Thursday, May 7, 2015

PARCC Practice Test Question 18 (Day 163)

Last night I tutored my geometry student. He just completed the test on Chapter 8 of the Glencoe text, on right triangles and trigonometry. He told me that the only questions he had trouble with were the ones on the Laws of Sines and Cosines -- and these made up a great portion of the test. This drives home my point that these Laws probably don't belong in a Geometry class at all -- instead, they should be saved for Precalculus (or Honors Algebra II/Trig).

Of course, Common Core appears to encourage pushing the Laws down into Geometry. We did a trig question for PARCC yesterday -- fortunately it only required the use of the sine ratio and not the Law of Sines. But we haven't finished the practice PARCC test yet -- there may still be a question needing one of the Laws yet on the test.

And speaking of Common Core, my student is now on Chapter 9 of the Glencoe text -- and this chapter is on transformations, which are strongly associated with Common Core. Here is an overview of Glencoe's Chapter 9:

Section 9-1: Reflections
Section 9-2: Translations
Section 9-3: Rotations
Section 9-4: Compositions of Reflections
Section 9-5: Symmetry
Section 9-6: Dilations

It should be easy to find the correspondences between the Glencoe sections and those of the U of Chicago text. Reflections are the topic of the entire fourth chapter of the U of Chicago, with most of Glencoe's 9-1 appearing in U of Chicago's 4-1 and 4-2. The second and third chapters of Glencoe's Chapter 9, or translations and rotations, correspond neatly to the same numbered sections in U of Chicago's Chapter 6. Glide reflections appear in both Glencoe's 9-4 and U of Chicago's 6-6. We see that Section 9-5 of Glencoe is on symmetry -- the main topic of the Mario Livio book that I'm reading now as well. Line symmetry is covered in U of Chicago's 4-7, but rotational symmetry -- which appears in Glencoe's 9-5 -- isn't adequately covered in the U of Chicago, which is a shame. I had to add rotational symmetry to my worksheets, as it's part of Common Core -- and we've already seen it appear on the PARCC! Three-dimensional reflection symmetry (plane symmetry) appears in Section 9-5 of both texts, but of course three-dimensional rotational symmetry (called axis symmetry by Glencoe) doesn't appear in the U of Chicago at all. Finally, most of the information on dilations covered in Glencoe's 9-6 appear in U of Chicago's 12-1 and 12-2.

There are key differences between Glencoe and the U of Chicago regarding how the various transformations are taught. To illustrate this, we look at the section that my student covered today in Glencoe -- Section 9-2, on translations. Glencoe gives the following definition: "A translation is a transformation that moves all points of a figure the same distance in the same direction." Then we notice that, rather than focus on the distance and direction of a translation, Glencoe goes straight to the coordinate plane and the horizontal and vertical components of a translation. The word "vector" doesn't appear here, but it's clear that vectors are the underlying concept here.

Later, Glencoe gives the U of Chicago definition of translation as the composition of two reflections over parallel lines. (For some reason, U of Chicago uses the word "composite," while Glencoe uses the far more common spelling "composition.") Actually, when I researched the Glencoe text on the web, I notice that newer versions of the Glencoe text actually use the word "vector" in this section, and the part about reflections in parallel lines doesn't appear at all.

So my student's version of Glencoe is a compromise between the new version of that text and my version of the U of Chicago. In my text, reflections in parallel lines are emphasized, and the coordinate plane appears only as an afterthought -- Questions 16 and 17 in Section 6-2 briefly mention coordinates used to find translations.

When we look at the PARCC practice test, we see that so far, translations do not appear on the PARCC -- perhaps because these are the easiest transformations to understand. But of our PARCC questions on rotations and dilations, all of them so far appear on the coordinate plane. Our one question on reflections technically doesn't use the coordinate plane -- although the mirrors r and s in Question 2 might as well be the x- and y-axes, and their intersection P might as well be the origin. So U of Chicago is doing a disservice to students by de-emphasizing the coordinate plane during its transformation questions. I think that the old Glencoe approach -- that is, the way that it's done in my students textbook -- will probably be the best.

Of course, the Glencoe text, as a pre-Common Core text, teaches transformations long after the notions of congruence and similarity. But Common Core states that these transformations be used to define congruence and similarity. A teacher in a Common Core course using the Glencoe text can simulate the Common Core approach by reordering the chapters -- specifically, Chapter 9 on transformations would precede Chapter 4 on congruence. Technically, the Dr. Hung-Hsi Wu approach would have us put transformations even before Chapter 3 on parallel lines, but this technique is awkward and not recommended unless one uses worksheets based on the Wu methods (such as the ones I post on the blog -- and I believe the New York curriculum is based on Wu as well).

My student was able to complete the lesson on translations easily -- his only problem was that he found it tedious to graph the translations on the coordinate plane. Then again, many students find any sort of graphing in any math class to be tedious. He will have to get used to graphing a lot as he moves through the ninth chapter of Glencoe, since the graphs make the chapter easier to understand.

Speaking of Chapter 9, I am now in that numbered chapter of The Equation That Couldn't Be Solved, titled "Requiem for a Romantic Genius." In this chapter, Livio discusses Galois's legacy, including the search to discover what makes certain people smarter than others. He writes that, just as with Einstein, Galois's brain was dissected after his death. But the autopsy didn't reveal anything unique about the size or shape of the mathematician's brain.

Later on Livio mentions the psychologist Howard Gardner. Gardner is well-known in the educational world for coming up with the idea of multiple intelligences. According to this link, these are Visual-Spatial, Bodily-Kinesthetic, Musical, Interpersonal, Intrapersonal, Linguistic, Logical-Mathematical:

Livio writes that many geniuses like Galois exhibited a combination of some of these traits -- but of course, Galois's strength was Logical-Mathematical. Well, let's use our logical-mathematical intelligence to answer the next PARCC question. Question 18 of the practice PARCC exam is on the volume of a cylinder:

Two cylinders each with a height of 50 inches are shown.

(Cylinder P is a right cylinder of radius x, while cylinder S is an oblique cylinder of radius y.)

Which statements about cylinders P and S are true?

Select all that apply.

(A) If x = y, the volume of cylinder P is greater than the volume of cylinder S, because cylinder P is a right cylinder.
(B) If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.
(C) If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted.
(D) If x < y, the area of a horizontal cross section of cylinder P is greater than the area of a horizontal cross section of cylinder S.
(E) If x < y, the area of a horizontal cross section of cylinder P is equal to the area of a horizontal cross section of cylinder S.
(F) If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.

The relevant section of U of Chicago is Section 10-5, on the volumes of prisms and cylinders. We can't help but notice that this is the section where Cavalieri's Principle is needed -- and even though Cavalieri is not mentioned in the question, the way the answer choices are worded hint at the Principle as the key to solving this problem. (In case you're curious, the Italian mathematician Cavalieri lived in the 17th century, so this was after his countrymen mentioned in Livio's book discovered the cubic formula. One of Cavalieri's teachers was Galileo.)

The first three choices state what happens when the radii are equal: x = y. Then all of the cross sections have equal areas, so by Cavalieri's Principle the volumes are equal, choice (B). The last three choices state what happens when the radii aren't equal: x < y. Then the cross sections don't have equal areas -- P's sections have area pi*x^2 and S's sections have area pi*y^2, and since x < y, we can conclude that pi*x^2 < pi*y^2. This is choice (F). So the two correct answers are (B) and (F).

Every time I write about Cavalieri's Principle, I can't help but think about Dr. Katharine Beals, because she had argued so strongly against Cavalieri's Principle and Common Core. I often mention Beals as an example of a traditionalist who opposes Common Core for mathematical -- as opposed to political -- reasons. (The other such traditionalist I've mentioned on the blog is Dr. Ze'ev Wurman.)

Most of Beals's recent anti-Core posts have been about math in grades 4-6 -- the range where I convert from pure traditionalism to a mixture of traditionalist and progressive philosophies. Here is a link to her post about Gardner's multiple intelligences:

And you’re a “verbal-linguistic” or a “logical mathematical” learner if you are, in what is still the most commonly used sense of the term, “smart.”

The implication in this post is that only these last two are true intelligences, and the other five are only used to justify progressive, rather than traditionalist, philosophy in the classroom.

It was Beals who argued that good tests should focus only on skill and content. Returning to today's PARCC question, we know that Beals would be opposed to this sort of question if the students had to generate their own explanations (the part after "because"). My student had to do some explanation questions in his homework last night, where he had to write things such as "these are translation images of each other because they have the same orientation."

By Beals's standards, a better set of answer choices would be:

(A) If x = y, the volume of cylinder P is greater than the volume of cylinder S
(B) If x = y, the volume of cylinder P is equal to the volume of cylinder S
(C) If x = y, the volume of cylinder P is less than the volume of cylinder S
(D) If x < y, the volume of cylinder P is greater than the volume of cylinder S
(E) If x < y, the volume of cylinder P is equal to the volume of cylinder S
(F) If x < y, the volume of cylinder P is less than the volume of cylinder S

and an even better question -- one that is purely skill-based -- would be simply:

Find the volume of cylinder S.
(The correct answer to this question would be 50*pi*y^2.)

But there's one more thing that I want to discuss here. Beals discusses a Calculus problem that she found on a high school exit exam from Finland:

The cheese is sold in a straight circular cylinder-shaped package. The cylinder has height h and base radius r. The cheese is first cut vertically into two parts. One half is then cut at an angle as by the smaller piece, which has height h. Calculate the volume of this piece of cheese using integration.

Now Beals was including this problem to argue that the Common Core tests should require students to solve problems such as this one in order to graduate from high school. Whether students should have to learn Calculus in order to graduate from high school is a topic for another day. What I wanted to point out is that for a special case of this problem, we can solve this problem using -- not integral calculus, but Cavalieri's Principle!

Based on this drawing, it appears that one base of the cylinder is centered at the origin. The base is in the xy-plane and has the equation x^2 + y^2 = r^2. The first cut appears to be along the yz-plane -- so it has the equation x = 0 -- and the second cut appears to have the equation xh - zr = 0, as it passes through the the origin and (r, 0, h) and contains the y-axis. So z = xh / r.

Now here's the special case we want to consider. Let the height of the cylinder equal the circumference of the base -- that is, h = 2pi * r.

We wish to consider cross sections of this piece of cheese. But instead of horizontal planes, we consider vertical planes parallel to the xz-plane -- that is, planes of equation y = constant, as this constant ranges from -r to r. Notice that cross sections of the original cylinder are all rectangles of the same height as the cylinder, r. But when we make the cuts, the cross sections are all right triangles -- these occur when we cut each rectangle by the line where z = xh / r intersects the plane.

We see that these triangles are all similar, since their legs are parallel to the x- and z-axes and their hypotenuses all have slope h / r. If we let x and z represent the legs that are parallel to these respective axes, then we calculate their values in the plane y = constant as follows: to find x, we first recall that x^2 + y^2 = r^2, so x = sqrt(r^2 - y^2). To find z, we note that z = xh / r -- and since h = 2pi * r, we have z = 2pi * x = 2 pi * sqrt(r^2 - y^2).

Now we can find the area of this triangular cross section as follows: the base is x = sqrt(r^2 - y^2) and the height is z = 2pi * sqrt(r^2 - y^2). So the area is 1/2 * xz = pi(r^2 - y^2).

Now consider a sphere centered at the origin of radius r. We notice that the cross section of a sphere with the plane y = constant is a circle of radius sqrt(r^2 - y^2), since this radius is the leg of a right triangle with the other leg y (actually the absolute value of r) and hypotenuse r. And so the area of this circle must be -- you guessed it -- pi(r^2 - y^2).

So all the cross sections of the cheese and the sphere have the same area. Therefore, by Cavalieri's Principle, they have the same volume -- namely 4/3 * pi * r^3.

Now this only works if the height is exactly 2pi * r. If the height isn't 2pi * r, then the area of each triangle is exactly h / (2pi * r) times the area of the cross section of the sphere. So we may think of the triangle as having the same area as h / (2pi * r) copies of that circle (which may be non-integer -- a sector of that circle), so the volume of the cheese is exactly h / (2pi * r) times the volume of the sphere -- so it's 2/3 * r^2 * h.

If we were to use Calculus, as originally intended, that cross section pi(r^2 - y^2) would actually be the integrand, and we integrate it as y ranges from -r to r. But Cavalieri's Principle takes the place of the integral (since Cavalieri is formally proved using that same integral). Now the problem has been reduced to one that, in theory, a Geometry student could solve.

So I decided to include this problem on my worksheet. Since this problem is difficult, it is the only problem on the page (except for a quick follow-up to the original problem of the day). I only covered the special case h = 2pi * r on this page. But even though it's on the difficult side, Beals seems to believe that it belongs on the Common Core test, so here it is.

PARCC Practice EOY Exam Question 18
U of Chicago Correspondence: Section 10-5, Volumes of Prisms and Cylinders
Key Theorem: Cavalieri's Principle

Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).

Common Core Standard:
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Commentary: The text gives a few problems which are directly based on Cavalieri's principle, though none are exactly like this PARCC problem. Such problems are under Properties in the SPUR section -- not Skills as most traditionalists prefer.

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