Thursday, May 26, 2016

PARCC Practice Test Question 27 (Day 170)

Chapter 27 of Morris Kline's Mathematics and the Physical World is "Mathematics and Nature." Yet another very famous name is the author of today's quote:

"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality?" -- Albert Einstein

Kline begins:

"There is a well-known story, largely apocryphal, concerning the visit of the French Encyclopedist, Denis Diderot, to the court of Catherine the Great...Diderot accepted [the invitation -- dw]. Euler appeared, rattled off some meaningless mathematical formulas, and concluded, 'So God exists.'"

If I remember correctly, the story was that Euler used his famous formula "e^(i*pi) = -1" in his proof of the existence of God. As we can see, this chapter of Kline is somewhat religious in nature. My earlier promise is that I would not discuss religion here on the blog unless it is in the context of calendars, which is why I added the "Calendar" label to this post.

Kline writes that many mathematicians, not just Euler, believed that the theorems of math were absolute truths, and so one could mathematically prove the existence of God. But he writes why this belief was flawed:

"The development of non-Euclidean geometry showed that man's mathematics did not speak for nature, much less lead to a proof of the existence of God."

So in a way, this is a continuation of the previous chapter. The fact that there exist geometries other than the familiar Euclidean geometry demonstrate that there is no absolute truth in mathematics -- truth is relative to the axioms and postulates that are chosen at a particular time. If we chose Euclid's Fifth Postulate, then the angles of a triangle add up to 180. If we choose a different postulate, then the angles of a triangle don't add up to 180. As Kline writes, even 2 + 2 = 4 is not absolutely true:

"But there are algebras, physically useful algebras, in which this statement does not hold."

Oh, and since I added the "Calendar" label to this post, I'd better say something about calendars and religion lest I break my own promise about writing about religion on the blog. Well, let's see:

-- On the Jewish Calendar, today is Lag B'Omer (33 days after Passover).
-- On the Christian Calendar, today is Corpus Christi (60 days after Easter). Yes, there's a Texas city named after this feast day.
-- On the Usher Calendar, Memorial Day weekend coincides with Pentecost (the 50th day after Usher Easter, which always falls April 5th-11th). I explained back on Leap Day that many American secular holidays and Christian holidays line up on the Usher Calendar.
-- On the Muslim Calendar, Ramadan will begin in just over a week (approx. June 6th).

Question 27 of the PARCC Practice Exam is about the diagonal of a screen:

27. A computer monitor is 20 inches wide. The aspect ratio, which is the ratio of the width of the screen to the height of the screen, is 16:9. What is the length of the diagonal of the screen, to the nearest whole inch?

This question involves similar rectangles and require us to set up a proportion. We'll let a be the altitude and b the base of the screen:

b/a = 16/9
20/a = 16/9
16a = 180
a = 11.25"

You may wonder, why did we use a for altitude and b for the base, rather than the more natural h for height and w for width? Oh, it's because we need to use the Pythagorean Theorem to find the diagonal:

a^2 + b^2 = c^2
(11.25)^2 + 20^2 = c^2
126.5625 + 400 = c^2
526.5625 = c^2
c = 22.9", which rounds to 23"

As is typical for measurement problems, PARCC will count either 22" or 23" as correct.

On one hand, we could bill this as a Pythagorean Theorem question from Lesson 8-7. But this question is very nearly like one that I included on the Chapter 12 Test (the lchapter on similarity) back in February. On that day, I alluded to the fact that TV's used to have a 4:3 aspect ratio, but now the standard is 16:9, just as in the PARCC problem.

But that question, as originally written in the SPUR section of the U of Chicago text, gave the ratio of a standard def TV as 9:7, referring to the diagonal-to-width ratio rather than width-to-length. This is approximately the same as a 4:3 width-to-length ratio. I converted this to an HD TV when I wrote the Chapter 12 Test, and I implied a diagonal-to width ratio of 8:7. This does correspond approximately to a 16:9 aspect ratio.

Notice that the 4:3 aspect ratio of the old TV sets should bring to mind the 3-4-5 Pythagorean triple and right triangle. Indeed, the ratio 9:7 is a poor excuse of an approximation to the diagonal-to-width ratio of the standard def TV's, considering that 5:4 is the exact value and is simpler than 9:7. (In music, we ordinarily don't play 9:7, a "septimal major third," instead of 5:4, the just major third.) On the other hand, the diagonal of a triangle with legs 9 and 16 is sqrt(337). I'd use 8:7 to approximate sqrt(337):16 well before using 9:7 to "approximate" 5:4.

Then again, the ratios 8:7 and 9:7 are easy to compare. The diagonal-to-width ratio is smaller for the HD TV. which makes sense, if you think about it. If we had a standard def TV of width 20", it's easy to calculate its diagonal as 25". Then if we played a widescreen video on this TV, black bars would appear near the top and bottom of the screen, so that the diagonal of the visible image is only 23" instead of 25". (This does not mean that the top and bottom bars are about 1" each. We must use the altitude, not the diagonal, to find the size of the black bars. We calculate the height of the 20" standard def TV as 15", and we already found the height of the widescreen image to be 11.25". This means that the top and bottom bars are closer to 2" each.)

PARCC Practice EOY Question 27
U of Chicago Correspondence: Lesson 8-7, The Pythagorean Theorem

Key Theorems: Pythagorean Theorem
In any right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2.

Common Core Standard:
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Commentary: The questions come from the old Pizzazz worksheet that I covered during the week I subbed in an Algebra II class. Oh why couldn't I have covered that class this week instead, when I'm discussing these questions on the blog?

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