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Alma College has offered monthly challenge problems for high school students since before I started teaching. Somehow, I got on their mailing list early in my career (yes they STILL send a physical copy of the problem) and I've been hooked.
The problems are a great way for a teacher to keep all their skills fresh as well as challenge students with problems that extend beyond the usual book problems. I also post the new problems for all my students to try, and I warn my Geometry students that they might not have a basis in the mathematics needed to completely solve the problems.
This month, the problem was geared towards students that have had at least a little trig, but I would say that students in Algebra 2 or Pre-Calculus have had enough math. At the end of the year, some Geometry students (after an intro to trigonometry) could handle it.
For those that wish to try this problem, I won't spoil the problem with my final proof, but I wanted to talk about something that we don't often allow our students the time to do; Try methods that don't work.
What were your initial thoughts about the problem?
My first thought was to divide the above regions into six right triangles and use a combination of Pythagorean's Theorem and right triangle trigonometry. It felt forced to me and the problem wasn't resolving itself the way I needed it to. Without giving anything away, I worked over several pages before I abandoned this tactic. I think that it could have worked, but there were so many pieces I kept getting lost in the direct I was heading.
So that didn't work? What's next?
My next thought was to use coordinate geometry and find lines that intersect. It wasn't as bad, but it still was a lot of bits and pieces and I feared that I was going to make a computational error that would lead to an incorrect solution and I would spend more time hunting for my error than proving the problem. I need to explore this situation.
Explore?
I took a side trip to Geogebra and quickly verified that the theorem should be true. I just wanted to play with Geogebra and the scientist in me could be sated knowing that it did work even if the mathematician in me didn't get the proof. Select point D in the image below to see how the distances change, but the quotient stays the same and even more importantly, the sum of the distances to the sides stay the same. This reminded me of something that I always tell my students and led me to the method that I would eventually use to prove this problem.
SOLVE A SIMPLER PROBLEM
What if P was at a vertex? What if P was on the line of symmetry? And, finally, what if P was anywhere in the triangle. (Drag point D to each of these areas) Because of this approach, I was able to generate a proof for all three situations with the last proof really working for the other two instances of the general problem. Because I started with an easier problem, I was able to build to a more complex solution for the given problem. Because I explored, I was able to see an easier approach.
So what? What's the point?
So what?! How often to students get to see their math teachers not know the answer right away? How often do we model confusion and missteps? This blog post is a small attempt to set the record straight that we don't always know the right 1st solution and that's ok. As someone who's taught both of my current assigned courses for at least 10 years each, I need to remind myself to let the students find their own way AND give them the time for that. They will be better for it.
[UPDATED] How the student's did it.
My solution worked and once I found how to approach it, it worked very nicely. However, the students that solved the problem approached it using three triangles. The areas of these triangles add to the area of the given equilateral triangle. It was elegant and required nothing more than Pythagorean's theorem, some algebra, and some area sum rules. It was very impressive and why I try to not tell students how to do math. They obviously need experience with multiple methods, but giving students a chance to find their own way can (and usually does) lead to great learning and longer lasting methodology. There isn't one way to do it and that's beautiful.
The problems are a great way for a teacher to keep all their skills fresh as well as challenge students with problems that extend beyond the usual book problems. I also post the new problems for all my students to try, and I warn my Geometry students that they might not have a basis in the mathematics needed to completely solve the problems.
This month, the problem was geared towards students that have had at least a little trig, but I would say that students in Algebra 2 or Pre-Calculus have had enough math. At the end of the year, some Geometry students (after an intro to trigonometry) could handle it.
For those that wish to try this problem, I won't spoil the problem with my final proof, but I wanted to talk about something that we don't often allow our students the time to do; Try methods that don't work.
What were your initial thoughts about the problem?
My first thought was to divide the above regions into six right triangles and use a combination of Pythagorean's Theorem and right triangle trigonometry. It felt forced to me and the problem wasn't resolving itself the way I needed it to. Without giving anything away, I worked over several pages before I abandoned this tactic. I think that it could have worked, but there were so many pieces I kept getting lost in the direct I was heading.
So that didn't work? What's next?
My next thought was to use coordinate geometry and find lines that intersect. It wasn't as bad, but it still was a lot of bits and pieces and I feared that I was going to make a computational error that would lead to an incorrect solution and I would spend more time hunting for my error than proving the problem. I need to explore this situation.
Explore?
I took a side trip to Geogebra and quickly verified that the theorem should be true. I just wanted to play with Geogebra and the scientist in me could be sated knowing that it did work even if the mathematician in me didn't get the proof. Select point D in the image below to see how the distances change, but the quotient stays the same and even more importantly, the sum of the distances to the sides stay the same. This reminded me of something that I always tell my students and led me to the method that I would eventually use to prove this problem.
SOLVE A SIMPLER PROBLEM
What if P was at a vertex? What if P was on the line of symmetry? And, finally, what if P was anywhere in the triangle. (Drag point D to each of these areas) Because of this approach, I was able to generate a proof for all three situations with the last proof really working for the other two instances of the general problem. Because I started with an easier problem, I was able to build to a more complex solution for the given problem. Because I explored, I was able to see an easier approach.
So what? What's the point?
So what?! How often to students get to see their math teachers not know the answer right away? How often do we model confusion and missteps? This blog post is a small attempt to set the record straight that we don't always know the right 1st solution and that's ok. As someone who's taught both of my current assigned courses for at least 10 years each, I need to remind myself to let the students find their own way AND give them the time for that. They will be better for it.
[UPDATED] How the student's did it.
My solution worked and once I found how to approach it, it worked very nicely. However, the students that solved the problem approached it using three triangles. The areas of these triangles add to the area of the given equilateral triangle. It was elegant and required nothing more than Pythagorean's theorem, some algebra, and some area sum rules. It was very impressive and why I try to not tell students how to do math. They obviously need experience with multiple methods, but giving students a chance to find their own way can (and usually does) lead to great learning and longer lasting methodology. There isn't one way to do it and that's beautiful.
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