As seen previously, I used polar equations of conics as an opportunity to talk about the solar system. Now it's time to see if I get the students to take something home about the idea, rather than just point to their pretty poster.
A few days after the completion of their solar systems, it was time for a test. I included a section the related to the discussion we had:
Results were mixed. I figured this might happen. Projects can be tricky. Students don't always see them as part of the curriculum and won't attempt to absorb the learning portion, focusing only on the product. Many saw our discussions as just random asides, not an answer to why someone might care about a polar conic.
I liked this question because it surfaced a big misconception. Student equated "eccentricity" with "size" and very frequently described ellipse B as least eccentric citing variations of "it's the smallest of the four." This was great to learn because it helped shaped the next day.
Which by the way, if an assessment isn't influencing your lesson plan, why are you giving it?
First, I explicitly called out the misconception. Size has nothing to do with an object's eccentricity. Size is a function of semi-major axis. They were given these four ellipses:
I used this tactic with cardiods and rose curves about a week before. My question to them is the same: "how did I make these?"
To structure the activity, they set this up in desmos:
I used variables because it would easier to adjust the two numbers involved. I instructed them to manually change a (semi-major axis) and k (eccentricity). To prevent endless trial and error, I gave them the scope of a and k. The semi-major axis is an integer between 5 and 20, eccentricity a decimal between 0.2 and 0.9 with a step size of 0.1.
I did not make any mention of using the slider that pops up once you give a and k a value. I suggested it would be easier to delete and retype values. Some chose to use the slider. I have nothing against the slider. But I want to make sure they gave some conscience input to what they were typing, rather than wiggle a slider and hope to get lucky.
Like before, it got quiet. Joy after solving one. Heartbreak and frustration when it's OH SO CLOSE. But after 20-25 minutes, an extremely accessible activity. When we reviewed answers, there was little to no disagreement. Everyone had dialed in the same values.
All of this to confront the misconception that surfaced on the test: eccentricity has nothing to do with size. Now that they had played with for a while, it was much more obvious WHY number 3 in this problem set is the most eccentric. And though number 2 and 4 have the greatest size (in fact they have the same a value of 15), it has nothing to do with how eccentric we might consider them.