This was the open question I posed to my Advanced Algebra 2 students today:
Write a system of inequalities that has (4, 3) as a part of its solution. It should have 2 or 3 inequalities. (I had thought about leaving the last sentence off, but I was trying not to overwhelm them.)
When I asked for their answers, I got crickets again. (Recall, I had tried this with them a couple of weeks ago and pretty much got no response from them.) So, I asked them for one inequality that would work. Crickets. More crickets. That, and a "I don't know how to work backward." After a few moments, someone gave me one inequality. So we graphed it and checked (4, 3) and it didn't work - (4, 3) was on the line. I asked him how to change it and he thought about it and came back with a different inequality. This time (4, 3) wasn't on the line, but he had the wrong inequality symbol. We flipped the inequality and had a working inequality. Yay! How about a second one? Crickets.... but for a shorter time. Same student, new inequality. Worked - success!
Can we come up with 2 different ones? Just try... Different student, new inequality. Got a working one and the student came up with a second one.
By the time we were done, we came up with 5 systems (look at the first 3 pages of the pdf below - first 5 slides). The first time we did an open question, I primarily had 3 students contributing. This time I had 6 or so contributing - 4 of them who had not last time. I'll take the improvement. Maybe next time, I'll have more.
Tomorrow I'm trying this question with Algebra 2 (similar to what I did with my Advanced Algebra 2's a couple of weeks ago):
I know that (2, 3) is the solution to a system of equations. Find two equations (in x and y) that have (2, 3) as their solution. How do you know that (2, 3) is the solution?
(I think that's how I phrased it.