“Teach A Level Maths” Yr1/AS Pure Maths Sample 1 “Teach A Level Maths” Yr1/AS Pure Maths © 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 7 of the 46 presentations that make up the work for Year 1/AS Pure Maths. Roots, Surds and Discriminant Simultaneous Equations and Intersections Linear and Quadratic Inequalities Stationary Points: applications Circle Problems Quadratic Trig Equations Indices and Laws of Logarithms

Roots, Surds and Discriminant 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.

The Discriminant of a Quadratic Function For the equation . . . . . . the discriminant There are no real roots as the function is never equal to zero y = x2 – 4x +7 If we try to solve , we get The square of any real number is positive so there are no real solutions to

Simultaneous Equations and Intersections The following slide shows an example of solving a linear and a quadratic equation simultaneously. The discriminant ( met earlier ) is revised and the solution to the equations is interpreted graphically.

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

Linear and Quadratic Inequalities 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.

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

Stationary Points: applications Mathematical modelling is one of the overarching themes in the 2017 A level. The following slides show an example of the application of stationary points to a simple problem. Students are asked to think about the assumptions made in setting up the model.

e. g. 2 I want to make a fruit cage from 2 pieces of netting e.g.2 I want to make a fruit cage from 2 pieces of netting. The smaller piece will make the top and I’ll use the longer piece, which is 10m long, for the sides. The cage is to be rectangular and to enclose the largest possible area. One side won’t need to be netted as there is a wall on that side. How long should the 3 sides be? Solution: We draw a diagram and choose letters for the unknown lengths and area. wall The lengths are x metres and y metres. x x A The area is A m2. y

(You may have thought of other assumptions) wall x y A We want to find the maximum value of A, so we need an expression for A that we can differentiate. The area of a rectangle = length  breadth We can’t differentiate with 3 variables so we need to substitute for one of them. I have assumed that there is no wastage at the wall or corners. The area I find will be too large. (You may have thought of other assumptions) Can you think of one assumption I have made in setting up this model? What effect will it have on the answer? Since the length of netting is 10m, we know that Rearranging,

5 2 × = x x A y wall Substituting for y in (1), We can now find the stationary points. Substitute in 5 2 × = x Substitute in (1),

x A y wall We have We know that we have found the maximum since and we recognise the shape of this quadratic.

Circle Problems There is also a greater emphasis on problem solving. Questions involving circles provide useful practice. Here students are reminded of the properties of circles, with worked examples encouraging them to solve problems and emphasising the need to draw diagrams.

Decide with your partner how you would solve this problem Diagrams are essential when solving problems involving circles. They don’t need to be accurate ! e.g.1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) x Decide with your partner how you would solve this problem (5, 7) x (2, 3) My reasoning (in order) is on the next slide. tangent

We know a point on the tangent so x1 = 5 and y1 = 7 e.g.1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) A tangent is a straight line so we need the equation of a straight line: gradient m x (2, 3) (5, 7) tangent gradient m1 We know a point on the tangent so x1 = 5 and y1 = 7 We need the gradient m The gradients of tangent and radius are perpendicular so We can find m1 using

We need to reverse the order of my logic to do the calculation. gradient m x (2, 3) (5, 7) tangent gradient m1 Multiply by 4: 4y – 28 = – 3(x – 5)  3x + 4y = 43

Quadratic Trig Equations By the time students meet quadratic trig equations they have practised solving linear trig equations and have seen a proof of a Pythagorean identity.

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

Solving for . Principal Solution: The 2nd solution is

Indices and Laws of Logarithms 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 do 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 giving the answer correct to 3 significant figures. x is the logarithm of 4 with a base of 10 We write ( 3 s.f. ) In general if then index = log

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