When working a mathematical problem, there are many sources of authority a pupil can reference to check the accuracy of their work:
Practically, I assign several book problems during class and tell students to work independently. After five to ten minutes, I interrupt the class. I tell them that they will have to combine all of their answers onto a single sheet that they will turn in. Each group is given a large sheet of paper already sectioned-off and numbered. There is one region for each of the assigned book problems. It is the groups' responsibility to agree on which work and answer to put on the one sheet they submit together. The combined pressures of needing to submit their work and to work together gets the students arguing about the correct solution to each problem - exactly what I want.
We spent several days in class discussing Pythagorean Triples. These are whole number solutions to the Pythagorean Theorem: if $a$ and $b$ are the legs of a right triangle and $c$ is its hypotenuse, then $a^2 + b^2 = c^2." In class, we discussed two patterns that can help us find Pythagorean Triples.
1. Any scale multiple of a Pythagorean Triple is also a Pythagorean Triple. For example $3, 4, 5$ is a Pythagorean Triple, and so are $6, 8, 10;$ $9, 12, 15;$ and $3k, 4k, 5k$ for any whole number $k.$
2. If we look at the Pythagorean Triples where $c$ is two greater than $b,$ the differences between successive values of $b$ and $c$ display linear growth as you can see in the examples below. From the first line to the second, $b$ and $c$ both increase by $7.$ From the second line to the third, $b$ and $c$ both increase by $9.$ From the third line to the fourth, $b$ and $c$ both increase by $11.$ This difference continues to increase by $2$ between each two lines.
On the most recent quiz, I gave two extra credit questions asking students to find Pythagorean Triples under certain parameters. Students used both of the above techniques, and one industrious student employed her own approach to find a solution to $22^2 + b^2 = c^2,$ where $b+2 = c$:
3. Substitute $b+2$ for $c$ and use algebra
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.
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 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.