## Wednesday, April 22, 2015

### PARCC Practice Test Question 7 (Day 152)

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, Section 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 Section 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. Yesterday, I wrote that thinking backwards questions should be given only when the concept being tested is easy. 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 Section 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 was the problem that my geometry student had when completing the similarity questions in Chapter 7 of the Glencoe text.

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 Section 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 a, D 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)

I mentioned something similar at the end of Monday's lesson, when we were trying to divide a segment into a given ratio. For today's problem, using vectors turns out to be easier, unlike Monday.

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. It might even be good to have a scale factor less than 1 in this case -- then this problem could be done almost the same way as PARCC Question 5 from Monday's post, so students can see how Question 5 and this question are related. 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: Section 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:
CCSS.MATH.CONTENT.HSG.SRT.A.1
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.