Monday, March 2, 2015

Section 8-1: Perimeter Formulas (Day 122)

We now proceed in the U of Chicago text with Chapter 8, which is on measurement formulas -- such as those for perimeter and area. Recall the distinction between metric geometry, or geometry with measurements, with non-metric geometry without measurements. Well, we are definitely in the metric chapters right now. I saved the harder metric geometry until now, since the measurement formulas are notoriously difficult to remember.

Here is my plan for Chapter 8:

Today, March 2nd -- Section 8-1: Perimeter Formulas
Tomorrow, March 3rd -- Section 8-2: Tiling the Plane
Wednesday, March 4th -- Section 8-3: Fundamental Properties of Area
Thursday, March 5th -- Section 8-4: Areas of Irregular Regions
Friday, March 6th -- Activity Day
Monday, March 9th -- Section 8-5: Areas of Triangles
Tuesday, March 10th -- Section 8-6: Areas of Trapezoids
Wednesday, March 11th -- Review for Chapter 8 Test
Thursday, March 12th -- Chapter 8 Test
Friday, March 13th -- Pi Day Special (based on Sections 8-8 and 8-9)

Notice that Section 8-7, on the Pythagorean Theorem, has been omitted. This is because the U of Chicago text uses area to prove the theorem here in Chapter 8 while the Common Core Standards prefer that we that we use similar triangles to prove it. This is why we've already discussed the Pythagorean Theorem while covering the similarity in Chapters 11 through 14.

On the other hand, now that we've reached the month of March, we notice that the most important day of this month has been highlighted. No, I don't mean St. Patrick's Day or the spring equinox -- of course, I mean Pi Day. This year Pi Day -- March 14th -- falls on a Saturday, so it's a day on which most students won't go to school. So the day that Pi Day will be observed in most math classes this year will be Friday, March 13th.

To me, Pi Day should be the highlight of the school year -- the very best day to be a student in any math class in which the constant pi is a part of the curriculum. I've mentioned before that if possible, I go through pains to make sure that the chapter of the text in which pi appears is taught during the month of March. So this is another reason why I waited until now to cover Chapter 8.

Notice that according to the pattern that I've established, Friday, March 13th ought to be the day of the Chapter 8 Test. But there's no way that I'm giving a test on Pi Day Observed. That day should be a party day, not a test day. And so I break my pattern and give a test on Thursday -- and this test will cover the first two-thirds of the chapter -- the parts of the chapter that have nothing to do with pi.

My Pi Day lessons will be based on the lessons of Drs. Franklin Mason and Hung-Hsi Wu. But there is a way to squeeze in one of Wu's lessons this week. You see, Wu discusses how to estimate the area of the unit disk by placing it on a rectangular grid -- essentially using the areas of the rectangles to approximate the area of the circle.

Notice that this is basically what happens in Section 8-4 of the U of Chicago text! In this section, square grids are used to approximate the areas of irregular regions -- most of these are either lakes or, eventually, triangles, in anticipation of Section 8-5. I'm very surprised that the U of Chicago doesn't place a circle on one of the grids to approximate its area! Of course, when I create the worksheet for that lesson, I will include a circle area problem, in anticipation of Pi Day and in accord with the way that Wu teaches the concept of pi.

Yes, I'm sure you can tell how excited I am about Pi Day! But it's not Pi Day yet, and we're not in Section 8-8 or even 8-4 yet. Section 8-1 of the U of Chicago text is on perimeter formulas. But this is so straightforward that there's nothing much to say -- which is why I felt that I could waste most of this post discussing Pi Day. There is only one definition in this section:

The perimeter of a polygon is the sum of the lengths of its sides.

And then there's only one formula. Notice that this is the first of several times that the important word "formula" appears in Chapters 8 through 10:

Equilateral Polygon Perimeter Formula:
In an equilateral polygon with n sides of length s, the perimeter p = ns.

Notice that the formula is stated for equilateral polygons. All regular polygons are equilateral, but not all equilateral polygons are regular. The text points out that a rhombus is an equilateral quadrilateral, but it isn't regular unless it's a square. Of course, all equilateral triangles are regular. The formula is stated for equilateral polygons because we don't care whether the angles are congruent or not -- all that matters is the congruence of the sides.

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