This was the beginning of a post from March. Kind of cheating and its not a complete post, but its a start with an idea or two to share.
I chose to place the equations of circles in our coordinate geometry unit. It made sense to me, sense the equation is derived using slope and distance in a sense.
G.GPE.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.)
We had a brief amount of time at the end of class yesterday. I quickly gave each student some graph paper and filled in our last 10 minutes by asking students to select 2 points in the plane and connect them with a segment. We used a “slope triangle” (student term, not mine) to connect the 2 points.
I asked for the length of the segment and many quickly used Pythagorean Theorem. Their assignment was simply to create more segments, add slope triangles and use PT to find the segment length. Then explain the process in their own words.
We began class by discussing their work. We chose a pair of points to show our ability to find the segment length. I walked around the room to check student work. I then gave them the distance formula and we used it with our pair of points. Some thought it was more work.
I asked them to compare the 2 methods. How are they alike/different? They shared what they saw with a classmate. Then discussed the pros & cons of each approach. Some preferred the graphing over the formula. One student stated if they had a graph, PT would be the way to go, but with only coordinates, they felt they would do better with the formula. Another student shared they would more likely forget the formula, but they could always sketch / plot the points and add a slope triangle.
What was worthwhile – is them realizing they have a couple of approaches in their tool belts.
We then moved on to deriving the equation of a circle.
I began with a circle centered at the origin. As we transformed the circle, I hoped for students to notice connections between the new center and the given equation, as they did.