Teach A Level Maths Vol. 1: AS Core Modules Demo Disc Christine Crisp Explanation of Clip-art images An important result, example or summary that students might want to note. It would be a good idea for students to check they can use their calculators correctly to get the result shown. An exercise for students to do without help. The slides that follow are samples from the 51 presentations that make up the work for the AS core modules C1 and C2. 6: Roots, Surds and

8: Discriminant Simultaneous Equations and Intersections 9: Linear and Quadratic Inequalities 11: The Rule for Differentiation 13: Stationary 18:Points Circle Problems 25: Definite Integration 26: Definite Integration and Areas 29: The Binomial Expansion 33: Geometric series Sum to

38: Infinity The Graph of tan 43: Quadratic Trig Equations 46: Indices and Laws of Logarithms 6: Roots, Surds and Discriminant Demo version note: Students have already met the discriminant in solving quadratic equations. On the following slide the calculation is shown and the link is made with the graph of the quadratic function. Roots, Surds and Discriminant The Discriminant of a Quadratic Function For the equationx 2 4 x 7 0 . . .

. . . the b 2 4ac 16 28 discriminant 12 0 y x2 4x 7 There are no real roots as the function is never equal to zero 2 If we try to solvex 4 x 7 0, we get 4 12 x 2 The square of any real number is positive so there are no real solutions 12 to 8: Simultaneous Equations and Intersections Demo version note: The following slide shows an example of solving a linear and a quadratic equation simultaneously. The discriminant ( met in presentation 6 )

is revised and the solution to the equations is interpreted graphically. Simultaneous Equations and Intersections y x 2 3 (1) y 4 x 1 (2) e.g. 2 x 2 3 4 x 1 y: x 2 4 x 4 0 The b 2 4ac 4 2 4(1)(4) 0 Eliminate discriminant, The quadratic equation has equal roots. 2 Solvi x 4 x 4 0

ng ( x 2)( x 2) 0 y x2 3 y 4 x 1 x 2 (twice) x 2 y 7 The line is a tangent to the 9: Linear and Quadratic Inequalities Demo version note: Students are shown how to solve quadratic inequalities using earlier work on sketching the quadratic function. The following slide shows one of the two types of solutions that arise. The notepad icon indicates that this is an important example that students may want to copy. Linear and Quadratic Inequalities

x 2 4 x 5 0 e.g.2 Find the values of x that satisfy Solution: 2 Find the zeros off ( x ) wheref ( x ) x 4 x 5 x 2 4 x 5 0 x 5 ( x 5)( x 1) 0 or x 1 x 2 4 x 5 is greater than equal to 0or above the xaxis are 2 sets of values of There x

x 1 or x 5 These represent 2 separate intervals and CANNOT be combined yy x 2 4 x 5 11: The Rule for Differentiation Demo version note: In this presentation, the rule for differentiation of a polynomial is developed by pattern spotting, working initially with the familiar quadratic function. A later presentation outlines the theory of differentiation. The Rule for

Differentiation The Gradient at a point on a Definition: Curve The gradient at a point on a curve equals the gradient of the tangent at that point. e.g. y x2 12 (2, 4)x 3 Tangent (2, 4) at The gradient of the tangent at (2, 4) is 12 m 3 4 So, the gradient of the curve at (2, 4) is 4

13: Stationary Points Demo version note: Stationary points are defined and the students practice solving equations to find them, using cubic functions, before going on to use the 2nd derivative to determine the nature of the points. The work is extended to other functions in a later presentation. Stationary Points The stationary points of a curve are the points where the gradient is zero e.g. y x3 3x2 9x A local maximum x

dy 0 dx x A local minimum The word local is usually omitted and the points called maximum and minimum points. 18: Circle Problems Demo version note: The specifications require students to know 3 properties of circles. Students are reminded of each and the worked examples, using them to solve problems, emphasise the need to draw diagrams. Circle Problems e.g.2 The centre of a circle is at the point C (-1,

2). The radius is 3. Find the length of the tangents from the point P ( 3, 0). Method: tangent Sketch! Use 1 tangent and join the radius. The required length is C (-1, 2) x Find AP. CP and use 20 Pythagoras theorem 3 for triangle CPA Solution: x tangent A 11 P (3,0) d ( x 2 x 1 ) 2 ( y 2 y1 ) 2

CP ( 3 ( 1)) 2 (0 2) 2 AP 2 PC 2 AC 2 CP 16 4 20 AP 20 9 11 25: Definite Integration Demo version note: The next slide shows a typical summary. The clip-art notepad indicates to students that they may want to take a note. Definite Integration SUMMAR Y The method for evaluating the definite integral is:

Find the indefinite integral but omit C Draw square the limits on Replace x with brackets and hang the end the top the limitbottom limit Subtract and evaluate 26: Definite Integration and Areas Demo version note: The presentations are frequently broken up with short exercises. The next slide shows the solution to part

of a harder exercise on finding areas. The students had been asked to find the points of intersection of the line and curve, sketch the graph and find the enclosed area. Definite Integration and Areas 2 y x 2 (b) y 4 x ; x 2 4 x 2 y 4 x 2 y x 2 2 x x 2 0 ( x 2)( x 1) 0 x 2 or x 1 Substitute iny x 2 :

x 2 y 0 , x 1 y 3 Area under the curve 1 x 4 x dx 4 x 3 2 3 2

1 9 2 1 9 3 3 Area of the triangle 2 2 Shaded area = area under curve area of triangle 9 2

29: The Binomial Expansion Demo version note: The following short exercise on Pascals triangle appears near the start of the development of the Binomial Expansion. Answers or full solutions are given to all exercises. The Binomial Expansion Exercis e Find the coefficients in the expansion (a of b) 6 Solution: We need 7 rows 1 1 1

1 1 Coefficien ts 1 6 2 3 4 5 1 1 3 6 10 15

1 1 4 10 20 1 5 15 1 6 1 33: Geometric series Sum to Infinity Demo version note: The students are shown an example to illustrate the general idea of a sum to infinity. A more formal

discussion follows with worked examples and exercises. Geometric series Sum to Infinity Suppose we have a 2 metre length of string . . . . . . which we cut in half 1m 1m We leave one half alone and cut the 2 nd in half again 1m 1 2 . . . and again cut the last piece in half 1m 1

2 1 2 m m 1 4 m m 1 4 m 38: The Graph of tan Demo version note: The next slide shows part way y tan

through the development of the sin y cos graphy of using and . The Graph of tan The graphs ofsin y for 0 360are andcos y sin tan This line, where is not defined is

called an asymptote. y tan x x y y cos Dividing by zero gives infinity tan 90 so is not defined when . sin 90 1 cos 90

0 x 43: Quadratic Trig Equations Demo version note: By the time students meet quadratic trig equations they have practised using both degrees and radians. Quadratic Trig Equations 2 cos 2 3 cos 2 0for e.g. 3 Solve the equation the interval0 2 , giving exact answers. c cos . Then, Solution: Let 2c 2 3c 2 0 ( 2c 1)(c 2) 0 c 12 or

Factorisin g: cos 12 or cos 2 y The graph ofy cos . . . 1 shows thatcos always lies between -1 0 cos + 1 so, 2 and -1 has no solutions for . c 2 2

y cos Quadratic Trig Equations Solvingcos 1 for 0 2. 2 60 3 Principal Solution: y 1 0 y 0 5 3

-1 5 3 y cos Ans : 5 , 3 3 2 46: Indices and Laws of Logarithms Demo version note: x

a b The approach to solving the equation started with a = 10 and b an integer power of 10. The word logarithm has been introduced and here the students are shown how to use their calculators to solve when x is not an integer. The calculator icon indicates that students should check the calculation. Indices and Laws of Logarithms A logarithm is just an index. To solve an equation where the index is unknown, we can use logarithms. e.g. Solve the equation 10 x 4 giving the answer correct to 3 significant figures. x is the logarithm of 4 with a base of We 10

10 x 4 x log 10 4 write 0 602 ( 3 s.f. ) In general if 10 x b then x log 10 b inde x log Full version available from:Chartwell-Yorke Ltd. 114 High Street, Belmont Village, Bolton, Lancashire, BL7 8AL England, tel (+44) (0)1204 811001, fax (+44) (0)1204 811008 [email protected] www.chartwellyorke.com