Anne Truitt Zelenka

Right before winter break my Calc BC class did a short unit on parametric and polar equations. They did fine on the parametric stuff but thinking in polar coordinates doesn’t come naturally, and they stumbled on sketching polar curves and finding area enclosed by them. So I am reteaching it as we move into vector functions.

Yesterday we played Polar Pictionary, a game I designed to get them more fluent in polar coordinates. This would work for a pre-calculus class too.

Here’s how you do it:

1. Start by giving an example of a Pictionary idea “drawn” with polar equations. I “drew” a butterfly with four very simple equations: a four-leafed rose, an ellipse with major axis along the y-axis, and two straight lines for antennae (the butterfly above has some wing designs added that the students suggested). Ask students to sketch the curves and guess what they represent.
2. Each student gets a card telling them what they need to “draw” with polar coordinate equations.
3. They have ten minutes to come up with a set of equations and, if necessary, limits on the domains of those equations, that draw a picture of the item on their card.
4. Each student writes his or her equations on the white board and the other students try to guess, either by thinking in their mind of what it draws or by sketching the curves.
5. A student gets a point for guessing correctly and the student who made the equations gets a point too.

I let the students use calculators and Geometer’s Sketchpad when they were developing their equations but during the guessing phase they had to do everything by hand.

This was a great activity. Students asked questions that hadn’t come up before: “how do I draw a line in polar coordinates?” “how do I shift a polar graph?” “how do I make two short curves that look like antennae?” etc. They came up with ideas for representing their items that were really creative and very different from what I would have done. I’d like to do this every year and keep a gallery of drawings.

Three out of my five Pre-Algebra Seminar kids passed their main math class this trimester. That means two failed. Not a great record. This trimester I hope to do better. I feel like I have a much better idea what to do and yet I feel scared. What if I just suck as a teacher?

When I start to think things like that, I remind myself that I should take a growth mindset with regard to my teaching, just like I try to instill a growth mindset into my students. It doesn’t matter if you get an F this trimester or if you were the worst, most boring, and least energetic teacher ever. Figure out what you need to do to improve. Check what other more proficient students/teachers do. Reflect on your results. Get better.

So here are some reflections on what seemed to work this past trimester with my algebra support class. Did they really work? I dunno — it’s not feasible to run a controlled experiment on my five students. But I know what engaged them and what seemed to promote improvement even if almost infinitesimal. And I know what was boring and disengaging to all of us.

What worked:

• Linear vs. exponential growth sticker activity — I had the students use stickers to explore the difference between linear growth and exponential during their unit on linear functions. Okay, they didn’t need to know about exponential growth; it’s not on the standards. But I want them to start realizing that the world isn’t made of nice linear processes even though people tend to think linearly. Population growth, compound interest, infectious disease — all these things require an understanding of exponential growth. Having to peel off 64 stickers and put them on a page vs. say, 8 stickers,  demonstrates how exponential growth gets crazy fast. And the kids like having their hands busy. I didn’t have to do anything to motivate them to put on the stickers. They just did it with zero complaints.
• Make-your-own tangrams — they’re starting geometry today and I wanted to get them in a geometric mood. So they did a paper folding/cutting activity in which they made their own tangram set. Then they explored the Pythagorean theorem with it. Again, it was good because it got their minds and hands engaged at the same time. My student who did worst on the trimester final made his tangram set the fastest and with the most accuracy — does he have some hidden geometro-spatial abilities I need to tap into? After we put out the squares to show the squared sides and hypotenuses of a right triangle one of the students yelled out, “I get it!” and her eyes were lit up like I’ve never seen.
• White board drills with gate problems — right before the final I drilled the kids hard on topics I knew would give them trouble on the exam. I told them that once they showed me they knew how to do a particular kind of problem they could work on their homework. One student pressed me for extra problems to make sure he had mastered them. It was like being in the calculus class — he genuinely wanted to learn. That’s where I want to get with these kids. I’m not starting from scratch: they come to class; they do their work; they want to succeed. Yet sometimes they need help finding that inner spark that drives them to really want to understand not just pass the class.
• M&M proportions — I don’t want to buy the kids off with sugary treats, but we did use M&Ms to calculate percentages and proportions and it’s an activity I’d definitely do again. They were so motivated to eat the candy that they jammed through the percentage calculations.
• Multiplication mad minutes — Most of us need help on our 12 times tables and my algebra students are no different. I saw them get steadily better at multiplication through the semester as we did weekly mad minutes focused on 8s, 9s, and 12s time tables. Next step I think is to do division because they could stand to improve their speed on that.

What didn’t work? A lot of things: letting them have unstructured time to work on homework, allowing disruptive students to derail the class, reteaching things in boring ways instead of hooking them up with new ways of experiencing math, moving slowly instead of quickly, generally following the last teacher’s recommendations instead of finding my own way with this group.