Expressing Geometric Properties with Equations
G-GPE.A Translate between the geometric description and the equation for a conic section
- 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.
How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.
Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?
It’s a right triangle.
The hypotenuse is always 5.
The legs change.
What else do you notice? What has to be true for these objects?
The Pythagorean Theorem works.
Leg squared plus leg squared equals five squared.
What do you notice about the legs? How can we represent the legs on the graph?
One leg is always horizontal.
One leg is always vertical.
How can we represent their lengths in the coordinate plane?
x and y?
(I think they thought that the obvious was too easy.)
What do x and y have to do with point P?
Oh! They’re the x- and y-coordinates of point P.
So what can we say is always true?
Is there an equation that is always true?
What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)
And then we traced point P as we moved it about coordinate plane.
So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².
I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.
And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.
Here are the results of the questions that they worked.
What would you do next?
What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.
We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.
And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.
How can we graph the circle, limited to functions?
How can we tell which points are correct?
I asked them to write the equation of a circle given its center and radius, practicing attend to precision.
54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.
(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)
I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).
But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.
When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.
We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?
And so the journey continues, one #AskDontTell learning episode at a time.