3D Vector Magnitude

After watching my current IB year 2 students struggle with vectors (I took over the class this year) and it was the 1's turn to push slog death march through vectors I was determined to slow down and make sure they understood the basics... Or at least give them as much time as possible to create some conceptual understanding (certainly not the same thing).

We started vectors by talking about different notations and their meanings. We talked about the magnitude of a 2D vector. They could see the clear connection with Pythagorean Theorem, a few even suggested its use, but the 3D formula was not clear to them. Not at all. Students suggested we could use a cube root or cube the coordinates. Both potentially valid extensions of the pattern for sure. I ended class by giving them the formula with no proof no justification - time was running out and I had problems I wanted them to finish. Oops.

Realizing the obvious error of my ways, I started the next class with a model of a 3D vector (I used a wire coat-hanger as the vector and large graph paper to form the X-Y plane). My model was just to show them what I wanted them to build. Their task was to use a meter stick as their vector - which required some engineering - and find the coordinates of the vector in 3D. A few students suggested aligning the vector with a coordinate axis... I nixed that idea.

Once they had the coordinates I asked them to prove the formula given in the last class or better yet  to ignore the formula and use only the coordinates to find the length of the vector - which they already knew from the meter stick. 10-15 minutes later folks had drawn right triangles and created a solution!

Once everyone was finished, I pulled the whole class over to one of the more neatly created models so I could point and formalize the ideas.Towards the end several students commented "oh that makes sense," or "now I get it - that formula does make sense." Voila! A simple but effective way to spend 40 minutes of class.