A Polar Prom
This has been a big week for my AP Calculus seniors as they prepared for their senior prom, frantically crammed in any last information for their AP classes, planned their senior skip day, submitted their final college decisions to guidance, and began creating plans for their graduation and graduation parties.
Even though AP exams are starting a week later than normal due to how the calendar falls this year, it doesn’t feel like we’re getting any extra time. For the first time possibly ever, the senior prom was scheduled for the Thursday before the AP exams begin. Prom at my school has always been on a Thursday; several different rationales exist as to why our school’s prom is on a Thursday rather than on a weekend. There was a long-standing rule when I first began teaching at my school that seniors needed to come to school the day after prom in order to walk the stage at graduation. Most came in just before 10:30 so it counted as a day while still allowing them to sleep in after staying up late with friends. Post-Covid, that rule has disappeared and it has become somewhat of a senior skip day. Last year’s seniors made the day after prom their Senior Skip Day. This year’s seniors decided that since the day after prom was already a skippable day, it didn’t really count as a proper Senior Skip Day. So they are making this upcoming Monday, the first day of AP testing, their Senior Skip Day. Since seniors left at 10:30 on Thursday to get ready for prom, they had somewhat of a 5-day weekend.
All of this has made reviewing for the AP exam and even just finishing the curriculum somewhat of a challenge. In one of my AP classes, we fully finished polar functions this week and had time to practice a polar FRQ. In my other AP class, we still need another day or two to practice polar area and slope - days which we won’t have with students missing school next week due to AP exams - aghhh! I’m planning on having an after-school session next Friday (the last school day before the AP Calculus exam) to go through three polar FRQs as well as a couple review FRQ types that will most likely appear on the exam.
Based on FRQ trends (click HERE for a Word document that includes the above list), I have a strong feeling that there will be a polar FRQ this year. There hasn’t been one since 2019; instead, there has been only parametric for the past three years in a row. My seniors last year were overjoyed when they opened their FRQ booklet and saw that FRQ #2 was parametric rather than polar. I’ve been telling my seniors this year my polar predictions, and they’ve been telling me to stop manifesting a polar FRQ . We’ll know in a week if I’m right!
I love polar graphs and the idea of the polar coordinate system. I really wish I had more time to delve deeper in the curriculum with my students rather than feeling like we’re rushing through introducing the polar coordinates, graphing polar curves, finding polar slope, and calculating polar area. Next year will be an entirely new experience since many of my BC students will have taken the new AP Pre-Calculus course already. The scope and sequence of this AP class includes polar coordinates, polar graphing, and polar slope, so they’ll already have a solid introduction to the concept.
Before getting to polar area, I introduce students to the polar coordinate system since most of them have never seen it before. Once they can confidently plot points with both positive and negative angles and radii, we practice graphing polar curves. To do this, I have them graph the polar function on the rectangular plane like they did in Pre-Calculus. We number the points in they order they appear, and then plot these points on a polar grid, again numbering the points in the correct sequence to help connect them.
Next, students watch my two videos on polar area (VIDEO 1 and VIDEO 2) and take all notes so they have background information on where the polar area integral comes from and how to find the bounds. Finding the bounds is the most important part, so when we practice in class, I have them try to come up with at least two integral expressions for each area problem. The visualization is the most important part with polar area, so here are a few things I used this year:
VISUAL 1: A simple Geogebra applet with a slider to show how polar area is found by adding up infinitely many sectors (as opposed to Riemann rectangles)
VIDUAL 2: A Desmos activity that I edited so that we can refer to it as we go through various examples. I have students go through the activity to generate the visual and then draw and shade the curve and create the integral expression on individual whiteboards in their groups. I pace the activity so we all stay together and make sure every group has the correct answer on their graphing calculator before moving to the next problem. They told me that this Desmos really helped them see how polar area is created by sweeping out area radically, NOT by shading up and down like in Riemann sum area approximations.
VISUAL 3: The most introductory that I show first before they even watch my instructional videos on polar area. I show them the picture of the pie and the rectangular cookie sheet and ask how they would cut them up. Next year my goal is to actually bring in a pie and a cookie sheet and cut them up the “wrong way” to really show them how fundamentally wrong it is to imagine cutting up a pie into rectangles - how it really makes sense to start at theta = 0 (the ‘x-axis’) and go around cutting slices, just like we go around counter-clockwise and add up little sectors to get an area. Just showing them the picture of the pies cut the ‘wrong’ way generated enough of a reaction, so I can’t wait to see how much it helps when I bring in actual pies!
After they had sufficient practice finding polar area bounded by one curve and between two curves, I had them try some of the polar FRQs from the past decade as well as THIS circuit written and shared publicly by Mark Kiraly. The circuit begins on page 5 of the PDF - the first few pages are a helpful overview of polar conversion equations and how to derive the polar area formula.
This is the first year I’ve used the circuit and I really like it. Even though it’s only 8 questions, it covers slope, basic polar area, and area between two polar curves. I will definitely keep this as part of my polar curriculum for future years!