It's summer! I can't wait to share all the awesome things I've been up to at PCMI, but first I'm going back to my final unit in PreCalculus which I never got a chance to share. Last year we did an exploration of polar functions but ran out of time to look into parametrics. This year when I asked the calculus teacher what her preference was she said to focus on parametric instead. I chatted with several people on twitter, someone (@dandersod I believe) showed me how to graph parametrics on Desmos and my colleague* shared her awesome materials with me. Put it all together and I got this:
Kids finish a test and the instructions on the board say:
1. Pick up the assignment papers
2. Sign out an iPad
3. Work silently (other people are testing!)
On the desks where I've spread out the assignment papers I write Introduction above the first page (in dry erase marker because I have dry erase desks) and Choose One above the remaining three pages. Most students finish the introduction (about half an hour) so I assign as homework completing that page and making a first attempt at the other assignment. The following class we discuss the intro, why parametrics exist and then they spend the rest of the period working on the context they chose. I like projects where there are similar options because students can still have conversations (they all have to graph and calculate) but each of them has to do their own work.
It was important to begin graphing by hand so students had an understanding of how parametrics work. Some students were concerned that the t value wasn't showing up on the graph and tried to include it in some rather creative ways. I've edited the instructions slightly (should've thought to post the original to ask for feedback on the adjustments...) so hopefully that will be a less prevalent error. Other students picked strange values for the second set of equations, ah radians. Two thoughts: 1) I'm glad I was able to incorporate a trig function since we hadn't used them much since first semester 2) I love graphing utilities - I was able to say, "Okay, you're not really sure what this graph was supposed to look like, that's fine. Graph it on Desmos and see what happens!" My box of helpful hints on how to type things into Desmos wasn't as visible in the first version, many kids skipped those steps. They were also unclear on what to type exactly as written and what to substitute with other information. Turns out (-4+3t, 1+2t) graphs just as well as f(t)=-4+3t, g(t)=1+2t and (f(t), g(t)), but I like the way we found to use Desmos as it shows that each value of t gives an (x,y) coordinate a bit clearer.
These activities are designed to be done on Desmos as well. Some students didn't appreciate when I wouldn't help them with technical issues until they got out the intro sheet and put it side by side with their iPad. But, they were able to correct their own issues that way so it was worth the eye rolling. They also struggled with graphing the obstacle/hoop. In response to those questions I asked them, "Where is the obstacle/hoop?" And continued asking variations on that question until they told me "At x=__" At which point I responded, "So type x=__ on the next line." Then I pointed to the next line of instructions (how to restrict the range) and walked away.
These context based questions require students to continuously switch among equations, graph and description. They have to know what t represents and what it means to land as well as solve quadratics and estimate values on a graph.
A student asked me if the equation took gravity into account. What an awesome question! I was proud that I remembered my physics to point out the -16t^2 (this is feet based physics, I remember 9.8 for meters even more clearly).
In my opinion this assignment shows why parametrics are useful - you can know horizontal distance, vertical distance and time using one set of equations. I failed to successfully convince my students that this was amazing. They obediently wrote that down and that the parameter allows them to restrict the function. But neither of these facts were impressive to them. Thoughts on how to convince students that parametrics are useful and different?
*Colleague O'Malley - I've yet to convince her to jump into our awesome online math community but she does recognize its power and occasionally asks me to ask twitter questions on her behalf. She found the equations and contexts in McDougal Littell Algebra 2 and then wrote up projects for TI. I only had to modify a bit so we could use Desmos. She very generously allowed me to share with all of you!