Walk the Line

I keep a close watch on this number line 

I keep my eyes wide open for the sign 

Making sense, connections all the time 

For number sense, I walk the line

Apologies to Johnny Cash.

At Math In Action two weeks ago, I presented on number talks in the middle grades. (Here's the handout/resources, including a link to the slides.) I was quoting Pam Harris, then the keynote speaker was quoting Pam, so I made a comment on Twitter, badda bing, she turned out to be coming to my neighborhood the next week. Muskegon Regional Math Science Center let me crash, thanks Kristin Frang, so I got to crash for 1.5/2 days. 

Pam was presenting on secondary number strings (thread 1 and thread 2 for my notes). Mini-review: I wouldn't hesitate to bring Pam in to work with teachers. Great energy, hilarious, solid ideas presented in a way that invites teacher access, and great modeling of instruction. All building on a great central message about giving learners an opportunity to mathematize.

I'm teaching Introduction to Mathematics this semester, a gen ed math class with a lot of freedom. I'm using Anna Weltman's This Is Not a Math Book as a text, mostly as a resource for learners and introduction activities.  In my head it's about redeeming mathematics for this successful students who have (mostly) learned to dislike mathematics. While we're mostly doing math and art, emphasizing problem posing and solving. But we take some time to redeem arithmetic, and we need to do algebra before we do patterning.

So, Pam's problem strings (modified because I can't help), followed by some clothesline math... rare day when I didn't have a way to take pictures. Sorry! The theme of the lesson is what else do we know?, which is one of my main understandings of math.

I drew a line, put on a hashmark, and labeled above it x, and below it 3. What else do we know? Some discussion about putting on a scale, vs knowing left or right. Finally someone shares -x is -3, and says it would go somewhere to the left. The someone said that means we know zero, which I encouraged as a good mathematician question.

First I did Pam's coordinates problem string.
(-2,5) show in graph, table, function notation
(2,-3)
(3,-9)
(1, __) which first brought up approximating with the line, then the idea of slope.
( __, 0) which brought a lot of people to a halt as they tried to remember an algorithm, then a remembered algorithm for solving equations. I made a bit of a joke how I was not interested in a memorized method, sorry, but only making sense.

Next, x is -2. Where is zero, left or right? What is x? Some discussion but pretty quick.

x-4 is 6. Is x left or right? What is x? How do you know? Quick discussion.

x+4 is -6. Heated discussion. -10 and -2 mentioned frequently. More and less start getting used more than left an d right. Interesting symmetry comparison between the last and this.

Then I introduced the clothesline, strung across the front of the room, with just an x in the middle. What else do we know? I had cards to ask them about. 0, -x, x+3, 5. We know where zero is, then dissuaded. What if it's negative? We know where -x is, then dissuaded. Then x+3, that has to be to the right. We don't know how far, but definitely right. What else? x-3 we would know. Excellent! Finally I put 0 up (left of x-3 by less than the distance to x) and ask what x could be? 

Then I gave them cards. Make some cards, figure out the order you're going to reveal them and the questions you'll ask. Two really interesting situations came up. 

One group had a couple of variable expressions, and then x+200. Really nicely subverting the sense of scale, and a great numeracy discussion of what x could be then. They wanted to just toss off a big number, but other learners argued for more precision.

Another group introduced a new variable, g, and then blew our minds with g^2. Has to be to the right, because squares are bigger. But what if it's negative? It's still to the right? Always? Then a lot of discussion if placement of it was setting the scale. Not until 0 is placed. It was amazing.

So that's my story. Thanks to Pam Harris, Chris Shore and these great learners.