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 knowandmaking their thinking visible? How might aCompare & Connectdiscussion 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:

(Note: The idea for this particular Math Talk came from a session I attended at NCSM 2016 led by Crystal Lancour & Jennifer Trievel of Delaware. Thanks, y’all!)

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:

**S’s Method**

**L’s Method**

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.)

Also consider:

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**.