Lesson 8-1: Perimeter Formulas (Day 81)

Today I subbed in a high school art class. It's not the same class as the one mentioned in Thursday's post -- indeed, that day was painting, while today is ceramics. In either case, since it's high school art, there's no need for "A Day in the Life."

The students are working on individual projects. Many of them are creating whistles, while some students are creating more interesting pieces, including animal shapes. One hard working guy is making a Pokemon whistle.

As far as behavior is concerned, the regular teacher warns me that there is a clean-up bell that rings five minutes early. This is a double-edged sword -- it's good to give the students time for cleaning (and not leave a mess), but if we give the students too much time to clean up, they might interpret the clean-up bell as a dismissal bell and try to leave early. (This was a problem in the art class that I subbed for last week as well.)

And in fact, in one class a few students start cleaning up with 15-20 minutes to go. This is on a Monday late day, and so 15-20 minutes is at least one-third of the period! (In all classes, there's also the usual problem where some students claim to be done with everything.) As usual, my solution involves songs. I sing "One Billion Is Big" to all classes -- and in the class where some students clean up early, I add in "Twelve Days of Christmath."

Speaking of Christmath -- um, Christmas -- today I begin passing out holiday pencils and candy. I give them out to the students with the best-looking art or appear to be hardest-working.

Today our Chapter 8 weirdness begins. The change in the district calendar requires that we cover Lessons 8-1 and 8-2 before the final. Chapter 8 is a tough chapter, but fortunately, these first two lessons are relatively easy.

Nowadays, we study Chapter 8 in January, and so the pi lesson no longer falls on Pi Day. And now we must study the first two lessons of Chapter 8 in December.

Of course, even though we're studying Lesson 8-1 today, this doesn't mean that students shouldn't still be preparing for the final. The worksheets I provided at the end of last week should provide sufficient preparation for the big test.

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, December 16th -- Lesson 8-1: Perimeter Formulas
Tomorrow, December 17th -- Lesson 8-2: Tiling the Plane
Wednesday-Friday, December 18th-20th -- Finals Week
Monday-Monday, December 23rd-January 6th -- Winter Break
Tuesday, January 7th -- Lesson 8-6: Areas of Trapezoids
Wednesday, January 8th -- Lesson 8-7: The Pythagorean Theorem
Thursday, January 9th -- Lesson 8-8: Arc Measure and Arc Length
Friday, January 10th -- Lesson 8-9: The Area of a Circle (plus Activity)
Monday, January 13th -- Chapter 8 Test

Notice that Lessons 8-3 through 8-5 have been omitted. Of these, the most important lesson is 8-5, on the areas of triangles. I'll find a way to squeeze this lesson in after winter break.

My pi lessons will be based on the lessons of Drs. Franklin Mason and Hung-Hsi Wu. 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! And since Lesson 8-4 is one of the skipped lessons this year, following the Wu lesson could allow us to squeeze Lesson 8-4 in after all.

But that's enough about pi -- let's get to today's Geometry lesson. 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 lessons. There is only one definition in this section:

Definition:
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.