Today's class started with looking at how to express the distance between a point (x, y) on a circle with radius 5 and the center located at (0 , 0). Because I had left expressions written out with the center point still being shown, many got to the equation (x - 0)2 + (y - 0)2 = 52. Some students wanted to swap out the zeroes but realized this wouldn't work.
I asked the class why this expression described points on the circle. It took some thought but, finally, students referenced the triangles being formed and the distance formula. I asked the class what would happen if we dilated the circle and now had a radius of 1? While some students were still unclear about this, most said we would only have to replace the 5 with a 1 in the equation, resulting in
(x - 0)2 + (y - 0)2 = 12. I next asked what would the equation be if the dilation resulted in a radius of 10? At this point, students quickly stated the equation would be (x - 0)2 + (y - 0)2 = 102.
Next, I asked what would happen if the center were translated three units to the right. While some students wanted to replace the x and y values with the new center of (3, 0), many realized the equation would be (x - 3)2 + (y - 0)2 = 52. I repeated this with translating the center to the point
(-2, 0). Students wrote out (x - -2)2 + (y - 0)2 = 52. I asked what happens when you subtract a negative value? Students hesitated but one student said it would result in adding the value. Other students agreed, so I wrote out (x + 2)2 + (y - 0)2 = 52.
I pointed out that as we move the center in the positive direction we subtract the value and as we move in the negative direction we add the value. A colleague had mentioned that she thinks of the addition and subtraction process from the perspective of what does it take to move the center back to (0, 0)? I mentioned this to the class and many students related to this idea.
At this point, students started asking what would happen if the center were translated along the y-axis? Others pointed out that it should then change the value of the y-coordinate of the center being used. I asked what the equation would be if the center were translated to the point (4, 2)? Most students wrote out (x - 4)2 + (y - 2)2 = 52. There were still some students that wanted to replace the x and y values with 4 and 2.
At this point, I passed out a practice sheet (pages 4 and 5 from Michelle Bousquet's Equation of a Circle lesson plan). Students started by writing out the general equation of the circle. A few students wanted to replace h and k with numbers, but most were able to write the general equation:
(x - h)2 + (y - k)2 = r2.
The other issue that came up in these problems was the first example. Students tried using the endpoints as the center or couldn't remember how to find the midpoint of a line segment. The other thing that some students failed to realize was they needed to calculate the length of the diameter in order to find the circle's radius.
Most students completed the four example problems by the end of class. We'll go through these next class and complete the remainder of problems on the practice sheet . The plan is to focus on working through these and another set of practice problems before moving on to determining a circle's properties from an expanded expression.
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