### By Jennifer Knudsen | March 7, 2018

I recently re-tweeted Christina Cho’s (@ccho01) tweet on her second graders’ conjecture wall. So inspiring! If second graders can conjecture, so can middle schoolers, don’t you think? We’ve found it to be true.

I thought I’d share a little bit about how we think about conjecturing.

A simple, particular conjecture could be a guess at what value of x makes 3x+35 =74 true. That would be a very particular conjecture. Students could approach it, before they know how to solve “two step equations,” by making their best guess. What if x is 10?

Would that work? Maybe x needs to be a little bigger than that. Justifications could be substituting the value for x and seeing if the equation is true. The whole process can lead to development of the standard methods for solving such equations—and why they work.

Two related general conjectures we explore in the book are about proportional relationships and their representations. We start by defining such relationships via equations and then ask students to establish properties of their graphs and tables.

Proportional relationships can be represented by equations of the form *y = kx*, where *k* is any constant number.

What are the properties of tables of a proportional relationship? Why?

What are the properties of graphs of a proportional relationship? Why?

It’s interesting to consider why, for example, you can “add” two rows in a proportional table to get a third, but not in more generalized linear ones (with non-zero intercepts)—reasoning from algebra. While often all these properties of proportional relationships are often taught as observations, they are an opportunity for argumentation.

In your own practice, we recommend keeping a balance of particular and general conjectures for students to justify.

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The image in this blog post can be found in our book: Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). *Mathematical Argumentation in Middle School—The What, Why, and How. *Thousand Oaks, CA: Corwin.