"This terror then and darkness of mind must be dispelled not by the rays of the sun and glittering shafts of light, but by the aspect and law of nature." -- Lucretius
This chapter is mostly about Copernicus and Kepler and their heliocentric theory. These two 17th century astronomers transformed the universe from one centered at the earth, with the sun and planets moving around it in "epicycles," to one with a focus at the sun and the earth and other planets moving around it in ellipses. An "epicycle" is like a circle going around another circle, such as the moon going around the sun (except that this is more like an ellipse upon an ellipse."
By the way, this weekend I plan on attending a local Renaissance Faire. One of the carnival games involves shooting water at a bulls-eye. The targets are the planets and sun, and the bulls-eye is the center of the universe -- the earth. Copernicus, of course, lived late in the Renaissance, so the geocentric universe was the prevailing opinion for most of the Renaissance.
I don't have much to write about this chapter, because I have an announcement to make. My employment situation is about to change. I am on the verge of accepting a teaching position for the 2016-7 school year at a charter middle school in Los Angeles. I'll have more to say about that on the blog as I receive more information.
Meanwhile, let's back to the traditionalists. Last week, I mentioned how some states are considering dropping Common Core, including Michigan. Well, here's yet another heated Common Core debate thread from that particular state.
One poster in particular definitely appears to be a traditionalist -- Kelley. In response to a comment that the Common Core doesn't dictate teaching methods, Kelley writes:
So we see that in addition to a traditionalist, Kelley is a strong Tenth Amendment supporter (her comments about the Core being unconstitutional). Remember that I'm torn between whether there should be national standards or 50 different state standards.
Continuing with her traditionalist criticism of the curriculum, she writes:
Later on, when another poster asked to look at the new proposed standards first:
So Kelley likes the pre-Core Massachusetts standards, which Michigan is considering adopting. But some Tenth Amendment supporters oppose even two states having the same standards -- they refer to it as an illegal interstate compact.
As for myself, I consider regional standards to be a good compromise between national standards and 50 separate state standards. But before Michigan adopts the Bay State's pre-Core standards, let's see whether Massachusetts itself will adopt the old standards. Then I'd like to see the standards expand to the other nearby New England states, as many of these have small populations. My own state of California, meanwhile, is so large that our state standards would a priori be regional standards!
Meanwhile, a pro-Core writer brings up an idea that I've mentioned before -- Presidential (or Gubernatorial, in this case) Consistency:
YEAH the government officials do not care if the kids are up to speed. Why? Simple they send their own kids to private school.
They just want to use a special test that will make them look good but in reality because will think we are up to international standards when in reality we are not. We finally got it right with common core and now going backwards again.
THEY SHOULD BE A LAW THAT IF YOU ARE GOING TO RUN FOR OFFICE IN MICHIGAN THAT YOUR KIDS MUST BE ENROLLED ONGOING IN THE PUBLIC SCHOOLS OF MICHIGAN!! Then and only then they may start to care.
Question 8 of the PARCC Practice Test is about the equation of a circle:
8. Part A
A circle in the xy-coordinate plane has the equation x^2 + y^2 + 6y - 4 = 0. If the equation of the circle is written in the form x^2 + (y + k)^2 = c, where k and c are constants, what is the value of k?
What is the radius of the circle?
And so this is our second circle equation with completing the square this week! Here is the answer:
x^2 + y^2 + 6y - 4 = 0
x^2 + y^2 + 6y = 4
x^2 + y^2 + 6y + 9 = 13
x^2 + (y + 3)^2 = 13.
This gives us the answers k = 3 and sqrt(13), or choice (C) as the answer to Part B. Again, the wrong answer choices for Part B give away the most common student errors. Choice (D) occurs when students give the value of r^2 rather than r. Choice (A) occurs if students stop after the second step, see 4 on the right side alone, and assume that this is r^2 without completing the square. Choice (B) occurs if students make both errors.
But my concern with this question is Part A. Notice that the center of the circle is (0, -3). But the question does not ask for the y-coordinate of the center -- it asks for the value of k in the special equation provided, x^2 + (y + k)^2 = c. Since this equation includes (y + k)^2 and not (y - k)^2, this means that k = +3, not -3.
Let me be honest -- I don't like this part of the question one bit. Students are expected to know the equation for the area of a circle, (x - h)^2 + (y - k)^2 = r^2. Using (y + k)^2 only ends up confusing students more than necessary.
To see what's wrong with this question, suppose Part B had been written like this:
If the equation of the circle is written in the form x^2 + (y + k)^2 = c, where k and c are constants, what is the value of c?
Now suddenly (D) is the correct answer, even though (C) is the radius of the circle. And the following version of Part B is even worse:
If the equation of the circle is written in the form x^2 + (y + k)^2 = r, where k and r are constants, what is the value of r?
Here the answer is again (D), even though (C) is still the radius. That is, r is not the radius. But almost every text is going to use r for the radius in this situation. So to me, using r to denote the square of the radius, rather than the radius itself, isn't merely confusing -- it's downright deceitful!
At this point, someone might point out that students might make the mistake I mentioned earlier, and think that the right hand side of the equation is the radius rather than its square. In this case, two wrongs actually make a right -- the students don't know that they need to take the square root to find the radius, and they don't know that the question is asking for the square of the radius!
Likewise, the original Part A as given in the text also has two wrongs making a right. A student could think that +3 is the y-coordinate of the center and not realize that the question is actually asking for the opposite of the y-coordinate.
In fact, some might argue that the purpose of this question is to protect student from having to give a negative sign. But then this defeats the purpose of giving this problem -- in such a problem, it's possible for a student to make 0, 1, or 2 errors. It's set up so that a student will make 0 errors -- that is, a student will likely make either 1 or 2 errors. Since two wrongs here make a right, a student gets credit for making 2 errors, but not for 1 error. A student is rewarded for making more mistakes! This is surely not the message we want to send our students!
Furthermore, suppose the following question were to appear on a PARCC Algebra I exam:
-- Give the general solution of the equation ax^2 + bx - c = 0.
The correct answer is x = (-b +/- sqrt(b^2 + 4ac))/2a. The negative sign on c in the original problem forces the discriminant to be b^2 + 4ac. Sure, the Quadratic Formula has b^2 - 4ac, but the question asks for a general solution to the given equation, not for the Quadratic Formula itself. And the following question is even worse:
-- Give the general solution of the equation ax^2 + bx = c
where the negative sign on c isn't as apparent. Here's yet another bad question:
-- Give the general solution of the equation cx^2 + bx + a = 0
where the correct answer is x = (-b +/- sqrt(b^2 - 4ac))/2c. The following problem isn't as bad:
-- Give the general solution of the equation ax^2 + bx = d
where at least the use of the letter d is a tip-off that something is different. We can solve this by using the Quadratic Formula with a = a, b = b, and c = -d. There's no reason to assume that c = d, while there's a great reason to assume that c = c!
One might argue that questions like ax^2 + bx = c force students to think, rather than allow them to write the Quadratic Formula without any thought at all. But once again, there's a fine line between challenging and deceiving, and the question ax^2 + bx = c does the latter.
There's no inherent reason that a must be the quadratic coefficient, b the linear coefficient, and c the constant in the equation ax^2 + bx + c = 0. But this is a convention -- virtually every single Algebra I text that teaches the Quadratic Formula uses these variables. Conventions are used not only in mathematics, where a, b, and c are coefficients, (h, k) is the center of the circle with radius r, but also in the sciences (a for acceleration, c for the speed of light), computer science, and so on.
Using variables for a purpose that directly oppose their conventional use not only deceives students, but also reinforces the notion that algebra is more about manipulating symbols rather than actually solving problems. This goes back to Kline and the true purpose of mathematics.
So I'd like to discard this sort of problem and throw it away. But of course, we can't because it appears on the PARCC. Therefore questions such as these fuel the assertions of Kelley and other traditionalists that yes, the Common Core really does dictate the curriculum.
Actually, this is a bit tricky. I've once said before that the only way to have a set of standards that does not dictate curriculum is for the test to ask higher-level questions, which are more likely to be made curriculum-independent. (Deceptive questions like today's shouldn't be on any sort of test no matter what the standards are.)
Questions that ask about reflections, rotations, and translations obviously require that congruence be developed using transformations. This is why I wonder whether it's possible to use area arguments in place of transformations, since area is something that the students should be learning anyway.
PARCC Practice EOY Question 8
U of Chicago Correspondence: Lesson 11-3, Equations for Circles
Key Theorem: Theorem (Equation for a Circle)
The circle with center (h, k) and radius r is the set of points (x, y) 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: It's the way this question is worded that makes it so tricky, so I just added in questions from last year's review worksheet. Today is an activity day, so the rest of the worksheet is based on yesterday's question, as it lends itself to activity more naturally.