How might we use a Math Talk to help students sense-make about solving equations? How might this same Math Talk help students work on showing what they know and making their thinking visible? How might a Compare & Connect discussion routine overlaid on this Math Talk help deepen our understanding of strategies for solving equations? Let’s explore these ideas further…!
Here is one of my favourite Math Talks:
Take a minute and think about how you would respond to this prompt. Can you solve this problem without first “translating” it into algebraic notation and solving via algebraic manipulation? Rather, can you solve this problem visually?
Also, can you (as Jill Gough and Jennifer Wilson encourage us to do) “show your work so that a reader understands without having to ask [you] a question”? (See more on Jill & Jennifer’s #ShowYourWork progressions here.)
How might we make our thinking visible for how we solved this problem?
Here is how S, a recent conference session participant, made her thinking visible:
She explained her thinking aloud as:
I removed 4 horses from each side. Then I was left with the fact that 2 elephants were as strong as 8 horses. I divided these 8 horses into two equal groups showing me that each elephant is as strong as 4 horses.
When asked “Did anyone else solve the problem this way?” the majority of the participants indicated that they solved this problem the same way as S had.
L, however, had a different approach. Here is how she made her thinking visible:
Can you understand what L did without asking her any questions? How is her approach similar to and different from S’s approach?
L explained her thinking to the group in this way:
I started by dividing the problem in half. I knew that if 2 elephants and 4 horses were as strong as 12 horses, then 1 elephant and 2 horses must be as strong as 6 horses. Knowing that 1 elephant and 2 horses were as strong as 6 horses, I removed 2 horses from both sides of this situation, leaving me with the fact that 1 elephant is as strong as 4 horses.
How did L look for and make use of structure? And, does her method work all the time?
We quickly considered what the corresponding algebraic manipulation might have looked like for S and L’s approaches. We decided to represent the strength of an elephant with the variable e and the strength of a horse with the numerical value of 1. Here is what we came up with for the corresponding algebraic methods for these two visual solution pathways:
Do both of these methods work every time? When might S’s method be advantageous? When might L’s method be advantageous?
Why might it be important for students to Compare & Connect these two methods in a Math Talk? (See Elham Kazemi & Allison Hintz’s book Intentional Talk: How to Structure and Lead Productive Mathematical Discussions for more information on the Compare & Connect discussion structure.)
How does L’s method push back on well-intentioned but over-structured “procedures” we might teach our students for solving equations?
More on this idea in an upcoming blog post on Sense-Making for Solving Equations.