Friday, May 1, 2015

PARCC Practice Test Question 14 (Day 159)

Last night I tutored my geometry student. He is now wrapping up Chapter 8 of the Glencoe text, with a test coming up. In the practice problems for the test, a right triangle ABC is given, where a = 15, b = 16, and c = 21. (No, 15-16-21 is not a Pythagorean triple. These numbers have obviously been rounded off to simply the problem.)

Question 10 asks the student to find cos B, which he found to be 15/21 or 5/7. Then Question 12 asks him to find sin A, which he also found to be 5/7. So I found my opportunity to tell my student that the word "cosine" means complementary sine. He was able to figure out the answer to PARCC Question 10 from the beginning of this week relatively quickly.

Chapter 5 of Mario Livio's The Equation That Couldn't Be Solved is all about the other co-discoverer of the nonexistence of the Quintic Formula, the French mathematician Evariste Galois. The chapter is titled "The Romantic Mathematician."

Livio begins by discussing Galois's childhood. His father was the mayor of a small Parisian suburb, and his mother came from a family of lawyers. Just like Abel's father, Galois's mother homeschooled her son until he was nearly 12, when he entered the Lycee Louis-le-Grand.

Just as with Abel, Livio contrasts some of Galois's teachers. His debate teacher, Pierre-Laurent Laborie, thought that Galois was too young for his classes and forced him to repeat a year -- but it was during this year that the boy discovered his passion for mathematics. Livio states that Galois's most inspiring teacher -- just as Holmboe had been for Abel was Louis-Paul-Emile Richard. Galois had already mastered an entire geometry text in just two days, and so Richard encourage the youngster to engage in his own research.

Livio states that this geometry text was Andre Marie Legendre's Elements of Geometry. This was the first widespread geometry text since Euclid's Elements. As this is a geometry blog, let me take some time to discuss Legendre's famous text. Here is a link to Legendre on Google books:

Legendre's first 21 propositions are in fact definitions. The first word that he defines is geometry -- it is "a science which has for its object the measure of extension. Extension has three dimensions, length, breadth, and thickness." Definition 17 contains the quadrilaterals, and these are all defined exclusively, as was common back then. So a rectangle "has its angles right angles, without having its sides equal."

Propositions 22-26 are Legendre's axioms. These are not the same as Euclid's. Indeed, Legendre's fifth axiom is: "Two magnitudes, whether they be lines, surfaces, or solids, are equal, when, being applied one to the other, they coincide with each other entirely; that is, when they exactly fill the same space." Notice that this is Euclid's Principle of Superposition, and is the basis for the Common Core definition of "congruent."

Proposition 27 is the first theorem, which states "all right angles are equal." This is Euclid's Fourth Postulate, but it's Legendre's first theorem as he provides a proof. Proposition 36 is SAS, and it is proved using Legendre's fifth postulate so it's proof is similar to Euclid's. Proposition 38, ASA, is also proved using superposition. But Proposition 43, SSS, is proved using the previous proposition, which is actually SAS Inequality (or Hinge Theorem).

Of course, what seems to be missing in Legendre is a Parallel Postulate. The phrase "Of Parallel Lines" appears at the top of page 13. On that page is Proposition 56, which is HL, and then Proposition 57, the Triangle Angle-Sum Theorem. The latter theorem famous requires a Parallel Postulate in Euclid. Legendre's proof appears to be much more complicated than the usual proof. I'm having trouble understanding it fully -- to think that a teenage Galois mastered not only this proof, but the entire book, in just two days.

Galois's goal was to attend the famous Ecole Polytechnique. The Polytechnique was the school for mathematics in those days -- all of the most famous mathematicians in the world were there. But Galois failed the entrance exam both times that he attempted it. Livio describes his second attempt: "Speculation has it that Galois's tendency to calculate mostly in his head and to commit only the final results to the blackboard left a bad impression in an oral examination in which he was supposed to show all of his deliberations. [Examiner] Dinet in particular had a reputation for posing relatively simple questions, but also of being utterly uncompromising when it came to the answers."

And once again, just as with Abel, we are reminded of the Common Core. Imagine replacing "blackboard" with "computer," "oral" with "typed," and "Dinet" with "PARCC (or SBAC) graders" in the passage above. A major concern with the Common Core is that geniuses like Abel and Galois could have trouble succeeding in class and passing the exams.

In between all of this, Galois -- after first trying, like Abel, to discover the Quintic Formula -- proved that no formula can solve all quintic equations. Compare this to the Quadratic Formula, which does solve all quadratic equations. But Galois took this a step further -- he demonstrated which quintics do have a formula to solve them, and which ones don't. Livio will describe this further in Chapter 6 of his book.

It's impossible to discuss Galois's life without considering the politics of his day. During the 18th and 19th centuries, France alternated between a kingdom ruled by the Louis dynasty, an empire ruled by the Napoleon dynasty, and a republic. Galois's family supported the republic. But many leaders at the school Galois ultimately attended -- the Ecole Normale -- were loyal to the crown. On July 14th -- the anniversary of the storming of the Bastille -- Galois and his friend Ernest Duchatelet were arrested for attempting to lead a republican revolt.

In great detail, Livio describes the tragic end for Galois. Two days before the end of his sentence -- by this time, he had been transferred to a halfway house -- the great mathematician was challenged to a duel. Galois ended up losing the duel when his opponent shot him in the stomach. He was rushed to the hospital, where he died the next day at the age of 20 -- even younger than Abel had lived. It's a mystery as to the identity of the killer. Livio has his own theory, but I won't spoil it here. You can read Livio's book and find out.

Once again, many of our students can relate to Galois's tragedy. How many of our students have been involved in violence? And the reason for the duel was, most likely, that Galois had been caught looking at a girl the wrong way -- hence the title's chapter "The Romantic Mathematician." Once again, our students may be able to relate.

Question 14 of the PARCC Practice Test is on equations of a circle. This is the second such problem appearing on the test. So we go straight from Galois and the quintic to the lowly quadratic:

The equation x^2 - 8x + y^2 = 9 defines a circle in the xy-coordinate plane. What is the radius of the circle?

Of course, we know that completing the square is the technique that we must use here:

x^2 - 8x + y^2 = 9
x^2 - 8x + 16 + y^2 = 9 + 16
(x - 4)^2 + y^2 = 25

So the square of the radius must be 25. Therefore the radius of the circle is 5.

I've already stated that I don't necessarily like these circle equation questions. But if we compare today's question to last week's PARCC Question 6, I find today's to be much more reasonable. There is only a single part to this question -- find the radius -- not multi-part like Question 6. Because the center lies on one of the axes, we only need to complete the square in x, not y. And finally, the question simply asks for the radius -- unlike the thinking backwards in Question 6 where the radius is given and the students had to find the right hand side of the equation that produces that radius.

Once again, Section 11-3 of the U of Chicago text is on equations for circles -- and once again, none of the questions in this section require students to complete the square. But even a few of the questions in this section, as written, ask students to identify the radius of the circle, and at least these problems remind students that the right hand side of the equation is r^2, not r -- the square root of the right hand side is the actual value of the radius.

I could essentially repeat the Question 6 worksheet and be done with it. But then again, today is actually an activity day, so let's try to make this into an activity instead. I wanted an activity that involved making circles, so I found the following website:

Yes, it's been a while since I've found a good activity on another blog, but here it is. Jeff Brenneman is a teacher in a Chicago high school. Unfortunately he hasn't posted since the first day of school, but nearly two years ago he created a Battleship game that involves circles and graphs.

Notice that one can play this game without using equations. But if we want to use this to prepare for PARCC, then one might point out that the circles' equations can be used to find their intersection:

Enemy sub is:
2 units away from (0, 1): x^2 + (y - 1)^2 = 4
1 unit away from (1, 3): (x - 1)^2 + (y - 3)^2 = 1
5 units away from (4, 0): (x - 4)^2 + y^2 = 25

To solve this system of quadratic equations, we expand them just as the PARCC test does:

x^2 + y^2 - 2y = 3
x^2 - 2x + y^2 - 6y = -9
x^2 - 8x + y^2 = 9

We subtract the second equation from the first, and then the second equation from the third:

2x + 4y = 12
-6x + 6y = 18

So we just reduced our system of three quadratic equations to two linear equations, a previously solved problem -- just as Abel and Galois tried to do with the quintic.

This technique, called triangulation, is mentioned on several episodes of Square One TV, particularly in the Mathnet segment that occurred at the end of every episode. Here's how Brenneman describes his game:

I would break the students up into teams of two or three; each team gets one H.I.D.D.EN. submarine and three naval stations. Teams get to place their submarine and naval stations at whatever coordinates they choose (within certain borders, of course).

After all submarines and stations are placed, I provide each team with information about how far away each enemy submarine is located from their stations. (This adds a layer of complexity to the original problem scenario, as teams now have information about multiple submarines and they have to mix & match circles in order to pinpoint them all.) The teams then race against each other to try and be the first to locate and destroy the other submarines. Winning team gets riches and glory. Well, just glory. Not much glory.

That is, as much glory as Galois received while he was alive. To end the story of Galois, Livio points out that activists wanted to use Galois's funeral to further their cause -- a rally for the republic. But instead, the rebellion was delayed three days, in order to mark the death of General Lamarque instead. As Livio goes on to write, "Fate thus robbed Galois even of the opportunity to incite a rebellion in death." Otherwise, it would have been Galois whose funeral sparked the climactic uprising at the end of the well-known play and movie Les Miserables.

PARCC Practice EOY Exam Question 14
U of Chicago Correspondence: Section 11-3, Equations for Circles
Key Theorem: Equation for a Circle

The circle with center (hk) and radius r is the set of points (xy) satisfying

(x - h)^2 + (y - k)^2 = r^2.

Common Core Standard:
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Commentary: I've already said the following after Question 6, but I repeat myself anyway. The U of Chicago gives equations of circles, but never equations where students have to complete the square to find the center and radius. The fact that this Common Core standard is immediately followed by standard involving the equations of conic sections should have been a red flag that this standard belongs in Algebra II, not Geometry. But this all goes back to the fact that the Common Core doesn't divide its standards into courses.

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