Lesson 12-2: Size Changes Without Coordinates (Day 122)

Today I subbed in a seventh grade PE class. It's in my first OC district. Of course, it's another class for which it's not worth doing "A Day in the Life."

It's my first time I've covered for this teacher since a two-day assignment in February 2020. This teacher likes to begin each period by having the students walk two laps. Last year when I covered for him, I sang "The Big March" while walking, since it was the first week of that dreadful period. And this year it's the Big March again, so naturally I sing this same song today. To reduce the monotony, I add another simple song to our long walk -- the "Row" parodies "Measures of Center" and "Same Sign Add and Keep."

The walk is followed by a weekly "Friday stretch" -- the students do ten different stretches with 10-15 seconds of rest between them, and then they measure their heart rate. This teacher did nothing like this last year (even though one of the days I subbed for him was a Friday). I want to keep in shape, so I do the Friday stretch along with them -- meaning that I do it three times today.

Today is Fiveday on the Eleven Calendar:

Resolution #5: We treat people who are great at math as heroes.

Obviously this is a bit tricky in a P.E. class. Indeed, I don't really mention heroes or pay homage to anyone at all today. After the stretch, the students take turns participating in a 50-yard dash, and so the default hero is sprinter Usain Bolt.

The fastest sprinters among today's seventh graders run sub-7 seconds for the 50. Some of the faster girls can run it in sub-8.

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

What is ED/FG? ABCD is a square. E and F are midpoints [of BC and AD respectively].

[There's one more thing that's given in the diagram -- G is the foot of the perpendicular from A to BF.]

Since we're not given the side length of the square, let's just call it s -- most likely, it will turn out that the exact length doesn't matter. Actually, let's call it 2s -- since midpoints appear in the problem, we most likely will prefer that extra factor of 2 to avoid annoying halves later on. (In fact, if the exact length ultimately doesn't matter, we can get away with simply calling it 2 and avoiding s altogether, but let's keep the s here.)

This means that in right triangle CDE, CD = 2x (side of a square) and CE = x (half of a side). Since we know two legs of a right triangle, we use the Pythagorean Theorem to find the hypotenuse:

a^2 + b^2 = c^2, where a = s, b = 2s, c = ED

s^2 + (2s)^2 = ED^2

5s^2 = ED^2

ED = s sqrt(5)

Now we have one of the sides we need, ED -- all that remains is to find the other side, FG. We notice that FB = ED = s sqrt(5), but this doesn't give us FG right away.

The fact that we're asked to find a ratio ED/FG suggests that the answer has something to do with similar triangles. And indeed, we can use a theorem found in Lesson 14-2:

Right Triangle Altitude Theorem:

b. each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

In the diagram in the U of Chicago text, this appears as a = sqrt(cx), where a is the leg, c is the hypotenuse, and x is the segment we're trying to find.

a = sqrt(cx), where a = s, c = s sqrt(5), x = GF

a^2 = cx

s^2 = s sqrt(5) x

s = x sqrt(5)

x = s/sqrt(5)

Thus we have ED = s sqrt(5) and GF = s/sqrt(5). It now remains to find the ratio:

ED/GF = s sqrt(5)/(s/sqrt(5)) = sqrt(5)^2 = 5

Therefore the desired ratio is 5 -- and of course, today's date is the fifth. We haven't quite reached this theorem since we're not in Chapter 14 yet. But the theorem ultimately goes back to similarity -- and that's what we're learning about now, similarity and dilations.

Lesson 12-2 of the U of Chicago text is called "Size Changes Without Coordinates." In the modern Third Edition of the text, size changes without coordinates don't appear on their own. The first lesson of the new text, Lesson 12-1, corresponds more closely to Lesson 12-3 of the old text. The opening dilation activity of the old Lesson 12-2 is nonetheless squeezed into the new 12-1.

OK, today is the day that I want to post a pandemic activity, which usually means Desmos. But it's tricky to find a dilation activity that doesn't contain coordinates. (While reflections and rotations are often considered without coordinates, translations and dilations tend to be taught on the number plane.)

I do finally find one to my liking:

https://teacher.desmos.com/activitybuilder/custom/5e13560bbefb180dcd92d427

This activity uses a "stretching machine" to perform the dilations -- and this stretching machine isn't dependent on a coordinate plane.

I'll post one of last year's worksheets to supplement this Desmos activity.