Friday, May 6, 2016

PARCC Practice Test Question 13 (Day 156)

Today I subbed in a high school math class -- in fact, this teacher is scheduled to be out for a full week, up to and including this upcoming Thursday. So I will have plenty of things to say about this class here on the blog.

This teacher has three sections of grandfathered Algebra II (1st, 4th, and 5th periods), as well as two sections of Honors Integrated Math I (2nd and 6th periods).

The Algebra II students are taking a quiz on statistics. But according to the plans for next week, the students will be working on trigonometry. I'm actually surprised that I'm seeing trig in a regular Algebra II class (as opposed to Honors). Then again, these worksheets appear to be the simpler trig of Chapter 14 of the U of Chicago Geometry text. We'll see how these students do on these Pizzazz worksheets next week.

By the way, during the teacher's 3rd period conference, I cover another math teacher, who had an Integrated Math II class. But this teacher tried to leave a video for the students to watch -- the video didn't work, and so all of the subs providing period coverage just make it a free period. It appeared, though, that the Math II students are also learning about trig --it's also Chapter 14 Trig.

Now the class I want to focus on here on the blog is Integrated Math I. The students in that class are starting a project that involves geometric constructions. Because today is an activity day anyway, I will post this assignment as today's activity. (Yes, I know that this project has nothing to do with PARCC Question 13, but activities that occur in the classroom always take priority over other plans for the blog.)

Here is the information from the worksheet:

Geometric Constructions Project
Overview: Your job is to construct any two of the attached figures using only a straightedge and a compass and then to create one more of your own design. Each of the three figures must be neat and decorated sufficiently to hang on the wall. For one of the three figures you will be required to turn in at least three partially completed constructions (illustrating the progression of steps required to create it) and a fully completed, but undecorated version (all construction marks remain, but it isn't colored). All of the work is due at the end of class on 5-13-16 [the day that the teacher returns -- dw].

This project is somewhat tricky. The students tell me that the they've seen their teacher perform a few basic constructions such as drawing a circle using a compass, but they haven't seen anything major yet such as constructing an equilateral triangle or hexagon.

Of course, these appear in the Common Core Standards:

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle

Today, I show the students how to perform these constructions, so that they can create the more difficult designs next week.

Before we get to our PARCC question, let me point out that this is a traditionalists-labeled post. But now that I'm getting ready to work at a school in the fall, I want to tone it down from having all these weekly posts where I quote comments posted on articles from various newspapers. The focus of any teacher blog should be what's happening in the classroom. It will still be a few more months before I get my own classroom, but I want to start thinking about my future classroom here on the blog.

(By the way, I have a special traditionalist post planned for this upcoming Wednesday. It will refer to something that will happen both in the class I'm subbing this week and the class that I will be teaching this fall.)

Still, I have several loose ends to tie up regarding traditionalists today:

-- Minnesota has adopted the pre-Core Massachusetts standards. Earlier, I'd written that they were considering the change -- since then that change has been made official.

-- The game show Jeopardy! made fun of Common Core again. On last night's episode, there was a category called Common "Core" -- the word "core" in quotes implies that this four-letter word will be a part of every correct question. One of the contestants, a history teacher, used this opportunity to criticize the Common Core Standards. She wasn't specific -- she just implied that the Common Core was bad, period.

Of course, we know that there are no Common Core Standards for history, so this teacher shouldn't be affected by the Core. Then again, there's a link between AP US History and Common Core in that David Coleman is considered the architect of both. Some people believe that Coleman introduced a liberal bias to the AP US History test (which students are taking today, in fact), and so many people link this bias to the Common Core itself.

-- First Daughter Malia has made her college decision. She'll be going to Harvard in fall 2017. I'd been writing about Malia because of Presidential Consistency -- many Common Core opponents don't like it when politicians, especially the president, promote Common Core yet insulate their own children from the ill effects of the Core.

In particular, I wasn't sure which of the three Sidwell Friends tracks Malia was placed. But we must assume that she was placed on the highest track since she's been admitted to an Ivy League school. Of course, this track leads to AP Calculus (presumably, Malia took her AP test yesterday).

This will draw the ire of the traditionalists and other Common Core opponents. President Obama promotes the Common Core, which doesn't encourage eighth grade Algebra I or senior year Calculus, yet he makes sure that his daughter is on the Calculus track at Sidwell. His own daughter will get into Harvard, but it will be harder for Common Core students to get into the Ivy League, which will be looking for Calculus on the transcript.

-- And traditionalist Dr. Katharine Beals is at it again. This question comes from the 8th grade Common Core test given in her home state of Pennsylvania:

Kelsey draws a series of right triangles with sides that have the lengths shown in the table below:

Lengths of Sides of Kelsey's Right Triangles (inches)
Triangle    First Leg    Second Leg    Hypotenuse
       A                1                   1                sqrt(2)
       B                1                   2                sqrt(5)
       C                1                   3                sqrt(10)
       D                1                   4                sqrt(17)
       E                 1                  5                sqrt(26)

Kelsey continues making right triangles following the same pattern she used to make the first five right triangles.

C. [Beals omits Parts A and B here, likely because she only has a problem with Part C -- dw] Explain why none of the right triangles Kelsey makes will have a hypotenuse with a rational number length.

Beals then provides an example of "a complete explanation," according to the scorer:

The only square roots that produce rationals are those that come out even or give an exact value. Like the square root of 9 or 16 or 25, unlike any of those that are listed. Kelsey's triangles all have lengths of hypotenuses that are not rational and produce numbers that do not come out even or give exact values. Like sqrt(17) which starts coming out as 4.12310562562... and never stops.

As usual with her "Math problems of the week" series, Beals asks several "extra credit questions":

Extra Credit: 1. In what sense is this a complete explanation? Hint: Consider the terms "even," "exact value," and "never stops." Consider as well the difference between answering a question and rephrasing the question as a statement. 2. Discuss the human element involved in scoring responses to open-ended questions on tests taken by millions of students. 3. Discuss the human element involved in scoring the verbal explanations submitted by tens of thousands of English-impaired students.

So let's look at each of these in turn:

1. I wonder what sort of explanation Beals would prefer to see. Let's attempt to rewrite the student response without using those three terms she wants us to avoid:

The only whole numbers whose square roots are rational are perfect squares, like 9, 16, or 25, unlike any of those that are listed. Kelsey's triangles all have lengths of hypotenuses that are irrational and whose decimals neither terminate nor repeat, like sqrt(17), which is approximately 4.12310562562.

Of course, stopping the decimal expansion at that point makes it appear that "562" might repeat, but in reality they don't. (In fact the last "2" is a rounded value -- it's actually a "1," but the next digit is "7," so "1" is rounded up to "2.")

But wait a minute -- we're avoiding phrases like "comes out even" and replacing them with more mathematically precise terms as "perfect square." Yet Beals often refers to mathematically precise terms as "labels" that should be avoided (for example, "number sentence," which is more precise than incorrectly calling an inequality an "equation"). So what does Beals really want here?

Let's save Question 2 for last and move on to Question 3:

3. When Beals worries about English learners, we've seen what she's really saying before -- she wants there to be a more symbolic answer. But we know that in reality, most students -- and probably most people -- dislike reliance on symbols.

If we use a, b, and c for the sides of a right triangle, then we can let a = 1 be the first leg of the triangle, to obtain c = sqrt(b^2 + 1). The real problem is that b^2 + 1 is never a perfect square, as b^2 obviously is a perfect square. The only two consecutive whole numbers that are perfect squares are 0 and 1, and 0 can't be the side length of a triangle. So c is always irrational.

The problem with an answer such as this one is eighth graders are unlikely to give it, because it requires algebraic expressions such as sqrt(b^2 + 1). Yet this is the sort of symbolism that Beals likes to see more of.

2. To some extent, I agree with Beals here. The real solution to this conundrum is simply not to include this sort of question on the test. I believe that if our Common Core tests are going to be given on the computer, then they should be able to report the scores instantly. So any question that can't be scored immediately, such as this Part C question here, should be thrown out. Presumably there's no problem with Parts A or B, so these can remain on the test.

By the way, in second period some students wonder why they have to learn classical constructions. I've stated before that the age-old question of "Why should we learn this?" or "When will we use this?" isn't asked when the lesson is easy, fun, or high-status even if they'll never use it in life. Though the drawing and coloring parts of our project are fun, the constructions aren't. And I obviously fail to show that constructions are easy when I have trouble figuring out how to use the teacher's "compass" (really a small ribbon tied to a marker) to draw a circle. (And we know that it's next-to-impossible for anything taught in math to lead to a high status.)

But since this is still a traditionalists-labeled topic, let's put it on them. Why, according to the traditionalists, should students learn classical constructions? How would they answer the question "Why should we learn this?" to students who will ask that question when any lesson isn't easy, fun, or of a high status?

I performed a Google search at Beals's website, but I couldn't find any mention of constructions, straightedges, or compasses. I tried another search for classical constructions using a different name, the known traditionalist, Barry Garelick. All I found were complaints that Common Core students must wait until high school to study constructions, rather than learn them in middle school (probably by 7th grade at the latest, since they want 8th graders to learn Algebra I).

So let's try to reconcile that -- traditionalists think these students should have learned constructions two years ago, while the students think they shouldn't have to learn constructions at all.
So for sixth period, I tried to prepare an answer the best I could. And here's what I wrote:

Why should we study classical constructions?

-- The ancient Greeks were sticklers for exactness. If it isn't exact, it isn't mathematics.
-- If done correctly (and that's a big if), straightedges and compasses allow you to construct circles, equilateral triangles, and regular hexagons exactly.
-- To the Greek mathematicians, the challenge of using only compasses and straightedges to construct exact geometric figures was fun, like a game or puzzle!
-- Doubling the cube, squaring the circle, and trisecting an angle were three ancient construction problems that even the Greeks found to be too difficult. After almost 2000 years, someone finally proved that these three problems are impossible.

At the end of the period, one student said that he enjoyed the lesson. So hopefully the rest of this activity will go well.

By the way, I saw two references to Theoni Pappas and her Mathematical Calendar today. First, in the Math II class I covered during conference period, apparently this teacher likes setting up calendars full of her own problems, with the date as the correct answer. All twelve months were posted around the classroom. (This is also something I might do in my own classroom next year.) The other reference is, today's question asks to give the number of lines of a certain figure -- which turns out to be a six-pointed star similar to the first question on today's project. Of course, that figure has six lines of symmetry -- and today is the 6th.

With everything else going on today at school, I hardly have time for Chapter 13 of Morris Kline's Mathematics and the Physical World, which is called "Motion on an Inclined Plane."

"But where the senses fail us reason must step in." -- Galileo

Yes, it was Galileo who first used an inclined plane for experiments. According to Kline, Galileo proved that neglecting friction, the amount of time it takes an object to roll down an inclined plane is the same as the time it takes the same object dropped from the same height to fall. My own high school physics teacher simplified this as:

Vectors operating at right angles are independent.

In this case, the vertical vectors are independent of the horizontal vectors -- which is why only the vertical vectors determine the time it takes the object to roll down.

Sorry, but we need to get into the PARCC question pronto:

13. Part A

The number of people who live in a unit of area is called the population density of the area. It is usually given as people "per square mile" or "per square kilometer."

A map of the Orchard Hill neighborhood is shown. The population of Orchard Hill is 360 people. The length of each block is the same and the length of 20 blocks is one mile.

What is the area in square miles of Orchard Hill?
A. 0.03 square mile
B. 0.15 square mile
C. 0.35 square mile
D. 0.60 square mile

Part B

What is the population density of the Orchard Hill neighborhood, given as the number of people per square mile?

For Part A, each block is 0.05 mile. The area of the neighborhood is 4 blocks by 3 blocks, or 0.2 mile by 0.15 mile, or 0.03 square mile, which is choice (A). For Part B, the density is 360 people/0.03 square mile, which equals 12,000 people per square mile. This may sound high, but actually most urban areas have a large population density -- the density of Washington DC is over 10,000 people per square mile.

PARCC Practice EOY Exam Question 13
U of Chicago Correspondence: Section 8-3, Fundamental Properties of Area
Key Theorem: Area Postulate

b. Rectangle Formula: The area of a rectangle with dimensions l and w is lw.

Common Core Standard:
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Commentary: The U of Chicago text discusses area, but not population density. The activity for today is not about population density, but about constructions.

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