Friday, January 19, 2018

Lesson 9-2: Prisms and Cylinders (Day 92)

Lesson 9-2 of the U of Chicago text is called "Prisms and Cylinders." Our text refers to both prisms and cylinders as "cylindric surfaces."

Before we begin the lesson, let me tell you that when I woke up this morning, the song playing on my radio was "No Scrubs." Exactly one year ago today, I sang the parody "No Drens" in my class. (I didn't post the song until January 24th, but I did originally sing it in class on the 19th, ahead of the 5's Dren Quiz on the 20th.)

Actually, this version was a remix by Unlike Pluto and Joanna Jones. There aren't any significant differences in lyrics between this and the original TLC version -- but still, I didn't know that a new version had been released until this morning. As I wrote back in my January 6th post, I need to be careful with how I talk about "drens" in class. Perhaps I should add another verse emphasizing that a dren is someone who is too lazy to do basic math, in order to avoid offending weaker students -- and instead, encourage those weaker students to keep working hard on their math.

The following is based on the third verse as sung by TLC and Joanna Jones. The lines that have been changed are in CAPS:

If you don't have a car and YOU WANNA AFFORD ONE,
Oh yes, I'm talking to you.
Oh yes, I'm talking to you.
If you have your MATH but you don't show no WORK,
Oh yes, I'm talking to you.
Wanna get HIRED TO EARN MORE money,
Oh yes, I'm talking to you.

Here I dropped "son" (as in "oh yes son") since unlike scrubs, drens can be of either gender. But the first line still seems a little off.

This lesson consists mainly of definitions. Terms defined in this lesson are surface, solid, box, rectangular solid, faces, opposite faces, edges, vertices, and skew lines -- and that's just the first page!

Another term defined in this lesson is parallelepiped. Recall from my January 6th post that Pappas uses this term on her calendar. (By the way, I haven't posted any Pappas Geometry problems in a while, but there will be some questions coming up next week.)

Today is also an activity day. I want to establish the habit of using the Exploration Questions in the current section as homework, so here they are for Lesson 10-2:

28. Some prisms have a special property relative to light? What is this property?
29. The cells in honeycombs of bees are in the shapes of hexagonal prisms. Why do bees use this shape?

So this is a good research activity for the students. For you teachers, here are links to the answers:

The second link contains a video. Of course, students can find videos to answer the first as well.

Let's get back to Euclid's Elements, since I've started discussing it yesterday. We can finally look at the first two definitions in Book XI:

Definition 1.
solid is that which has length, breadth, and depth.
Definition 2.
A face of a solid is a surface.

Both Euclid and the U of Chicago distinguish between the boundary and the interior.

Here are the definitions of the two surfaces mentioned in the lesson title:

Definition 13.
prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms.
Definition 21.
When a rectangular parallelogram with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder.
Definition 22.
The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.
Definition 23.
And the bases are the circles described by the two sides opposite to one another which are carried round.

It's interesting to compare Euclid's definitions to the U of Chicago's. Euclid imagines a cylinder as a solid of revolution -- we take a rectangle and rotate it around one of its sides. (Don't forget that a rotation in 3D has an axis, not a center.) But in the U of Chicago text, we perform a very different isometry -- we begin with a circle and translate it out of the plane. David Joyce points out that Euclid's definition doesn't generalize -- it describes only right cylinders. Not only does the U of Chicago definition include oblique cylinders, but the idea of cylindric solids (or surfaces) extends to prisms as well.

By the way, Euclid defines a prism in terms of its faces. Since the lateral faces of Euclid's prisms are parallelograms (and not necessarily rectangles), oblique prisms are included.

Let's return to the theorems. Since we looked at Proposition 4 yesterday, let's try Proposition 5 today:

Proposition 5.
If a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane.

Here's a modern version of the proof. Notice that Euclid writes "For suppose they do not...," which implies that this is an indirect proof.

Given: Line AB perpendicular to BC, BD, and BE.
Prove: Lines BC, BD, and BE are coplanar.

Indirect Proof:
Assume to the contrary that they aren't. By the Point-Line-Plane Postulate part f, any three points are coplanar, and so let P be the plane containing B, D, E, and let Q be the plane containing A, B, C.

By Point-Line-Plane part g, since planes P and Q intersect (in B), they intersect in a line. Let F be another point on this line. So lines AB, BC, and BF are coplanar as they all lie in plane Q.

We are given that AB is perpendicular to both BD and BE, and so by yesterday's Proposition 4, AB is perpendicular to the entire plane containing BD and BE, namely P. By the definition of a line perpendicular to a plane, AB is perpendicular to every line in P through B, which includes BF.

But we are given that AB is also perpendicular to BC. Therefore AB is perpendicular to two lines, BC and BF, both in plane Q. This is a contradiction, since through a point on a line, there is exactly one line in the plane perpendicular to the line. Thus the assumption that BC, BD, and BE aren't coplanar is false. Therefore BC, BD, and BE are coplanar. QED

This proof isn't as difficult as yesterday's, but it is an indirect proof -- and we don't cover indirect proofs in our text until Lesson 13-4. It also requires yesterday's Proposition 4 in order to prove -- and Proposition 4 has a difficult proof.

Here is the new worksheet for today's vocab and activity:

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