PARCC Practice Test Question 6 (Day 151)

Last night I tutored my geometry student. Section 8-5 of the Glencoe text is specifically on Angles of Elevation and Depression.

I mentioned that the U of Chicago text doesn't have a separate section for angles of elevation. The term "angle of elevation" is mentioned in Section 14-3 of the U of Chicago, on the Tangent Ratio:

Example 2. At a location 50 m from the base of a tree, the angle of elevation of the tree is 33 degrees. Determine the height of the tree.

We notice in the Questions that there are several problems where the phrase "angle of elevation" could appear, but the U of Chicago avoids this term. For example, we look at Question 4:

4. When the sun is 32 degrees up from the horizon, the wall of a store casts a shadow 25 meters long. How high is the wall?

This question could be rewritten as:

4'. When the angle of elevation of the sun is 32 degrees, the wall of a store casts a shadow 25 meters long. How high is the wall?

Similarly, we notice Question 10:

10. From eye level 5' off the ground and 20' away from a flagpole, a person has to look up at a 40 degree angle to see the top of the pole. How high is the pole?

Rewritten, this becomes:

10'. From eye level 5' off the ground and 20' away from a flagpole, the angle of elevation is 40 degrees to the top of the pole. How high is the pole?

It makes sense for angles of elevation to appear in the lesson on tangents rather than sines and cosines since, as I pointed out, most angle of elevation problems involve vertical height and horizontal distance, so the hypotenuse of the triangle doesn't appear. In the Glencoe text, most of the problems in Section 8-5 involve the tangent ratio, but a few use sine instead. I must also point out that several of the problems involve inverse sine or tangent, while the U of Chicago text mentions only the inverse tangent among the arcfunctions. I consider this to be a deficiency of the U of Chicago text.

In the U of Chicago, the phrase angle of depression appears in Section 14-5 on sines and cosines:

14. From the top of a building, you look down at an object on the ground. If your eyes are 55 feet above the ground, and the angle of sight, called the angle of depression, is 50 degrees below the horizontal, how far is the object from you?

Notice that this question is written so that the cosine is needed, rather than sine or tangent.

In the SPUR review section at the end of Chapter 14, angles of elevation and depression are certainly under the section U for USES -- and yes, my student groaned slightly when I told him that all of the questions in this section are word problems The four questions under Objective I (the ninth letter of the alphabet), namely Questions 52 through 55, could be written use the phrase angle of elevation (depression) yet none of them do.

I will have more to say on angles of elevation and depression, and trigonometry in general, when we get to the PARCC trig questions. But today's PARCC question isn't on trig. Question 6 of the PARCC Practice Test is on completing the square:

Use the information provided to answer Part A and Part B for question 6.

The equation x^2 + y^2 - 4x + 2y = b describes a circle.

Part A
Determine the y-coordinate of the center of the circle.

Part B
The radius of the circle is 7 units. What is the value of b in the equation?

So far on this PARCC Practice Test, I pointed out that some traditionalists may object to some of the questions, but I myself have no problem with them -- even though I acknowledge that some students may be tricked on questions where may need to choose more than one answer. But all of this changes with this problem. Question 6 is the first question that I believe doesn't belong on a Geometry test.

To solve this question, we must complete the square:

x^2 + y^2 - 4x + 2y = b
x^2 - 4x + y^2 + 2y = b
x^2 - 4x + 4 + y^2 + 2y + 1 = b + 4 + 1
(x - 2)^2 + (y + 1)^2 = b + 5

Comparing this with the equation (x - h)^2 + (y - k)^2 = r^2, we see that the center of the circle must be the point (2, -1). So we get k = -1 as the answer to Part A.

For Part B, we notice that the right-hand side of the question, b + 5, must equal r^2. Since r = 7, we have r^2 = 49, so b + 5 = 49. Subtracting 5 gives us b = 44 as the answer to Part B.

The first objection I have is the need to complete the square. A few weeks ago, I saw that equations for circles appeared on the PARCC and so I quickly included Section 11-3 from the U of Chicago on a worksheet. But Section 11-3 doesn't contain any completing the square questions. When I created my worksheet for 11-3, I squeezed in completing the square questions because I knew that they were coming on the PARCC.

Many students haven't seen completing the square since Algebra I -- and many haven't seen it at all since many Algebra I teachers skip it. This is why I'd prefer this question to be on an Algebra II test, not a Geometry test.

Of course, a common error to be made on this sort of question, even by Algebra II students, would be to mix up the sign. Had this been a multiple choice question, I can almost guarantee that one of the wrong choices for Part A would be 1. And the other two wrong choices may be 2 and -2 -- the test writers banking on students confusing the x- and y-coordinates of the center.

Part B just makes this problem even worse. This question provides the radius of the circle and students are asked to find the value of b. It would have been much more straightforward to give the value of b -- that is, begin with the equation x^2 + y^2 - 4x + 2y = 44 -- and then ask the students for the radius. I have no problem with having questions with a Part B on the test, and this would've made a better Part B than what is actually included here. Notice that I will organize today's worksheet differently for multi-part questions -- there will be just one or two questions divided into parts similarly to the PARCC question.

Such questions that force the students to think backwards appear several times on the PARCC. I have no problem with this thinking backwards questions, and would've had no problem with it here in Question 6 had this not been an Algebra II question masquerading as a Geometry question. If we are going to have thinking backwards questions on the test, then they should be saved for questions when the underlying concepts are simple. If we are going to have completing the square to find the equation for a circle on a Geometry test, then there should only be straightforward questions, such as just to identify the center and radius of the circle.

Before we leave this question, let me point out that if this had been an Integrated Math course, this might have been an excellent question to ask. Integrated Math courses ought to focus on the connection between algebra and geometry. I could envision an Integrated Math II or III course in which one covers circles -- their circumferences, areas, tangent lines, inscribed angles, and equations, and then segue from there to the equations of the other conic sections.

But as a question on a traditionalist Geometry test, this question is terrible. Notice that Dr. Franklin Mason steers away from equations for circles completely in his text. Unfortunately, this is only the first of several terrible questions on the PARCC.

PARCC Practice EOY Exam Question 6

U of Chicago Correspondence: Section 11-3, Equations for Circles

Key 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:

CCSS.MATH.CONTENT.HSG.GPE.A.1
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: The U of Chicago gives equations of circles, but never equations where students have to complete the square to find the center and radius. The fact that this Common Core standard is immediately followed by standard involving the equations of conic sections should have been a red flag that this standard belongs in Algebra II, not Geometry. But this all goes back to the fact that the Common Core doesn't divide its standards into courses.