Lesson 12-1: Size Changes on a Coordinate Plane (Day 121)

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

The surface area of a sphere of radius r divided by the area of a circle of radius r.

This is straightforward provided we know the formulas. The sphere surface area formula is 4pi r^2, and the circle area formula is pi r^2. Thus the desired ratio is 4 -- and of course, today's date is the fourth. In fact, Lesson 10-9 of the U of Chicago text points this out:

"Surprisingly, this formula indicates that the surface area of a sphere is equal to 4 times the area of a great circle of the sphere."

Lesson 12-1 of the U of Chicago text is called "Size Changes on a Coordinate Plane." In the modern Third Edition of the text, size changes on a coordinate plane appear in Lesson 3-7. Yes, Chapter 12 is officially the same in both editions, but for some reason, the new edition introduces transformations involving size ("dilations") very early in the text. Beginning with the old Lesson 12-3, most of the old Chapter 12 material does indeed appear in the new Chapter 12 as well.

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

In the past, I skipped over Lesson 12-1. This is because I was mainly concerned with circularity -- dilations are used to prove some of the properties of coordinates, but right in this lesson, coordinates are used to prove the properties of dilations.

But last year, I was fed up with juggling the order of the U of Chicago text (and I got in trouble trying to juggle the Illinois State text as well). This year I want to stick to the order as intended by the authors of the U of Chicago text. And furthermore, we've seen that the actual dilation problems on the PARCC and SBAC involve performing dilations on a coordinate plane -- not using dilations to prove properties of coordinates! So Lesson 12-1 is more in line with PARCC and SBAC.

Here is the main theorem of Lesson 12-1 along with its coordinate proof:

Theorem:

Let S_k be the transformation mapping (x, y) onto (kx, ky).

Let P' = S_k(P) and Q' = S_k(Q). Then

(1) Line P'Q' | | line PQ, and

(2) P'Q' = k * PQ.

Proof:

Let P = (a, b) and Q = (c, d) be the preimages.

Then P' = (ka, kb) and Q' = (kc, kd).

(1) Line P'Q' is parallel to line PQ if the slopes are the same.

slope of line P'Q' = (kd - kb) / (kc - ka) = k(d - b) / k(c - a) = (d - b) / (c - a)

slope of line PQ = (d - b) / (c - a)

Thus line PQ | | line P'Q'.

(2) The goal is to show that P'Q' = k * PQ.

From the Distance Formula,

PQ = sqrt((c - a)^2 + (d - b)^2).

Also from the Distance Formula,

P'Q' = sqrt((kc - ka)^2 + (kd - kb)^2)

        = sqrt((k(c - a))^2 + (k(d - b))^2)      (Distributive Property)

        = sqrt(k^2(c - a)^2 + k^2(d - b)^2)    (Power of a Product)

        = sqrt(k^2((c - a)^2 + (d - b)^2))       (Distributive Property)

        = sqrt(k^2)sqrt((c - a)^2 + (d - b)^2) (Square Root of a Product)

        = ksqrt((c - a)^2 + (d - b)^2)              (Since k > 0, sqrt(k^2) = k)

        = k * PQ                                             (Substitution) QED

At the end of this post, it's back to posting worksheets based on the U of Chicago text. This time, I post an activity from last year where students dilate cartoon characters. This activity makes more sense this year than last year since it requires using coordinates.

Oh, and before you ask, my next pandemic-friendly activity won't be until tomorrow's post.

This lesson could've actually helped my eighth graders as well. We were supposed to cover dilations earlier but we ran out of time. For that matter, slope and the Distance Formula are also part of the eighth grade curriculum. I wouldn't make eighth graders perform the two preceding proofs with so many variables, but specific numerical examples are within the reach of eighth graders.