You won’t know this, but this is my 2nd draft of this post. (I even changed the title.) I went running half way through what I was trying to write and everything that I really wanted to say came to me so clearly. I should have stopped and recorded the words that were so fluidly coming to mind, but I didn’t.
It made me wonder, however, about why the words that I was struggling to string together for the original post came to me so easily while out running. I decided that my mind was free of the clutter of trying to say the right things and in the right way–allowing me the ability get at the heart of what I wanted to communicate.
This–I decided–is the reason that balance problems are a great way for students to interact with the reasoning of equation solving. They are free of the clutter of the procedures and notation and get at what is really going on.
Yesterday, I read Michael Pershan’s post about the use of balance problems (which I consider a sort of puzzle) as part of a series on solving equations.
He is able to so clearly articulate all of the reasons that I believe that these sorts of tasks are a valuable tool in reinforcing the thinking of solving equations. (I would read his post on this because he is really good at thinking through these sorts of things.)
I used this “balance reasoning” that he referred to in order to make the connection between what students know how to do naturally and the skill that I wanted them to walk away from 8th grade being fluent in–solving equations.
8th graders have done a ton of work solving equations, but still, many are not fluent. I used these puzzles during warm-ups without expecting any sort of equation writing/solving. Over the course of time…I wanted the students to begin to ask why we were doing them when they were so easy. I used the balance problems from EDC, but I also made great use of some of the resources found here.
One resource I used from this site is Mystery Numbers. Students were able to use guess and check, along with “balance reasoning”, to find the correct value.
The “mystery” of these problems can be ramped up rather easily and foster the need for a more sophisticated strategy. I began having the students simply write the equation that could be represented by the mystery number equation. The students had been wrapped up in the fact that these were at once so easy, but now not so much. I used the mystery number problems written side by side with its algebraic representation in order to show the students how the “balance reasoning” related to the algebraic manipulation.
Students seemed to invest in this process because it was connected to something that they knew they could do. I talked with them a lot about how math is all about learning more and more sophisticated ways to represent problems–even ones that they could do with an elementary level of understanding. I believe it was the connection to a task that they found doable-that allowed them to find the algebra less intimidating.
Additionally, I used problems like this from Sarah Rubin.
I’m very interested in following the rest of the train of thought over at Problem Problems…