Friday, February 6, 2015

Activity: Logic Problems and Euler Lines (Day 108)

Merry Christmas! Actually, right around the time I was posting about Calendar Reform, I found an article on Slate which proposed that this weekend, February 7th, should be Christmas:

The author, L.V. Anderson, suggested the change because she wanted there to be more time between Thanksgiving and Christmas. One commenter pointed out that it would make more sense to change Thanksgiving, say, to the Canadian date in October, rather than expect all the churches to change Christmas. (As it turns out, I decided to calculate what date is as many days after American Thanksgiving as December 25th is after Canadian Thanksgiving. The answer turned out to be -- February 8th!)

Another commenter pointed out that an early February Christmas would fit better between the two school semesters. After all, the reason for the early start calendar with school starting in August is that it's easier to start school a month earlier than it is to move Christmas a month later!

As it turns out, under our early start calendar, today -- Anderson Christmas Eve -- marks the end of the fifth quaver of the year. And for math buffs, February 7th is a day to celebrate -- not because it's Christmas, but because it's e Day -- just like Pi Day, except for the number e. One of the students I was tutoring -- an Algebra II student -- is just now beginning the chapter on Exponential Functions and the number e -- named after the 18th century Swiss mathematician Leonhard Euler. He was the one who reminded me that tomorrow is e Day, since e is approximately 2.7... for February 7th!

Speaking of tutoring, let's get back to my geometry student. I spent most of the time last night reviewing the indirect proofs in Glencoe's Section 5-3. I showed him my worksheet from yesterday, and I think he was able to figure out the indirect proofs that I included. He quickly figured out the Triangle Inequality and how to determine whether three numbers can be the sides of a triangle. Also, he saw my proof of the Triangle Inequality and noticed that the Glencoe text will ask him to prove it in the exercises, so my proof sketch should help him, even though he has to fill in some steps. Then again, I don't know whether the teacher will actually assign him the proof question from the text.

I think that for my next worksheet for this upcoming Tuesday -- most likely the day that I will tutor him again, I will review the Triangle Inequality and use it to prove the SAS Inequality, even though we already covered it at the end of first semester. I said that I will move SAS Inequality to second semester next year, but it's so much easier just to include it right now. And besides, its converse, the SSS Inequality, hasn't appeared yet as it doesn't even appear in the U of Chicago.

Now for today's activities. First, I post a worksheet similar to what I showed my student back on Wednesday, with the two triangles with vertices well-chosen as to minimize the use of complicated fractions when calculating the centroid, circumcenter, and orthocenter. Only the centroid of the second triangle has a fraction as its y-coordinate, and the only fractional slope that appears is 1/2 -- but it's its opposite reciprocal, -2, that is needed to find the circumcenter and orthocenter.

At this point, some of you may ask, why shield the students from fractional coordinates, when the PARCC or SBAC exams may have fractions on them? It's because I want the students to understand the concepts first, rather than be intimidated by the fractions. My student, back on Tuesday, saw the questions from Glencoe where fractions appeared in intermediate steps (that is, where y is a fraction and this value of y must be plugged in to find x). I want him to think less about the fractions that might appear and more about what steps he needs to take to find the three centers and why.

Notice that for any triangle, the centroid, circumcenter, and orthocenter are collinear. The proof is somewhat complex -- one might really be tempted just to use a coordinate proof here. This fact was first discovered by the mathematician Euler -- that's right, the same Euler after whom e was named. I also remind you that Euler was also one who solved the Bridges of Konigsberg problem from the first day of school. (Yes, Euler was a very prolific mathematician!) And so we also named the line on which these three centers lie after him -- the Euler line of the triangle.

In my first activity, the students hopefully discover that the three centers lie on the Euler line. This is the ultimate goal.

Now for the other activity, I promised some logic problems, since this is the topic of Section 13-3 of the U of Chicago. Instead of including problems from the text, I chose some problems I found on a message board:;wap2

The author of these logic problems goes by the online name "Fireball." The website is devoted to my favorite game show, The Price Is Right. But in the summer, the game show is in reruns, and so those who post there find other topics to discuss. So "Fireball" came up with some logic puzzles. As usual, teachers can decide to include either the logic puzzles or the Euler lines, or a combination of both.

Speaking of The Price Is Right, I wanted to include an activity earlier based on that game show -- "The Triangle Is Right." Students are given right triangles with two legs (or possibly one leg and one hypotenuse) given, and they must guess the length of the remaining side. The winner is the student who comes the closest to the correct length -- without going over, of course. But I didn't include it as an activity, because I liked my puzzle-proof of the Pythagorean Theorem better.

The logic problems from the U of Chicago are about the same difficulty as the Beginner Level problem from "Fireball." Even his Easy Level may take the students some time to solve. I wasn't going to include the Intermediate Level problem, but I did only because it refers to the presidents, and so I honor the Lincoln's birthday holiday on Monday by including it.

Let me conclude this post by wishing L.V. Anderson a Merry Christmas, mathematicians a Happy e Day, and students a Happy Lincoln's Three-Day Weekend. See you for my next post on Tuesday.

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