From CCSS-M: Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
I like the way this standard is worded. I am glad that it is not enough for our students to write an equation of a circle – instead, we must provide an opportunity for our students to derive the equation of the circle (that will be another post), and we have a reason to complete the square (to find the center and radius, which is the topic of this post).
As I was preparing this lesson, I wracked my brain to think of how my students could discover “completing the square” instead of me just giving them an algorithm to follow. I decided to try starting with a circle in center-radius form, expanding that to the general form (or standard form, depending on which textbook you are using) – and then letting them figure out how to go backwards. Of course we had to spend the first few minutes of the lesson going back over how to expand binomials (they were only in algebra last year) – but I didn’t just tell them that, either. I posed a quick poll to start the lesson:
While they were answering, I was monitoring the student responses.
I often go on and show students the correct answer (indicated by a green bar) when I show the results to a Quick Poll. In this case, however, I unchecked “show correct answer” before showing the class results. When the vote is split like this in our classroom, my students have learned this means that they have to get up, find another student in the room, and try to convince that student why they chose what they did. I send the poll again to see if anyone has been convinced to change the response.
And as you can tell, the results were still not that great. So they had to find a different person to convince of their answer. And I sent the poll one more time. And all but a few very stubborn boys got it correct.
After the Quick Poll, we used the TI-Nspire CAS – for students to enter into the practice of look for regularity in repeated reasoning.
I felt like they really needed to have a handle on expanding binomials before they could be proficient completing the square.
Once they had determined how to go backwards (the phrase “divide by 2 and square” was their phrase, not mine),
we looked at a visual representing of completing the square from MathNspired.
My final Quick Poll was evidence that students were beginning to make sense of why and how we were completing the square to find the center and radius of a circle.
Even though they will eventually need to do this without technology, we use the technology while we are learning to make more sense of the mathematics.
And so the journey continues ….