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

Today is the second day of subbing in a high school Spanish classroom. Once again, there is no real instruction today, though I do help students with their math assignments in all classes.

(Also, tomorrow is Dr. Seuss Day, celebrated on the author's birthday. Since most elementary schools are closed on Saturdays, I assume that Dr. Seuss was observed today by most young students.)

Chapter 12 of the U of Chicago text is called "Similarity." I've written a little on similarity lately, since students in my new district have just finished Chapter 7 of the Glencoe text, also on similarity. (As I find out today while helping the students out in the classes today, they're now in Glencoe Chapter 8, on trigonometry.)

This is also the chapter from which I started posting two years ago, after I left my old job. Yes, it's now officially two years since I left.

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. [2019 update: And today's an activity day, so I'll actually keep this one.]

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.