Lesson 2-7: Terms Associated with Polygons (Day 16)

Today I subbed in an eighth grade math classroom. I often like to write about the texts that eighth graders are now using, especially since eighth grade is the first where Common Core transformational geometry makes an appearance.

The text is called California Math, referring to our state, and is published by Glencoe McGraw Hill -- and of course we've discussed Glencoe texts many times here on the blog. Now this text, unlike the Glencoe Geometry text that I've mentioned previously, is obviously geared towards Common Core standards. Here is the table of contents:

1. Real Numbers
2. Equations in One Variable
3. Equations in Two Variables
4. Functions
5. Triangles and the Pythagorean Theorem
6. Transformations
7. Congruence and Similarity
8. Volume and Surface Area
9. Scatter Plots and Data Analysis

Like many Common Core texts, this book is divided into two consumable volumes corresponding roughly to the semesters, with Volume 1 consisting of the first four chapters. The students were currently working on Lesson 1-4, Powers of Monomials. The corresponding standard appears to be:

CCSS.MATH.CONTENT.8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

Here are a few things that I've noticed about the Glencoe's new eighth grade text. First of all, we see that the first semester of this course is almost the full first semester of Algebra I. The only first semester Algebra I topic missing is inequalities. I presume that inequalities would appear in an Integrated Math I course, which is otherwise not much different from Math 8.

As this is a Common Core text, we see a full chapter devoted to transformations, with a chapter on congruence and similarity following the transformations chapter. Lesson 7-6 shows the derivation of slope from similar triangles, which is a big Common Core topic. Then again, slope first appears in this text in Lesson 3-2 -- recall that I once tried right here on the blog to design a course which cleanly moves directly from 7-6 to 3-2, but I found it a bit difficult (especially if we want to keep this course at the eighth-grade level). Also note that similarity in Chapter 7 appears after the Pythagorean Theorem in Chapter 5, but I believe that this connection isn't required until High School Geometry under the Common Core.

Last night there was another presidential debate. There was a brief mention of Common Core (courtesy of the Washington Post debate transcripts):

TRUMP: ... that they come into our country as an act of love.

With all of the problems we that we have, in so many instances -- we have wonderful people coming in. But with all of the problems -- this is not an act of love. He's [Jeb Bush -- dw] weak on immigration -- by the way, in favor of Common Core, which is also a disaster, but weak on immigration.

Meanwhile, this is what I wrote last year about today's lesson:

Lesson 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 (Lesson 2-5, Definitions) in terms of unions (Lesson 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.