Thursday, May 12, 2016

PARCC Practice Test Question 17 (Day 160)

Today is the fifth day that I am subbing in the Algebra II and Integrated Math I classes. Actually, during sixth period I find out that the teacher is not returning tomorrow, so I will be covering this class for one more day.

Meanwhile, it is the second block day, with even-numbered periods, including both Math I classes, meeting today. Because of this, today is the second post with the "traditionalists" label. Tomorrow, the post will not be labeled "traditionalists" -- yesterday's and today's posts are it for the week.

I've been continuing to think about the block schedule debate. I'm starting to wonder whether, as usual, a compromise -- the Hybrid Block Schedule -- is the best. I'm hoping that the "Modified Block Schedule" that traditionalist Jeff Lindsay cites as one of the three best schedules (along with the traditional seven- and six-period days) refers to the Hybrid Block Schedule.

Recall occasional blocks are great for science labs, and it's reasonable that teachers can come up with labs or other extended activities once a week. It's only when we get into the Common and Pure Block Schedules where teachers have to figure out how to fill a block period multiple times a week. And this is when teachers add more activities to cover the time, which leads to the lower achievement results that Lindsay mentions on his blog.

Most traditionalists other than Lindsay don't discuss the block schedule that much. One traditionalist, Barry Garelick, writes that he was once a long-term sub at a block schedule school. But he definitely liked the Pure Block Schedule his school had -- he only had to teach three periods, all of them even, so he had every other day off!

Even though the Hybrid Block Schedule may be the best, I don't criticize the Common Block Schedule, with its four block days and one all-classes day per week. This is because it's the schedule my own high school had when I was in Grades 10-12, and it's the schedule at the school where I will teach in the fall. Now that I'll be starting a full-time job soon, I must be careful not to bite the hand that feeds me and criticize anything particular to my new school here on the blog. I can, and do, criticize the Pure Block Schedule -- the lack of correspondence between days of the week and periods of the day make this schedule too confusing.

So how do I handle the block classes today? Well, in the Integrated Math I classes, the students continue work on the Geometric Constructions Project. Today I decide to help the students out with specific parts of their project -- including the figure where they need to show three steps of the construction, and how to create their own figure.

By the way, looking at the specific steps, I've seen various students use both methods for constructing the hexagon -- six arcs around the hexagon, which is the method mentioned on the worksheet, and starting with a diameter, which is mentioned in both YouTube video links (and is an extension of the equilateral triangle method given on the worksheet). I've always found the first method easier, but one of the videos argues that the second method is better as it avoids cumulative error -- with the first method, a mistake made with one arc leads to several arcs being incorrect.

One benefit of the block schedule is that it allows for time for projects such as this one. But this project ends up lasting an entire week, spanning both traditional and block periods. Recall that a few days ago, I spoke to another Honors Integrated Math I teacher about this project. She told me that she won't give the students as much time to complete the project -- it might make sense just to give them a single block period. My students get so much time simply because the teacher knew that he would be absent for a week, and so the project is to cover his entire absence.

Most of the students are busy working on the project for most of this block. But a few students finish the project today. The real problem is not what to do with these students to cover the block today, but rather what to do with the early finishers tomorrow. Some students end up going to their biology class to study from that text. (Actually, since I'm curious as to whether the science classes do labs on block days, I ask some of the students about their science class. Apparently, some of the students have a special Biomed elective in addition to their bio class, and today they performed a lab where they prepare a gel with real human DNA samples. I wonder whether this lab is possible under a traditional bell schedule. Anyway, today they are working on a lab report.)

Meanwhile, in 4th period Algebra II, today the seventh quaver progress reports are handed out. I notice that slightly more than half of this class are earning grades of D or F. This is to be expected since there is a huge jump in difficulty from Geometry to Algebra II -- especially the second semester of Algebra II. There are two seniors in the class, and one is failing. Meanwhile, there are two sophomores in the class, and of course both are doing great. One of them has the only A in the class, and the other has a B+, tied for the second-best grade. Only the top students get to take Algebra I in 8th grade, leading to Algebra II as a sophomore and AP Calculus as a senior.

For those who wonder whether we cover twice as much work on block days, I actually do assign two Pizzazz worksheets today, both on the Pythagorean Theorem. One asks, "What Do Two Bullets Have When They Get Married?" (answer: a BB) and the other asks, "What Did Lancelot Say To The Beautiful Ellen?" (answer: I-C-U-R-A-B-U-T-L-N). The Lancelot side is more challenging, as it often requires the students to find a leg, rather than just the hypotenuse (and this was the most common error, to answer that a leg is longer than the hypotenuse). But to inspire the students to work on both worksheets, I play my usual Conjectures/"Who Am I?" game.

Chapter 17 of Morris Kline's Mathematics and the Physical World is called "More Light on Light." In this chapter, as the title implies, Kline discusses more about the physics of light.

"My design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which I shall premise the following Definitions and Axioms." -- Newton, in his Opticks

So we continue with Newton's experiments on light. The science of light is still known today as "Opticks," or as we spell it now, optics. Kline begins:

"More pervasive than motion is the phenomenon of light. Since ancient times scientists have undertaken to investigate the nature of light, its behavior as it passes through a medium such as air or water, and its associated process of vision."

Kline begins by considering light reflecting off of a spherical lens or mirror. This case is also considered in Lesson 15-3 of the U of Chicago text. He begins by letting O be the source of the light, P be any point where the light ray strikes the mirror, and C be the center of the sphere. I is a point on line OC where the reflected ray hits. Then he labels the following angles: Angle 1 is POA, Angle 2 is PCA, Angle 3 is PIA, i is the angle of incidence OPC, and r is the angle of reflection CPI. Then he performs a little Geometry. According to Lesson 13-7, the measure of an exterior angle of a triangle is the sum of the two remote interior angles:

Angle 2 = Angle 1 + Angle i
Angle i = Angle 2 - Angle 1


Angle 3 = Angle 2 + Angle r
Angle r = Angle 3 - Angle 2

Since the angle of incidence equals the angle of reflection, of course:

Angle 2 - Angle 1 = Angle 3 - Angle 2
Angle 1 + Angle 3 = 2 Angle 2

Then Kline approximates the measures of these angles as the ratio of the arclength PA divided by a segment, which he admits isn't exact, but is close enough. (He states that this is just like the idea behind the radian measure of an angle -- the arclength divided by the radius, except that the known lengths aren't exactly radii.)

Angle 1 + Angle 3 = 2 Angle 2
Arc PA/OA + Arc PA/IA = 2 Arc PA/CA

Dividing by the arclength PA:

1/OA + 1/IA = 2/CA

Kline rewrites this as 1/p + 1/q = 2/R. I've seen this equation given before in Algebra II classes, usually as an example of a rational equation in the real world.

Question 17 of the PARCC Practice Exam is on translations:

17. Triangle ABC is shown in the coordinate plane. The triangle will be translated 2 units down and 3 units right to create triangle A'B'C'. Indicate whether each of the listed parts of the image will or will not be the same as the corresponding part in the preimage (triangle ABC) by selecting the appropriate box in the table.

                                                  Will be the Same          Will Not be the Same
The coordinates of A'
The coordinates of C'
The perimeter of triangle A'B'C'
The area of triangle A'B'C'
The measure of angle B'
The slope of A'C'

To answer this, we note that translations are isometries. According to Chapter 6 of the U of Chicago text, all isometries preserve the four A-B-C-D properties: Angle measure, Betweenness, Collinearity, and Distance. The preservation of angle measure allows us to check "Will be the Same" in the fifth row, and the preservation of distance allows us to check "Will be the Same" in the third row, as a perimeter is just a sum of distances.

As it turns out, area is preserved by all isometries, but we must wait until we get to the Area Postulate of Lesson 8-3, where we learn that congruent figures (like the preimage and image of an isometry) have the same area. Slope, meanwhile, isn't preserved by all isometries, but it is preserved by all translations, which are the subject of this question. The proof was somewhat tricky, but we learned on the blog that all lines are either identical to or parallel to their translation images -- so in either case they have the same slope.

So far, we have four checks in the "Will be the Same" column. But the first two rows -- the coordinates of A' and C', must be in the "Will Not be the Same" column. This is because translations don't have fixed points.

PARCC Practice EOY Question 17
U of Chicago Correspondence: Lesson 6-2, Translations
Key Theorem: Two Reflections 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 two lines.

Common Core Standard:
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Commentary: Unlike dilations, translations do appear in Integrated Math I.

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