I came up with a clever test question last year:
I wanted to tie vectors together with solving triangles for a couple reasons. Primarily, to give purpose to calculating the angle between two vectors. If you go the mathematical route using dot products and all that it's just all so...fake. Second, to connect the idea of vector magnitude to representing something in a geometric shape. In the above problem, the idea is to find the magnitude and angle of side b in order to find the angle between a and b. After all that you can find the area of the triangle or anything else about it.
The issue was I came at this idea at the end of our vector discussion and it generated a lot of confusion. Primarily they weren't making the connection that vectors could actually mean something other than be arbitrary arrows. This year I was going to lead with the idea. In the future, I hope this helps me lay the groundwork for changing Pre-Cal from a random list of topics into a concrete set of applications.
I open with a discussion of computer graphics. All the favorite images they love to like and heart are usually rendered as bitmaps, hard coded pixels in a grid. I find a random .jpg from the internet, open it in Photoshop and zoom:
I do the same with a vector image that's been rendered out as a PDF. No grid of dots. What gives? To keep things simple (and because I don't know all the gritty details) we discuss that image files contain information about relatively defined coordinates which the computer uses to dynamically draw whatever lines they see. If you ever zoom in on a PDF and catch it go fuzzy before smoothing out again, this is kinda sorta what's going on, the computer/phone is redrawing everything on the fly. There was a side note that computers are trained to avoid showing you the pixel grid as much as possible through approximation, ie why bitmaps get blurry when blown up.
Create Your Own
Since we aren't computers, our ability to render and re-render vector based images is quite slow. We'll need to define our coordinates by hand and then determine the relative horizontal and vertical moves necessary to move from point to point. Here we introduce i-hat and j-hat notation.
Their task here was to define a series of points, connect the dots, and define the vectors that move them around the perimeter. In phase two, we'll talk about calculating the lengths of those lines, how to scale our shapes, and use the grid paper to approximate the area of the shapes. It all slowly builds towards the problem I wanted to tackle in the first place.
What I like about this activity is it shows the absolute nature of a vector. The same line can be defined two different ways depending on direction of travel. Would -3i + 4j have a different length than 3i - 4j ? Most of them say no, because duh, same line. Can we prove that?
Another interesting "hmm..." moment is showing them that in a shape of n + 1, the first n vectors sum to the value of the final one. Why might that be?