The heart of mathematical discovery is solving new problems. This is worthy, challenging work requiring creativity and perseverance. As you search for an answer, the end result is very much in question. Sometimes, however, it can be a welcome change to complete a task with explicit instructions. My students have been learning about the four most common triangle centers, and look what beautiful work they produced! Be sure to notice the beautifully written, precise language used to describe each center.
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Vietnamese, like other languages, varies depending on what region of the country you are in. The most prominent division is between the Vietnamese of the North and that of the South, as I experienced in a recent trip to Da Nang. As with the soda/pop and hoagie/submarine disputes in the US, many of the variations in Vietnamese center around food. A waiter was quick to correct me when asking for “chanh leo” (passion fruit). He was insistent that I call it “chanh dây.” Another regional language is about a word that is printed on every banknote the government prints. The word “nghìn” (thousand) is appears on every bill of the Vietnamese Dong. However, the currency is produced by the federal government in Hanoi in the North, where I live. Even though “nghìn” is on the physical bill, everybody in the South uses the word “ngàn” for thousand when quoting prices. By refusing to use the word “nghìn,” it appears that the South is continually thumbing its nose at the North. Given the history of conflict between the North and the South, this is a reasonable interpretation of the nghìn/ngàn dispute. Even as someone who has only spent three months learning northern Vietnamese, I couldn’t help but feel that the Southerners were saying the word “ngàn” just to mess with me. A brief interaction with a local in Da Nang sheds some light on the situation. I pointed out that I was familiar the word “nghìn” in Hanoi, and that hearing “ngàn” was throwing me off. The vendor just smiled and laughed. Perhaps her amusement at my attempts at language masked a deep resentment for the vocabulary of the North. However, I find a more likely explanation is that the difference in terminology isn’t that big a deal to her. Difference in language is one of the inconveniences we put up with for the benefit of interacting with each other.
CONTAINS AN INTERACTIVE APPLET. BEST READ ON A LAPTOP OR DESKTOP.
The Greeks set the standard for mathematical precision with their study geometric constructions using only compass and straightedge. Students worldwide have access to the digital versions of the same tools through the free software Geogebra.
When the Greeks were making their constructions, they knew how to construct a triangle with sidelengths 3, 4, and 5, say. However, if a Greek then wanted to construct the slightly different 3, 4, 6 triangle, they would have to start from scratch. With Geogebra software, modern students are able to overcome this limitation and complete one a single construction for all triangles. Watch the animation and note how the sidelengths of the triangle change and the segments on the left grow and shrink. Then recreate the same effect using the Geogebra applet. When using the Segment with Given Length and Circle with Center and Radius tools, type in "f," "g," or "h," depending on the length desired.
I give the same assignment to my 9th grade Geometry students. One reason that I like this problem is because the ultimate judgement of a student’s success lies outside of the teacher’s authority. The student can judge themselves if their solution is correct using the software. It’s amazing to watch students’ internal motivation drive them to complete the challenge. Even though I did not attach a grade to the problem, a student emailed me at 10 pm on a Friday to share their solution with me.
I’ve been teaching at an international school in Hanoi, Vietnam for a little over a month now. I love my work helping students develop their ability to think on their own. Realistically, very few students will use the subject matter of graphs and equations outside of their formal education. However, using facts to build an argument is a valuable skill in many areas of life. I view mathematics as a vessel to deliver habits of thought and argument that are so broadly applicable. When I began teaching in Hanoi, I brought all of the tricks of the trade I’d developed in the States. Largely, my teaching techniques have carried over: keep students busy, refocus student complaints, and never disabuse students of the misconception that you know something you actually don't. There is one area, however, where my experience in the US has not fully prepared me for teaching here. All students need the opportunity to share their ideas with the class. In the US, I’ve generally been able to convince even the shyer students to put their work on the board or to answer a question every once in a while. To do this, I have two approaches. I create awkward silences in class and make eye contact with a student who has not shared an opinion recently. Eventually the painful awkwardness becomes worse than the idea of going to the board, and the student will put their work up for the class. Other times, I prepare the student beforehand to share their idea. When students are working, I ask the student individually if they could put their answer on the board in a few minutes. Then I call on that specific student when the time comes to share their work. These two approaches have gotten me by so far. In my school in Hanoi, most students are of either Korean or Vietnamese nationality. Whether it is due to cultural differences, a language barrier, or a combination of both, my techniques to cajole students to speak have not had much success. I have a large percentage of students who giggle and look away when I make eye contact and who laugh and vigorously shake their heads when I suggest that they come to the board to answer a problem. It is so prevalent among my students, that I fear I may break the internal workings of a student if I were to compel them to answer. One way I've found to circumvent this difficulty is to give all students a lowstakes chance to share what they know, which gives both them and me feedback about what topics to focus on. In my Pencil & Pen activity, I give all students a slip of paper with a math problem and five minutes to solve it in pencil. After the five minutes, I solve the problem at the board and ask students to correct – if necessary – their own work in pen. Then I collect collect their work, and see how many students were able to solve the problem without help and how many needed a crutch. I’ve done this activity twice now and both times have received useful feedback in designing the next week’s lessons. I’m looking forward to engaging all of my students throughout the year and hearing their voices. I hope you’ll check back often for updates. Recently a friend posed a Geometry problem to me: $\angle BAC = 120^{\circ}$, and $AX$ bisects $\angle BAC.$ If $BY$ and $CZ$ bisect angles $\angle ABX$ and $\angle XCA$, respectively, then $\angle ZXY$ is a right angle. After standard anglechasing failed to produce an answer, I turned to my bag of problem solving techniques: Look at a Special Case; Investigate the Converse; and Ask a Related Question. Special Case: An easy way to simplify the problem is to assume that $\Delta ABC$ is isosceles. In this case, we can use special right triangles to prove our result. The symmetry of $\Delta ABC$ shows that $ZY  BC \perp AX.$ Because $BY$ is an angle bisector, $\angle ZBY = \angle YBX = 15^{\circ}.$ This shows that $\Delta ZBY$ is isosceles with base $BY$. As a result, $BZ = ZY.$ $\Delta BZM$ is a 306090 triangle, so $PX = ZM = BZ / 2$, and symmetry gives $PZ = ZY / 2 = BZ / 2.$ Thus $\Delta PZX$ is an isosceles right triangle, and, similarly, so is $\Delta PYX.$ Thus the result is proved. The Converse makes the right angle the assumed and attempts to prove that two segments are angle bisectors. $\angle BAC = 120^{\circ}$, and $AX$ bisects $\angle BAC.$ If $\angle ZXY$ is a right angle, then $BY$ and $CZ$ bisect angles $\angle ABX$ and $\angle XCA$, respectively. The converse is clearly proved false by the animation, because $\angle ABY$ grows and $\angle YBX$ shrinks. However, the animation lead to a related question, which I have not yet found a sufficient answer to. I am still investigating both the original and the related problems. Though I have not produced a complete answer, I do not believe I have wasted my efforts, and I've learned a lot along the way. I will take a break and return to the problem in a little while.

About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
September 2017
