I displayed a graph with two circles drawn on it. One circle was centered at (4.3) with a radius of 5 units. The second circle was centered at (-6, 4) and had a radius of 2. I asked the class to consider what they knew about these two circles and how they could determine, using just the centers and radii, if the two circles overlapped.
This took quite a while. Students were not overly focused and had to be redirected several times to focus on the question. I eventually placed at circle centered at (-2, -8) with a radius of 10 on the graph. I asked how they knew the new circle overlapped the circle with radius 5?
After much thought, one group said you could see if the circles had common points. I responded that, yes, finding if the circles had points of intersection would work. I still wondered if there were other ways to do this.
Finally, somebody brought up the idea that you could compare the distance between the circles' centers and the size of the radii. We clarified what was meant and people agreed that this could determine if the circles did not overlap. I wrote:
r1 + r2 < distance between centers --> circles do not overlap
I then asked what would determine if the circles were externally tangent or intersected. At this point students talked about the distance between centers equaling the sum of the radii or being less than the sum of the radii. Our list was now:
r1 + r2 < distance between centers --> circles do not overlap
r1 + r2 = distance between centers --> circles are externally tangent
r1 + r2 > distance between centers --> circles overlap
The question I posed next dealt with how much overlap was occurring. Specifically, when are two circles internally tangent and when is one circle completely enclosed within another circle.
This took some time, but did proceed ahead a bit better than the first investigation. Several groups were coming up with good ideas but were struggling to express these mathematically. After a while, I let one group describe their idea. I demonstrated their thinking using the graphic on the board.
After a bit more time, a group expressed the idea of comparing the radius of the larger circle with how close the centers were with each other. We discussed at what point the circles become internally tangent. I used this to expand to what conditions would need to exist to indicate that the smaller circle was completely contained within the larger circle.
Our list expanded to:
r1 + r2 < distance between centers --> circles do not overlap
The question I posed next dealt with how much overlap was occurring. Specifically, when are two circles internally tangent and when is one circle completely enclosed within another circle.
This took some time, but did proceed ahead a bit better than the first investigation. Several groups were coming up with good ideas but were struggling to express these mathematically. After a while, I let one group describe their idea. I demonstrated their thinking using the graphic on the board.
After a bit more time, a group expressed the idea of comparing the radius of the larger circle with how close the centers were with each other. We discussed at what point the circles become internally tangent. I used this to expand to what conditions would need to exist to indicate that the smaller circle was completely contained within the larger circle.
Our list expanded to:
r1 + r2 < distance between centers --> circles do not overlap
r1 + r2 = distance between centers --> circles are externally tangent
r1 + r2 > distance between centers --> circles overlap
r1 - r2 = distance between centers --> circles are internally tangent
r1 - r2 = distance between centers --> circles are internally tangent
r1 - r2 > distance between centers --> smaller circle completely contained within larger circle
This seemed to make sense to the class. A few people were still pondering the results but most felt comfortable with the relationships displayed.
At this point we were at the end of the period. I pointed out that we'll be interested in how much area is contained in the overlap. We'll dive into this idea next class.
This seemed to make sense to the class. A few people were still pondering the results but most felt comfortable with the relationships displayed.
At this point we were at the end of the period. I pointed out that we'll be interested in how much area is contained in the overlap. We'll dive into this idea next class.
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