Tuesday, March 10, 2015

Section 8-6: Areas of Trapezoids (Day 128)

Last night I tutored my geometry student. Over the weekend, his teacher assigned him a take-home exam for Chapter 6 of the Glencoe text.

My student told me that he had trouble with just one question on this test -- and it was about special right triangles, like the 30-60-90 triangle. This question was as surprising to me as it was to him, since 30-60-90 triangles don't ever appear in Chapter 6 of the Glencoe text! And so both of us wondered, why did the teacher include this item on the test?

I suspect that the teacher obtained the test from another source -- perhaps it was another textbook in which special right triangles appear in the same chapter as quadrilaterals. I'm not sure what text would present the material in that particular order. For example, in the U of Chicago text, we know that quadrilaterals appear in Chapter 5, with some info in Chapter 7, while special right triangles must wait all the way until Section 14-1.

It's possible that the teacher's test comes is based on the Prentice-Hall text. Indeed, David Joyce writes that 45-45-90 and 30-60-90 triangles appear as early as Chapter 5. Ironically, the main topic in Prentice-Hall's Chapter 5 is area -- the topic of U of Chicago's Chapter 8, the chapter that we are currently discussing on the blog. Both Prentice-Hall Chapter 5 and U of Chicago Chapter 8 use area to derive the Pythagorean Theorem. But U of Chicago waits six more chapters before using the Pythagorean Theorem to derive the 45-45-90 and 30-60-90 theorems while Prentice-Hall derives them right away.

Quadrilaterals, meanwhile, don't appear until Chapter 9 in the Prentice-Hall text. So I can easily see a teacher taking a test based on Chapter 9 of the Prentice-Hall text and assigning it to a class right after that class finished Chapter 6 of the Glencoe text, since Prentice-Hall Chapter 9 and Glencoe Chapter 6 seem to correspond. I can easily see a teacher doing this if the school recently switched from the Prentice-Hall to the Glencoe text, The textbooks are new but the teacher still has tests based on the old text on file.

Many math test writers like to include questions from prior chapters on their tests, so a question from the quadrilateral chapter may require knowledge of a prior chapter, This is a good practice -- math isn't a set of isolated facts,. Instead the information ultimately connects -- and many questions that appear on PARCC or SBAC tests may require knowledge that comes from several different lessons, not a single lesson, in a text. The problem is that what defines a prior lesson is textbook-dependent -- a lesson that may be prior in one text may not be prior in another text. And so a test designed for a text in which special triangles are taught prior to quadrilaterals will confuse a student who learned from a text in which special triangles aren't taught prior to quadrilaterals.

I don't know the exact question that my student had to answer. Perhaps he had to use a 30-60-90 to find the hypotenuse, which happened to be the side of a parallelogram as well. At any rate, since it was a take-home test, he ended up searching for info about 30-60-90 triangles online.

After turning in the Chapter 6 Test, my student began the next chapter in Glencoe, which is on proportions and similarity. Let's see how it is organized:

Section 7-1: Ratios and Proportions
Section 7-2: Similar Polygons
Section 7-3: Similar Triangles
Section 7-4: Parallel Lines and Proportional Parts
Section 7-5: Parts of Similar Triangles
Section 7-6: Similarity Transformations
Section 7-7: Scale Drawings and Models

Clearly, this corresponds to Chapter 12 of the U of Chicago text. The first two sections in Glencoe correspond to Sections 12-4 and 12-5 in the U of Chicago. All three Similarity statements (AA, SAS, and SSS) appear in Glencoe's 7-3, so this is both 12-8 and 12-9 for U of Chicago. Glencoe's 7-4 introduces a Triangle Proportionality Theorem -- this turns out to be the same as the Side-Splitting Theorem in U of Chicago's 12-10. Glencoe's 7-5 introduces a Triangle Angle Bisector Theorem -- I've seen references to this theorem before, but it doesn't appear in many other texts, including the U of Chicago, so 7-5 has no correspondence. Notice that Glencoe's 7-6 has similarity transformations, including dilations -- and it actually uses the term "dilation," unlike the U of Chicago which simply calls them "size changes." So Glencoe's 7-6 is like U of Chicago's 12-1, except, of course, that 12-1 reflects the Common Core's use of dilations to define similarity. Finally, the last section in Glencoe is incorporated into U of Chicago's 12-5 and 12-6.

My student just barely started Section 7-1 in Glencoe. But notice that I didn't have a worksheet from U of Chicago's Section 12-4, on proportions. This is because we needed to spend so much time on deriving the properties of similarity from dilations following Dr. Hung-Hsi Wu that we ran out of time simply working on proportions, as Glencoe's 7-1 and U of Chicago's 12-4 do.

I now regret my decision to skip Section 12-4 -- especially seeing that my student seemed to understand the concept of similarity well, but had trouble setting up and solving the proportions. As the ability to set up and solve proportions is critical to success in the similarity chapter, I plan on including 12-4 in my similarity unit next year. The other error that my student discovered when I gave him my 12-5 worksheet (in preparation of Glencoe 7-2) is how I sometimes wrote "Review 9." at the beginning of a question if Question 9 is review, but "9. Review." is more elegant. I also gave my student the 14-1 lesson so he could learn about 30-60-90 and 45-45-90 triangles.

Meanwhile, today I subbed in a middle school math class. Most of the classes were sixth grade classes ,but during the conference period, I filled in for a seventh grade math teacher. Indeed, all of the math teachers were at a meeting to preview new texts for Common Core math.

The seventh graders were studying percents. Here is a seventh grade standard for percents:

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

The sixth graders, meanwhile, were studying simple one-step equations. Here is the relevant Common Core standard:

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which pq and x are all nonnegative rational numbers.

I played my usual points game, with the questions in both the sixth and seventh grade classes coming from the popular Pizzazz worksheets, where the answers correspond to letters that spell the answer to a riddle. Now the sixth graders did well with most of the questions, with these questions being of the form x + p = q or x - p = q, with p and q whole numbers.

But there was one type of question on which they struggled: 5 - x = 2. In Algebra I, one learns that to solve this, one must subtract five from both sides to obtain -x = -3, and then multiply or divide both sides by -1 to obtain the solution x = 3. But this question is too advanced for a sixth grader, We notice that word "nonnegative" appearing in the standard above. Some sixth graders have heard of negative numbers -- one group with zero points was afraid that I'd deduct a point from them, so they'd have a score of -1 -- and indeed, they are mentioned in the Common Core standards:

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

But applying negative numbers to the equation 5 - x = 2 only confused them. Last night's homework assignment contained problems such as 5 + x = 12, which can be rewritten as x + 5 = 12 since addition is commutative, but 5 - x = 2 can't be written as x - 5 = 2. Naturally, some students tried to do this anyway, and couldn't understand why x = 7 isn't a valid solution to 5 - x = 2. Even if we allowed p and q to be negative, 5 - x = 2 differs from the others in that the coefficient of x is not 1 but -1, so that 5 - x = 2 is strictly a two-step, not one-step, equation.

The problem with 5 - x = 2 occurred in third period. So during subsequent classes, I just skipped questions such as that altogether -- I just told them the correct answer so that they can obtain the letter for the Pizzazz puzzle. I even skipped questions like 5 + x = 12, since doing such questions would only lead to the students overgeneralizing about 5 - x = 2. To make sure that I had the students' attention while explaining why I was skipping some of the problems, I reordered the questions of my game so that the "guess my weight" question occurred before the the first problem to be skipped, and I told them that I was skipping the question before revealing my weight. Recall that the purpose of the "guess my weight" question is to grab the kids' attention, and I was using that question for that exact purpose today.

In some ways, both my geometry student and the sixth graders had the exact same problem -- they were completing worksheets that didn't correspond to what was taught in class. My geometry student had 30-60-90 triangles when he hadn't learned them yet, and my sixth graders had to solve two-step equations with -x when they hadn't learned them yet. It's possible that the sixth graders had to solve such equations as 5 - x = 2 under the old California standards, where Algebra I is expected in eighth grade, but relaxed under Common Core where the normative age for Algebra I is ninth grade.

Once again, today's officially my lesson for 8-6, but I'm spending more time discussing stuff that has nothing to do with 8-6 (but at least I'm discussing geometry, unlike yesterday). Well, section 8-6 of the U of Chicago text is on areas of trapezoids. Once again, this is pretty straightforward, but here are some of the key issues:

-- The text derives the trapezoid area by dividing it into triangles. The text refers to dividing a polygon into triangles as triangulating the polygon.

-- The text uses inclusive definitions, so a parallelogram is a trapezoid. If you're wondering why there's a section for areas of trapezoids but not of parallelograms, this is why. Recall that the most useful fact about a trapezoid that isn't isosceles is its area formula.

Actually, here's another issue about inclusive definitions. Sometimes, it appears that students more easily recognize that squares and rhombuses are both kites (as I mentioned in yesterday's lesson) than that parallelograms are trapezoids. My student saw a picture of a baseball diamond on my 14-1 worksheet and kept referring to it as "kite" and "rhombus." This isn't the first time that I've heard a student call a rhombus a "kite," despite the students using texts with exclusive definitions.

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