Monday, February 22, 2021

Lesson 11-3: Equations for Circles (Day 113)

This week the Big March starts in earnest. As I've written before, the first week of the Big March isn't terrible, since at least it's a four-day week. The second week starts the string of five-day weeks -- these are what actually make the Big March the Big March.

Let's get to today's lesson. I've written above that one of the most difficult units always seems to begin right around the start of the Big March. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the equation of a circle, which just a few years ago was part of Algebra II! (Meanwhile, English classes tend to read Shakespeare during the Big March, for example.)

Lesson 11-3 of the U of Chicago text is called "Equations for Circles." In the modern Third Edition of the text, equations for circles appear in Lesson 11-6.

This is what I wrote last year about today's lesson:

The first circle lesson is on Lesson 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam.

Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.

In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!

I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson.

Here are the worksheets for today. (Yes, today is a rare short post from me.)


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