REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011

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1 REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011

2 Version 1: 28 April 10

3 Version 1: 28 April 10 Unit T3

4 Unit T3 This is a working document for teachers to adapt for their own needs. Knowledge of the content of Units T1 and T2 is assumed. Topic No. Topic Subject Content 1 Geometry and Measures Understanding and applying Pythagoras s Theorem 2 Algebra Solving linear equations 3 Number Percentage change 4 Geometry and Measures Upper and lower bounds 5 Algebra Factorisation and removing brackets 6 Algebra Graphs of linear functions 7 Number Repeated proportional change 8 Algebra Simultaneous equations 9 Statistics Statistical charts, diagrams and tables 10 Algebra Algebraic fractions 11 Geometry and Measures Trigonometry 12 Statistics Measures of central tendency 13 Statistics Measures of dispersion 14 Algebra Quadratic equations 15 Geometry and Measures Perimeters and areas of compound shapes Version 1: 28 April 10 2

5 TOPIC 1: GEOMETRY and MEASURES Understanding and Applying Pythagoras s Theorem understand and apply Pythagoras theorem. Investigative approach in pairs ; for example Pythagorean triples Individual work on set questions Calculate a side of a right-angled triangle when the other two sides are known SAM T3 Q. 13(b) Version 1: 28 April 10 3

6 TOPIC 2: ALGEBRA Linear Equations formulate, use and solve linear equations. Introduce solving linear equations by the notion of reversing the operations. Look at examples like x + 4 = 23, 3x = 18. Progress to two stage equations where reversing the order is important also, e.g. 2x 5 = 17. SAM T3 Q. 4, Q.12(c), Q. 17(b) Use algebra to solve a problem such as If I double a number, then add 1 and the result is 49, what is the number? Include questions where there are brackets, e.g. 2(x + 5) = 16 Progress to 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. e.g. 4x 3 = x + 12 Solve equations such as 1 1 ( x 3) ( x 2) Version 1: 28 April 10 4

7 TOPIC 3: NUMBER Percentage Change calculate percentage change; Work out the cost of a computer which is offered at 15% discount in a sale. Introduce the following types of problem: Find a certain percentage of an amount e.g. 25% of 52. Proceed by either finding 1% of the amount by dividing by 100 (starting amount is equivalent to 100%), then multiply by 25 or use a multiplying factor, which is in this case 0.25 SAM T3 Q. 1(b) SAM T3 Q. 16 Introduce problems that incorporate increases or decreases of an amount. When the multiplying factor is greater than 1 there is an increase e.g. increase an amount by 25%, means that the initial amount is now (100+25)% and the multiplying factor would be Similarly a decrease of 25% would mean (100-25)% and the multiplying factor applied would be 0.75 Percentage change. Express the change in value as a percentage as follows; percentage change is equal to the change divided by the original amount multiplied by 100. A similar approach can be used for percentage profit. Change could relate to profit, loss, appreciation, depreciation, error, discount etc. These terms should be discussed. express one number as a percentage of another. Express one number as a percentage of another, by taking the first number, dividing by the second number and multiplying by 100 Version 1: 28 April 10 5

8 TOPIC 4: GEOMETRY and MEASURES Upper and lower bounds develop an understanding of the continuous nature of measure and approximate nature of measurement; understand and calculate the upper and lower bounds of the values of expressions involving numbers expressed to a given degree of accuracy. Class discussion and individual work on set questions Know the difference between 4.60 and 4.6 as measurements; realise that a length of l written as 9.7 cm correct to one decimal place means that 9.65 l < 9.75 Individual work on set questions Give the upper/lower bound for a given length up to 3 s.f. and 2 d.p. Version 1: 28 April 10 6

9 TOPIC 5: ALGEBRA Removing Brackets and Factorising understand key concepts and terms, for example, factorise, generalise, n th term; simplify, remove brackets, do simple factorising; use these techniques with a range of more complex expressions. Introduce the notion of term and how to collect them. Show examples of linear and non-linear terms to be collected. Highlight cases like x and x 2, xy 2 and x 2 y. Move on to multiplying out single brackets. Know that 2(a + b) is the same as 2a + 2b and converse. Introduce multiplying out two brackets by using the box method. Expand and simplify (x + 4)(x 2) and find the difference of two squares. Know that (a b) 2 = a 2 2ab + b 2. Play algebra whist in groups of four set of 50 cards with a to 10a, b to 10b, ab to 10ab, a 2 to 10a 2 and b 2 to 10b 2. Begin by looking at examples of factorising when the common factor is a number and then when it is a letter. Move on to examples with numbers and letters as factors. Set of algebra cards SAM T3 Q. 12(a) (b) To include: factorising, such as: x 2 3x = x(x 3) 2x 2 4x = 2x(x 2) Version 1: 28 April 10 7

10 TOPIC 5: ALGEBRA Removing Brackets and Factorising (cont.) Factorise simple quadratic expressions to produce two linear brackets as factors. T3 Q. 17(a)(i) To include: factorising, such as: x 2 8x + 15 = (x 3)(x 5) x 2 16 = (x 4)(x + 4) Version 1: 28 April 10 8

11 TOPIC 6: ALGEBRA Graphs of Linear Functions explore the properties of linear functions; express a function in words, in tabular form, graphically and symbolically; make tables of such functions, sketch and interpret their graphs using graphical calculators and computers to understand their behaviour; Introduce the idea of gradient by looking at the steepness of lines in a variety of contexts. Use the data projector to interpret lines of the form y = mx where m is positive. Pupils can draw the graph of such equations by using Omnigraph. Calculate the gradient of a straight line given two points using the formula Gradient = vertical change horizontal change Include drawing graphs of y = x + c and connect the equation to the y-intercept. Pupils can draw the graph of such equations by using Omnigraph. Derive a linear relationship from a straight line graph in the form y = mx + c. Introduce the concept of negative gradients and give examples of when these would arise in real life. Highlight what happens when two lines have the same gradient parallel lines. e.g. y = 5x and y = 5x + 3 represent parallel lines with gradient 5. Data projector Graphic calculators Computer package such as omnigraph SAM T3 Q. 3 Version 1: 28 April 10 9

12 TOPIC 6: ALGEBRA Graphs of Linear Functions (cont.) 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. Introduce linear equations in the form x + y = c. Look at the technique for drawing lines of the form ax + by = c. e.g. draw the graph of 3x 4y = 7 and understand this is a linear equation and therefore the graph will be a straight line. Version 1: 28 April 10 10

13 TOPIC 7: NUMBER Repeated proportional change understand and use repeated proportional change, including the calculation of compound interest restricted to a maximum of three iterations. Repeated proportional change increases or decreases an amount by a percentage more than once. Introduce repeated proportional change by applying the method of percentage change more than once. When the method has been applied for the first time the solution obtained is used as the starting point for the second calculation. This is restricted to three calculations so the term iteration must be discussed. Compound interest includes interest on interest already paid or charged. The calculation of compound interest uses the same approach as repeated proportional change. SAM T3 Q. 11 To calculate compound interest two methods can be applied: The first requires each year (time period) to be calculated in turn with the amount obtained at the end of one year being used for the beginning of the next. The second uses a formula: initial amount multiplied by the multiplying factor which includes the increase or decrease raised to the power of the number of years (time period) of the investment. The power will be restricted to a maximum of three. Version 1: 28 April 10 11

14 TOPIC 8: ALGEBRA Simultaneous Equations use algebraic and graphical methods to solve simultaneous linear equations 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 co-ordinates work in both equations before moving on. Given two equations, find points on each equation, by putting x = 0 and finding y, and then putting y = 0 and finding x. (Or use other appropriate values for x and/or y.) 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. Finally show examples of simultaneous equations where the two equations need to be multiplied to make the coefficients equal in size. Always highlight the notion of checking that the solutions fit into the two original simultaneous equations. Computer package such as Omnigraph SAM T3 Q. 17(c) Version 1: 28 April 10 12

15 TOPIC 9: STATISTICS Statistical charts, diagrams and tables grouped and ungrouped data Interpret and display information in a variety of ways, using paper and ICT, including: Discuss 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. box plots and stem-and-leaf diagrams; Illustrate the form of a box plot as a box with whiskers, one on either side (box and whisker diagram). State clearly that the box plot highlights the quartiles and extreme values of a set of data. Note that the middle 50% of the data is the box with the median represented by a vertical line within the box. Discuss further that the box plot is useful for visualising the spread of data and that it can be easily drawn under a cumulative frequency curve. SAM T3 Q. 8 Introduce a stem-and-leaf diagram as a diagram that may be used to show the shape of a distribution of data. Discuss that a stem-and-leaf diagram shows all the original data and can be used to display discrete or continuous data. Note that the stem is often the first digit of the numbers (usually tens) with the leaves representing the units. A key is necessary and attaches a meaning to the data. The stem-and-leaf diagram is normally sorted into order. This allows the median and quartiles to be obtained more easily. Version 1: 28 April 10 13

16 TOPIC 9: STATISTICS Statistical charts, diagrams and tables grouped and ungrouped data (cont.) frequency tables and charts for ungrouped or grouped discrete data and continuous data (including frequency polygons); Introduce frequency tables as a form of representing data which enables the data to be more easily analysed. Show the structure of a frequency table with three columns which include group labels or categories, tally marks and frequency. Introduce a frequency diagram as being similar to a bar chart except the bars are drawn with no spaces and the horizontal axis has a continuous scale (not distinct categories). The frequency diagram uses bars to display grouped data on a continuous scale. Progress to the frequency polygon which uses the midpoints of the bars of a bar chart. Note that a frequency polygon requires the midpoints of the data to be joined with straight lines. A frequency diagram is not required prior to a frequency polygon being drawn. SAM T3 Q. 9(a) Version 1: 28 April 10 14

17 TOPIC 10: ALGEBRA Algebraic Fractions simplify more complex expressions such as algebraic fractions. Introduce algebraic fractions by looking at simplifying fractions such as 2 6x y 3x 3 2 8xy 4y Move on to multiplying the numerator and denominator by the same amount to look at equivalent algebraic fractions. If algebraic fractions have the same denominator then they can be compared. Look at adding and subtracting algebraic fractions. e.g. adding fractions such as a b d c ad bc bd Introduce pupils to multiplying simple algebraic fractions by noting that you must multiply the numerators and the denominators. Look at examples that require cancellation before multiplying (common factor on the top and bottom lines). e.g. multiplying fractions such as a b Move on to dividing algebraic fractions by multiplying by the reciprocal. c d ac bd Version 1: 28 April 10 15

18 TOPIC 11: GEOMETRY and MEASURES - Trigonometry understand and apply the sine, cosine and tangent to right-angled triangles in 2-D; understand and apply the sine, cosine and tangent to right-angled triangles in 3-D; extend their understanding of trigonometry to the application of trigonometry to the solution of problems in 3-D. Introduce the student to the concepts of sine, cosine and tangent of an angle. Students can analyse a problem and identify the appropriate ratio to find an unknown angle/side of a right-angled triangle. Use sine, cosine and tangent to then calculate the side or angle of a triangle. 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 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, 2. questions where the pupils have to sketch the problem before attempting to solve it. 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. SAM T3 Q. 18(a) Version 1: 28 April 10 16

19 TOPIC 12: STATISTICS Measures of central tendency calculate or estimate and use the mean and range of sets of discrete then continuous data; find the mean of grouped data; find the median and modal classes of grouped data; Prepare tables; calculate the mean: (i) Measurement of Heights: Use 10 cm intervals from cm class intervals defined as: Interval Mid interval point 120< h 130 cm 125 cm 130< h 140 cm 135 cm 140< h 150 cm 145 cm SAM T3 Q. 9(b), Q.14(b) h = height (centimetres) (ii) Examination marks: Range 0-100, intervals of 10 marks 0-9 midpoint midpoint midpoint midpoint 34.5 etc Version 1: 28 April 10 17

20 TOPIC 12: STATISTICS Measures of central tendency (cont.) compare sets of data by making appropriate use of mean, mode, median and range. Calculate the mean score for each of two teams who have played different numbers of games over a season to compare their performance. When calculating the mean from a grouped frequency a further column must be calculated which is found by multiplying the midpoint (mid class value) to the frequency. The sum of this column divided by the total frequency will give the mean. SAM T3 Q.14(a) Version 1: 28 April 10 18

21 TOPIC 13: STATISTICS Measures of dispersion construct cumulative frequency tables; construct a cumulative frequency curve; find the median, the upper quartile, the lower quartile and the interquartile range; describe the dispersion of data. Introduce a cumulative frequency table as an extension to a frequency table. The cumulative frequency is a running total of the frequencies. The cumulative frequency is plotted against the highest value in each class. Plot the cumulative frequency on the vertical axis with the highest value in each class (upper class boundary) along the horizontal axis. It should be noted that the cumulative frequency curve (ogive) requires the points to be joined not as a series of straight lines but as a smooth curve. It is useful to show the connection between the cumulative frequency curve and a box plot. Both illustrate the dispersion of the data with the lower quartile (quarter of the total frequency), median (half of the total frequency) and upper quartile (three quarters of the total frequency) being identified. As the quartiles split the data up into four equal parts, when the total frequency is sufficiently large the following approximation can be made; one quarter of the total frequency is the lower quartile, half the total frequency is the median and three-quarters of the total frequency is the upper quartile. Construction lines should be shown on the cumulative frequency curve when finding the median and quartiles. Introduce the interquartile range as a measurement of spread which can be found by subtracting the lower quartile from the upper quartile. A large value for the interquartile range would indicate that the data is more dispersed/spread out. SAM T3 Q. 19 Version 1: 28 April 10 19

22 TOPIC 14: ALGEBRA Quadratic Equations Know the forms of graphs of quadratic functions; use factors to solve quadratic equations. Introduce quadratic functions by drawing the graph of x 2. The parabola must be a smooth curve. Move on to graphs of y = ax 2 + bx + c. Revision of substituting into formulae may be advantageous here. Look at graphs where the coefficient of x 2 is negative. Revise over multiplying out brackets by the box method. Move onto factorising expressions where the coefficient of x 2 is 1. Relate this to solving quadratic equations with two linear brackets. Graph paper Calculators SAM T3 Q. 17(b) Show more complicated quadratic equations where the coefficient of x 2 is other than 1. Look at the difference of two squares. Version 1: 28 April 10 20

23 TOPIC 15: GEOMETRY and MEASURES Perimeters and areas of compound shapes calculate lengths of circular arcs and perimeter of composite shapes; calculate areas of shapes whose perimeters include circular arcs. 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. SAM T3 Q. 5(a), 6 Version 1: 28 April 10 21

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