2 Additions to my Conceptual Understanding

I just graphed these inequalities for the first time. I was filling out the end of year survey for Math for America and the last section asked us to identify the common misunderstandings that might arise and how we would address these misunderstandings.
I've been thinking a lot this week about how to teach for conceptual understanding, how to get kids to use skills as their inherent pattern seeking mechanism activates. How to create curricula that gets them generalizing useful and true patterns (rather than, for example, that every function is linear), and how to offer accessible depth in mathematical thinking.
So I was excited to play with these inequalities, partly because they were new to me, partly because they demand conceptual understanding and resist procedural memory. Rather than go on, I leave you to play with them yourself if you're not already familiar. I'm going to use them with my 10th graders next week!

Then I discovered that jd2718 came up with this activity back in March, and there's a nice discussion of them in the comments on that post. Check it out! Thank you jd2718 for being so creative, brilliant, infectious!

Also, I did some catch up last night on Ben's blog and was really inspired by his discussion of revising how we teach and present negative numbers. He writes beautifully and at length about this, and I seriously encourage you to take 10 minutes and read his post. He's been reading some awesome primary texts and the history they tell has convinced him that we should teach negatives a bit differently. First, we can we ask the question (perhaps often) do negative numbers even exist? Then let's introduce negative numbers first as solutions to things, like 5-7. Let arithmetic necessitate these new creatures. Once we're comfy, we can start using those things as objects with which to do arithmetic, as solutions to equations, etc. Last we can use them as coefficients (and exponents?!). I love this!