Tuesday, May 10, 2016

PARCC Practice Test Question 15 (Day 158)

Today is the third of five days I'm spending in the Algebra II and Integrated Math I class.

Today I actually want to discuss the Algebra II class in more detail, even though our focus on the blog will continue to be the Integrated Math I class. This class is not assigned any written homework this week -- instead, the students were assigned to watch the following videos:

Notice that the first video is called "Gettin' Triggy wit It," a parody of Will Smith's song "Gettin' Jiggy wit It." We've seen several parodies of songs for Pi Day on the blog before -- well, there exist other math-based parodies that have nothing to do with pi, and this is one of them.

Today the students worked on another Pizzazz worksheet. This time, it is page 167, "Why Did the Saltine Lock Itself in the Bank Vault?" Students are given a right triangle and asked to find the sine, cosine, and tangent of one of the angles. All three sides are given, so the students only have to know which ratio is correct. This is the subject of the other videos posted above -- SOH-CAH-TOA, which is a common mnemonic.

Most students appear to figure out the worksheet without any problems. The final answer is, "It wanted to be a safe cracker." Those who struggle with this assignment in first period most likely confuse sin A with sin B  as both of these appear on the worksheet. We know that for sin A and sin B, the hypotenuse is the same, but for sin A we consider the side opposite of Angle A, and for sin B we consider the side opposite of Angle B. In fourth period, I see fewer problems with sin B -- but I wonder whether it's because they understand it better or because they simply copy "It wanted to be a safe cracker" from someone else's worksheet.

But actually, I really have a problem with one of the triangles on the worksheet. The sides are given as 1-sqrt(2)-3 -- and we see that this violates the Pythagorean Theorem. Indeed, not only are these not the sides of a right triangle, they can't be the sides of any triangle, as they violate Triangle Inequality.

Most likely, Pizzazz intended the sides to be 1-sqrt(3)-2, not 1-sqrt(2)-3. Sure, there are other ways to fix the triangle -- perhaps the author forgot the radical sign on 3 for a 1-sqrt(2)-sqrt(3) triangle, or maybe he/she omitted the factor of 2 on sqrt(2) for a 1-2sqrt(2)-3 triangle. But I measure angle A with a protractor and see that its measure is 30 degrees. So the triangle is a 30-60-90 triangle -- now we know that its intended side lengths are 1-sqrt(3)-2. Still, in order to complete the puzzle we must answer the questions as they are indicated on the triangle -- so sin A = 1/3, not 1/2.

I actually mention the real 1-sqrt(3)-2 triangle in class to make a point. I explain to the class what they have accomplished -- let's say you have a triangle with Angle A = 30 degrees, Angle C = 90 degrees, BC = 1, and AB = 2, then what is sin A? A clueless student would say -.988, but you guys who are smart know that the answer is just 1/2. (I don't say this, but the wrong answer -.988 comes from blindly finding the sine of 30 radians on a calculator!)

In Integrated Math I, students continue working on the project. Naturally, no one likes the fact that now it must be turned in by every individual. The questions like "When do we have to do constructions in real life?" come up again the instant the students find out that the project must be submitted individually. Other than that, most students are able to complete at least two of the constructions needed for the project.

Chapter 15 of Morris Kline's Mathematics and the Physical World is called "From Projectiles to Planet and Satellite." In this chapter, Kline tells us that the same force that governs projectile motion also keeps the planets in orbit.

"What? You think Isaac Newton told a lie? Where do you hope to go when you die?" -- Anonymous

Kline begins the chapter with one last word from Galileo:

"Toward the end of his Two New Sciences Galileo wrote, 'The principles which are set forth in this little treatise will, when taken up by speculative minds, lead to many another more remarkable result.'"

But, as the opening quote implies, this chapter isn't about Galileo at all. Instead it's all about another famous scientist -- the 17th century Englishman Isaac Newton.

This chapter chronologically begins right where the previous chapter ends. Indeed, Kline writes:

"Newton was born in 1642, the year of Galileo's death."

As we already know, Newton made many contributions to mathematics, as Kline describes:

"Because a plague was raging in London, Newton returned home -- to think. His creative faculties blossomed in the years immediately following his graduation from college. In a few years he made first-class contributions to algebra; he formulated and applied the law of gravitation, which proved to be the key to unifying earthly and heavenly motions; he set forth basic procedures in the calculus that entitle him to the honor of being one of the two founders of the subject...."

Yes, so those students who took the AP Calculus exam last week have Newton to thank. Indeed, Kline writes, "But Newton is ranked with Archimedes and Gauss as one of the three greatest mathematicians in all history."

But of course, it's his law of gravitation for which Newton is best-known. Kline writes that his discovery of the law can't be as simple as his observing an apple fall from a tree since "the fall of an apple involves only the pull of the earth on objects nearby."

Instead, as the title of this chapter implies, Newton's insight was that the same laws that determine the paths of projectiles also keep the planets in orbit. His law can be written in symbols as:

F = GmM / (r^2)

According to Kline, Newton verified this formula by considering the acceleration of the moon as it travels in its orbit. He calculated this using trigonometry:

cos E = EM / EP

where EM is the distance from the earth to the moon, and EP is the distance to the moon if it were to travel in a straight line rather than an orbit for a time of one minute. Kline performs part of the calculation and states that the calculated acceleration agrees with the value derived from his law of gravitation, thereby verifying the law.

Kline concludes the chapter by pointing out that the planets in orbit around the sun follow the same laws as projectiles shot out the ground because the planets once were projectiles themselves:

"Newton applied this very reasoning to answer the question of how the planets originated. He argued that they must have been shot out of the sun at some angle...Hence the mathematics of projectile motion led among other things to a theory about the origin of the planets that is still accepted."

Question 15 of the PARCC Practice Exam is on volume:

15. The given cylindrical container is used to fill the rectangular prism fish tank with water.

(Here are the dimensions of the solids: the cylinder has height 8" and diameter 6", while the prism has dimensions 12" by 24" by 24".)

What is the least number of full cylindrical containers needed to completely fill the fish tank?

This is an excellent calculator question. It is clearly a question about volume, so we must calculate the volumes of the solids. Here is the volume of the cylinder:

V_cyl = pi r^2 h
V_cyl = pi (3)^2 (8)
V_cyl = 72pi
V_cyl = 226.2 cubic inches

And here is the volume of the rectangular prism:

V_box = lwh
V_box = (12)(24)(24)
V_box = 6912 cubic inches

So the ratio of the volumes is:

V_box / V_cyl = 6912/(72pi)
V_box / V_cyl = 96/pi
V_box / V_cyl = 30.6
V_box / V_cyl = 31 containers

The most common error will be to substitute in the diameter of 6 inches into the cylinder volume formula where the radius belongs.

Meanwhile, our luck from yesterday has run out. There is no Kuta worksheet lying around the Integrated Math I classroom that I can post today on volume -- and with good reason. Volume is usually considered to be an Integrated Math II (possibly even Math III) topic. There is no volume anywhere in the Math I curriculum.

As I've said before, if I must make a decision between posting something that occurs in the classroom and sticking to my original plans, priority should go to the classroom. But it's awkward to break up this PARCC review, when my plans are to post one question per day. Yesterday I was able to cheat by posting that Kuta worksheet, which was related to both the classroom and the PARCC. But today, there's nothing I can do -- volume, the subject of today's PARCC question, has absolutely nothing to do with anything that's occurring in the class.

I was originally just going to post a few volume questions and ignore the classroom. But I just couldn't resist creating a new version of the Pizzazz worksheet -- one where that mistake with the sides 1-sqrt(2)-3 has been corrected to 1-sqrt(3)-2.

PARCC Practice EOY Question 15
U of Chicago Correspondence: Lesson 10-5, Volumes of Prisms and Cylinders
Key Theorem: Prism-Cylinder Volume Formula

The volume V of any prism or cylinder is the product of its height h and the area B of its base.
V = Bh

Common Core Standard:
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Commentary: Yes, I know that this trig worksheet has nothing to do with the PARCC. But once again, the classroom always takes priority over my previous plans. And besides, there will probably be a trig question later on the PARCC.

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