# Parametrics and Fractions

I had a pretty awesome multivariable calculus class. We didn’t do anything out-of-the-ordinary, but I let go of my “planned” curriculum and tried to do some more freestyling stuff. Notably, I asked kids “what would they need to (uniquely) define a line in 3-space” and from there, we built up the idea of needing parametrics. What was awesome was that kids developed the idea, but didn’t even know that what they were doing was parametrics. In fact, it was only until I switched a variable they had defined (“a”) to a new letter (“t”) did I get the “ooooooh!”

I need to remember to allow the class to lead where we go more often, while doing a lot of maneuvering so I’m making sure they’re exploring some of the dead ends (“we need the slope of a line in 3-space” “wait, what would a slope mean in 3-space?”) while still making sure they’re also traveling down some of the more fruitful paths.

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After school today, we had our first TaLL Tuesday meeting. That’s an in-house professional development program my school has developed. I joined a group on looking vertically at lower and middle school math. We had a lot of fun playing with ideas today. My second favorite part was playing “four 4s.” My first favorite part was thinking how I could best get 5th grade kids to understand how to add 1/2 + 4/5, without telling them how to and by exploiting what they already know.

It was fun and a pretty engrossing intellectual challenge to think about all the pitfalls one could have in leading that lesson. One huge thing that dawned on me was that the idea of “one whole” is incredibly tricky. And that, along with equivalent fractions, are the two huge conceptual parts that you need to build up.