# Year 11 Maths enrichment Gradients Contents This enrichment

• Slides: 10

Contents This enrichment work is meant to develop your understanding of some GCSE topics and extend them into more advanced ideas. All the methods used are ones that you have learnt or will learn in the Higher GCSE course, but the outcome is part of the A level. Each section should take about 30 mins. Click on the links below to go straight to the section. 1. The gradient at a point 2. The gradient function 3. Pattern hunting 4. Proving the results 1/5/2022 Maths @ Paston © P. R. Brayne 2

1. The gradient at a point How can you calculate the gradient at a point on a graph? We’ll start with the curve y=x 2, and try to find the gradient at (1, 1). You could draw a tangent, and measure the gradient (Now click) 1. 8 1 but this would not be accurate. 1/5/2022 Maths @ Paston © P. R. Brayne 3

1. The gradient at a point We can calculate exactly the gradient of a chord (a line between two points on the curve) Look at the graph and check the calculations below – click to move the position of B. The graph shows the chord from (1, 1) to (1. 2, (2, 4)1. 2 (1. 5, 1. 5 on 2)the on the graph of of y=x 2 which has gradient: To find the gradient we bring B in closer to A and hope to spot a pattern… 1/5/2022 Maths @ Paston © P. R. Brayne 4

Does it make sense? Try for yourself. Complete the table below – use your calculator to work out the gradients from the point in the table to the point (1, 1). x y=x 2 2 1. 5 1. 4 1. 3 1. 2 1. 1 22=4 1/5/2022 Gradient of chord 3 Compare the values of the gradient with the values of x. What do you think the gradient will be at x=1? Maths @ Paston © P. R. Brayne 5

2. The gradient function Here, you are going to try to find a formula for the gradient from the formula of the curve. Find the gradient of the curve for a number of values of x, put these in a table. Can you spot the formula that connects the values of x to the values of the gradient? The spreadsheet on the left will do all the calculations for you. Click on it to open it – you should get the “enable macros” option. If you don’t, click here! 1/5/2022 Maths @ Paston © P. R. Brayne 6