Monday, April 27, 2015

PARCC Practice Test Question 10 (Day 155)

Over the weekend, I finally got those Mario Livio books that I was discussing on the blog a couple of weeks ago. Though I didn't get the book that was mentioned in the NOVA episode -- Is God a Mathematician? -- I did buy the one that I wanted to mention here on the blog -- The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. I also got checked out from the library another Livio book, an audiobook this time -- Brilliant Blunders, From Darwin to Einstein: Colossal Mistakes by Great Scientists That Changed Our Understanding of Life and the Universe.

In Brilliant Blunders, I listened to the first two chapters and a short part of the third. Chapter 1 is introduction, and Chapter 2 is the first scientist Livio describes in his book, British naturalist Charles Darwin, and his theory of evolution. Chapter 3, based on the short part that I heard, will reveal that Darwin's error is that, while he knew that natural selection occurred, he didn't know why. That would have to wait until the discovery of genetics, by the Czech scientist Gregor Mendel.

So far, I have read Chapter 1 of The Equation That Couldn't Be Solved. This first chapter is simply titled "Symmetry." In this chapter, Livio describes four types of symmetry -- mirror-reflection symmetry (his example is the bilateral symmetry of animals), rotational symmetry (snowflakes), translational symmetry (19th century British artist William Morris and 18th century Austrian musician Wolfgang Amadeus Mozart) and glide-reflection symmetry (more art plus the motion of humans walking and snakes slithering). And these are, of course, the four isometries that appear on the Common Core tests. Livio even mentions a fifth symmetry that occurs in 3D -- screw symmetry (leaves of certain flowers). I mentioned the 3D screw isometry back during spring break.

Livio also mentions the 20th century American mathematician George David Birkhoff. I mentioned Birkhoff earlier on this blog as the originator of the Ruler Postulate, the first postulate in many geometry texts (and part of the Point-Line-Postulate in the U of Chicago). Here Livio credits Birkhoff with creating a formula, M = O/C, where M is the aesthetic measure of a work of art, O is the order, or symmetry, present in the work, and C is the complexity of the work.

Question 10 from the PARCC Practice test is on trigonometry. Fortunately, this will be one of the simpler trig questions if students know what they are doing.

In right triangle ABC (and with the right angle at A), Angles B and C are not equal. Let sin B = r and cos B = s. What is sin C - cos C?

(A) r + s
(B) r - s
(C) s - r
(D) r/s

We think back to Section 14-4 of the U of Chicago text, since this is on sines and cosines. When we discussed 14-4 on the blog, I mentioned how I often tell my students where the name "cosine" actually comes from. Let me repeat the explanation given in the U of Chicago text -- this refers to Example 1 from the text, where the right angle is C and A is an acute angle. Since in the PARCC question the right angle is A instead, I will change the letter A to C in order to avoid confusion:

...notice that sin C = cos B and sin B = cos C. This is because the leg opposite either angle is the leg adjacent to the other. Angles C and B are also complementary. This is the origin of the term "cosine"; cosine is short for complement's sine.
[emphasis U of Chicago's]

(As an aside, often Precalculus students wonder why, once they learn about the cotangent, we do have cot B = 1/tan B, but not cos B = 1/sin B. This is because, if C and B are complementary, we do have tan C = 1/tan B, but not sin C = 1/sin B. The "co" in cosine, cotangent, and cosecant stands for "complement," not "reciprocal.")

Once we remember this, the question becomes trivial. So sin C = cos B = s, and cos C = sin B = r, and so sin C - cos C = s - r, which is choice (C). The trick is that if the students think that the question is asking for sine and cosine of B instead of C, they might choose (B) instead of (C) -- an easy mistake to make since choice (B) precedes choice (C).

So far, of the three questions we've seen from the calculator section of the PARCC Practice Test, two of the questions don't require any calculations at all! Thursday's question involved two angles that add up to 90 degrees, and we had to calculate one of the measures. Friday's question involved two angles that add up to 90 degrees, but we only had to figure out which angles those were, not any of their individual measures. Today's question involves two angles that add up to 90 degrees, but we only have to find the relationship between their sines and cosines, not calculate those values.

One more thing I want to point out is that extra mention in the problem that Angles B and C do not have equal measure. I suspect the reason for this mention is that if Angle B = C, then both (B) and (C) would be correct answers as r - s = s - r = 0. So that line is just to avoid the loophole that someone may choose (B), then claim that we can't rule it out in case that B = C.

I haven't had the opportunity yet to mention this relationship between sines and cosines to the geometry student I tutor. We had to jump into the Law of Sines so fast that I didn't have time to talk about cosine as the complement's sine. Hopefully I'll be able to mention it by the end of this week.

PARCC Practice EOY Exam Question 10
U of Chicago Correspondence: Section 14-4, The Sine and Cosine Ratios
Key Theorem: Definition of sine and cosine

In right triangle ABC with right angle C,
the sine of Angle A, written sin A, is leg opposite Angle A / hypotenuse
the cosine of Angle A, written cos A, is leg adjacent to A / hypotenuse

Common Core Standard:
Explain and use the relationship between the sine and cosine of complementary angles.

Commentary: That "cosine" means "complement's sine" is mentioned in U of Chicago, but it is not emphasized there as strongly as in the Common Core Standards. We see that Question 3 asks for cos F and sin G in triangle FGH, and Question 6 asks for sin 48 and cos 42. And technically speaking, Question 5 asks for sin 45 and cos 45 -- and these are equal because 45 degrees is its own complement. In the SPUR section, Questions 13-14 ask for sin A and cos B, as part of Objective C in the Skills section.

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