I am planning this multivariable calculus class that has taken me down this strange and circuitous rabbit hole of mathematics, (which I will not outline because zzzzzz), but here’s one thing that just popped up in this rabbit hole:
I’m having my kids think about alternative questions they can ask about a system of lines, other than “here is a system of lines… find the solution.” And I was brainstorming things they might come up with. And I came up with this:
Write a system of lines with solution (2,5).
I haven’t taught Algebra II in years, so I’m unsure if I’ve asked this before … but I kinda love it. My favorite types of questions tend to be the “backwards question” (given the conclusion, come up with the givens). And I suspect that this kind of question for kids in Algebra I or Algebra II would generate some awesome discussions and help the teacher figure out misconceptions, and draw some neat graphical connections.
(For example: I imagine the idea of having one equation fixed, and changing the second equation shows the graphical connection that the point of intersection remains constant… and vice versa.)
 I know I’ve asked students to write a system of lines with no solution before, but I’m not sure about this one!