Tuesday, March 1, 2016

Lesson 8-1: Perimeter Formulas (Day 113)

This is what I wrote last year about today's lesson:

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 1st -- Lesson 8-1: Perimeter Formulas
Tomorrow, March 2nd -- Lesson 8-2: Tiling the Plane
Thursday, March 3rd -- Lesson 8-3: Fundamental Properties of Area
Friday, March 4th -- Activity Day
Monday, March 7th -- Lesson 8-4: Areas of Irregular Regions
Tuesday, March 8th -- Lesson 8-5: Areas of Triangles
Wednesday, March 9th -- Lesson 8-6: Areas of Trapezoids
Thursday, March 10th -- Review for Chapter 8 Test
Friday, March 11th -- Chapter 8 Test
Monday, March 14th -- Pi Day Special (based on Lessons 8-8 and 8-9)

Notice that Lesson 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 Monday.

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.

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 next 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 Lesson 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 Lesson 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 Lesson 8-8 or even 8-4 yet. Lesson 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.

I have made one change to this worksheet since I first posted it last year. Notice that I am still mourning the loss of my grandmother. She lived much of her life in Kansas City, MO.

Last year's worksheet included a road map to various Midwestern cities, including the distances and driving times between the cities. Although several Missouri cities were listed, Kansas City was not among them. Today I honor my grandmother by adding Kansas City to the map.

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