Wednesday, December 16, 2015

Review for Chapter 6 Test (Day 73)

This is the second post this month which I must begin with current events, since it directly affects schools right here in Southern California. Yesterday, the largest school district in the state, the Los Angeles Unified School District, canceled classes due to a bomb threat. It's now believed that the threat was a hoax, but the district decided to err on the side of caution. Yes, perhaps the LAUSD overreacted in canceling classes, but with the San Bernardino incident still fresh in our minds, no one wanted to take any chances with our children.

We know that the LAUSD operates on an Early Start Calendar -- meaning that winter break divides the year into semesters. So this week is the last week of the first semester -- finals week. Many high schools were originally scheduled to start finals yesterday, and now they are scrambling to come up with amended testing schedules. Many high schools are now forced to give three finals in one day, especially since some LAUSD schools have seven- or eight-period days.

Some people believe that high school students set up the hoax in order to have a day off from finals. I doubt this conspiracy theory, and here's why -- two districts were threatened yesterday, with the other being the largest in the nation, New York City. While LAUSD has an Early Start calendar, NYCPS has a traditionalist Labor Day Start Calendar, meaning that this isn't finals week in the Big Apple. If high school students were behind the threat, it's more likely that another district having finals this week would have been mentioned.

Note that today there was another school closure here in Southern California -- at a single high school in the city of Fullerton. I was able to confirm that the first day of school was on August 10th, thus making it an Early Start School. Because no other schools are named in the threat, there's a somewhat greater chance that this really was just a copycat, a student trying to avoid finals for a day. As it turns out, that school will now have students take four finals tomorrow, since Friday is originally scheduled to be a staff development day.

Today I am posting review for a test. This is not a final, because this blog is following the calendar of another district, not LAUSD or Fullerton UHSD. My district is on a Middle Start Calendar -- one where school starts in August, but not early enough to end the first semester by Christmas. Last year I followed the calendar of another district that was on the Early Start Calendar, and so I did post the final before winter break.

This is the Chapter 6 Review worksheet. Originally, my plans were to post the review for Chapter 6 today, then the actual Chapter 6 Test today (unchanged from last year's), and finally on Friday the graphing worksheet that I mentioned at the end of last week -- the one that the teacher I was subbing for assigned his eighth graders.

But I've made -- you guessed it -- yet another change. Why am I changing it up again? I just found out that on Friday, I'll be subbing for the continuation school again -- that's right, the same class that plays Jeopardy games on Fridays. Every time I sub for that class, I post my own version of the Jeopardy game -- with Geometry questions thrown in -- here on the blog. And so I'm rearranging this week's schedule in order to accommodate Jeopardy.

I still think that there can never be too many graphing assignments, considering how many such questions appear on Common Core tests such as the PARCC. And so I decided to post the graphing worksheet today instead. In case you can't figure it out by the title "Frosted Friendship," the graph is supposed to be a boy building a snowman. For simplicity, I'm going to call the snowman "Frosty," even though this day and age, the most popular snowman is not Frosty, but Olaf.

On one hand, this worksheet is so long that I almost want to make it the entire test and forget about the 20 test questions I wrote last year. On the other hand, that would be especially time-consuming to grade -- and besides, this is just a simple graphing worksheet, having nothing to do with any of the transformations (since eighth graders won't see them until the second semester). So it would be completely inappropriate to make this worksheet the test for high school Geometry -- especially not a test for Chapter 6 on Transformations and Congruence.

But there are ways to sneak transformations into this worksheet. First of all, notice that even though this graph is mostly asymmetrical -- including the snowman's facial features, Frosty's torso in fact is perfectly symmetrical. We see that his body begins near the bottom of the third column at (-3, 10) and continues through the rest of that column, at (-2, 18). Since the word "Stop" does not appear at the bottom of that column, we must continue with (-13, 18) down to (-12, 10) in the fourth column.

Notice that if we only draw Frosty's body, then he ends up without the top of his head. This is because the top of his head is covered by the (asymmetrical) hat that we aren't drawing. We can fix this just by connecting the last point (-12, 10) back to the starting point (-3, 10) -- although now the top of his head may appear too flat.

I say that Frosty's torso has bilateral symmetry, so there must exist some reflection that maps the right half to the left half. As it turns out, this reflection maps (x, y) to (-15 - x, y) -- that is, the mirror for the reflection is x = -7.5. This becomes obvious when we see that the mirror image of (0, -4) is (-15, -4), so the mirror must be halfway between x = 0 and x = -15 -- that is, x = -7.5. Notice that when the mirror contains a half-integer, it still maps points with integer coordinates to other points with such coordinates.

Also, I pointed out earlier that unlike Tom Turkey from last month, this worksheet uses points from all four quadrants. But let's say that we want to avoid negative coordinates. To do this, we can simply perform a translation. The translation mapping (x, y) to (x + 17, y + 18) will make all of the coordinates nonnegative. If we want to avoid points on the axes (since, a bit surprisingly, students get more confused when there are zeros as coordinate), then (x + 18, y + 19) will be better. It may be simpler just to use (x + 19, y + 19) or even (x + 20, y + 20).

So we have a reflection mapping Frosty's right half to his left half, and then we have a translation mapping him into the first quadrant. If we compose these, we get a glide reflection -- of course, the representation isn't canonical because the translation isn't in the same direction as the mirror -- if we use (x + 19, y + 19), then the translation is parallel to y = x while the mirror is vertical. If we want to emphasize the canonical representation of the glide reflection, then the translation must be vertical, so we'd have to do something like (x, y + 19). Or we can just reflect over y = x instead, but then poor Frosty will be sideways!

For tomorrow's test, I will give students the coordinates for half of Frosty, and then they must reflect the points to give the other half and then translate him. The practice sheet for today includes only the first 16 questions from last year. The final four questions last year were graphing questions, and so these have been replaced with Frosty.

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