Tuesday, May 5, 2015

PARCC Practice Test Question 16 (Day 161)

Chapter 7 of Mario Livio's The Equation That Couldn't Be Solved is called "Symmetry Rules." This chapter is about how symmetry rules physics.

According to Livio, every major leap in the advancement of physics has occurred when another symmetry rule was discovered. He begins with the famous British scientist Isaac Newton, who realized that the laws of motion are the same no matter where one is in the universe. The force that caused an apple to fall on Newton's head is the same as the force that keeps the planets in motion. Of course, this is the force that Newton called gravity.

Then, of course, we have the most well-known scientist of all -- Albert Einstein. He discovered another symmetry rule -- that the speed of light is the same for all observers. It is this symmetry that led to Einstein's theory of relativity. According to this theory, we should not consider the three spatial dimensions as being separate from space, but as four dimensions of spacetime. In this theory, gravity is not a distinct force, but instead the result of spacetime being distorted by mass. Because of this, the geometry of four-dimensional spacetime is non-Euclidean.

Several mathematicians and physicists who have been mentioned here on the blog also appear in this chapter of Livio's book. The Princeton mathematician John Horton Conway is mentioned. On my blog, I discussed how Conway categorizes quadrilaterals by their symmetry group -- so names like kite, parallelogram, and rectangle refer to quadrilaterals with different symmetry groups. Livio, meanwhile, describes how Conway categorizes, not quadrilaterals, but frieze patterns. Unlike finite quadrilaterals, where only reflections and rotations are possible, frieze patterns are infinite and so allow translations and glide reflections as well. Just as there are seven different types of quadrilaterals based on their symmetry, there are seven types of frieze patterns based on their symmetry. According to Livio, Conway names these hop, step, jump, sidle, spinning hop, spinning sidle, spinning jump.

Livio moves on to discuss another major theory -- quantum mechanics. As it turns out, the laws of quantum mechanics are, in essence, symmetry laws. And I've already mentioned on the blog the discoverer of these laws -- Emmy Noether, the German mathematician whose Google Doodle appeared on her birthday about a month and a half ago.

The author points out that the search for symmetry drove the search for unification laws. He credits Newton with discovering the first unification law -- the force that causes apples to fall is the same as the force that keeps the earth orbiting the sun (gravity). The second unification combines electricity and magnetism to obtain electromagnetism. The next unification combines electromagnetism with the weak nuclear force to obtain the electroweak force. Every force has a particle associated with it, which may also be viewed as a wave that transmits the force. Livio describes the brilliant physicists who made all of these findings.

But what remains elusive is the theory that unites all of the forces -- a theory that incorporates both relativity and the standard model of quantum mechanics. This is the "theory of everything" after which last year's Stephen Hawking movie was named.

Livio ends the chapter by mentioning the most promising theory of everything -- string theory. This theory is difficult to understand and describe, but according to Livio, string theory predicts that every particle has a property, "spin," that is preserved under another type of symmetry -- supersymmetry.

We now move on to the next PARCC problem. Question 16 of the PARCC Practice Test is another question on dilations:

In the xy-coordinate plane, Triangle ABC has vertices A(-4, 6), B(2, 6), and C(2, 2). Triangle DEF is shown in the plane [to have vertices D(-2, 3), E(1, 3), and F(1, 1)].

What is the scale factor and the center of dilation that maps Triangle ABC to Triangle DEF?

(A) The scale factor is 2, and the center of dilation is point B.
(B) The scale factor is 2, and the center of dilation is the origin.
(C) The scale factor is 1/2, and the center of dilation is point B.
(D) The scale factor is 1/2, and the center of dilation is the origin.

Question 7 is also on dilations, but today's question is much, much easier. First of all, this question has only four choices while Question 7 had nine choices, as there were combinations of three possible scale factors and three possible centers of dilation. Furthermore, students need only select a single letter to choose one of the four possibilities, while Question 7 required that two letters be selected, so this question has less room for error in marking the correct letter.

Looking at our choices, we see that the possibilities of the center are B and the origin. But clearly B is not the center -- for if B were the center, it would be a fixed point, so the image of B would have to be the point itself. Yet the image of B is E, and one can tell from the graph that the coordinates of E are (1, 3) and not (2, 6). So the center can't be B, so by process of elimination it must be the origin. As the dilation is centered at the origin, the coordinates of the image are obtained, according to Section 12-1 of the U of Chicago text, simply by multiplying each coordinate of the preimage by the scale factor of the dilation. The coordinates of E are exactly half those of B, so the scale factor is 1/2. Once again, we distinguish between the choices of 2 and 1/2 by noting that ABC is the preimage and DEF is the image -- not vice versa, as often confused the geometry student I tutor. The answer is (D).

Even if this problem were not multiple choice, we can still identify the center as the origin by noting that there exists a constant such that each coordinate of the image equals the corresponding coordinate multiplied by that constant. Of course, that constant is 1/2.

And if one even considers that to be cheating as that trick doesn't generalize to centers of dilation other than (0, 0), we can use methods that do so generalize. We notice that BC = 4 and EF = 2 -- these distances are easy to find because the sides are vertical. (We could also have found the horizontal distances AB = 6 and DE = 3). This gives us the scale factor as 1/2 since EF is half the distance of BC. Then we must find O such that OE is half of OB:

o - e = 1/2(o - b)
2o - 2e = o - b
o = 2e - b
o = 2(1, 1) - (2, 2)
o = (2, 2) - (2, 2)
o = (0, 0)

We can also use find the equations of lines BE and CF using the Point-Slope Formula to see where they intersect. The equation of line BE is y = 3x and the equation of line CF is y = x, and these lines intersect at (0, 0).

No matter what method we use, the question is much easier to solve. I prefer dilations centered at the origin to those not centered there for this type of question. And preimages with sides that are vertical or horizontal are always easier to figure out. Therefore I'd rather see questions like 16 and not 7 on the actual PARCC.

At last, we are halfway through the practice PARCC. I repeat the notes that I wrote for Question 7:

PARCC Practice EOY Exam Question 16
U of Chicago Correspondence: Section 12-2, Size Changes Without Coordinates
Key Theorem: Definition of Size Change (Dilation)

Let O be a point and k be a positive real number. For any point P, let S(P) = P' be the point on Ray OP with OP' = k * OP. Then S is the size change [dilation] with center O and magnitude [scale factor] k.

Common Core Standard:
Verify experimentally the properties of dilations given by a center and a scale factor.

Commentary: This dilation, unlike that of Question 7, is centered at the origin. So except for the task of finding out that it is indeed centered at the origin, most of today's problem corresponds to Section 12-1 of the U of Chicago text.

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