# Three Conjectures…

In class today, students introduced three conjecture that are on the table for our consideration over the next few weeks.

- Justice’s Conjecture: For any trip, the average speed will be greater than (or equal to) the average velocity
- Many have thought through this, are convinced. Most of them have arguments that are well reasoned plausibility, but we’re not near a proof.

- Renshell’s Conjecture: On a position vs. time graph, the slope of a line would seem to give you the velocity
- Many are beginning to articulate how the algebra connects to to graph, and some are beginning to see this as obvious

- Justin’s Conjecture: You can’t find the average speed by simply adding up the different speeds and dividing by the number of different speeds
- We have a couple of examples where this didn’t seem to work. A few have articulated the idea of a weighted average, but

On Tuesday, We’ll work specifically on, “How would you explain to someone else in a different section of our class what Renshell’s rule is, and more importantly WHY it works?”

We might also do: “Draw a position vs. time graph where average velocity is equal to average speed… draw a position vs time graph where average velocity is less than average speed… then either prove Justice wrong by drawing a position vs. time graph where average velocity is greater than average speed, or try to explain why such a graph is impossible.”

Unfortunately, we have to move on the accelerated motion… but I’d love to spend another day or two doing experiments with two-body constant velocity situations, hashing out explanations for rules that seem to work (or not), making connections among representations, and working to prove or disprove conjectures. There are those who need more practice and time on the basics, and plenty of fodder for those who ready to move on.

I’m remembering why this class (one where I teach students within the constraints of a heavy pacing guide and third party exams) is both a joy and a pain.