Friday, May 20, 2016

PARCC Practice Test Question 23 (Day 166)

Today is a traditionalist-labeled post. I used last week's traditionalists posts to discuss the block schedule that my new school will use in the fall. Today's post will discuss the other aspect of my new school -- the fact that it is a charter school.

What exactly is a charter school? Well, why don't we ask the president? He has declared the first week of May each year as National Charter Schools week. This is what he said in this week's presidential proclamation about two weeks ago:

"During National Charter Schools Week, we celebrate the role of high-quality public charter schools in helping to ensure students are prepared and able to seize their piece of the American dream, and we honor the dedicated professionals across America who make this calling their life's work by serving in charter schools," Obama said in a presidential proclamation. "With the flexibility to develop new methods for educating our youth, and to develop remedies that could help underperforming schools, these innovative and autonomous public schools often offer lessons that can be applied in other institutions of learning across our country, including in traditional public schools."

Charter schools are not without controversy, though. Once again, I want to avoid biting the hand that feeds me, and so I don't discuss this controversy on the blog. Instead, let's go to see what the traditionalists have to say about charter schools.

Before I was hired to work at one, I've only mentioned charter schools thrice on the blog. The first example I gave was a middle school that starts in fifth grade, and the second example I gave was a Waldorf-inspired school. The third example I gave was BASIS -- a charter school which prides itself on accelerating students as much as possible. We know that the traditionalists especially like BASIS, since traditionalists also emphasize acceleration.

No, the school I'll be working at in the fall isn't any of those three schools. My middle school begins at the more common sixth grade, not fifth (though it is a K-8 campus), it isn't a Waldorf school, and it certainly isn't BASIS (as there are no BASIS charters in California) -- otherwise I'd be preparing to teach Saxon Math 87, Algebra I, and Algebra II to my middle school students! But this does show that there are many different types of charters.

Charters are associated with choice. A quick glance at some traditionalist websites tells me that some traditionalists favor charters because they give teachers choices such as pedagogy -- meaning, of course, that they like charters that choose the traditionalist pedagogy over the more dominant progressive pedagogy. (I wonder whether they'd be as enthusiastic about charters if the dominant pedagogy were traditionalism and the progressives were the ones who wanted an alternative choice!)

Oh, and by the way, I subbed in at another class on the block schedule due to SBAC testing. This is the same school that I was at last week, with a Hybrid Block Schedule -- but as I pointed out last week, the blocks are now two full hours since school can't be out a half hour early everyday. This time it's a health class, but once again, the students watch a video (on the danger of STD's) and then study for a test coming up on Monday. Again, I can see traditionalists like Jeff Lindsay who oppose the block schedule point out that teachers use blocks just to play videos -- but once again, a counterargument is that teachers use blocks to have the sub play videos. Because I'm a sub, I can't be sure what these teachers do on block days when there's no sub. (And notice that the math teacher last week didn't play any videos on block days.)

Chapter 23 of Morris Kline's Mathematics and the Physical World is called "The Integral Calculus." I covered derivatives yesterday, so now it's time for integrals.

"As God calculates, so the world is made." -- Leibniz

Kline begins:

"One of the fascinating facts about the development of mathematics is that an idea that is created to solve one type of problem often solves another which on superficial examination seems to be totally unrelated. This turn of events occurred in the history of the calculus."

Notice that Kline quotes our other co-inventor, Leibniz. And the connection that he and Newton discovered is that the derivative and the integral are related -- the Fundamental Theorem of Calculus.

Kline's example to find the area of the curve y = x^2 between x = 3 and x = 5. We now think of this as the Riemann sum, but Newton and Leibniz originated the idea. Kline writes:

y_1 h + y_2 h + ... + y_n h

to approximate the area by n rectangles.

Actually, I won't write the rest of Kline's equations here, since the summation and integral signs look messy when converted into ASCII (and because I took up so much of this post with the discussion about traditionalists).

Question 23 of the PARCC Practice Exam is on translations on the coordinate plane:

23. Quadrilateral ABCD is shown graphed in the xy-coordinate plane.

Part A
Quadrilateral ABCD will be translated according to the rule (x, y) -> (x + 3, y - 4) to form A'B'C'D'.
Select the correct orientation of A'B'C'D' and place it correctly in the plane.

Part B
Quadrilateral ABCD maps onto A"B"C"D". It will undergo a different transformation that will map A(-6, 3) to A", B(-4, 5) to B", C(-1, 6) to C", and D(-3, 2) to D". The transformation will consist of a reflection over the y-axis followed by a translation. Point D" is shown plotted in the plane after the transformation.
Plot the point A" in the plane.

[The coordinates of D" are (6, 2) -- dw]

Part A is straightforward. The transformation is a translation, so we know that the image has the same orientation as the preimage. So we only need to translate a single point, say A(-6, 3) to A'(-3, -1), and place the quadrilateral so that A' is at the correct point.

Part B appears to be a glide reflection -- the composite of a reflection and a translation. Since we're given the image D", it's best just to reflect D first in the y-axis, to obtain (3, 2). The final point we need for D" is (6, 2), which is three units to the right of the mirror image. Thus the translation must be three units right. Transforming A in the same manner gives us (6, 3) as the mirror image and then (9, 3) as the final image.

Notice that for glide reflections, the translation is usually taken to be parallel to the mirror, but this time the translation is perpendicular to the mirror. We've discussed this in the past -- when the translation is perpendicular to the mirror, the composite turns out to be a simple reflection -- not a glide reflection at all! The mirror for this new reflection is found by taking the original mirror and sliding it half the distance of the translation. So we translate the y-axis 3/2 units to the right, giving us the line x = 3/2 as the new mirror. Reflecting A in this new mirror does indeed give us (9, 3), but it's actually easier to reflect in the y-axis and then translate.

PARCC Practice EOY Question 23
U of Chicago Correspondence: Lesson 6-2, Translations

Key Theorem: Two Reflection Theorem for Translations

If m | | l, the translation r_m o r_l slides figures two times the distance between l and m, in the direction from l to m perpendicular to those lines.

Common Core Standard:
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Commentary: After not providing much of an activity last week, this week I decide to repeat an activity from last year that is all about compositions of transformations.

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