Students constructed parallel lines using a Geometry page of a TI-Nspire document. As students work on any construction, we ask what segments, angles, arcs, and triangles are always congruent in the construction.

I’ve had Smarter Balanced Assessment Consortium Sample Item MAT.HS.SR.1.00GCO.O.244 saved for several years now. (The original link where I got the SBAC item is no longer active, but the item can be found in this PDF.) Before we talked about their construction results as a whole class, I sent the SBAC item to students as a Quick Poll.

Watch the Parallel Line movie. The steps in the construction result in a line through the given point that is parallel to the given line. Which statement justifies why the constructed line is parallel to the given line?

A. When two lines are each perpendicular to a third line, the lines are parallel.

B. When two lines are each parallel to a third line, the lines are parallel.

C. When two lines are intersected by a transversal and alternate interior angles are congruent, the lines are parallel.

D. When two lines are intersected by a transversal and corresponding angles are congruent, the lines are parallel.

They watched the video and answered with the following results. (There was a video linked in the original document where I found the sample item. This video is similar, and it can be set to auto-repeat.)

I wrote recently about Conditional Statements and Instructional Adjustments. When it’s not so obvious what is the correct answer, I will sometimes get students to find a person at a different table that answered differently and have them convince the person that they are right. This time, students formed teams with a couple of D’s and a C. Listen to the C argument. Listen to the D argument. Change your answer if the other argument convinced you. Keep your answer if the other argument didn’t convince you. Send in your response a second time.

So as I was looking at the second round of student responses, I was disconcerted to find that the results were the same. I thought surely that the D’s would convince the C’s. I had to quickly make an instructional adjustment after the second Quick Poll with results that appeared to be the same as the first.

What would you do next?

It turns out that the results weren’t actually the same as the first time:

1 student didn’t answer the first time and answered incorrectly the second time.

1 student didn’t answer the first time and answered correctly the second time.

2 students answered correctly the first time and didn’t answer at all the second time.

4 students answered incorrectly the first time answered correctly the second time.

3 students answered correctly the first time and incorrectly the second time.

4 students answered incorrectly both times.

(I’m glad I don’t have to show that progression in a Venn Diagram, and I obviously didn’t have time to figure all of that out in class while students were working.)

So we are back to the question what would you do next?

Would your answer change if you have 9 minutes left in class?

I asked a student who answered incorrectly both times to share his argument with the class. Another student critiqued his argument to show why it was invalid.

And then, of course, we were foiled (or saved, depending on your perspective) by the bell.

And so the journey continues …