Friday, April 22, 2016

PARCC Practice Test Question 3 (Day 146)

Chapter 3 of Morris Kline's Mathematics and the Physical World is "The Science of Arithmetic." In this chapter, Kline, as the title implies, applies the simplest mathematics to nature.

As usual, Kline begins every chapter with a quote:

I had been to school...and could say the multiplication table up to 6 x 7 = 35, and I don't reckon I could ever get any further than that if I was to live forever. I don't take stock in no mathematics, anyway. -- Huckleberry Finn

I've seen this Mark Twain quote before. It is often mentioned in progressive math books to emphasize how many students don't work hard to learn math as they see no point in it. Kline continues by describing why many students exhibit Huck Finn's attitude towards math:

"The Greeks had a word for it. They called the boring manipulations of arithmetic that are used in commerce, trade, calendar-reckoning, and military problems logistica. Most young people would endorse this Greek deprecation of arithmetic for many reasons. But the reason that is pertinent is that we are obliged to learn the subject before we can really understand it. Because the knowledge of counting, adding, subtracting, and the like is regarded as a preparation for 'life' we are taught it mechanically from early childhood. The practice takes precedence over the principles. No doubt this introduction to life is not especially cheering. It enables us to handle money efficiently and perhaps serves to arouse our cupidity, but it hardly inspires us."

And of course, this sounds exactly like the traditionalist philosophy of teaching. So here Kline is directly attacking traditionalism -- and because of this, I am labeling this post as "traditionalists."

Kline writes that as opposed to boring logistica, the Greeks were interested in the concept of number and its applications to nature, which they called arithmetica. And so Kline devotes the rest of this chapter with using the natural numbers to describe the natural world.

Again, let's recall my definition of dren -- a reverse-nerd who can't even do third grade math. By this definition, Huck Finn is undoubtedly a dren, as 6 x 7 -- which equals 42, not 35 -- counts as a third grade math problem. But let's say that Huck at least knows the tables to 5 x 5. Arguably, this would allow him to satisfy the following second grade Common Core standard:

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Again, Huck says that even if he were immortal, he could never master third grade math -- much less Algebra, let alone the Calculus that traditionalists want students to take. Instead, he feels that this second grade standard represents all the math he'll ever know.

There have been many stories in the news lately about states that are dropping Common Core, including Huck Finn's home state of Missouri. I've mentioned before that the Show-Me State was considering ending Common Core -- well, it's official:

Traditionalists may appreciate the fact that the Missouri standards explicitly divide the high school math standards into classes Algebra I, Geometry, and Algebra II.

Meanwhile, New York State isn't dropping Common Core yet, but the opt-out movement is as strong as ever in the Empire State:

Finally, Massachusetts is also considering dropping Common Core Standards and replacing them with the old pre-Core standards -- and Michigan may adopt the Bay State's old standards too:

This last link leads to a heated 2000+ comment debate thread. Many of the posts are off-topic, but here is one from an apparent traditionalist:

I've dealt with this very issue as a result of a move and was smart enough to marry a math wiz who's primary area of study was aeronautical engineering. Our primary concern was our oldest child being challenged enough in the new school, which was properly addressed with advanced math class options. 
If the issue was left up to me, I would consult with the right people on a local level and make the proper decision. The last thing I would need is the federal government being involved.  

Now try looking at this post from Huck Finn's perspective -- Huck Finn, of course, would avoid the advanced classes mentioned in this post.

(The rest of that comment turned into the federal vs. state vs. local control debate. As I've mentioned in previous posts, I'm torn between favoring federal and state control as many other countries have national standards while others have the equivalent of state standards. Purely local standards, on the other hand, deviate too far from what other countries do.)

Question 3 of the PARCC Practice Test is on dilations:

3. A dilation with a center at P(0, 0) and a scale factor k is applied to MN. Let M'N' represent the image of MN after the dilation.

Select each correct statement.
A. If k > 0, then M'N' > MN.
B. If k > 1, then M'N' > MN.
C. If 0 < k < 1, then M'N' < MN.
D. If 0.5 < k < 1.5, then M'N' < MN.
E. If k = 1, then M'N' = MN.
F. If k = 0.5, then M'N' = 0.5(MN).

This question is a simple dilation problem. It can be solved using the following theorem from Lesson 12-1 of the U of Chicago text:

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' | | PQ,
(2) P'Q' = k * PQ.

This theorem is stated again in Lesson 12-3 as "Size Change Distance Theorem." We know that "size change" means dilation, so we could call it the Dilation Distance Theorem. On the blog, we actually call it the Dilation Postulate in order to avoid circularity down the road.

Whatever we call this result, this is what we use to answer the question. If the scale factor k is more than 1, then the dilation is an expansion, and M'N' > MN, so (B) is correct. If the scale factor k is between 0 and 1, then the dilation is a contraction, and M'N' < MN, so (C) is correct. If the scale factor k is exactly 1, then the dilation is the identity, and M'N' = MN, so (E) is correct. And choice (F) is exactly the Dilation Postulate in the case k = 0.5, so (F) is correct.

Choice (A) is wrong, because it is directly refuted by (C). And choice (D) is wrong, because it is directly refuted by (B). So the correct answers are (B), (C), (E), and (F). For the second day in a row, we have four correct answers, and so for the second day in a row, the most common student error will be the failure to mark all four answers.

Today is an activity day. Recall that we had an entire link of lessons to try out back in February. Well, here is a lesson on dilations:

The author of this blog is Julia Finneyfrock, a North Carolina private high school teacher. Finneyfrock refers to herself as the "Designated Deriver."

Like many of the lessons I linked to back in February, Finneyfrock's is highly computerized. She mentions three different apps -- PearDeck, Desmos, and polygraph -- in her lesson (and she even mentions a fourth app, Haiku, just to check homework). Here is an excerpt of the lesson:

When they are finished checking their homework, my students then enter our class code in onPeardDeck . PearDeck creates interactive lectures. I embedded a Desmos Activity into the first part of my lesson for every transformation. As soon as my students log into PearDeck they are taken directly to this activity and are able to start at their own pace.

Next, students were able to practice dilations on PearDeck together. I gave my students a shapes and the dilation and they drew the dilation on their slide (I let them draw pictures if they finish early). I was then able to overlay all of the drawings to see if any of the students were off, and I could show individual students work as well.

We practiced 4 dilations together, then they were taken to another Desmos Activity. This activity is a polygraph game that I created for students to practice describing all of the transformations they learned so far (reflections, translations, dilations). Students played this game for about 10-15 minutes at the end of class. Students LOVE polygraph. I literally had to kick students out of my class because they didn’t want to leave.

I have nothing against computerized lessons -- after all, the PARCC and SBAC are on computers, so students must be familiar with them in order to be successful. And of course, one application of dilations is computer graphics -- contracting objects so that they fit on the screen.

The problem I have is that we can't have a computerized lesson unless we have the software. All the PearDeck this and Desmos that in the world means nothing unless we have PearDeck and Desmos installed on enough computers in the classroom for the students to enjoy them. As a sub, I've seen a few (mostly eighth grade) math classes with computers. But these were used mostly as online texts, not for anything described in Finneyfrock's post. The best I can do is convert her lesson into a more paper-based lesson and post it onto the blog.

How did Finneyfrock's students react to this assignment? Let's find out:

“Class is over already? I don’t want to go to my next class!”
“I love this! Can we do this more often?”
“This is so much fun”

Unfortunately, I doubt that my paper version of this assignment will draw the same response. Any interactive computer assignment will be more fun than a paper version. Even Huck Finn, if he lived in the computer age, might actually enjoy this math assignment for once! On the other hand, let's see whether any assignment endorsed by traditionalists can elicit the same response from students in a Geometry class.

It's clear that Finneyfrock has completely embraced the use of computers and other forms of technology in her classroom. If you have access to some of the same software that she does, you may wish to read about the assignment directly from her blog at the link above.

Otherwise, I am posting my paper version below. One thing I noticed about most of Finneyfrock's transformations is that she often scales by a different factor in the x- and y-directions. I've mentioned before that while such a transformation is a linear transformation, it is not a true dilation. The Dilation Postulate/Dilation Distance Theorem, the main result of this lesson, only applies to true dilations with the same scale factor in all directions.

PARCC Practice EOY Question3
U of Chicago Correspondence: Lesson 12-3, Properties of Size Changes
Key Theorem: Size Change Distance Theorem (Dilation Postulate)

Under a size change with magnitude k > 0, the distance between any two image points is k times the distance between their preimages.

Common Core Standard:
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Commentary: The first three lessons of Chapter 12 all cover basic properties of dilations. In the posted activity, I focus on true dilations, not mere horizontal/vertical stretches. But I maintain the spirit of Finneyfrock's lesson by providing an answer sheet. If this answer sheet is printed directly onto a transparency, then it can be placed directly over a student sheet in order to check the answers, just like Finneyfrock's lesson does, except ours is comparatively low-tech.

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