Wednesday, March 22, 2017

Lesson 12-4: Proportions (Day 124)

Lesson 12-4 of the U of Chicago text is on proportions. It's easy to figure out why there would a lesson on proportions in a chapter on dilations and similarity.

But I don't have any worksheet from last year to post. This is because I was always pressed for time whenever we reached this lesson, so I always omitted Lesson 12-4 in order to reach the lessons containing actual geometry.

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:

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:

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:

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

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

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