1. Demo Disc “Teach A Level Maths” Vol. 1: AS Core Modules © Christine Crisp

2. 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.

3. 38: The Graph of tan  43: Quadratic Trig Equations 29: The Binomial Expansion 33: Geometric series – Sum to Infinity 25: Definite Integration 13: Stationary Points 11: The Rule for Differentiation 9: Linear and Quadratic Inequalities 8: Simultaneous Equations and Intersections 6: Roots, Surds and Discriminant The slides that follow are samples from the 51 presentations that make up the work for the AS core modules C1 and C2. 18: Circle Problems 46: Indices and Laws of Logarithms 26: Definite Integration and Areas

4. 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.

5. For the equation...... the discriminant There are no real roots as the function is never equal to zero The Discriminant of a Quadratic Function If we try to solve, we get The square of any real number is positive so there are no real solutions to Roots, Surds and Discriminant

6. 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.

7. e.g. 2 Eliminate y : The discriminant, The quadratic equation has equal roots. The line is a tangent to the curve. Solving Simultaneous Equations and Intersections

8. 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.

9. Solution: e.g.2 Find the values of x that satisfy or There are 2 sets of values of x Find the zeros of where is greater than or equal to 0 above the x -axis or These represent 2 separate intervals and CANNOT be combined Linear and Quadratic Inequalities

10. 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.

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

12. 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 2 nd derivative to determine the nature of the points. The work is extended to other functions in a later presentation.

13. The stationary points of a curve are the points where the gradient is zero A local maximum A local minimum x x The word local is usually omitted and the points called maximum and minimum points. e.g. Stationary Points

14. 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.

15. 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). x C (-1, 2) Solution: P (3,0) x Method: Sketch! Find CP and use Pythagoras’ theorem for triangle CPA A tangent Use 1 tangent and join the radius. The required length is AP. Circle Problems 3

16. 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.

17. SUMMARY  Find the indefinite integral but omit C  Draw square brackets and hang the limits on the end  Replace x with the top limit the bottom limit  Subtract and evaluate The method for evaluating the definite integral is: Definite Integration

18. 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.

19. Area of the triangle or Substitute in : Area under the curve (b) ; Shaded area = area under curve – area of triangle Definite Integration and Areas

20. 29: The Binomial Expansion Demo version note: The following short exercise on Pascal’s triangle appears near the start of the development of the Binomial Expansion. Answers or full solutions are given to all exercises.

21. Exercise Find the coefficients in the expansion of Solution: We need 7 rows 121 1331 11 1 14641 1 5 101 5 1 6 15 1 20156 Coefficients The Binomial Expansion

22. 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.

23. Suppose we have a 2 metre length of string...... which we cut in half We leave one half alone and cut the 2 nd in half again... and again cut the last piece in half Geometric series – Sum to Infinity

24. 38: The Graph of tan  Demo version note: The next slide shows part way through the development of the graph of using and.

25. x The graphs of and for are x x This line, where is not defined is called an asymptote. Dividing by zero gives infinity so is not defined when. The Graph of tan 

26. 43: Quadratic Trig Equations Demo version note: By the time students meet quadratic trig equations they have practised using both degrees and radians.

27. e.g. 3 Solve the equation for the interval, giving exact answers. or Factorising: The graph of... Solution: Let. Then, shows that always lies between -1 and + 1 so, has no solutions for. 1 Quadratic Trig Equations

28. 1 Principal Solution: Solving for. Ans: Quadratic Trig Equations

29. 46: Indices and Laws of Logarithms Demo version note: 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.

30. A logarithm is just an index. To solve an equation where the index is unknown, we can use logarithms. e.g. Solve the equation giving the answer correct to 3 significant figures. x is the logarithm of 4 with a base of 10 We write In general if then  log index ( 3 s.f. ) Indices and Laws of Logarithms

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