Thursday, April 16, 2015

PARCC Practice Test Question 3 (Day 148)

Question 3 of the PARCC Practice EOY is on arc length of a circle. Let's take a look at the details of this problem:

The circle with center F is divided into sectors. In circle F, EB is a diameter. The length of FB is 3 units.

(In the figure, the measures of three central angles are given. Angle AFB = 120, BFC = 45, and CFD = 30 degrees.)

Select the correct expression that represents the arc length of arc AED.

(A) pi
(B) 11pi/4
(C) 13pi/4
(D) 7pi/4

If students know what they are doing, then they should be able to figure this one out. We notice that the given length of 3 units refers to the radius of the circle, so the circumference must be 2pi * 3, or 6pi units. The sum of the measures of the three angles is 120 + 45 + 30 = 195 degrees. This means that the remaining central angle, AFD, is 360 - 195 = 165 degrees, and therefore the measure of the minor arc AED must also be 165 degrees. So the arc length is 165/360 * 6pi, or 11pi/4, which is (B).

Officially, we discussed arc length as part of the Pi Day lesson. But now I wonder whether I spent too much time on trying to derive the circumference of a circle from the area and not enough time on the arc lengths. Well, this is why I'm going through all of the PARCC questions, to make sure that every question that appears on the PARCC is covered on this blog.

We notice that of all the arc length problems in the U of Chicago text give the degree measure of either the arc itself or the central angle it subtends. But this PARCC question requires students to calculate the angle by subtracting the sum of the three angle measures from 360.

Students may also have problem with finding 165/360 * 6pi, as this is a non-calculator question. I was able to perform this calculation first by cancelling the factor 6, to obtain 165pi/60. It might not be obvious that 15 is the factor that needs to be cancelled -- except it was to me, only because 165/60 is clearly not 1 and the other three choices all have 4 in the denominator. On a multiple choice question, often looking at the choices can give clues as to how to simplify the fractions.

And of course, another thing about multiple choice questions is that the wrong answers -- the "distractors" -- often correspond to plausible answers. Notice that all three distractors for this question are the lengths of other arcs in the problem -- choice (A) is the length of arc AE, choice (D) is the length of arc ED, and choice (C) is the length of the major arc ABD, when the question asks for the length of the minor arc AED,

Traditionalists shouldn't have any problem with this question. I would definitely consider this to be a content-based skill question, and it requires students to make some calculations in order to find out first the arc measure, and then the arc length. Yesterday's question was objectionable to most traditionalists, but not today's.

PARCC Practice Test Question 3
U of Chicago Correspondence: Section 8-8, Arc Measure and Arc Length
Key Theorem: Circle Circumference Formula

If a circle has circumference C and diameter d, then C = pi * d.

Common Core Standard:
Find arc lengths and areas of sectors of circles
CCSS.MATH.CONTENT.HSG.C.B.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Commentary: The questions in Section 8-8 of the U of Chicago on arc length are easier because the measure of the central angle is always given, not calculated. The Common Core Standard relevant is basically the heading "Find arc lengths...," not the content of HSG.C.B.5. In particular, radian measure doesn't appear until Precalculus Trigonometry.