In a recent post, I discussed story tables, mostly in the context of teaching about functions in high school, and as a springboard to discuss priorities in tool selection. I first heard about story tables from Shira Helft, who I believe was the first to appreciate the power of that representation. Today, a conversation with Shira.
Story Tables and Basic Equation Solving
by Shira Helft (@MsHelft) and Henri Picciotto (@hpicciotto)
Shira: Henri, thank you so much for inviting me to join you in this conversation. A lot of the thinking I’ve done already with story tables has been in a high school context. In today’s post, let’s consider what story tables might look like in middle school. More specifically, how might they be helpful as a way to understand solving equations and other pre-algebra skills?
Henri: As I explained in this post, I do not believe in “this is how you solve one-step equations, memorize these cases; this is how to solve two-step equations, etc.” I would rather have students learn some big concepts, and be in charge of applying those to problems of various levels of complexity. In other words, less memorizing, more thinking.
One worksheet that is useful early on is this one, which I’ve used to introduce the cover-up method, a good follow-up to the more important and more basic idea of solving by trial and error. You should take a look at it before reading on. Notice that I get to multiple-step equations very quickly. Also notice that in all of the equations, the unknown appears exactly once, so clearly things can and will get more complicated! Still, I think that set can be used to lead students to important insights.
Shira: I used the cover up method with my Algebra 1 students for the first time this year! A friend of mine, Anna Blinstein (@ablinstein), used those problems with her middle school students, and commented (on Twitter):
Really love this progression of problems from @hpicciotto for equation solving. They’ve resulted in conversations with students focused on strategy and key differences that students should notice when they solve equations, not mindless or memorized approaches.
This made us wonder – how might a story table illuminate these problems in a different way? Let’s look at the first equation, and set up a story table for it: Good math stories (or at least these ones) begin with a variable and follow the order of operations until you have built the entire story [3(x – 10)]. In this story, we know that the output should be 15.
Henri: Building on our past experience of trial and error, we decide to try 0, then 1 as the value for x:
Shira: Hmm…we are very far from the output we want but they do seem to be getting closer to 15. Let’s try 10 for the x and see what happens.
Henri: Better, but not great. But wait, what if we put 15 in the last column, where we want it, and work backwards from there?
Shira: Yes! The solution is x = 15! Many folks I have encountered have some way of thinking about “working backwards.” What I love about story tables is that they can both be a space for trial and error OR working backwards. They are also a helpful way to think of lots of other cool algebraic things, but we will leave that for another time. See if you can use story tables to help solve more equations! You can use trial and error in the table, or work backwards if you prefer.
Henri: Almost certainly, #3 will be problematic: 18/x + 12 = 15. How do we represent this situation in a story table, with x in the denominator? If no student has a good plan, I would suggest going from x to 18/x in two steps:
Shira: Ooh! The solution is x = 6. The reciprocal here reminds me about how story tables help me dig into all of the nuances in algebraic symbols with my students. I can imagine this being a way to highlight the difference between 18/x and x/18.
Henri: Story tables will work well in problems where x appears only once. Students can set up their own tables as needed. But that is not the end of the road. After doing this for a while, the time is ripe for a discussion of the cover-up method, which is based on the same ideas, but is a bit less laborious (see page 1 of the worksheet.) Understanding “cover-up” makes it possible to generalize the key idea: each step should undo what was done, in reverse order. We work from the given expression to “x by itself” as students put it. This, in turn, is an insight which will be very useful when working on more complicated equations.
Shira: I love that with story tables there is very little memorization beyond order of operations and knowing what the symbols mean. In building the tables, we are also reviewing those skills, which I have found to be some of the biggest gatekeepers for students being successful at high school math. Henri, thanks for thinking with me about this.
Henri: Thanks for introducing me to story tables! Dear readers, if you try this approach, or find other ways to use story tables, let us know how it turns out!