# Day 128: Revisiting Linear Motion

How do you connect “old” ideas (that is, previous learning) with “new” or unfamiliar situations?  I’m always trying to build new dendrite connections between the old and the new.  Linear motion via parametrics is just one example.  As our opening activity, I asked students this question:  Write the vector equation and the parametric equations of the line through (-7, 4) and perpendicular to 2x + 3y = 12.  Draw a diagram of the situation.

I purposefully left out the t-interval to force a discussion of why it is important and how do we write parametric equations when there is no clear one-unit increment of time.  And what a rich discussion we had!  I found out what “stuck” from yesterday’s Geogebra exploration, drew in vectors (yay for review!) and really looked deeply at the variety of parametric equations that are correct even if they look different.

I was soooo thrilled when two students shared different x(t) equations (green vs. red).  Immediately, students took sides (Creating a fight, as Dan Meyer would say).

As a class we compared and contrasted what was written: same starting x-coordinate, different rates of change.  This led to a discussion of how one was 2 and the other was 7….it came down to the arbitrarily chosen “end” point.  One student used the slope and found the next point along the line; the other student used the y-intercept.  Cool, huh?!  This  drove us to explore whether both could be correct and why.  Can we use both to determine a specific location, and how are they related.  You can see the results of our queries.

How have you facilitated a deep discussion based on student responses?  And what turned the discussion into a deep one?