Last year’s lesson using the Illustrative Mathematics task Angle Bisection and Midpoints of Line Segments had plenty of room for improvement. This year, students left with a better understanding of proof and giving feedback on proof.

Our goal? SMP3: I can construct a viable argument **and** critique the reasoning of others.

Students started by reading through both parts of the proof, noticing and wondering.

I’ll admit, I really wanted someone to notice that parts a and b were converses. (I didn’t expect them to use that language … I was just looking for anything about the parts being “opposite”.) I wasn’t ready to tell them, so I specifically asked, “what is the difference between parts a and b”.

In triangle a thhey already give you the midpoint of line QR and asking you to draw the angle bisector, but in triangle b they are giving you the angle bisector and are asking you to find the midpoint of line QR. 1

In part a, you’re trying to find the angle bisector from the midpoint, but in part b, you’re trying to find the midpoint using the angle bisector. So they’re basically the opposite of each other, but you have the same point and the same line. They were just found in different ways. 1

Part a starts of with finding the midpoint to segment QR and then creates a line from P to go through the midpoint while part b starts with an angle bisector PS then goes to see if it intersects the midpoint to of segment QR. 1

in part a your contructing a midpoint, in part b you are constructing a bisector 1

In part a you are justifying that PM is a bisector of QPR, but in part b you are justifying that PS meets QR at its midpoint. 1

The difference is that part a to show that the bisector will go through the midpoint, while part b is asking to show that the bisector does go through the midpoint rather than just some random point. 1

In part A the midpoint is labeled M and in part B the midpoint is labeled S, but it is the same point. Also part A and part B make the same image, but the just switch the order they made the image. like finding the midpoint first then the bisector, vice versa 1

Students spent a few minutes creating an argument for part a. Then we looked at some of the student work from last year to critique the arguments.

In Embedding Formative Assessment, Dylan Wiliam suggests that students learning how to give feedback should start with anonymous student work … and eventually move towards student work from peers in the same class. This seemed to work well for this task. Additionally, I had the opportunity to purposefully **select **and **sequence** the work for giving feedback ahead of time, which gave us more time for learning during class.

My geometry students are 1:1 this year with MacBook Airs, and so I sent a PDF of the student work samples through TI-Nspire Navigator for Networked Computers, which gave them an up-close look at the student work instead of my having to stand at the copy machine for a while or students trying to decipher from it only being displayed on the board at the front of the room.

We looked at one student work sample at a time using Think-Pair-Share to make student thinking visible. What feedback would you give this student?

M is the same distance from Q and R, but points on the angle bisector are the same distance from the sides of the angle. How do you know M is the same distance from ray PQ as it is from ray PR? We represent distance from a point to a line as the length of the segment perpendicular from the point to the line.

What is a perpendicular bisector of an angle?

What is the difference in saying segment QR is a perpendicular bisector of ray PM and saying ray PM is a perpendicular bisector of segment PM?

Before we looked at the next student work sample, I asked students to practice **look for and make use of structure**, asking what they saw when segment QR was drawn.

An angle bisector.

A midpoint.

Triangles.

How many triangles?

3 triangles.

What kind of triangles?

The big one is isosceles.

What do you know about isosceles triangles?

They have two congruent angles.

Eventually we showed that the two triangles were congruent using SAS.

Then we looked at another student work sample.

This student noted that the triangle is isosceles, but jumped from one pair of corresponding congruent sides to the angle bisector.

And one other student work sample, where the student noted that the triangles were congruent, but didn’t give a reason why.

Students looked at part b for a few minutes. Then we looked at one last student work sample. What do you wonder about this argument?

Does S have to be the midpoint?

After working for a few more minutes, students gave each other feedback and then revised their argument based on the feedback.

Are we going to look at a correct argument for this?

Will you check mine to be sure that it is right?

Last year, students didn’t care so much whether their argument was correct, nor did they care about seeing a “viable argument”. Somehow, figuring out how to improve some of the arguments for part a got them more interested in their argument for part b.

We plan to look at the following five arguments tomorrow.

With what do you agree?

With what do you disagree?

And so the journey continues … thankful for do-overs from one year to the next.