A couple years ago I had an idea for a display piece for vectors. Take some straws, create some vectors in each of the four quadrants, cut the straws to length, and demonstrate the right triangle nature of a vector. It was ok. But there was a lot of struggle. The wrong kind of struggle. The "I'm going to do this because you said so but I may not be learning anything" kind of struggle. It happened two years in a row, so I figured it was a time for a change.
Since I shifted my focus on vectors to a bigger, broader category like Images, I thought that would be the starting point for this display piece. Creating images has yielded all the aspects you'd want to cover with vectors: operations, scaling, magnitude, and direction. And edge cases like vectors with 0i and 0j come up naturally, I don't have to force it or attempt to single it out as one of those capital letter Named Issues.
given: colored paper, markers, rulers, full sheet of grid paper
- construct a 7-sided (at a minimum) shape
- identify the vectors that define the sides of the shape
- calculate the length of each vector
- scale the shape by a factor of 2 or 3
- identify the vectors that identify the sides of the scaled shape
- prove that the N+1 side of the shape is equivalent to the first N vectors added together
- calculate the "true" angles for each of the sides
- compare the side lengths of your original shape to the scaled shape
- compare the angles of your original shape to the scaled shape
Lots of good work here. It can be a little tedious once they start digging into the calculations, but the creativity on display here was great.
The great discussion point came when it was time to calculate angles. At first some confusion, do I calculate my "true" angle (basically a +180 or +360 correction if you're pointed at the II, III, or IV quadrant) based on the physical location of the vector or the orientation of that vector?
Many kids saw through the simplistic nature of the two questions. Of course the lengths will double if I scaled the whole thing by 2, duh! Why would the angles change? Everything's pointed the same way, duh!
I enjoyed this version of the project a lot better. My kids continue to amaze me with how well they handle tasks with some structure but not a script. More to come on this.