This time, I decided to first make sure that students were able to see whether a line passes through two points. I designed a small matching activity: given some cards with equations on them, and some cards with pairs of points on them, match the equations with the pairs of points. After they matched everything that it was possible to match, some odd and some empty cards would remain. I was hoping students would use the understanding they developed/formalized during the matching activity to come up with suitable matches for the odd cards.
You can find the cards here.
Easy as pie? No. It turns out that not one of the students in my class (11th grade) were able to match the equations with the points. They had simply no idea of how the x and y in the equation related to the x- and y-coordinates of the points. What the hell have they been learning for four years?
Some approaches that students tried were:
- Getting the gradient by using two points, then comparing this gradient to the one in the equation. Fine, as long as there is just one equation with that gradient.
- Graphing the equations ("but we don't remember how to graph lines from equations") and see if they pass through the pair of points. Fine, if they understood how to graph the lines and if the scale of the graph was appropriate.
- Making a table of values, to see if the pair of points would come up as a pair of values in the table. Fine, as long as the points have integer coordinates and the person has a lot of time and patience.