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.

## 4 comments:

I was really struck by the fact that one student said, "I don't know how to work backwards." It struck me because that student was thinking about how to think!

It seems like your students can totally do these open-ended problems but struggle to organize their thoughts and feel like they have something to try. Does that seem like a fair characterization?

I wonder if, after you've solved one together, if it would help them to analyze how they thought. You could interview the student who gave the first answer: "what did you notice about the problem? What did you try first? Did you ever get stuck?" as well as asking those who didn't answer, "What did you want to try first? Did you think about what she thought? What made the problem hard?"

I could even imagine putting up a problem and saying, "I don't want us to solve this today. I don't care if we get to the right answer. I just want to think about analyzing it. What does it seem to be about? What skills might we use? What strategies could we try?" Try to make as long a list as possible! Later, if you do solve it, you could see how many of your initial ideas made sense!

Another thought is to think about the "work backwards" strategy -- my favorite question for work backwards is, "if this happened, what's one thing that must have happened before?" e.g. if (4, 3) is a solution, what's one think that must have been true about the problem? [For example, it's between the two lines].

Thanks for making your classroom experiences public! I hope this was the kind of thinking you were hoping to spark the rest of us into...

PS -- I didn't mean to be anonymous! I'm really Max, of @maxmathforum on Twitter!

Well said Mr Max Ray you really explained it well detailed.

Sounds like a promising result to me. Your students are only going to get better at it. What about if as a follow up you had asked small groups of kids to complete the tasks, perhaps with different points given as the solution requirement. Those who had contributed to the discussion could have formed a group, and been asked to do a larger system, perhaps with 4 or more inequalities involved.

I hope you feel pleased with your result, I think the increased participation, plus the successful outcome are good signs of things to come with this.

@LizDk

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