Lesson 10-1: Surface Area of Prisms and Cylinders (Day 101)

Chapter 10 of the U of Chicago text is on surface areas and volumes. Measurement is usually the focus of the three-dimensional chapters in a Geometry text, not Euclid's propositions that we've been discussing the past two weeks.

Lesson 10-1 of the U of Chicago text is called "Surface Areas of Prisms and Cylinders." In the modern Third Edition of the text, surface areas of prisms and cylinders appear in Lesson 9-9.

Three years ago, I actually borrowed a lesson from the (former) King of the MTBoS, Dan Meyer, and I repeated this post last year.

This means that if I were to repeat that activity this year, this would be yet another three-activity week, with the 2016 activity today, the 2018 activity tomorrow, and then this year's brand new activity on Friday. This year I'm taking charge of how many activities I'm including this year, and so the old activities from 2016 and 2018 will be dropped.

Once I drop the old activity, I don't have much to say about today's worksheet. I will say that I include the Exploration Questions as a bonus. One of them is open-ended -- don't let traditionalists see that problem, as they'll complain it's ill-posed. Everything else I have to say about lateral and surface area I mentioned in last week's parody song "All About That Base (and Height)."

Even though we're in Chapter 10 now, we might as well continue with Euclid. After all, David Joyce implies that he wouldn't mind teaching only "the basics of solid geometry" and throwing out surface area and volume altogether.

Proposition 14.

Planes to which the same straight line is at right angles are parallel.

This is another version of the Two Perpendiculars Theorem. Earlier, in Proposition 9, we had two line perpendicular to one plane, and now we have one line perpendicular to two planes. In all three theorems, two objects perpendicular to the same object are parallel.

Euclid's proof, once again, is indirect.

Indirect proof:
Point A lies in plane P and point B lies in plane Q, with AB perpendicular to both. Assume that planes P and Q intersect in point K. By the definition of a line perpendicular to a plane, AB is perp. to AK, and for the same reason, AB is perp. to BK. Then Triangle ABK would have two right angles, which is a contradiction since a triangle can have at most one right angle. (This is essentially Triangle Sum -- the two right angles add up to 180, so all three angles would be more than 180.) Therefore the planes P and Q can't intersect -- that is, they are parallel. QED

Euclid mentions a line GH where P and Q intersect, but Joyce tells us it's not necessary. Joyce also adds that Euclid forgot to mention the case where K, a point common to both planes, is actually on the original line that's perpendicular to them both. He doesn't tell us how to prove this case, but here's what I'm thinking -- consider a plane R that contains the original line. Planes P and R intersect in some line through K, and planes Q and R intersect in another such line, since the intersection of two planes is a line. By definition of a line perpendicular to a plane, these two new lines are both perp. to the original line. So in a single plane R, we have two lines perpendicular to a line through the same point K on the line, a contradiction.

Before I end this post, no, I'm not going to label three traditionalists' posts in a row. But as so often happens, with so much going on at Barry Garelick's blog, sometimes I forget to check the other websites where traditionalists often post:

https://www.joannejacobs.com/2019/01/wealthy-district-cuts-grade-level-math/

Apparently, a Northern California school district is on the verge of eliminating Common Core Math 8 for all eighth graders, thereby forcing all eighth graders to take Algebra I.

Many math teachers in the district are opposed to the change, since they know full well that many of their middle school students won't be able to handle Algebra I. The only people in favor of the change are -- you guessed it -- traditionalists like Ze'ev Wurman, who posted in the comments. Here's an excerpt of what he writes:

When California abandoned its 1997 standards in 2013, 2/3 of its 8th grade cohort took Algebra 1, with high and ever-increasing success rates. But if the goal is to have everyone with identical outcomes, the majority of students must be dragged down to the level of the worst 5-10%, which is what California has been busily doing.

Hmm, if 2/3 of the students are taking Algebra I, it follows that 1/3 is taking Math 8 -- and that's a lot more than the 5-10% that Wurman quotes. He then mentions a district that's doing the opposite of what the traditionalists want -- San Francisco, where all eighth graders take Math 8.

I have nothing more to say on this. We already know what the traditionalists want, and we've already had too many traditionalists' posts this week. Still, I wanted to link to this nearly week-old post before I forget about it.

Here is the worksheet for today's lesson: