Monday, April 18, 2016

My Most Popular Lesson (Day 142)

Our coverage of the U of Chicago text is complete. I timed it this way because many schools here in California are beginning the SBAC, and I was always planning on finishing all of the relevant sections of the text before the SBAC. Every lesson from this point on will be basically a review for the Common Core Tests.

Last year, the first entry of this review period of the year was my posting my most popular lesson (by hit count). But in that particular post, I happened to write about the BASIS school curriculum. As BASIS came up again in the comments I quoted in my last post, I'll rewrite what I wrote last year about the BASIS curriculum:

One school system well known for accelerating students is BASIS Charter Schools. This charter network does not exist in California -- its schools are in Arizona, Texas, and DC.

One notices that there is no Geometry course in the BASIS system. But the Saxon text is mentioned. I have mentioned the Saxon high school texts earlier on the blog -- we discovered that Saxon is actually an integrated course where the three traditionalist courses Algebra I, Geometry, and Algebra II are consolidated into two courses, which Saxon calls "Algebra 1" and "Algebra 2." So BASIS students actually do receive Geometry instruction after all. According to the link, the goal at BASIS is actually to get the students to BC Calculus by junior year:

5th grade: Saxon 76 or 87
6th grade: Pre-Algebra
7th grade: Algebra 1
8th grade: Algebra 2
9th grade: Pre-Calculus
10th grade: Calculus AB
11th grade: Calculus BC
12th grade: Post AP or "Capstone" Course

This is definitely a very accelerated pathway. Notice that the one Saxon text that I own -- Saxon 65 -- would be considered a fourth grade course at BASIS -- and even kindergartners are assigned the first grade Saxon text. Then again, notice that BASIS is a charter school -- that is, that parents choose to send their students there. So students go in fully aware that all classes, not just math, are well above grade level.

Personally, I'm not sure whether I could have handled the BASIS course load. I might have been able to survive BASIS math, but I'm not sure about other courses. For example, AP English appears in sophomore year, and students take three science classes -- biology, chemistry, and physics -- beginning in sixth grade! I know I'd struggle in those classes -- and that's assuming that I could even survive the elementary school, where every student takes Mandarin. It's hard enough to learn another language that uses the familiar letters A, B, C, but it's so much more difficult to speak a language that doesn't use our alphabet.

Getting back to Geometry, I point out that there are schools that don't require students to take Geometry before (traditionalist, not Saxon) Algebra II. Here in California, many of our community colleges have only Algebra I (or a placement test score) as the prerequisite for Algebra II, which in turn is the prerequisite for transfer-level courses. So this opens the door for schools to go from Common Core in 8th grade to Algebra I in 9th, skip over Geometry (or offer it as a non-required course in summer) to Algebra II in 10th, and then on to Precalculus and Calculus.

Last year, I noticed that by view count, my most popular post by far, with 48 views, was Lesson 5-2 of the U of Chicago text, "Types of Quadrilaterals," posted back in October 2014. I'm not quite sure why this lesson had the most hits -- I wondered whether it was due to readers searching for the inclusive definition of trapezoid, which we are starting to hear about more and more often.

But as of this year, that post is now only my second most popular post -- since then, another entry has surpassed it. With 435 views, a post from almost exactly a year ago is my most read entry. This post is part of last year's discussion of PARCC Released Test Questions. Starting the middle of this week, I will begin writing about the test questions that were released this year -- so for the most part, I won't be going back to last year's questions.

Yet I do want to go back to my post from last year with the most hits. The PARCC question I discuss in this post is about dilations. I wrote that I didn't like how the question was worded, and I suspect the popularity of that post is due to other teachers and students feeling confused with that same question.

Well, here is the question that frustrated so many of us:

Question 7 of the PARCC Practice Test is on dilations on a coordinate plane. It is the last question in the non-calculator section:

In the xy-coordinate plane, Triangle ABC has vertices at A(1, -2), B(1, 0.5), and C(2, 1) and Triangle DEF has vertices at D(4, -3), E(4, 2), and F(6, 3).

The triangles are similar because Triangle DEF is the image of Triangle ABC under a dilation. What is the center and scale factor of this dilation?

Select the two true statements.

(A) The center of the dilation is at (-2, -1).
(B) The center of the dilation is at (-1, -2).
(C) The center of the dilation is at (0, 0).
(D) The scale factor is 1/2.
(E) The scale factor is 2.
(F) The scale factor is 4.

Since we are dealing with dilations on a coordinate plane, Lesson 12-1 of the U of Chicago text, "Size Changes on a Coordinate Plane," seems appropriate here. So let's attempt to find the scale factor based on what is taught in Section 12-1. We see that point B has coordinates (1, 0.5), and we notice that its image E has coordinates (4, 2). Since the coordinates of E are exactly four times those of B, we are tempted to say that the scale factor is 4 -- which happens to be one of the choices.

But then point C has coordinates (2, 1), and its image F has coordinates (6, 3), so the coordinates of the image are only thrice those of the preimage. Not only does the scale factor 3 not match the earlier calculated scale factor 4, but 3 isn't even one of the choices. And moreover, we can't multiply the coordinates of A, (1, -2), by anything to obtain the coordinates of its image D, (4, -3) So what gives?

The problem is that the formula given in Lesson 12-1 works only when the center of the dilation is the origin, that is (0, 0). As it turns out, we just proved that the center of the dilation must not be the origin as soon as we obtained different scale factors for different points! The next section, 12-2, discusses how to find the center and scale factor of a dilation given a preimage and image. But none of these questions appear on the coordinate plane.

In short, the center of every dilation that appears in the U of Chicago text has either coordinates (0, 0) or no coordinates at all. But this PARCC question involves a dilation whose center has coordinates other than (0, 0). So this problem has no analog in the U of Chicago text.

Thus once again, we have another PARCC problem that I don't necessarily like. Many traditionalists don't like the Common Core transformations at all. I have no problem with dilations, but notice that this is another thinking backwards problem. Instead of giving the center, scale factor, and preimage and asking students to produce the image, this question provides the preimage and image and asks for the center and scale factor. I wouldn't mind this question had the center been at the origin, but I disagree with the presence on the PARCC of a thinking backwards question involving a dilation centered at a point other than the origin.

So how exactly would we answer this problem and complete this PARCC question? We look at Lesson 12-2 and the directions for Questions 14 through 16:

In 14-16, trace each figure. Use a ruler to determine the center and the scale factor k for each size transformation represented. (The image is blue.)

But just "use a ruler" is no help to solve the PARCC question. We're given no graph, but just a list of the coordinates. So we back up to the examples and find out how to draw a dilation image:

Step 1. Measure OA.
Step 2. On Ray OA, locate A' so that OA' = 2.5 * OA. That is, A' is 2.5 times [in this problem, the scale factor is 2.5 -- dw] as far from the center as A. Point A' is the size change [dilation] image of A.

In this example, the center O, scale factor 2.5, and preimage A are given. In our problem, we are given the preimage A and image D and wish to find the center O and the scale factor. We can convert this algorithm into one to find the center O as follows:

Step 1. Choose any point A and its image A' (or D), and draw line AA'.
Step 2. Choose another point B and its image B' (or E), and draw line BB'.
Step 3. Then O must be the point where lines AA' and BB' intersect.

At this point, one may ask, how do we know that lines AA' and BB' intersect? Well, because of the definition of dilation, we can prove that O must lie on both lines. Therefore, lines AA' and BB' are provably not parallel, unless they are identical. So all we have to do is choose B to be any point on the preimage that's not on line AA'. Since in general we're transforming figures like triangles and rectangles rather than lines, it should be easy to find a point B that's not on line AA'. For the best results, B should be as far away from line AA' as possible. (If we're doing Question 14 from the U of Chicago's 12-2, and we choose A to be upper-left corner of the rectangle, then choosing B to be the lower-right corner is a bad idea since this point would be too close to line AA', even if it's not exactly on line AA'. Choosing B to be the upper-right corner is a much better strategy.)

Step 4. Measure OA' and OA, and divide to find the scale factor.

And that's it! If we are given pictures as in Questions 14 through 16 of the U of Chicago, then we can use a ruler, as directed, to perform Steps 1 through 3 above. But to answer PARCC Question 7, we can perform steps 1 through 3 algebraically.

Step 1. Point A has coordinates (1, -2) and its image D has coordinates (4, -3). This is a classic Algebra I problem, to find the equation of a line given two points. We must first find the slope:

slope of line AD = (-3 + 2)/(4 - 1) = -1/3

Then we use the Point-Slope Formula for the equation of a line:

y + 2 = -1/3(x - 1)
y = (-1/3)x - 5/3

Step 2. Point B has coordinates (1, 0.5) and point E has coordinates (4, 2). So we have:

slope of line BE = (2 - 0.5)/(4 - 1) = 1.5/3 = 1/2
y - 1/2 = 1/2(x - 1)
y = (1/2)x

We could have chosen points A and C instead of A and B, if we want to avoid the decimal 0.5. But we cannot choose points B and C, because point C lies on line BE.

Step 3. To find out where these lines intersect, we solve the system of equations:

y = (-1/3)x - 5/3
y = (1/2)x

(1/2)x = (-1/3)x - 5/3

3x = -2x - 10
5x = -10
x = -2

y = (1/2)(-2)
y = -1

And so we conclude that (-2, -1) is the center of the dilation, which is choice (A). All that remains is to find the scale factor of the dilation.

Step 4. The scale factor is equal to OD/OA. To find this distance, we use the Distance Formula:

OA = sqrt((-2 - 1)^2 + (-1 + 2)^2) = sqrt(10)
OD = sqrt((-2 - 4)^2 + (-1 + 3)^2) = sqrt(40) = 2sqrt(10)
OD/OA = 2sqrt(10)/sqrt(10) = 2

So 2 is the scale factor of the dilation, which is choice (E). Notice that one advantage to using dilations for this problem is that there's a definite preimage ABC and image DEF. So we know that the scale factor must be OD/OA, not OA/OD. This is important since we see that OA/OD = 1/2 is one of the wrong choices. If we were simply given ABC ~ DEF, then there would be no way to tell whether (D) or (E) is the correct answer.

This method gives the correct answer, but it is definitely a lot of work. Not only that, but it involves too much Algebra I. Many Geometry students won't remember how to solve all of these problems -- and even if they do, they are not likely to see this problem on the PARCC and reason, "To solve this problem, I have to use Point-Slope to write two equations, solve the system of equations to find the center, and use the Distance Formula and division to find the scale factor."

Fortunately, there are many ways to simplify some of the steps. For starters, to find the scale factor in Step 4, we can divide DE/AB rather than OD/OA. This follows from a theorem in Lesson 12-3:

Size Change Distance Theorem:
Under a size change [dilation] with magnitude [scale factor] k > 0, the distance between any two image points is k times the distance between their preimages.

And it's easier to find AB and DE than OA or OD, because AB and DE are vertical. When solving any problem that involves finding distance and there is a choice of which distance to find, we should always choose a distance that is vertical or horizontal -- that is, segments whose endpoints have matching x- or y-coordinates -- whenever possible. So we have:

A(1, -2), B(1, 0.5), AB = 0.5 - (-2) = 2.5
D(4, -3), E(4, 2), DE = 2 - (-3) = 5
DE/AB = 5/2.5 = 2

This avoids the Distance Formula and the need for square roots -- though unfortunately, the division 5/2.5 may be tricky since this is the non-calculator section.

But even if this is easier, we still have Steps 1 through 3, with the Point-Slope Formula to remember and a system of equations to solve. As it turns out, there's a way to avoid this as well.

We begin by finding the scale factor first. Our DE/AB trick shows is that it's possible to find the scale factor to be 2 without having to know what point O is. Then we have to find what the point O must be in order to have OD = 2 * OA. To do this, we use vectors. If we think of all the points as vectors, so point A is the vector aD is the vector d, and O is the unknown vector o, then we are trying to solve:

o - d = 2(o - a)
o - d = 2o - 2a
2a - d = o

Since o equals 2a - d, we only need to substitute:

o = 2a - d
o = 2(1, -2) - (4, -3)
o = (2, -4) + (-4, 3)
o = (-2, -1)

But even this method may still be confusing to students. Notice that for this particular problem, there's still another way to find the center O, using a process of elimination. Recall that we began today's post by noting that the center of dilation can't be the origin, since the scale factors for BE and CF aren't equal. So we've already eliminated choice (C). Then we look at the coordinates B(1, 0.5) and E(4, 2), and note that in each case, the y-coordinate is exactly half of the x-coordinate. So the line passing through these points has equation y = (1/2)x -- and we know that O must lie on this line (as it must lie on AD and CF as well). Choice (B) doesn't satisfy y = (1/2)x, and choice (C) has already been eliminated, leaving choice (A) as the correct center. But this trick only works for certain problems of this type -- it doesn't work in general.

Once again, I don't like this PARCC question one bit. There are better ways to determine whether students really understand what dilations are. We could keep the thinking backwards format, but make sure the center of dilation is the origin. Then to find the center, our simple division at the beginning of this post will work. The other way to do it is to have the center be a point other than the origin, but give the scale factor and preimage and ask for the image. In this case, vectors will probably be the best way to do it. This is how I designed today's worksheet -- I included a thinking backwards dilation problem centered at the origin, a straightforward dilation problem centered elsewhere, and finally a question at the PARCC level.

Notice that Dr. Frankin Mason not only dropped dilations completely, but the coordinate plane does not appear any more in his chapter on similarity (which is his Chapter 7, just like Glencoe). We know that anything Dr. M drops from his text is because he lacked the time to teach it and decided that it wasn't as important. That makes this PARCC question look even worse.

PARCC Practice EOY Exam Question 7
U of Chicago Correspondence: Lesson 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: Even though this problem includes coordinates, we state 12-2 to be the corresponding section rather than 12-1 since all of the dilations from 12-1 are centered at the origin, while today's dilation is not centered at the origin. There are no dilations in the U of Chicago text where the center is given coordinates other than (0, 0). Because of its difficulty, today's question is another horrible question to include on the PARCC.

I conclude this post by reminding you that this was one of last year's Released Test Questions. I definitely hope that this year's Released Test Questions don't include anything like this. Dilations centered at points other than the origin don't belong on the PARCC -- or SBAC for that matter.

No comments:

Post a Comment