Circles on the Coordinate Plane
Lesson 1 of 17
Objective: SWBAT understand the derivation for the equation of a circle on the coordinate plane.
Opener: How Many Points?
Today's Opener starts off easy, but continues to add complexity as students explore it. All students should be able to give what I call the “four easy points”:
(5,0), (0,5), (-5,0), (0,-5)
All should recognize that these aren’t the only points, however, and once they do, that’s where the fun starts. (It is a judgment call from year to year whether I give students more or less challenging versions of the same problem.)
I expect my students to come to the table with different knowledge. Some will immediately recall the locus definition of a circle. Others will quickly come up with the idea that you can move five units diagonally by sliding over 3 and up 4 (or some similar permutation of that).
Hopefully, a student will confidently state that the point (5,5) is 5 units from (0,0). This leads to a rich discussion that ties in knowledge of 45/45/90 triangles, among other previous topics. Throughout the warm up I am on the lookout for these sorts of opportunities, moments to tie student thinking back to previously covered topics!
One possible teaching move here is to remind students of the distance formula. However, I find that it’s useful to state clearly to students that some distances on the coordinate are “easy” to determine: anything that’s simply horizontal or vertical means you just have to count. Diagonal lines require a little more work, but because we know the Pythagorean Theorem, that’s not so bad. And, I leave it at that.
During this segment of the lesson, I reveal these Discussion Questions to the class one at a time. As I do, I emphasize that students are not being graded on their responses. After giving my students a short amount of time to answer each question, we look at the results together. As we do, I ask students to explain why they answered how they did. What’s awesome about this use of technology is that questions and ideas really roll out of kids. They like to respond to the data, to offer explanations, justifications, and conjectures about answers.
Teacher's Note: The Cornell Good Questions Project has some interesting resources related to using discussion questions in the math classroom.
Guided Notes and Exercises
I plan to work through the first two pages of the Equations of Circles Packet with my students. I prefer to use a chalk board, and to encourage students to come up and help.
The big idea here is that I want all of my students to see the connection between the Pythagorean Theorem and the Equation of a Circle. I also want students to remember what it means to graph an equation on the coordinate plane. This packet starts there, then moves into some guided notes, example exercises, and problem solving.
Tonight's homework is to get started on a Delta Math assignment about the equations of circles.
When I introduce this assignment, I provide examples for how each module works.
Unit 2 Lesson 1 Exit Slip (see page 2): Write a sentence or two explaining why the center coordinates are being subtracted in the equation of a circle.
This is a question that I expect students will be asking often over the next few lessons. Now is a nice chance to pre-empt them, and, to see what they understand so far. This is one of those questions where it’s ok if few students answer “perfectly” - in contrast, I expect a lot of partially correct answers that reveal a lot about student conceptions and intuitions.
Possible Extension Questions:
- How would I graph this circle on a graphing calculator like the TI-83?
- How many points are on the line y = x?
- How many points are on a circle?
- Are there different levels of infinity?