Category Archives: Tools of Geometry

Angle Bisection and Midpoints of Line Segments, Take Two

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.

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

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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”.

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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?

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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.

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What is a perpendicular bisector of an angle?

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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.

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An angle bisector.

A midpoint.


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.

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

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Then we looked at another student work sample.

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

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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?

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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.

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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?

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


Posted by on September 20, 2015 in Angles & Triangles, Geometry, Tools of Geometry


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Placing a Fire Hydrant

Placing a Fire Hydrant

We’ve used the Illustrative Mathematics task Placing a Fire Hydrant for several years now. Each year, the task plays out a bit differently because of the questions that the students ask and the mathematics that students notice. Which is, honestly, why I continue to teach.

I set up our work for the day as practicing I can make sense of problems and persevere in solving them and also I can attend to precision. If you don’t know how to start at Level 3, use Levels 1 and 2 to help you get there.



In an effort not to articulate all of the requirements ahead of time, I simply asked: where would you place a fire hydrant to serve buildings A, B, and C. Students dropped a point at the location they thought best.

It was then obvious from the students’ choices that they thought equidistant was important.

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This year I didn’t put out tools that students might choose to use. Instead, I set the timer for them to work alone on paper for a few minutes and told them to ask for what they needed. Before I could get from the front of the room to the back, almost every hand was raised to request either a ruler or a protractor. (No one asked for a compass this year. Last year, when I had them out on the tables, lots of students used them.)

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I gave students a few more minutes to work individually with the option, this time, of working with the TI-Nspire software to show their thinking. And at the end of that, I added a few more minutes, asking students to focus on how they could justify that their solution always works. Then I gave them a few minutes to discuss their thinking with a partner.

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I watched (or monitored, according to Smith & Stein’s 5 Practices) while they worked using the Class Capture feature of TI-Nspire Navigator. During that time I also selected and sequenced for our whole class discussion. I wanted some of the vocabulary associated with special segments in triangles to come out of our discussion, so I didn’t immediately start with the correct solution.

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We started with Autumn, who had constructed the midpoints of the sides and then created both a midsegment of the triangle and some medians of the triangle. She could tell that the intersection of the midsegment and medians was “too high”.

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C chimed in that she had constructed lots of midsegments. In fact, she had created several midsegment triangles, one inside the other.

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Next we went to Addison, who not only had created all three medians of the triangle but had also measured to show that the medians weren’t the answer.

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That led to S, who had been trying to figure out when the intersection of the medians would be a good location for the fire hydrant.

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Arienne told us about her approach next. She had placed a point inside of the buildings, measured from the point to each building, and she was moving the point around to a location that would be equidistant from the buildings.

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Reagan talked with us about her solution next. She had constructed the perpendicular bisectors and measured from their intersection to each vertex to show that it always worked.

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I wonder what that point has to do with the vertices. What do you see in the diagram? (I was expecting students to “see” a circle. But they didn’t. They saw a triangular prism.) I wasn’t ready to show them the circle, though. How could I help make the circle visible without telling them? A new question came to me: What if we had a 4th building? Where could we place the building so that the fire hydrant served it, too?

I quickly collected Reagan’s file and sent it out to all of the students so that they could create a 4th building that was the same distance from the fire hydrant as A, B, and C.

While they were working, Janie said, “I have a 4th building the same distance, but how do I place it so that it always works?” (On the inside, I was thrilled that Janie asked this question. It is exciting for students to realize this early on in the course that we are about generalizing and proving so that something always works and not just for one case.)

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How do you place the 4th building so that it always works? What is significant about the location of the 3 buildings and the fire hydrant?

Sofia volunteered that her 4th building always works. (I have to admit that I was skeptical, but I made her the Live Presenter and asked how she made it.) Sofia had rotated building C about the fire hydrant to get d. (How many degrees? Does the number of degrees matter? Would rotating always work? Why would it work?) She rotated C again to get a 5th building between A and B. What is significant about the location of the 5 buildings and the fire hydrant?

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And then they saw it. It wasn’t yet pictured, but it had become visible. All of the buildings would form a circle around the fire hydrant! The fire hydrant is the circumcenter of ∆ABC. The circle is circumscribed about the triangle.

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And so the journey continues … every once in a while finding a more beautiful question.


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Placing a Fire Hydrant (2014)

I gave a talk at ASSM back in April entitled The Slow Math Movement. The following is an excerpt from that talk that describes how the Illustrative Mathematics Placing a Fire Hydrant task played out in my classroom last year:

Towards the beginning of our geometry course, we give students a task from Illustrative Mathematics called Placing a Fire Hydrant. Where would you place a fire hydrant to serve all three buildings?

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Through a Quick Poll, students drop a point at the location they think is best. Then we introduce the requirement that the fire hydrant should be equidistant from all three buildings.

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They started on paper, using rulers, folding, and compasses.

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Several of them realized that if they could find the circle that contained all three locations,

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the center would be equidistant (and thus the location of the fire hydrant).

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However, their methods for finding a circle to contain all three points were not very precise

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(which meant they didn’t already know everything they needed to know about triangle centers).

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Next they moved to technology. I watched while they worked using the Class Capture feature of our technology, and using what I learned from Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions to monitor, select, and sequence the student work for our whole class discussion.

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My students didn’t come into this lesson knowing the vocabulary associated with special segments in triangles, so I purposefully included some incorrect solutions for placing the fire hydrant equidistant from the buildings to bring out that new vocabulary.

Kolton had constructed the midpoints of the sides of the triangle. I made him the Live Presenter so that he could discuss his solution and so that students could learn what a median of the triangle was. His measurements showed that his solution didn’t always work,

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but the dynamic feature of our software let him move the buildings around and begin to consider when the intersection of the medians would be equidistant from the sides of the building.

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Chaney had constructed a midsegment of the triangle, and so we looked at hers next to learn that new term.

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Jameria had constructed the three midsegments of the triangle, creating a midsegment triangle. She was able to tell from her measurements that her solution didn’t always work, either, but we looked at anyway, and I told students that we would learn more about the midsegment triangle later in the course.

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We moved next to Sawyer, who recognized that the correct placement of the fire hydrant should be the center of a circle that contained all three buildings, but we could see from his work that he hadn’t yet figured out how to get a circle through all three buildings.

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Quinn had fashioned a circle through the three points, but still hadn’t actually constructed it.

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Caroline had constructed the perpendicular bisectors of each side of the triangle. She had measured from their intersection, the circumcenter, to each building to show that they were equidistant.

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As the Live Presenter, she started moving the buildings around to show that her solution always worked.

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Then we asked her to construct the circumscribed circle to emphasize that the intersection of the perpendicular bisectors is the circumcenter.

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As Caroline continued to move around the buildings, Gabe asked, “Why would we put the fire hydrant there?” Caroline stopped, and we took a good look at the setup.

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She moved the buildings again, to exaggerate how ridiculous it would be to place a fire hydrant that far away. Our dynamic technology made the students realize that the circumcenter isn’t always the most efficient place for the fire hydrant, even if it is equidistant from the three buildings. And so we began to explore when it makes sense to put the fire hydrant equidistant from the buildings and when it no longer makes sense.

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Take just a moment to contrast the Fire Hydrant task with how I used to teach special segments in triangles. Which one of these is the “Fast Math” option? Which one furthers the Slow Math Movement?


In his book The Falconer, Grant Licthman says, Questions are waypoints on the path of wisdom. Each question leads to one or more new questions or answers. Sometimes answers are dead ends; they don’t lead anywhere. Questions are never dead ends. Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

The technology that we use provide the impetus for students to ask questions, which leads to more questions and some answers, from and by the students. I get to watch and listen and push and probe my students by asking more questions.

What can you do this week to further The Slow Math Movement?

[Cross posted on The Slow Math Movement]


Posted by on September 9, 2015 in Angles & Triangles, Geometry, Tools of Geometry


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Constructing Parallel Lines & Instructional Adjustments

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.

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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.

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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.

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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 …


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Angle Bisection and Midpoints of Line Segments

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 found Angle Bisection and Midpoints of Line Segments.

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.

The task:

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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:

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Student B:

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Student C:

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Student D:

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Student E:

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Student F:

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Student G:

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Student H:

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Student I:

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Student J:

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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.)

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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”.


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Origami Regular Octagon

We folded a square piece of paper as described in the Illustrative Mathematics task, Origami Regular Octagon. I didn’t want students to know ahead of time that they were creating an octagon, so I changed the wording a bit. We folded (and refolded … luckily, there was not a 1-1 correspondence between paper squares and students). Students worked individually to write down a few observations and then we talked all together.

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It’s an octagon.

There are 8 equal sides.

There are 8 equal angles.

It’s a regular octagon (this is the first year my students have come to me knowing what it means for a polygon to be regular).

How do you know there are 8 equal sides and 8 equal angles?

Because we folded it that way.

How do you know there are 8 equal sides and 8 equal angles?

Because one side is a reflection of its opposite side about the line that we folded.

What is the significance of the lines that you folded?

They are lines of symmetry.

There are 8 of them.

The opposite sides are parallel.

How can you tell?

This took a while. Maybe longer than it needed to.

Another student raised his hand.

I figured out that the sum of the angles in the octagon is 540˚.

(I don’t have the sum of the interior angles of an octagon memorized since I can calculate it, but I did know that 540˚ was too small.)

How did you get that?
I made an octagon and measured the angle. Then I multiplied by 8.

On your handheld?

Okay. Let’s see what you have. I made him Live Presenter.

He showed us the angles he measured that were 67.5˚.

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It might help if we can see the sides of your angles. Will you use the segment tool to draw them?

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Other students argued that we needed to double 540 to get the sum of the angles in the octagon, 1080˚.

What else do you notice?


Congruent triangles.

Right triangles.

Students noticed different numbers of triangles.

And they recognized that we knew about congruence because of reflections.

Somehow we asked the question about the value of the angle (x).

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I set up a Quick Poll to collect student responses.

Almost everyone got the correct answer of 22.5˚.

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Can you tell me how you got your answer?

One student used the ¼ square with a 90˚ angle that had been bisected by the folded line to be 45˚ and the bisected again by the folded line to argue that x was 22.5˚.

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Did anyone do something different? Hands went up all around the room.

AC hasn’t talked to the whole class yet today, so I asked what she did.

I saw a circle with 360˚ and divided by 16.

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Then DC’s hand went up. 360/16 is equivalent to 180/8. I saw a line divided into 8 equal parts (or straight angle).

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Then TC showed us the isosceles triangle she used with the 62.5˚ base angles.

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Then someone else showed us the right triangle he used with the complementary acute angles.

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Before we knew it, we had spent almost an hour talking about a regular octagon. And learning math using quite a few Math Practices: construct a viable argument and critique the reasoning of others, look for and make use of structure, use appropriate tools strategically.

I’ve wondered before how much longer we will need to talk about generalizing relationships for interior and exterior angles in polygons. Today I got a glimpse of students being able to figure out those relationships by looking for and making use of structure. The only concern that remains is the length of time it would take to do that on a high stakes standardized test such as the ACT or SAT. And so the journey to do what is best for my students continues …


Posted by on September 21, 2014 in Geometry, Rigid Motions, Tools of Geometry


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Locating a Warehouse

We changed the Learning Mode to individual. Where would you place a warehouse that needed to be equidistant from all three roads? (From Illustrative Mathematics.)

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Students started sketching on paper, and I set up a Quick Poll so that we could see everyone’s conjecture at the same time.

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We changed the Learning Mode to whole class. With whom do you agree?

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I didn’t intend for us to talk in detail at this point. I wanted students to be able to test their conjecture using their dynamic geometry software. But we had done that the day before for Placing a Fire Hydrant (post to come), and class was cut short during this lesson because of lock-down and evacuation drills. So we did talk in more detail than I had planned. Is the point outside of the triangle equidistant from the three roads? One student vehemently defended her point: I drew a circle with that point as center that touched all three roads. (We have been talking about the distance from a point to a line.) How do you know that the roads are the same distance from the center? They are all radii of the circle. They are perpendicular to the road from the center.

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Could a point inside the triangle of roads be correct? If so, which? We started drawing distances from the points to the lines. Some points were about the same distance from two of the roads but obviously to close to the third road. What’s significant about the point that will be the same distance from all three sides of a triangle? Several students wondered about drawing perpendicular bisectors. Another student vehemently insisted that the point needed to lie on an angle bisector. Would that always work?

Are you going to let us try it ourselves? Well of course! So with about 12 minutes left, students began to construct.

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With about 3 minutes left, I made a student the Live Presenter who showed us that the angle bisectors are concurrent, and used the length measurement tool to show us that the point is equidistant to the sides of the triangle.

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With about 2 minutes left I made another student the Live Presenter who had made a circle inside the triangle. How did you get that circle? What is significant about the circle? It’s inscribed. The center is the where the angle bisectors intersect. So we call that point the incenter. It’s the center of the inscribed circle of a triangle, and the point of concurrency for the angle bisectors. How is this point different from the circumcenter?

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And with 1 minute left: Do you understand what we mean when we say that every point on an angle bisector is equidistant from the two sides?

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And so while I have some record of what every student did during class through Quick Polls and Class Capture and collecting their TNS document once the bell rang, my efforts at closure are foiled again. Maybe one day I’ll actually send one of the Exit Quick Polls that I have made for every lesson.


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The Center of a Circle

The Center of a Circle

I wrote about last year’s lesson here, but decided to write a new post since the lesson played out differently this year.

From NAEP, 1996, Grade 12 Mathematics:

This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

Describe a procedure for locating the point that is the center of a circular paper disk. Use geometric definitions, properties, or principles to explain why your procedure is correct. Use the disk provided to help you formulate your procedure. You may write on it or fold it in any way that you find helpful, but it will not be collected.

The NAEP results were not hugely promising in 1996.

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In the past, I’ve had my students start by thinking about and then doing a procedure for locating the center of a circular paper disk. This year, I had them write their procedure like the NAEP question requires. I didn’t really know what I was going to do after I had them write it. Since they were writing on paper, I didn’t know what responses I had, so I couldn’t exactly select and sequence student work as noted by Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions.

I noticed, though, that without my asking them, several students had started trying their procedure using a geometry page on TI-Nspire. So I decided we should all try that. Once everyone had finished describing their procedure, they tried it on a circle using our dynamic geometry software.

Now I was able to select and sequence using Class Capture. Whose would you choose to discuss with the whole class?

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We started by making S.N. the Live Presenter, who without knowing it, had the traditional construction for finding the center of the circle. She created a chord, and then its perpendicular bisector, and then the midpoint of the segment of the perpendicular bisectors with endpoints on the circle. When we first looked at hers, the chord was “centered”. It was easy to show using our dynamic geometry software that the location of the chord does not matter.

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Another student inscribed a rectangle in the circle and found the intersection of its diagonals.

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Or almost. Sometimes what happens isn’t what we think happens. In this case, the student was off just a bit when forming the diagonals. Our dynamic geometry software helps us learn to attend to precision.

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And another student circumscribed a square about the circle, and found the intersection of the perpendicular bisectors of the sides.

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Another student circumscribed a square about the circle and found the intersection of the diagonals of the square.

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And another student made us wonder whether the center of any equilateral triangle inscribed in the circle would always coincide with the center of the circle.

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I read through the student work after class.

I had missed one interesting approach, but the beauty of using the TI-Nspire Navigator System meant that I still have the student work. I collected the document at the end of class and saved it to my Portfolio. We had an opportunity to talk about this relationship later in the unit.

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I found an approach by a student new to our school from Iran, who only began learning English a few months ago. She attended to precision better than 99% of the grade 12 students who had this question on NAEP in 1996.

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I found a pretty creative approach that took seriously use appropriate tools strategically.

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It was interesting how many students thought to circumscribe a square about the circle. I’ve been using this task for about 10 years now, and my students haven’t used that approach before.

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Finally, a few students thought to inscribe a rectangle in the circle, something my students had not suggested before.

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And so the journey continues, learning about structure from my students year after year in different ways …


Posted by on March 2, 2014 in Circles, Geometry, Tools of Geometry


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The Shipwrecked Surfer

I had saved this task from the August 2006 Mathematics Teacher to use at some point. One of the authors has a link to the article/paper here.

The problem:

A surfer, shipwrecked on an island in the shape of an equilateral triangle, wants to build a hut so that the sum of its distances to the three beaches is minimal. Where should the hut be located? (See Figure 1.)

I like how the problem is worded; students first have to make sense of the problem, and most have to persevere in solving it. I also like that students have to think about what we mean when we say the distance from a point (hut) to a line (beach).

Students started their work on paper. Many students chose the centroid as the point of interest. Others chose randomly. They used rulers and compasses to test their conjectures. One student figured out pretty quickly that it didn’t matter where you put the point. It doesn’t matter where you put the point? No – the sum of the distances from the point to the sides is the same no matter where you put the point.

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We moved to our dynamic geometry software. We placed a point inside the triangle. We constructed the perpendicular from the point to each side. We measured the length of each segment on the perpendicular from the point to the side. We calculated the sum of the lengths. We moved the point around inside the triangle and verified that the sum is the same no matter where we put the point.

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So we can put the point anywhere. Now what? What’s the significance of placing the point anywhere? What’s the significance of the sum of the distances?

What’s significant about the current location of the point of interest for this student?

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Someone figured out that the sum of the distances equals the length of the altitude. Is that important? The altitude is the length of the segment perpendicular from any vertex to the opposite side.

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Look what happens if we redefine our point of interest on the altitude. What type of polygon is IBXC? Compare the value of XI to the sum of CI and BI.

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Somewhere around here the bell rang.

And that was it. We haven’t revisited the task, although a few students reflected on the task as part of their problem solving points for this quarter:

“The addition of the length to the sides from any random point in a triangle is equal to the altitude because the altitude is exactly that. If you put a point at the top corner of a triangle and add the distance from there to all three sides you get the altitude. The altitude is just adding all the distances to the sides at a certain point.”

“The entire reason the point inside the triangle was equidistance from the sides was because the triangle was equilateral. I’m not sure if this is more complicated than it seems, but if all of the angles are the same, all the sides are the same, and any way you rotate and turn the triangle it still looks the same, then a point inside the triangle is going to be equidistance from the sides because every point on the plane is equidistance. Think about it this way: If I move my point .2 cm to the left, then the distance from the side on its left is shortened by .2 cm, but the distance from the side on right is lengthened by.2 cm. Up, down, left, right, every distance’s total is equal.”

One of the best parts about teaching is that we get “do-overs”. Not always this year and with these students…but definitely next year with new students. How could I make this conversation more meaningful – and get to the mathematics sooner? I could pose the task the class period before. Students can do their paper work on the task outside of class – and come into class ready to share their conjectures. That would get us to the technology sooner – and to the deep exploration for which the technology provides the opportunity.

And so the journey continues…


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Balancing a Triangle

On the first two days of our Tools of Geometry unit, we talked about perpendicular bisectors and angle bisectors in the context of a task from Illustrative Mathematics. On the third day, we talked about medians and altitudes, absent of a context, except to call the centroid of a triangle its balancing point.

We did emphasize two Mathematical Practices.

1. Attend to precision.

I showed students the definition of a median and asked them to sketch one on paper:

A median of a triangle is a segment that goes from a vertex to the midpoint of the opposite side.

Do you know how many students would prefer that we just show them a median instead of having them make sense of the definition? We are getting good at waiting.

Then I had them construct a median using dynamic geometry software. Of course several students went on and constructed all three, because they predicted that the medians are concurrent at a point.

2. Look for regularity in repeated reasoning.

Is the location of the centroid significant?

I sent a Quick Poll to see what students predicted about the location (the diagram is on the left side of the picture).

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16 of the 30 were correct – and while I have the correct answer showing now, I didn’t show the students the correct answer.

We used our dynamic geometry software to explore the measurements. Students observed the measurements and predicted the relationship.

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Then without confirming our conjectures, I sent the poll again. This time 21 (22 if you count the last submission) were correct. I still didn’t show them the correct answer.

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The document was set up for automatic data collection. While students were moving the points of the triangle, the spreadsheet on the next page was filling up with data. Someone stated a conjecture. We tested the conjecture in the spreadsheet using a formula.

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Then I sent a different poll to assess student understanding. We were up to 24 correct. What question did the students who got 12 answer? An opportunity to both construct a viable argument and critique the reasoning of others. An opportunity to learn from our mistakes.

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We have yet to actually prove that the medians meet at a point. One proof of this assumes information about midsegments (which we will prove later). Another proof of this uses coordinate geometry (which we will get to later). So for now, we have observed that the medians meet at a point, and we know how that point partitions each median.

After medians, we spent some time on altitudes.

I gave them the definition and had them draw one on paper.

Then we constructed an altitude using technology.

Then we figured out whether the location of the orthocenter could give us information about the type of triangle.

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Then we worked a problem involving an altitude.

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Then we used technology to explore a triangle with all four special segments drawn from one vertex. Can you deduce which segment is which? How?

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For what kind of triangle are all four special segments the same triangle? Equilateral. How do you know?

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We used the angle measurement tool. Is the triangle equilateral? No. How do you know?

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What would happen if the triangle is equilateral? How do you know?


Note: I used the Geometry Nspired Special Segments in Triangles as a guide for this lesson.


Was this lesson important?

I haven’t figured that out yet. I do know that I don’t want to waste my students’ time.

And so the journey continues …


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