## Angle Bisection and Midpoints of Line Segments

17 Nov

As we finished Unit 2 on Tools of Geometry this year, I looked back at Illustrative Mathematics to see if a new task had been posted that we might use on our “put it all together” day before the summative assessment.

I had recently read Jessica Murk’s blog post on an introduction to peer feedback, and so I decided to incorporate the feedback template that she used with the task.

What misconceptions do you anticipate that students will have while working on this task?

What can you find right about the arguments below? What do you question about the arguments below?

Student A:

Student B:

Student C:

Student D:

Student E:

Student F:

Student G:

Student H:

Student I:

Student J:

The misconception that stuck out to me the most is that students didn’t recognize the difference between parts (a) and (b). I’ve wondered before whether we should still give students the opportunity to recognize differences and similarities between a conditional statement, its converse, inverse, contrapositive, and biconditional. We decided as a geometry team to continue including some work on building our deductive system using logic, even though our standards don’t explicitly include this work. We know that our standards are the “floor, not the ceiling”. We did this task before our work on conditional statements in Unit 3, and so students didn’t realize that, essentially, one statement was the converse of the other. Which means that what we start with (our given information) in part (a) is what we are trying to prove in part(b). And vice versa.

The feedback that students gave was tainted by this misconception.

Another misconception I noticed more than once is that while every point on an angle bisector is equidistant from the sides of the angle, students carelessly talked about the distance from a point to a line, not requiring the length of the segment perpendicular from the point to the line and instead just noting that that the lengths of two segments from two lines to a point are equal.

It occurred to me mid-lesson that maybe we should look at some student work together to give feedback. (This happened after I saw the “What he said” feedback given by one of the students.)

I have the Reflector App on my iPad, but between the wireless infrastructure in my room for large files like images and my fumbling around on the iPad, it takes too long to get student work displayed on the board. A document camera would be helpful. But I don’t have one. And I’m not sure how I’d get the work we do through the document camera into the student notes for the day. So I actually did take a picture or too, use Dropbox to get the pictures from my iPad to my computer, and then displayed them on the board using my Promethean ActivInspire flipchart so that we could write on them. And then a few of those were so light because of the pencil (and/or maybe lack of confidence that students had while writing) that the time spent wasn’t helpful for student learning.

Looking back at Jessica’s post, I see that her students partnered to give feedback, since they were just learning to give feedback. That might have helped some, but I’m not sure that would have “fixed” this lesson.

So while I can’t say with confidence that this was a great lesson, I can say with confidence that next year will be better. Next year, I’ll give students time to write their own arguments, and then I’ll show them some of the arguments shown here and ask them to provide feedback together to improve them. Maybe next year, too, I’ll add a question to the opener that gives a true conditional statement and a converse and ask whether the true conditional statement implies that the converse must be true, just so they have some experience with recognizing the difference between conditional statements and converses before we try this task.

And so the journey continues, this time with gratefulness for “do-overs”.

### 3 responses to “Angle Bisection and Midpoints of Line Segments”

1. November 19, 2014 at 9:05 am

In geometry proving the obvious is probably one of the more difficult things to do. Thinking about mathematicians and proof, they don’t begin the development of the subject by proving things, they
do so by observing things, guessing things, being bold and daring. Then having found something which looks very likely to be true they try to prove it. School math seems to go about it the wrong way. So …

Here’s a diagram.. your diagram..with M stated as the mid point of BC
Then some questions: What can you see in the diagram, anything obvious? What might be true?
….Can you give your reasons, informally to begin with?

I would hope(!) that some would think they had an isosceles triangle, some might guess that triangles BAM and CAM were equal (congruent), some might see symmetry. Some might see that the line AM bisected the angle.
All valuable stuff, and leading towards proof of the original statement.

Another thing that came to mind is that the students are all trying to get directly from the facts to the conclusion. No-one has asked the question “What do I need to know when I want to show two angles equal?”. It is very useful to look at a proving problem from both ends.

2. November 19, 2014 at 10:35 am

Hi, Howard. Thank you for your suggestions. I can see the advantage of only giving the given at first … and having a conversation about “what you see” before just having them jump into writing a proof. This was so early in the course … my students would approach this problem so differently now.