This is a Geogebra page for exploring the graph of y=k^x and it’s gradient function, which allows you to adjust the value of k. You can use it here without downloading anything. The idea is to guide students to find the value of k for which the function and the gradient function overlap each other. You can zoom in on the graph to get a pretty accurate value. Before you talk about e and why it is useful, your students should have had time to think about it for themselves first. I have written a blog post about it here. Please leave a comment on the blog if you have feelings about this activity. Perhaps you will be the first person to review one of my thingies on TES.com - wouldn’t that be something?

This is an interactive GeoGebra page. You can use the online version here without downloading anything if you like. It is a similar sort of idea to the “Introducing e^x”, which has a full description on my blog. I’ll summarise quickly how to use this one… Display the graph of y=sin kx with k=1 and ask them to draw the gradient function. They may guess that it looks like y=cosx Display the gradient function by clicking on the checkbox. Try some different values of k, and notice how the “amplitude” of the gradient function increases. *You can zoom in with just the x-axis by holding down Shift and click-dragging a point on the x-axis. * Try different values of k. You will find that and k=~57, the gradient function is pretty close to cosine. There is a secret checkbox next to the “Show y=f(x)” checkbox, which will reveal the y=cos(kx) graph. Display it and zoom in on the two graphs which seem to overlap. Adjust the value of k and zoom in, repeatedly so that you improve the accuracy of k. You will find that the k-value is around 57.2957795. Ask students to reflect on what they just saw. You want them to realise that when angles are measured in radians, the gradient of y=sin x is y=cos x, and that unless the angle is measured in radians, this is not true. I hope this makes sense and it is useful for you. If you have ideas for using this which I haven’t written about or ideas for making it better please get in touch.

This PPT is a proof-almost-without-words of the compound angle formulae for sine, cosine, tangent. You might use it to introduce the formulae. I would recommend doing a starter activity before-hand where students have to find the opposite and adjacent sides of a triangle given the hypotenuse and the angle. Why not also throw in a mechanics question like: write down the components of the 10N force which acts on the particle at 30 degrees above the horizontal, say. (Cue discussion about defining their own x and y directions.) You can elicit all the side lengths before they appear on the screen. Then when all the lengths are shown (when it moves up to the top of the screen), you can get them to make a neat copy and write down equations for sin (a+b) and cos (a+b). Try tan if they are confident.