## Friday, August 4, 2017

### Clothesline Math - Relational Thinking

So I’ve dabbled in the idea of clothesline math (using a clothesline as an open number line) before--read a lot of blog posts about it, done a square root clothesline activity with my students, and experienced it at a couple of workshops-- but Chris Shore’s TMC session got me excited about it in a new way. Although this was partially because of his charisma, and the enthusiasm of people around me, I think ultimately it was because I had some mathematical realizations that I didn’t anticipate.

So I like that clothesline math is hands-on, gets kids out of their seats, and feels different than a lot of the math that we do in my classroom. But I want to investigate a little bit more in what it has to offer mathematically. In thinking about geometry, we started with this problem:

With benchmarks of 0 and 180, we are able to place x clothespinned to a and b & c clothespinned to each other. A & x were the same distance from 0 as b & c were from 180, and we moved x, there we a line of symmetry at 90 degrees. This was cool, but the realization that b & c needed to be x units away from 180 degrees wasn’t mind-blowing for me. We are essentially taking the equation x + b = 180 and then turning it into b = 180 – x. Certainly an important understanding for kids, but something that could be figured out algebraically on paper and have almost the same weight as on the clothesline.

But then we got to this clothesline prompt:

At first, I thought ok, fine. This was going to be basically the same. 2x + 9 and 3x – 6 would be clothespinned together, and 7a + 1 as their reflection over the “line” at 90 degrees. But then Chris took out 2x and 3x to be placed after 2x + 9 and 3x – 6, and that’s when I started to get excited.

Let’s say that kids are given the equation 2x + 9 = 3x – 6 to solve. Here’s one possibility of what I might expect from my eighth grade students (they might deal with the variables first, or choose to eliminate different terms, but something along these lines).

At best, this demonstrates an understanding of how to isolate a variable. In my class we talk about what “moves” we can make to simplify an equation—in this case we’re looking for terms that we can “turn into 0’s” using the additive inverse and coefficients we can “scale to 1’s” using the multiplicative inverse.

And yet, with essentially the same prompt on the clothesline (we have 2x + 9 equivalent to 3x – 6), a whole new world of thinking opens up. Here’s a representation on paper of the thinking that my group did in placing 3x, 2x, and then x onto the clothesline.

In this progression of thought we had to do a lot of relational thinking, which I would argue is absent in the solving of the equation above. 2x + 9 and 3x – 6 is somewhere between 0 and 180 degrees—more specifically, it’s definitely between 2x and 3x. And in fact, the value of 2x + 9 and 3x – 6 is 9 more than 2x and 6 less than 3x. And then this is the amazing part. That distance is 15 and it’s also x. I want to think more about how to prompt this realization (either through a series of clothesline tasks that build this thinking before we get to something this complicated, or in the moment with this take), because this type of relational thinking isn’t something I’ve given my students a lot of experience with. And then once we know x is 15 degrees, then we can find the value of 2x, 3x, and the two original expressions.

Placing these expressions and solving on the open number line is dependent on the idea that all expressions we deal with represent a value (the location on the number line) that can be seen in relation to other expressions (the distances on the number line). This is something that we spend a lot of time on with numerical values in elementary, but that I haven’t leveraged very much with my students when they start to deal with algebraic expressions and equations. So it’s for this reason that I’m sold on doing some algebraic work with expressions and equations on the clothesline this year.

Some other clothesline math tips from Chris:
• Have benchmarks on another color to establish scale, but only put the number of benchmarks necessary to communicate your idea
• Have blank cards of another color for any other values or expressions that might need to be added
• First place the cards in the proper order, then proportionally space
• Students can give a thumb up or down for correct placement, then point the direction it should be moved, and clap when they think the card is in the correct place
• In order to have whole class engagement, students participate at their desks on whiteboards while 1 group is at the board. Then record final answer on paper when done with discussion, deductions, and decisions
• If kids estimate the value of a variable, can put an equivalent (approximate) value on with a clothespin
• Equivalency is better on the single line using a clothespin to attach the equivalent expressions. Save a double number line for rates