Yesterday in class, we were doing a task made popular by Fawn Nguyen. Let's call it similar golf. Students create their own miniature golf course and then need to do a bank shot off one of the walls to get a hole in one.
I expect student solution methods to end up using similar triangles. Here's a solution method I did not expect:
I expect student solution methods to end up using similar triangles. Here's a solution method I did not expect:
Student: "Mr. Busch, I think I found a solution!"
Me: "Show me what you got."
Student: "Well, I drew perpendiculars to the wall from the Ball and the Hole.
Me: "Show me what you got."
Student: "Well, I drew perpendiculars to the wall from the Ball and the Hole.
Given a line and points A and B on the same side of line and not on the line, draw perpendiculars from A and B to the line. Call the intersections C and D respectively.
Student: "Next, I drew the diagonals from the ball to the place where the perpendicular from the hole hit the wall. Then I drew the line from the hole to where the ball's line hits the wall."
Draw line segment AD and line segment BC. Call the intersection E.
Student: "Then from where the two diagonals intersected each other I drew another perpendicular to the wall."
Create the perpendicular from point E to line CD. Call the point of intersection F.
Student: "Where the new perpendicular hits the wall, I drew lines to the ball and the hole. This is where you should aim to make a hole in one."
Draw segment AF and segment BF.
Me: "That's really interesting. How do you know that it works?"
Student: "Well, I measured it and all the proportions check out."
Me: "Hmm. I don't know why that would be true. Can you try it with a different location of the ball and the hole and see if it still works?"
Student: "Did I just come up with something that works and you didn't know about it?!"
Me: "Maybe, but you'll need to convince me that it might be true. Maybe you just got lucky with your ball and hole placements."
Student: "Okay I'll check"
The student tried another variation and it still worked. After school, I put the situation into a Geogebra file and it does seem to hold true. At least I cant' find a counter example. So, here's the question, are triangle AFC and triangle BFD similar? Why or why not?
Student: "Well, I measured it and all the proportions check out."
Me: "Hmm. I don't know why that would be true. Can you try it with a different location of the ball and the hole and see if it still works?"
Student: "Did I just come up with something that works and you didn't know about it?!"
Me: "Maybe, but you'll need to convince me that it might be true. Maybe you just got lucky with your ball and hole placements."
Student: "Okay I'll check"
The student tried another variation and it still worked. After school, I put the situation into a Geogebra file and it does seem to hold true. At least I cant' find a counter example. So, here's the question, are triangle AFC and triangle BFD similar? Why or why not?
Here's a link to the Geogebra file I created if you want to play around with it yourself.
__EDIT__
We threw out the lesson plan in Geometry today in favor of solving this problem. Students used Geogebra to construct the situation and convince themselves of it "true-ness". Afterwards students self-clumped and went to work. Some used big-honking whiteboards. Some worked furiously in their notebooks. Good times. A couple student groups think they have a solution but the bell rang before they could write it up or present it. We'll see tomorrow.
Behold the power of Twitter. I put the question out to the #MTBoS and got multiple responses.David Butler is the easiest to follow. And in less than 140 characters. Most impressive.
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