Find Perimeter, Circumference, and Area (Day 18)

Today I subbed in a high school math class. This teacher had two sections of Statistics and three periods of Geometry. Naturally, the stats classes are mostly seniors. Meanwhile, this district is in the second year of its transition to Integrated Math, with both Math I and Math II now offered. So now the Geometry classes are mostly juniors grandfathered in to the old traditionalist Geometry class.

This is a Geometry blog, so of course I want to focus on the Geometry classes, but I will briefly discuss the stats classes. Note that these are regular Statistics classes, so they're not as rigorous as AP Stats, but they're well beyond simply finding the mean, median, and mode of a data set (thinking back to last week's Final Jeopardy question at the continuation high school).

The stats text is The Practice of Statistics, published by W.H. Freeman and Company. The students are working on a Chapter 1 Review worksheet -- presumably to prepare for an upcoming test. They had to describe distributions with numbers -- in addition to the mean and median, they also had to calculate the variance, standard deviation, five-number summary, and outliers. Most of them did well on the worksheet, but they did struggle with this one (excuse my ASCII here):

find sigma _ i=1 ^ n (x_i - x)^2

This means to find the sum of the squared deviations -- the first step to calculating the variance or standard deviation of the set. The text is geared towards using a TI-83 calculator (which shows how old this book is -- most students use TI-84 nowadays), but still this problem requires much work. One of the deviations is .02, with its square .0004 -- which the TI-84 writes as 4E-4, which only confused the students further. Another tricky question for the students is:

a distribution of 10 quiz sores has a mean of 22. If the highest score increases by 5 points, then the new mean will be _____.

Notice that if one score out of ten increases by five points, the mean must increase by 5/10 points. I had the students imagine that all of the scores were 22 and then one moved up to 27. Hopefully that made it easier for the students to visualize.

Before I leave stats, notice that some of this material now appears in Common Core Algebra I -- maybe not standard deviation, but measures of central tendency, five-number summary, outliers, and boxplots appear. Even the eighth grade text I mentioned last week had a chapter on Data Analysis. In some ways, Statistics is more relevant for students to learn than, say, the Quadratic Formula.

Now let's get to the Geometry classes. The students are working out of the McDougall Littell Geometry text -- this is definitely a pre-Core text, as this is a grandfathered class. Here is the table of contents for the McDougall Littell text:

1. Essentials of Geometry
2. Reasoning and Proof
3. Parallel and Perpendicular Lines
4. Congruent Triangles
5. Relationships within Triangles
6. Similarity
7. Right Triangles and Trigonometry
8. Quadrilaterals
9. Properties of Transformations
10. Properties of Circles
11. Measuring Length and Area
12. Surface Area and Volume of Solids

This chapter sequence is very nearly identical to that of the Glencoe text that I've discussed many times on the blog. The main difference is that quadrilaterals, Chapter 6 in the Glencoe text, are delayed to Chapter 8 in the McDougal Littell text. This pushes similarity and trig up a chapter -- if the goal is to complete six chapters each semester, then similarity appears at the end of the first semester and trig at the start of the second semester. I'd much rather keep quadrilaterals, which should be easier, in the first semester.

The students are working on Lesson 1.7 today. Let's look at the first chapter in more detail:

1.1 Identify Points, Lines, and Planes
1.2 Use Segments and Congruence
1.3 Use Midpoint and Distance Formulas
1.4 Measure and Classify Angles
1.5 Describe Angle-Pair Relationships
1.6 Classify Polygons
1.7 Find Perimeter, Circumference, and Area

And so the students are working on perimeter, circumference, and area. Once again, this shows how rigorous Chapter 1 is in many Geometry texts -- even though area doesn't formally appear until Chapter 11, there's still a lesson on it in here in the first chapter. Fortunately, only the formulas for rectangles (including squares), triangles, and circles appear in this lesson.

I wasn't originally going to post a lesson on perimeter and area today. But one purpose of having a math teaching blog is to discuss how I interact with the students -- which means that the lesson that I teach in an actual classroom takes precedence over my original pacing plans. If I could have chosen any lesson from this chapter to teach today, it would be Lesson 1.4 -- after all, I did say last week that I'm getting ready to teach angles this week. Lessons 1.4 and 1.5 of the McDougal Littell text correspond to Lessons 3-1 through 3-3 of the U of Chicago text. Recall that the U of Chicago has the opposite problem -- it waits until Chapter 3 to teach angles when it appears in Chapter 1 in most other Geometry texts. This explains why this blog is about to start Chapter 3 when actual classrooms are still in Chapter 1 -- Chapter 3 of my own text has the same content as Chapter 1 of the district's text.

Then again, it could be worse. The McDougall text is yet another text that presents the Distance Formula in Chapter 1 -- well before the Pythagorean Theorem appears in Chapter 7. Let's remind ourselves what Dr. David Joyce has to say about texts that teach the Distance Formula in Chapter 1:

Also in chapter 1 there is an introduction to plane coordinate geometry. Unfortunately, there is no connection made with plane synthetic geometry. Here in chapter 1, a distance formula is asserted with neither logical nor intuitive justification. Of course, the justification is the Pythagorean theorem, and that's not discussed until chapter 5. In that chapter there is an exercise to prove the distance formula from the Pythagorean theorem. The Pythagorean theorem itself gets proved in yet a later chapter.

In summary, the constructions should be postponed until they can be justified, and then they should be justified. The same for coordinate geometry.

I find it maddening whenever I see yet another text give the Distance Formula in Chapter 1. Indeed, I wonder why so many publishers think that this is a good idea. I suspect it's because distance -- that is to say, length -- is actually a basic concept that belongs in Chapter 1, and textbook writers feel that they haven't adequately covered length unless they present the Distance Formula. I'm certainly grateful that the U of Chicago waits until Chapter 11 to teach the Distance Formula -- well after the Pythagorean Theorem appears in Chapter 8.

And we've seen that the Glencoe text contains yet another lesson in its first chapter -- there's actually a lesson on three-dimensional figures, surface area, and volume! Fortunately for my students today, the McDougal Littell text doesn't contain that section -- they only have to deal with two dimensions.

So all things being considered, today's lesson isn't terrible. It's not as if the students have never seen these formulas before -- indeed, they are included in the middle school curriculum. Common errors I saw today include more or less what you expect -- forgetting to multiply by 1/2 to find the area of the triangle, forgetting to multiply by 1/2 when the diameter is given to find the radius, and, of course, trouble with units, forgetting square units for the area.

Therefore going around to help students becomes a little tricky. The easiest thing for me to notice is when someone doesn't put the units on correctly. But then I'm spending so much time telling students to write mi.^2 and cm^2 that others may be having trouble with the formula and I don't notice. One student wrote mi.^2 for the area but forgot to include units for the perimeter. But then I told her this, she hypercorrected the error by writing mi.^2 for the perimeter as well! Unfortunately, by the time I noticed this, it was time for me to collect the papers.

My posting of McDougall Littell's Lesson 1.7 means that my lesson on angles will be pushed back until tomorrow. I should have no problem compressing Lessons 3-1, 3-2, and 3-3 into two lessons -- after all, McDougall Littell needs only 1.4 and 1.5 to present the same material.

Today I post both the notes given by the teacher and the worksheet itself. Notice that the students only find the area of the triangle, not the perimeter -- which is good, since finding the perimeter of a triangle often requires the Distance Formula.