Log 2: April 24, 2002

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

Lesson Activities

What I Expected Students to Learn

Students had been briefly introduced to polar coordinates during the previous class, but I wanted them to have more of an opportunity to become comfortable with plotting points. I also wanted them to begin developing two-dimensional "number sense" in polar coordinates, a seat-of-the-pants sense of where a point is given the coordinates, by having them plot the points and tie it back to Cartesian coordinates, with which they were long-familiar.

I also thought that having them check answers with each other would reinforce the message that the teacher is not the fount of all knowledge in the classroom, and that many of their classmates are better resources than they necessarily realize.

Lesson Activities & Objectives

The lesson activities are described in detail in the Curriculum Unit.

Due to STAR testing earlier in the day, I knew students would be tired; and even more, the period was shortened to 50 minutes from the usual 100 minutes. (It is a testimony to my acclimation to the A-B block schedule that 50 minutes seems unconscionably short to me now.)

The day's lesson consisted of having students plot polar coordinates, then estimate their conversions into Cartesian coordinates. Students had to check their results with each other. This would prepare them later for converting Cartesian coordinates "exactly" (well, minus round-off error) as a way of double-checking results. They also had to check their answers with their classmates and resolve any discrepancies that resulted.

Connections between Lessons

I "sold" polar coordinates to students as an example of a well-known and often-used alternate coordinate system to Cartesian coordinates. Also, most of them found that converting polar coordinates to Cartesian coordinates made sense as they found the latter more familiar.

In subsequent lessons, students continued making polar estimates as they learned how to use trigonometry to make (more) exact conversions so they could check their answers, so it was a skill that they continued to use.

Teacher Actions

Instructional Strategies

Most of the class consisted of my writing up points on the board, then circulating around the room to answer questions and see what students were doing. I did not offer any specific method for making "estimates," but students developed some on their own, from "eyeballing" the points to drawing auxiliary lines or using rulers to visually "align" the point with the Cartesian axes.

Interactions & Promoting Participation

When students asked me a question, I urged them to check with one or two neighbors first. Of course, there are times when everyone in the room is confused, but I've learned that it makes far more sense to explain to the first few students who ask, then refer other students in the class to those newly "expert" students. The "experts" then have an opportunity to attempt to solidify their knowledge, it lessens the class' dependence on me, and I remain, of course, available if they continue to run into difficulties.

Student Actions

The majority of students, despite being exhausted from testing, were willing to give the activity a try. After I gave clarifications to a few students and started referring questions to them, they began swapping papers as I wandered around checking up on people who seemed to be having more difficulty.

Reflections

Assessment of Student Learning

Looking over students' papers revealed that a few of them were still confused about how polar coordinates work, which I hoped to address in subsequent classes. Also, comments and questions during class revealed that the notion of "estimating" was somewhat foreign. This was not surprising, as estimation is a topic that comes up all the time in "real life" yet is rarely even mentioned in the traditional math classroom (or at least the ones I have seen). Most students were, after a brief clarification, able to devise their own methods for doing so.

Future Adjustments

In the future I think I would be much more alert to particular misconceptions that showed up, such as confusion over negative angle measurements, attempts to convert to degrees by shifting decimal points on the y-coordinate. Also, I noticed that even though I asked students to initial papers to show that they'd checked for completion, some papers with almost no work on them were initialed. In later classes I warned that I would penalize if someone signed an incomplete (as opposed to incorrect) paper. This is something I would check for more closely during class, and warn about earlier.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 1 June 2002