from a point to a line, the perpendicular from a point on a line and the bisector of an angle; loci. Define language such as equilateral, perpendicular, and bisect. Carry the class through each of the standard constructions in steps and practise several examples of each. Explain to pupils what is meant by loci giving standard examples: The path followed by a body moving according to the rule whereby: it must always be a given distance from a fixed point it must always be a fixed distance from two fixed points it must always be a fixed distance from a given straight line it must always be a fixed distance from two fixed points it must always be a fixed distance from two straight lines Move on to use loci and construction to define regions. Region bounded by a circle and an intersecting line. For example worded problems involving maps. A prepared worksheet is particularly useful in this instance. SAM T62 Q. 6 Version 1: 28 April 10 12
TOPIC 10: ALGEBRA Indices use the rules of indices for integral and fractional values. 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. SAM T61 Q. 7 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: 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. SAM T61 Q. 13(b) a m = 1/a m A fractional power is a root of the base number a ⅓ = 3 a Version 1: 28 April 10 13
TOPIC 10: ALGEBRA Indices (Cont.) Use x o = 1, y 3 x ½ x 1½ = x 2 1 2, x / x 3 3 y 1 = x 1 x 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 14
TOPIC 11: NUMBER Surds use surds including rationalise a denominator; Surds are expressions with irrational square roots (they include roots that are not equivalent to integers or fractions). SAM T61 Q. 13(c) use surds and in exact calculations, without a calculator. Rules to learn: a b ab and a b To simply a surd try to find a square number, e.g. 12 = 4 3 = 2 3 a b To rationalise a surd, remove the surd from the denominator by multiplying the numerator and denominator by the value of the surd in the denominator. e.g. 1 3 1 3 3 3 3 3 If asked to give the exact answer, leave it in surd form. When undertaking calculations involving surds and without a calculator follow the order of precedence and the rules for surds. Version 1: 28 April 10 15
TOPIC 12: GEOMETRY and MEASURES Volume Students should be able to: calculate the volume of cylinders and other simple right prisms; calculate the volume of cones and spheres; Calculate the volume of a tent. Individual work on set questions Understand the properties of a prism. Calculate the volume of prisms including cylinders. find volumes, [for example, by dissection methods]; Calculate the volume of a tent Calculate the radius of a cylinder given the volume Calculate the height of a prism given the volume and other dimensions. SAM T62 Q. 11 Calculate the surface area and volume of compound solids, including examples in everyday use. Calculate the surface area and volume of a hot water tank whose shape consists of a cylinder and a hemisphere. Version 1: 28 April 10 16
TOPIC 13: GEOMETRY and MEASURES Perimeters involving arcs and areas calculate lengths of circular arcs and perimeter of composite shapes; calculate areas of shapes whose perimeters include circular arcs; calculate surface areas of cylinders, cones and spheres. At this stage pupils can recall and use the equation for the circumference of a circle. Discuss the direct relationship between the angle of the sector and the arc length, this can also be investigated in pairs. Individual work on set questions involving getting arc lengths, perimeters of sectors and perimeters of shapes which involve arc lengths. Understand that a cylinder can be made using a sheet of card and what is originally the length and width of the card becomes the circumference and height of the cylinder respectively. Pupils will then understand that the curved surface area of the cylinder with a given radius or diameter is the circumference times the height. Do set applications on the surface area of cylinders. Given the curved surface area of a cylinder solve for the radius/diameter/height Do set applications on the surface area of cones, stressing that the height used is the slant height of the cone. Given the curved surface area of a cylinder solve for the radius/diameter/height Do set applications on the surface area of and spheres. Given the curved surface area of a cylinder solve for the radius/diameter Version 1: 28 April 10 17
TOPIC 14: GEOMETRY and MEASURES Units and dimensions distinguish between formulae for length, area and volume by considering dimensions. Review with pupils the dimensions for length, area and volume with which they are already familiar. Recognise that d is a linear measurement and that r 2 is an area measurement. Identify from a range of formulae those which denote (say) volume: 4 r 2, 4 r 3 3, 1 r 3 2 h, r 2 Remind pupils that like with algebra you can only collect like terms; you can only add a length to another length not to an area for example. Do set questions where pupils have to decide if the equation is for an area length or volume, by substituting in dimensions instead of numbers, and simplifying the expression. Pupils often respond well to examples and exemplars when learning this topic. Version 1: 28 April 10 18
TOPIC 15: ALGEBRA Graphs of Reciprocal Functions Students should be able to: explore the properties of reciprocal functions To include drawing graphs of: y = x a where a 0 and x 0 Graphic calculators Computer package such as Omnigraph know the forms of graphs of reciprocal functions make tables of such functions, sketch and interpret their graphs using graphical calculators and computers to understand their behaviour. Highlight the fact that this is made up of two smooth curves and is not defined when x = 0. It has both line and rotational symmetry. It may be an advantage to pupils to have seen the shape of the reciprocal graph before they attempt to draw it. Care needs to be taken when filling in the table of values. Highlight the fact that this is made up of two smooth curves and is not defined when x = 0. It has both line and rotational symmetry. It may be an advantage to pupils to have seen the shape of the reciprocal graph before they attempt to draw it. Care needs to be taken when filling in the table of values. SAM T61 Q. 10 Version 1: 28 April 10 19
TOPIC 16: ALGEBRA Cubic, Trig. & Exponential Functions explore the properties of cubic, trig. and exponential functions; make tables of such functions, sketch and interpret their graphs using graphical calculators and computers to understand their behaviour; know the form of exponential functions; use growth and decay rates and display these graphically; solve polynomial equations by graphical methods. Pupils need to be able to work out coordinates of given functions by tabulating values. Once this has been achieved, graphs may be plotted on a computer and interpreted. Begin by discussing how a graph could show the answer to f(x) = 0 i.e. y = f(x) cuts the x-axis. Progress to show that the equation f(x) = a is solved by looking at where the graph of y = f(x) cuts the line at y = a. Show that by rearranging the equation, a graph can be used to solve a whole family of equations. Introduce exponential functions by explaining that they are functions where the unknown is the power. To include drawing graphs of: y = a x where a = 2, 3, 4 These examples all show exponential growth. Progress to examples showing exponential decay. Know about rates of economic growth and decline and the half-life of radioactive elements. Trig. Functions are restricted to y = sin x and y = cos x for 0 o < x < 360 o. Computer graph plotting package such as Omnigraph Graphic calculators SAM T61 Q. 15 Version 1: 28 April 10 20
TOPIC 17: PROBABILITY Mutually exclusive and exhaustive events know that if there are several possible outcomes of a event (exhaustive and mutually exclusive), the total of these probabilities is 1; understand that the probability of something happening is 1 minus the probability of it not happening; understand and apply the addition of probabilities for mutually exclusive events Review probability as a measure of how likely an event is of happening. Discuss the probability scale: 0 for impossible and 1 for certain. Define the terms event and outcome. Discuss the form of a sample space diagram and possibility space. Note that the probability (theoretical probability) of an event is the number of favourable outcomes divided by the total number of outcomes. Define mutually exclusive and exhaustive events: Events are mutually exclusive if they cannot occur at the same time. Note that the sum of the probabilities of all mutually exclusive events is 1. Events are exhaustive if together they include all possible outcomes ie. at least one of them must occur. Therefore it should be noted that the probabilities of exhaustive mutually exclusive events always add up to 1. Recognise that if the probability of a machine failing is 0.05 then the probability of it not failing is 0.95 Introduce the appropriate representation of probability as a capital P with the event written in brackets e.g. P(A). Therefore the probability of an event not happening can be written as follows; P(Ā) = 1 P(A) For mutually exclusive events use the addition law of probability to find the probability of either one or other of the two events happening; P(A or B) = P(A) + P(B) SAM T61 Q. 2 SAM T62 Q. 9(b) Version 1: 28 April 10 21
TOPIC 18: PROBABILITY Independent events and tree diagrams understand that when dealing with two independent events, the probability of them both happening is less than the probability of either of them happening (unless the probability is 0 or 1); Know that the probability of getting two consecutive sunny days over a weekend is less than the probability of getting a sunny Saturday or Sunday. Discuss that two events are independent if the outcome of one does not affect the outcome of the other. For independent events the multiplication rule of probability is applied; P(A and B) = P(A) P(B) It should be noted that when multiplying probabilities the resulting answer is smaller than either of the separate probabilities. SAM T62 Q. 9(a) calculate the probability of a combined event given the probability of two independent events and illustrate combined probability of several events using tabulation or a tree diagram; Given that there are 2 sets of traffic lights on the way to school and the probability of getting straight through the lights without having to stop are 0.6 and 0.4 respectively, find the probability of a cyclist having to stop at one set of lights, using a tree diagram or otherwise. This is usually referred to as a tree diagram with replacement as the events can then be observed as being independent. To obtain the appropriate answer use the following procedure: multiply along the branches to get the end results; on any set of branches which meet at a point, the numbers must always add up to 1; check that the end results also add up to 1. To obtain the relevant answer to a problem simply add up the relevant end points. SAM T61 Q. 12 Conditional probability. Version 1: 28 April 10 22
TOPIC 18: PROBABILITY Independent events and tree diagrams(cont.) Draw a tree diagram or use a tabulation to define all of the possible outcomes eg. tossing a coin 3 times. produce a tree diagram to illustrate the combined probability of several events which are not independent An operation has a 60% success rate the first time it is attempted. If it is unsuccessful it can be repeated, but with a success rate of only 30%. The probability of success the third time is so low that surgeons are unwilling to operate. What is the probability that the operation will fail twice? Discuss that sometimes the events illustrated by a tree diagram are not independent. The outcome of the first event may affect subsequent events. This type of tree diagram is sometimes referred to as one without replacement. Version 1: 28 April 10 23