This activity can brighten up the introduction of the scalar product formula: u.v = |u| |v| cos (theta) …using pattern spotting and allowing every student to make a contribution. I have used this activity over several years and spent time refining it to get it right. You can save yourself all this time for less than the price of a pint of beer or a cup of coffee. I promise I won’t spend the money on coffee. The full details of the lesson can be found here. I have improved my resources since the blog post was written: There are more vectors on the accurate graph and more vector-pair cards. I have used proper vector notation on the diagram and labelled the axes. If you need busy-work for fast finishers, they can work out the scalar products of more vector pairs involving the (sqrt(5), 2sqrt(5)) with all of the other vector pairs*. Ask them to find the answers to 3 sig fig.** (* The only vector I have paired it with so far is (5,12). ) (** If they finish that they can work out the angles if the (sqrt(5), 2sqrt(5)) is reflected in the x&y-axes, the line y=x etc… If they finish that ask them how they know that they have finished. There must be a permutations question in that somewhere…)

This is alternative proof-without-words for the A-level double angle formulae. If you ask me, this is a nicer way to derive them than just say alpha = beta = theta in the compound angle formulae. I suggest getting the students to use the substitution method first and show the circle method to the students who find it easy. If you have a better idea, please tweet me. This proof relies on the circle theorem: “angle at the centre is double the at the circumference” and right-angled trig.

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.

Perhaps you remember this old TV programme? Well now you can set up your own version for your classroom. Its a strategy game for teams. The team who answers the question correctly wins the tile and each team is trying to build a bridge across the board, so they can block each other. This is version 2 so I’ve made some improvements. Details here if you are interested I’ve included an example with terminology from A-level Mechanics M1. Click once to change a hexagon red and click again to make it white. Keep clicking and it will cycle through the different options. Unfortunately it doesn’t do that flashy thing when you win or play music, but of course you can sing the theme tune when theres a winner.

This is a random thing I made. It’s pretty straightforward to set up. set up the formula, say for B2: “=A1+A3” and copy and paste it into all the cells underneath. Choose a cell at the top in the middle somewhere. Type “1”. It should fill in with Pascal’s Triangle automatically. Set the conditional formatting to make empty cells white and colour in multiples of the contents of one cell (I used cell I8) something like this: “=AND(MOD(A2,$I$8)=0,A2<>0)” …or you could download my copy.