# Great Problems

Question today from Five Triangles on ratios, areas and perimeter.

1. Really fun problem. You should try it, I won’t give away the answer.

2. Spurred some great thinking on exponents. Let me explain.

I taught simplifying exponential expressions earlier this year. Questions like this one

Progressing up into more complex expressions with parentheses, several variables, and negative exponents. Leads to some fun problem solving, but beyond puzzley questions the unit was lacking in motivation.

For some reason, that Five Triangles question made me think of an awesome application of exponential expressions that my students struggled with:

One side length of a cube has a length 2x. What is its volume?

For some reason, my students continually distributed exponents to variables and not to constants. I still haven’t figured out where this particular misconception started, but exponential expressions always seemed like a bunch of magic to many of my students, and it didn’t trouble them all too much. But this is a concrete application that they can check more intuitively, and leads to a ton of interesting questions that not only reinforce exponential expressions, but push students to reason algebraically and gain comfort with variables, which has been another one of my struggles this year.

Excited to teach a lesson around these ideas! I’m thinking Thursday.