Log 3: April 26, 2002

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

Lesson Activities

What I Expected Students to Learn

During this lesson I expected students to gain some working knowledge of how to perform (more) exact polar-to-Cartesian conversions using sine and cosine, and to see a practical use for estimation skills. Despite some misgivings, I did not seek to develop sine and cosine beyond the ability to use them. Students had encountered trigonometric functions before, and I suspected most of them were only vaguely familiar with their significance as functions beyond being tools. On the other hand, I'm not completely averse to students being able to use math whose conceptual significance they don't quite grasp while pursuing some other set of concepts altogether.

Lesson Activities & Objectives

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

After students made estimates of several polar-to-Cartesian cooordinate conversions, I showed them an example of how to use trigonometric ratios to find "exact" Cartesian conversions. I also went through one example where the conversion would have been redundant, as the point lay on the x-axis. Then students were asked to do the rest and check them against their estimates and other students' work; they were to initial papers that they checked. I warned them that papers that didn't show some effort would not only result in a penalty for the person handing it in, but for the person initialing it.

Connections between Lessons

In the previous lesson, students had practiced coordinate estimations, so this gave them the opportunity for more practice and to see the usefulness of estimations in checking answers. In subsequent lessons, they would also see Cartesian-to-polar conversions, lending a sense of symmetry and also subtly conveying the idea that the two different coordinate systems describe the same underlying space.

Teacher Actions

Instructional Strategies

My strategies in this lesson consisted mainly of giving students a model to follow, which I did by lecture at the board while soliciting student responses ("So what do we do next?"). As usual, I circulated to answer questions and nudge students to consult each other.

Interactions & Promoting Participation

As I've mentioned before, I dislike lecturing from the front of the room, but I felt it would be helpful to some students to see an example worked, even by mass-dissemination. I solicited input from students wherever possible, including guesses. I make it a point to discuss as many answers as I hear, whether they're "right" or not. This way students feel safe enough to call out their thoughts and examine them so everyone can benefit from seeing different approaches. I also consider the metacognitive aspect of checking answers to be beneficial.

Student Actions

Students participated to varying degrees in the interactive parts of the examples worked at the board, and some of them began to take the initiative in checking their answers with others. Otherwise, much of the work was individual in nature. Some began to take the "shortcut" of computing x and y directly from the formulae once they were comfortable with them.

As I had miswritten sine and cosine, one student finally asked if they were backwards because the answers he was getting from the formulae didn't match his estimates. Other students spoke up in agreement once the dam had been breached. Yes, indeed, I confused x and y. It led to an unplanned, but near-perfect opportunity to point out the utility of estimates.

Reflections

Assessment of Student Learning

I noticed that students initially had trouble with the trigonometric ratios because they had trouble seeing "cos(theta)" (for instance) as a number for any given value of theta. My conjecture is that they don't yet think automatically of sine or cosine as functions that can be evaluated. However, as they found that the formula gave them numbers that visually and numerically matched their plots and estimates, they became more confident in what they were doing.

Future Adjustments

I don't know that I'd deliberately give the wrong formulae for sine and cosine again; besides, future classes might remember them more readily and point out the error. However, I'm sure someone will run into a calculator set to radians instead of degrees (or vice versa) at some point, and that would be another way to point out the utility of estimates.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 2 June 2002