To introduce the dot product to the precalc class, I had them first find the angle between two vectors using our old trig methods. Interestingly they used right triangle trig to get the answer; I didn’t anticipate that. Lovely!
To link it in better with the future (how did they not anticipate the future !!!????!!!), I asked them to solve it with law of cosines as well:
Then I had them work in pairs on the following Desmos Activity: Vector Investigation ?????????????????? where they were answering questions based on the dot product but it was called B in the activity. There are a couple of reasons that I decided to not tell them what was being measured, and after doing the activity I think it was the right move. Still if you want to try out the activity with the labels dot product, here is a link to the same activity. Some great observations and guesses from the students, many went down the path of thinking about it in terms of quadrants instead of angles, but I think they were able to build some intuition for what the dot product was measuring.
Here are some sample responses from the first slide.
After the first three slides, the groups had a very good idea that the mysterious B thing was somehow related to the angle between the two vectors (and not related, directly, to the quadrants). We still have a bit of work left to nail down dot product, but it was a nice start. To finish off the day we solved the original problem, but with dot products. (Quite a bit easier). I’m still pondering if I’d like to show the proof of how the dot product is related to the law of cosines, or to bag it.