The figure shows lines

*r*,

*n*, and

*p*intersecting to form angles numbered 1, 2, 3, 4, 5, and 6. All three lines lie in the same plane.

(In the figure, the three lines are concurrent, and they are lettered

*r*,

*n*, and

*p*in clockwise order. The angle numbered 1 is formed between lines

*r*and

*n*, and the other angles are numbered clockwise.)

Based on the figure, which of the individual statements would provide enough information to conclude that line

*r*is perpendicular to line

*p*?

Select

**all**that apply.

(A) Angle 2 = 90

(B) Angle 6 = 90

(C) Angle 3 = Angle 6

(D) Angle 1 + Angle 6 = 90

(E) Angle 3 + Angle 4 = 90

(F) Angle 4 + Angle 5 = 90

This question is straightforward. If we draw in boxes to denote right angles between

*r*and

*p*, we see that the following combinations of angles would have to be 90 degrees:

Angle 1 + Angle 2

Angle 3

Angle 4 + Angle 5 (choice F)

Angle 6 (choice B)

We eliminate choices (A), (D), and (E) as these would make

*p*perpendicular to

*n*rather than

*r*. And we can eliminate choice (C) as Angles 3 and 6 are vertical angles, so they would have equal measure regardless of whether any of the lines were perpendicular. The correct answers are (B) and (F).

The question itself is not difficult. If we had to show which section of the U of Chicago text the question corresponds to, then I'd have to say Section 3-5, since this is on Perpendicular Lines. But the problem the students will have most likely wouldn't be that they don't know which angle is 90, but that they won't select more than one answer. Choice (B) is correct, since it correct states that if Angle 6 is a right angle then lines

*r*and

*p*are perpendicular. So then students will just mark (B) and not check the rest of the answers to see whether a second answer is correct.

Another possible issue could be confusing a statement with its converse. The question asks students to tell which statements imply that

*r*is perpendicular to

*p*-- not which statements the perpendicularity of

*r*and

*p*imply. If it were the latter, (C) would be correct, since if

*r*were perpendicular to

*p*, then Angles 3 and 6 would indeed be equal as both would be 90 degrees. The problem is that we can have Angles 3 and 6 equal without

*r*and

*p*being perpendicular. But I expect the far more common mistake for students would be to omit (F) as a correct answer, rather than include (C).

I'm in the mood to post some Square One TV videos. Let's begin with a song that's definitely relevant to today's PARCC, "Perpendicular Lines":

Now today is an activity day. I've decided that I want this activity to focus not on perpendicular lines, but on the idea that sometimes more than one answer can be correct.

I rarely post entire episodes of Square One TV, but I couldn't find this as an isolated clip -- but fortunately, what I want is at the beginning of the show, right after the theme song. This is a game show called "Square One Challenge" -- a parody of

*To Tell the Truth*. The difference here is that sometimes they're

*both*telling the truth -- and other times they're both bluffing:

My activity page is based on this game. Notice even before clicking to start the video that the first question is based on folding paper and reflections, a Common Core transformation.

**PARCC Practice EOY Exam Question 9****U of Chicago Correspondence: Section 3-5, Perpendicular Lines**

**Key Theorem: Definition of perpendicular**

**Two segments, rays, or lines are perpendicular if and only if the lines containing them form a 90-degree angle.**

**Common Core Standard:**

CCSS.MATH.CONTENT.HSG.CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

**Commentary: Because perpendicular lines is such a basic concept, I argue that this question is more about logic and knowing how to choose more than one correct answer than about perpendicular lines per se. There are no logic questions like this one anywhere in the U of Chicago text.**

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