Friday, November 20, 2015

Activity: Transformations -- Based on Lessons 6-1 and 12-1 (Day 60)

Here it is, the last day before Thanksgiving vacation. This means that I'm taking all of next week off on this blog and wait to post Day 61 on Monday, November 30th.

This is what I wrote last year in explaining why many schools take an entire week off for Turkey Day:

Traditionally schools had only two long breaks during the school year -- winter break near Christmas, and spring break near Easter. Those are in addition to the long summer break. But recently, more and more schools are having a third long break during the school year, Thanksgiving vacation. The first time that I heard of the phrase was in the old Peanuts comics (yes, the Peanuts are still on my mind after watching the movie a few weeks ago) -- and immortalized in the second Christmas special, "It's Christmas Time Again Charlie Brown." In the December 27, 1990, comic, Charles Schulz wrote the following exchange:

Peppermint Patty: Marcie, what book were we supposed to read during Thanksgiving vacation?
Marcie: This is Christmas vacation, sir..
Peppermint Patty: Christmas vacation?! How can I read something during Christmas vacation when I didn't read what I was supposed to read during Thanksgiving vacation?
Marcie: Duck, sir! Easter is coming!!

Now this comic came out when I was in the fourth grade -- and up until then, Thanksgiving was merely a four-day weekend, from Thursday to Sunday. Of course, Thanksgiving is on a Thursday, but schools have been also closing on Friday since before I was born. We discussed back on Veteran's Day how many students and teachers alike feel entitled to an extra day off when Veteran's Day falls on Tuesday or Thursday, so I can easily see how Black Friday and the four-day Thanksgiving weekend for schools first began.

Of course, since I was born, air travel has become more common, and families often live in different states on opposite coasts. Some news reports began to identify Wednesday, the day before Thanksgiving, as the biggest travel day of the year. Families that travel on Wednesday obviously can't send their children to school that day. So in the 1990's, some districts began to observe a five-day weekend from Wednesday to Sunday, including the largest district in the area, LAUSD, for a few years around this time. On my way home from subbing last year, I drove past a school that took a five-day weekend, with Tuesday, the last day before the holiday, to be a minimum day (but this year that district switched to taking the whole week off).

The schools I attended as a student always held school the day before Thanksgiving, but for a few years, when I was in the sixth through ninth grades, a staff development day was held on the Monday after Thanksgiving (the day now called Cyber Monday, but this was back when the Internet was still in its infancy).

When I was a college student at UCLA, we had only the four-day weekend off. But I once read a Daily Bruin article suggest that the school take the entire week off. The following 2008 editorial published by the school paper (The Daily Bruin) points out that since UCLA began classes on a Thursday, called "Zero Week." So the suggestion is just to start classes the previous Monday so that the entire Thanksgiving week could be taken off. Naturally, travel is given as the top reason for having the whole week off:

As of today, UCLA still has only a four-day weekend for Thanksgiving. But shortly after that 2008 article was written, here in California, the budget crisis began. Many schools started having furlough days, and school years were fewer than 180 days. When districts chose which days to take off for the furlough days, the first three dates chosen were invariably the three days before Thanksgiving.

Then, of course, once funding for schools was restored, students and teachers alike decided that they liked having the entire week off. And so the Thanksgiving week stuck around at many schools districts, including LAUSD as well as both of the districts where I work.

In a way, the entire week off is the next logical step after a five-day weekend. Wednesday may be the biggest travel day of the year -- and so in order to get the jump on the crowds, flyers may leave on Tuesday instead. And once we take Tuesday off from school, we might as well take Monday off as well, since no one wants a lone day, a one-day week. And so the entire week is taken off.

But the week off still causes problems. Not everyone travels for Thanksgiving -- after all, someone has to host all of the big turkey dinners. And parents who don't travel may still put in full days of work Monday, Tuesday, and Wednesday, leaving their children who go to schools that take the full week off without daycare. So there is no holiday schedule that will satisfy everyone.

Notice that today is Day 60 on the school calendar. So we are now exactly one-third of the way through the year -- that is, one trimester. So we notice that even though winter break doesn't divide the semesters in this district, Thanksgiving break does divide the trimesters. And now that I'm thinking about it, I'm wondering whether this wasn't an intentional decision by the district administrators -- the first day of school was chosen by counting backward 60 school days from Thanksgiving, so that there would one whole trimester completed by the holiday.

Recall that trimesters are common in elementary schools and rare in high schools. They used to be common in middle schools, but now more and more middle schools are switching to semesters. My own district, therefore, uses trimesters only at the elementary level.

Also, recall that my district originally wanted to end the first semester by Christmas, but the union fought the idea since teachers didn't want to start school early enough in August. So I wouldn't be surprised if the compromise Middle Start calendar was created simply by saying that if we can't end the first semester by Christmas, let's at least end the first trimester by Thanksgiving.

A rule of thumb is that a trimester is approximately three months long. We observe that in my district, the first day of school was on August 26th and Thanksgiving is November 26th. So we conclude that if we start school exactly three months before Turkey Day -- on the fourth Wednesday in August -- then there will be exactly 60 days of school before the holiday break. The first trimester then consists of 12 weeks (as sixty divided by five is twelve), with three extra days to make up for the three holidays Labor Day, Columbus Day, and Veterans Day.

I posted the Chapter 5 Test yesterday in deference to schools that have a minimum day as the last day before the holiday. My own district doesn't have a minimum day before Thanksgiving, but I ended up subbing in a German class, and the students in that class took a test yesterday -- today they are taking notes on a video. On the other hand, one of the students in the German class is taking a Precalculus test today, on basic trig. He says that he's ready for the test, but he scared me a little when he told me that he had memorized the entire unit square -- when he should have said unit circle! Another student had to take a test today for his Integrated Math I class. Well, that shows the difference between the foreign language department and the math department!

And so today becomes an activity day instead. Last year, I posted a preview activity for Chapter 7, since under last year's Early Start Calendar, that was the first chapter after Thanksgiving. This year, Chapter 6 is ostensibly the first chapter after the break, but we've already covered two lessons from that chapter on the blog. But we haven't officially covered Lesson 6-1 yet.

Actually, Lesson 6-1 is a great lesson to cover. It's all about general transformations, with many of these transformations being done on the coordinate plane. Since the most transformation questions on the Common Core tests take place on the coordinate plane, I really want to emphasize them.

Many of the questions from this section of the U of Chicago text have students draw a figure on the plane, transform its vertices, and then identify the transformation as either a reflection, slide, turn, or size change. (The terms "slide" and "turn" are used since this in the text precedes Lessons 6-2 and 6-3 on translations and rotations.)

The previous page in the text gives a strange transformation, S(x, y) = (2x, x + y). This transformation is clearly not an isometry, as we can see that if the preimage is a house, the image is still a house, but a distorted house, not one where anyone would want to live.

What sort of transformation is S(xy) = (2xx + y)? Well, as it turns out, the transformation that maps (x, y) to (x, x + y) is the transformation I mentioned yesterday -- a transvection or shear. Its fixed line is the y-axis and its shear factor is 1.

Yesterday, I briefly mentioned that transvections preserve area, and we can use this fact to show that two triangles with the same base and height must have the same area, but I never gave the proof. So here's a paragraph proof:

Given: AB = DE, drop perpendiculars C to AB (meeting at M) and F to DE (meeting at N), CM = FN.
Prove: Triangles ABC and DEF have the same area.

We know that some isometry maps AB to DE, and so we begin our proof the same way that we started our proof of the SAS (and other) Congruence Theorem. As usual, if C and F are on opposite sides of DE, reflect C over DE. But the image of C isn't necessarily F. What we hope to show is that some transvection maps C to F.

We see that both CM and FN are perpendicular to DE, and they are congruent. And so CMNF is once again a Saccheri quadrilateral (a rectangle, since by now we're in Euclidean geometry), and since a rectangle is a trapezoid, CF is parallel to DE, so CF is an invariant line of any transvection with a fixed line DE. And so now we've found our transvection -- we choose a transvection with fixed line DE and shear factor CF/CM. This transvection maps C to F.

So DEF is the image of ABC under the composite of an isometry and a transvection. Since both isometries and transvections preserve area, ABC and DEF have the same area. QED

Then we see that the preimage and image in the U of Chicago text don't have the same area. This is because S doesn't map (x, y) to (x, x + y), but rather to (2x, x + y). So S is not a pure transvection, but rather the composite of a transvection and some other transformation. This second transformation merely doubles the x-coordinate. It can't be a dilation because a dilation would have to double both of the coordinates. Many Precalculus texts would call this a "horizontal stretch." And while transvection preserves area, horizontal stretch by a scale factor of 2 obviously doubles area. And so the image of the figure under S has exactly twice the area of the preimage.

By the way, the statement that transvections preserve area only works in Euclidean geometry. We can see eventually that in non-Euclidean geometry, area is deeply related to angle measure, and since transvections don't preserve angle measures, they can't preserve area either.

I was wondering whether it's possible to teach transvections in our Geometry classes. Under the Common Core, we're already teaching reflections, translations, and rotations for congruence, as well as dilations for similarity. So the natural step would be to teach transvections -- maybe even as part of teaching area, as transvections preserve area. The first unit of the second semester will be on dilations and so the next unit -- when it's time to teach area -- could be on transvections.

But I won't -- the Common Core doesn't require anyone to teach transvections. It may be possible to have an activity, similar to the one in the Japanese classroom we were discussing yesterday, where students can see how because of translations, triangles with the same base and height will also have the same area, just before showing them the formula. But the formula should still be proved using the Area Postulate and decomposition, as is done traditionally.

That's enough about a later activity -- what about today's activity? Here's another tradition that I've seen in math classes (besides taking a test) -- the day before Thanksgiving break, the students are going to graph points on a coordinate plane in the shape of a turkey.

Now I'm not an artist, so I couldn't come up with the points for the turkey's shape myself. So I decided to search for a turkey on the Internet. Although there are many turkeys around, I actually found one that not only involves graphing, but also transformations! The following worksheet comes from a school district in Fort Bend, Texas (which isn't even a Common Core state):

And after all of that discussion on transvections, the transformation turns out to be an ordinary dilation, with a scale factor of one-half. And after billing this as a preview of Chapter 6, the activity actually fits better with Lesson 12-1 of the U of Chicago text on dilations. Well, it's still a preview, but just a preview of the second semester, not next week.

This worksheet is somewhat tricky. Many of the points on Tom Turkey's preimage have odd coordinates, meaning that a dilation of scale factor 1/2 maps these to fractional points. And a few of the points on Tom's preimage even have fractional coordinates already. But at least all of the points are in the first quadrant (or the axes bounding that quadrant), so that we don't have to worry about sign errors.

It's possible to modify this worksheet to cover other transformations. On the coordinate plane, translations are the easiest. Reflections and rotations usually involve negative signs (unless the mirror isn't an axis or the rotation center isn't the origin), and so we'd need graph paper with a window of -30 to 30 on at least one of the axes. It's even possible to have a transvection -- it might be fun to see what Tom looks like sheared. I recommend using only 1 as the shear factor. Mapping (x, y) to (x, x + y) is easy but involves large coordinates. We could map (x, y) to (x, x - y) instead -- this avoids large coordinates but includes negative coordinates.

Notice that the Tom's waddle is a simple parallelogram. If you want students to perform different transformations, you can have them practice on the waddle. (One of the four possible transvections maps the waddle to a square.)

This activity may be a bit long, and so if your school has a minimum day, you can divide the class into partners -- one partner draws the preimage and the other the image. Both partners should work on calculating the image points first.

Thus ends this post and the first trimester. As I usually do during holiday periods, I plan on making two special posts during Thanksgiving break. The next school day post will be on November 30th.

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