## Thursday, March 8, 2018

### Lesson 12-4: Proportions (Day 124)

Lesson 12-4 of the U of Chicago text is called "Proportions." In the new Third Edition of the text, proportions appear in Lesson 12-2.

Today I subbed at a high school. It's not a math class, and so there's no "Day in the Life." What makes today strange is that I actually subbed in my old district -- the one whose calendar we're observing on the blog. (So today really is Day 124 in this district!) Recall that I never said that I was leaving my old district -- only that I could get far more calls by working in a new district.

This school follows a block schedule, with even classes on Tuesdays and Thursdays. (Odd classes are Wednesdays and Fridays, while Mondays are Common Planning/All Classes days.) It's a special ed class with second period English, fourth period Economics, and sixth period conference. The classes are extremely small -- I see only seven students total today (including one student common to both second and fourth periods).

Actually, fourth period is Econ only for the two seniors in the class -- the other students treat the class as a study hall. One of these students is working on math -- so of course I help him out. He is learning to factor in Algebra I. The aide tells me that he is falling well behind in his class -- his packet of work is blank, while the rest of the class has moved on to a new packet. And since today is Day 124, third quarter grades are right around the corner. I'm able to help him start to catch up today.

Meanwhile, today on her Mathematics Calendar 2018, Theoni Pappas writes:

If BC | | AD, find the area of ABCD = ?

(Other info given in the picture: AB = 2, BC = 10, AD = 6, Angle ABC = 150.)

Let's see. Since quadrilateral ABCD has two parallel sides, it's by definition a trapezoid. Thus we can use the formula for the area of a trapezoid, A = (1/2)h(b_1 + b_2). The lengths of the two bases are given as 6 and 10, but we don't know the height yet.

Drop a perpendicular to AD from side BC and label its foot E, so BE is the height we need. Since Angle ABC = 150, Angle ABE = 60, and so Triangle ABE is a 30-60-90 triangle. As its hypotenuse AB is 2, BE, the side opposite the 30-degree angle, must be 1. Now we can find the area:

A = (1/2)(1)(6 + 10)
A = 16/2
A = 8

Therefore the area of the trapezoid is 8 square units -- and of course, today's date is the eighth.

In the U of Chicago text, the area of a trapezoid appears in Lesson 8-6. But finding a height requires solving a 30-60-90 triangle, and special right triangles don't appear until Lesson 14-1. Meanwhile, for students at my new district working out of the Glencoe text, they might be currently working in Chapter 8 on special right triangles -- but then area doesn't appear until later in the text. So none of our students will be able to solve this problem yet.

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

It isn't too difficult to find proportion worksheets online, of course. A Google search for a proportions worksheet gives the following Kuta page as the fourth result:

https://cdn.kutasoftware.com/Worksheets/PreAlg/Proportions.pdf

Notice that Kuta considers this to be a Pre-Algebra topic, and in fact I did cover proportions earlier in some of my middle school classes. So I could give the above worksheet in my class.

But when I taught proportions to my sixth graders, I didn't teach cross-multiplication. Instead, I used tape diagrams and double number lines in accordance with the following Common Core standard:

CCSS.MATH.CONTENT.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Also notice that if we follow the standards in naive order, as the Illinois State Student Journal does, then it would be awkward to show the sixth graders equations for Standard RP3 before they see equations for the very first time in Standard EE5.

By the way, I don't necessarily have time to get into a big traditionalist debate today, but the traditionalist Barry Garelick, quoting the late Ralph Raimi, has something to say about teaching proportions using methods other than equations:

https://traditionalmath.wordpress.com/2017/02/12/ralph-raimi-on-proportionality-whatever-that-means/

There are some topics that even the most bleeding-heart progressive will use equations to teach -- for example, no one is going to avoid a^2 + b^2 = c^2 when teaching the Pythagorean Theorem. But proportions is one topic where there is a strong progressive resistance to equations, especially in the early middle school years.

Still, Raimi, like most traditionalists, preferred using equations and wished that his teachers used equations from the very beginning:

What was proportional to what no longer needed
to concern me; the relations dictated by the problem led to airtight equations
about the meaning of which I no longer had to think. It was, to me, a
liberation.

Of course, for high school Geometry lesson, the students obviously should have seen proportion equations by now, so equations are preferred for this lesson.

Last year I only linked to the worksheet but didn't post it. This year I'm actually posting it.