Wednesday, August 27, 2014

Section 2-7: Terms Associated with Polygons (Day 15)

Section 2-7 of the U of Chicago text deals with polygons. Notice that this lesson consists almost entirely of definitions and examples. But this chapter was setting up for this lesson, since a polygon is defined (Section 2-5, Definitions) in terms of unions (Section 2-6, Unions and Intersections) of segments:

A polygon is the union of three or more segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.

It follows that this section will be very tough on -- but very important for -- English learners. I made sure that there is plenty of room for the students to include both examples and non-examples of polygons. The names of n-gons for various values of n -- given as a list in the text -- will be given in a chart on my worksheet.

The text moves on to define a polygonal region. Many people -- students and teachers alike -- often abuse the term polygon by using it to refer to both the polygon and the polygonal region (which contains both the polygon and its interior). Indeed, even this book does it -- when we reach the chapter on area. Technically, triangles don't have areas -- triangular regions have areas -- but nearly every textbook refers to the "area of a triangle," not the "area of a triangular region." Our text mentions polygonal regions to define the convexity of a polygon -- in particular, if the polygonal region is convex (that is, if any segment whose endpoints lie in the region lies entirely in the region), then the polygon itself is convex.

The text then proceeds to define equilateral, isosceles, and scalene triangles. A triangle hierarchy is shown -- probably to prepare students for the more complicated quadrilateral hierarchy in a later chapter.

Many math teachers who write blogs say that they sometimes show YouTube videos in class. Here is one that gives a song about the three types of triangle. It comes from a TV show from my youth -- a PBS show called "Square One TV." This show contains several songs that may be appropriate for various levels of math, but I don't believe that I've ever seen any teacher recommend them for the classroom. I suspect it's because a teacher has to be exactly the correct age to have been in the target demographic when the show first aired and therefore have fond memories of the show. So let me be the first to recommend this link:

Another song from Square One TV that's relevant to this lesson is "Shape Up." Notice that many geometric figures appear on the singer's head -- though not every shape appearing on her head is a polygon:

With so many definitions in this important section, I decided to leave out the Review Exercises, but I did include the Exploration as a Bonus Question. There are many obsolete names for polygons, such as enneagon and pentadecagon, whose definitions the students are supposed to guess. (Notice that the text gives dodecagon as an obsolete name, yet I've seen modern texts that include it. But duodecagon is definitely obsolete.) An interesting case is trigon. Believe it or not, we still use the word trigon -- the study of trigons is called trigonometry!

The text suggests that students check the guesses in a large dictionary -- that's so 1991! Of course students should try a Google search to find these terms. I was able to find all of them online -- but trigon was a bit difficult because most of the results referred to the DC Comics character Trigon. I had to scroll down towards the bottom of the first page before I found a relevant result.

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