Sum of Angles in a Triangle

I’ve talked before about providing students an opportunity to look for and make use of structure while trying to prove the Triangle Sum Theorem. This summer I ran across a task on Illustrative Mathematics with a proof of the Triangle Sum Theorem using transformational geometry. The task is scaffolded quite a bit, and so while I didn’t give my students the task as-is with the scaffolded instructions, those questions ended up playing a big role in our whole class conversation. I asked the question the same way I have before – I just knew because of reading through the task and learning from the task that I wanted my students to recognize another way to prove the Triangle Sum Theorem.

We talked for a moment about our Learning Progression for look for and make use of structure. In geometry, we often have to ask, “What do you see that’s not pictured?” We often have to draw auxiliary lines to help make sense of a figure.

I gave students a diagram of a triangle. And I gave them the following words:

Triangle Sum

Given: ∆ABC

Prove: m∠A+m∠B+m∠C=180°

I then asked the class to think back through our progression in building our deductive system. What do we know? What have we proved? What have we allowed into our systems as postulates? They thought back through our units of study:

Triangles – medians, altitudes, angle bisectors, perpendicular bisectors

Vertical angles are congruent; Angle & Segment Addition Postulates

Parallel Line postulate & theorems – corresponding angles, alternate interior angles, …

Transformations – reflections, rotations, translations

I set the timer for 3 minutes, moved the Learning Mode clip to “Individual”, and watched (monitored) as they thought. Some sat for three minutes thinking without drawing anything. Some drew in an altitude for the triangle. Some composed the triangle into a rectangle or a non-special parallelogram… not all the same way.

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Students continued to work alone for several more minutes before they even noticed that the timer had finished. I couldn’t believe what I saw besides the traditional responses. On SC’s paper, I saw three triangles: the original, and two images of the original triangles that had been rotated about the sides. I’ve had students proving the Triangle Sum Theorem for years now and never once has someone thought to transform the triangle by rotating it. I asked students to share their work with their team. I listened. And I asked a few probing questions, especially to SC. SC needed to cut out a triangle congruent to the image so that she could describe the resulting rotations. Her first thought was that the triangle had been rotated around one of its vertices.

2014-10-14 09.45.28     2014-10-14 09.44.50

I drew a few of the diagrams that students had created on the board & asked students to take a few more minutes to see if they could justify the Triangle Sum Theorem using one of the diagrams.

Then we talked all together. Several students had used a rectangle to show why the sum of the measures of the angles of the triangle has to equal 180˚. (A few tried to use the interior angles of a quadrilateral summing to 360˚ in their reasoning, so we talked about being unable to use that, since it is really a result of what we are trying to prove.)

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Others used the side of the rectangle parallel to the base of the triangle showing that alternate interior angles congruent and then used the Angle Addition Postulate to finalize their result.

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Next we moved to the diagram of the rotated triangles.

How can we describe the rotation that resulted in the triangle on the left?

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Students suggested rotating 180˚ about the midpoint of segment AB. Point A = Point B’, and Point B = Point A’. We loosely used 1’, 2’, and 3’ to name the angles in the image. We know that ∠1’ is congruent to ∠1 because a rotation preserves congruence. And so then we know that segment A’B is parallel to segment AC since alternate interior angles are congruent.

Similarity, we can rotate the triangle 180˚ about the midpoint of segment BC. Point B = Point C’’ and Point C = Point B’’. We loosely used 1’’, 2’’, and 3’’ to name the angles in the image. We know that ∠3’’ is congruent to ∠3 because a rotation preserves congruence. And so then we know that segment A’’B is parallel to segment AC since alternate interior angles are congruent.

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So how do we know that A’’, B, and A’ are collinear?

Because if they are, then A’A’’ is parallel to AC, and m∠1’+m∠2+m∠3’’=180, which means m∠1+m∠2+m∠3=180.

For one of the first times in class, we actually used the parallel postulate to explain why A’’, B, and A’ are collinear (through a point not on a line, there is exactly one line through the point parallel to the given line). We are still studying Euclidean geometry, after all.

We are always running out of time, and so I was just using the rotation tools on my Promethean Board ActivInspire software in our conversation.

Next year, we will add our dynamic geometry software to help verify and make sense of our results.

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In my last CMC-S session yesterday I gave participants just a few minutes to come up with a way to prove the Triangle Sum Theorem using transformations. Of course I was expecting the rotation solution. I’m not sure when I’ll ever quit being surprised by solutions I don’t expect. One participant suggested that we translate ∆ABC using vector AB. We labeled the resulting image ∆A’B’C’. We know that ∠1 ≅∠1’ because a translation preserves angle congruence. Then BC’ is parallel to segment AC because corresponding angles are congruent.

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Similarly, we translated ∆ABC using vector CB. We labeled the resulting image ∆A’’B’’C’’. We know that ∠2 ≅∠2’’ because a translation preserves angle congruence. Then BA’’ is parallel to segment AC because corresponding angles are congruent.

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A’’, B, and C’ are collinear by the Parallel Postulate since there can be only one line through B parallel to segment AC. ∠3 ≅∠B’’BB’ because vertical angles are congruent. The Angle Addition Postulate gets us m∠2’’+m∠B’’BB’+m∠1’=180, and then with substitution and the definition of congruent angles, we can conclude that the sum of the measures of the angles of the triangle is 180˚.

And so the journey continues … ever grateful for resources like Illustrative Mathematics, that push me to keep learning – and keep me pushing my students to make connections that we haven’t previously been making, and ever grateful for the educators who attend conferences like CMC-South eager and willing to learn alongside other educators.

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Posted by on October 26, 2014 in Angles & Triangles, Geometry, Rigid Motions


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I have learned …

Students aren’t kidding when they ask, “When will we ever need to know this?” In How the Brain Learns Mathematics, David Sousa suggests that students need a reason to move information from short-term memory to long-term memory. What opportunity do we give our students to reflect on what they are learning and why during class?

Many teachers give Exit Tickets, which can give teachers good information about what students have learned. However, I’ve observed many exit tickets that are more useful for teachers than they are for students. If the exit ticket requires a calculation, when do students find out whether what they’ve submitted is correct? Immediately? Or the next time class meets? How many students then complete homework using wrong ideas?

Exit Tickets can be good formative assessment. In fact, Sousa also notes that closure in a lesson shouldn’t be students packing up their backpacks and walking out of the door. Closure needs to be a cognitive process – students need to think about what they have learned and what questions they have, connecting what they have learned in class today with what they have previously learned and maybe even to what they will learn. Exit Tickets can provide students an opportunity to cognitively think about what they are learning.

My question is what types of formative assessment are we using throughout the class period, instead of just at the end of class?

Are you familiar with Dr. Sousa’s brain research on the Primacy/Recency Effect? In essence, it shows that we remember best what we learn first in a learning episode; we remember second best what we learn last in a learning episode; and we remember least what’s in the middle of the learning episode. Think about how the typical math class has been set up. Students come in, and teachers go over homework (prime learning time). At the end of class, students practice (second prime learning time). In the middle of class, teachers teach the new material for the lesson (least prime learning time).

His research shows that 20 minutes is the ideal length for a learning episode. I teach on a block schedule, and so I find that I must be deliberate about planning shorter (20 minute, when possible) learning episodes within the block.

We were finishing up a unit on Angles & Triangles in geometry earlier this week. We begin each class with an opener of questions that students work through with their teams. I collect their responses, show them the solutions, they try to correct misconceptions with their teams, and then we talk all together about any remaining misconceptions. After the opener (first learning episode) each day, students glance through our learning goals for the unit so that they can think about what they know and what they still need to know.

photo 1

Learning goals:

I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

I can use and prove theorems about special pairs of angles. G-CO 9

I can solve problems using triangles. G-CO 10

I can prove theorems about angles in triangles. G-CO 10

I can solve problems using parallel lines. G-CO 9

I can prove theorems about parallel lines. G-CO 9

I can solve problems using congruent triangles. G-CO 8

I can explain criteria for triangle congruence. G-CO 8

Because it was the last day of the unit, I asked students to answer a Quick Poll letting me know what they have learned and what they still need to know. The more I use “I can” statements for learning goals, the more I notice that they give us a common language for talking about what we can already do and what we can’t do yet.

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i have learned that i can solve problems using triangles, i still need to know how to prove theorems          1

i still need to touch up on the statemfnts and postulates            1

A.) I have learned to make conclusions, find the measures of angles, and etc..

B.) I still need to know the process in constructing parallel lines on the calc. in greater detail           1

about vertical angles            1

i have learned parrell line

i still need to know alot        1

i still have trouble ex triangle congruences           1

1.i learned prove theorems about special angles

2.i need to work on inductive and deductive statements, theorems about angles in triangles           1

i have learned conditional statements. i still need to go back over them                       1

learned how to prove why things are what they are

still need to know how to correctly prove anything from a given           1

i have learned the conditional statements

CO9 G-CO10 G-CO9,8,8

i need work on the true false charts           1

I have learned how to construct parallel lines using a point.      1

L symbolic logic

NTK proofs    1

how to find exterior angles of triangles           how to form theorems   1

i learned how to do ratios in a triangle.

need to know how to prove theroms          1

i have learned more about parallel lines cut by a transversal i still need to know more about constructing my own proofs         1

i learned about the types of hypothesis. i still need to know the different angle terms.          1

i have learned how to work with ratios.

i stll need to know how to form theorms on my own.       1

I have learned converse, inverse, conditional, and contrapositive statements. I need to learn when to use certain postulates in order to complete proofs.            1

I have learned how to prove statements using postulates. I still need to know how to explain criteria for triangle congruence.  1

i can prove theorems about angles in triangles. explain criteria for triangle.   1

how differemt types of angles are equal anb the different type of statements

i need to know the difference between converse inverse and contrapositive statements      1

learned conditional, converse, invese, and contrapositives.         1

I have learned how to solve problems using triangles. I still need to know how to do well on tables.           1

how to construct parrelel lines

how to write a hypothesis and conclusion in its different forms and determine their truth value    1

how to identify logical statements

how to do proofs       1

i need to work on converse inverse conditional contrapositive   1

i have learned how to construct parallel lines.

i still need to know how to prove the truth value of a statement.           1

learned-how to construct parallel lines

need to know-idk      1

i have learned to solve proplems using parallel lines. i still need to learn how to prove problems.   1

what aternate interior angles are;

how to figure out truth talbes.        1

i have learned if p then        1

I took the information about what students still need to know and used it to structure the rest of the class period, instead of just going through review problems in the order I happened to put them together.

A few years ago, Jill Gough and her colleagues experimented with students and faculty taking a brain break every 20 minutes to tweet what they are learning … you can read more about it here.

What will you do to ensure that you are maximizing the learning episodes in your classes?

And so the journey continues, with thanks to @jgough for making me reflect on how often I do formative assessment throughout a class period.


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Lesson and Assessment Design – #T3Learns

What are we intentional about in our planning, process, and implementation?

  • Are the learning targets clear and explicit?
  • What are important check points and questions to guide the community to know if learning is occurring?
  • Is there a plan for actions needed when we learn we must pivot?

On Saturday, a small cadre of T3 Instructors gathered to learn together, to explore learning progressions, and to dive deeper in understanding of the Standards for Mathematical Practice.

The pitch:

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Jill and I fleshed out the essential learning in more detail:

  • I can design lessons anchored in CCSS or NGSS.
    • I can design a lesson incorporating national standards, an interactive TI-Nspire document, a learning progression, and a formative assessment plan.
    • I can anticipate Standards for Mathematical Practice that learners will employ during this lesson.
  • I can design a learning progression for a skill, competency, or process.
    • I can use student-friendly language when writing “I can…” statements.
    • I can design a leveled assessment for students based on a learning progression.
  • I can collaborate with colleagues to design and refine lessons and assessments.
    • I can calibrate learning progressions with CCSS and/or NGSS.
    • I can calibrate learning progressions with colleagues by giving and receiving growth mindset oriented feedback, i.e. I can offer actionable feedback to colleagues using I like… I wonder… what if…
    • I can refine my learning progressions and assessments using feedback from colleagues.

The first morning session offered our friends and colleagues an opportunity to experience a low-floor-high-ceiling task from Jo Boaler combined with a SMP learning progression. After the break, we transitioned to explore the Standards for Mathematical Practice in community. The afternoon session’s challenge was to redesign a lesson to incorporate the design components experienced in the morning session.

Don’t miss the tweets from this session.

Here are snippets of the feedback:

I came expecting…

  • To learn about good pedagogy and experience in real time examples of the same. To improve my own skills with lesson design and good pedagogy.
  • Actually, I came expecting a great workshop. I was not disappointed. I came expecting that there would be more focus using the TI-Nspire technology (directly). However, the structure and design was like none other…challenging at first…but then stimulating!
  • to learn how to be more deliberate in creating lessons. Both for the students I mentor and for T3 workshops.
  • I came expecting to deepen my knowledge of lesson design and assessment and to be challenged to incorporate more of this type of teaching into my classes.

I have gotten…

  • so much more than I anticipated. I learned how to begin writing clear “I can” statements. I also have been enriched by those around me. Picking the brains of others has always been a win!
  • More than I bargained. The PD was more of an institute. It seemed to have break-out sessions where I could learn through collaboration, participation, and then challenging direct instruction, … and more!
  • a clear mind map of the process involved in designing lessons. A clarification of what learning progressions are. Modeling skills for when I present trainings. Strengthening my understanding of the 8 math practices.
  • a better idea of a learning progression within a single goal. I think I had not really thought about progressions within a single lesson before. Thanks for opening my eyes to applying it to individual lesson goals.

I still need (or want)…

  • To keep practicing to gain a higher level of expertise and comfort with good lesson design. Seeing how seamlessly these high quality practices can be integrated into lessons inspires me to delve into the resources provided and learn more about them. I appreciate the opportunity to stay connected as I continue to learn.
  • days like this where I can collaborate and get feedback on activities that will improve my teaching and delivery of professional development
  • I want to get better at writing the “I can” statements that are specific to a lesson.
  • I want to keep learning about the use of the five practices and formative assessment.

We want to see more collaborative productive struggle, pathways for success, opportunities for self- and formative assessment, productive conversation to learn. and more.

And so the journey continues…

[Cross-posted on Experiments in Learning by Doing]


Posted by on October 19, 2014 in Professional Development, Uncategorized


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Systems of Linear Equations

We are working on Systems of Linear Equations in our Algebra 1 class.

The first day, our lesson goals were the following:

Level 4: I can create a system of linear equations to solve a problem.

Level 3: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We used the Mathematics Assessment Project formative assessment lesson Solving Linear Equations in Two Variables to introduce systems.

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Then we asked students to draw a sketch of a system of equations with one solution, a sketch of a system of equations with no solution, and a sketch of a system of equations with many solutions.

We used the Math Nspired activity How Many Solutions to a System to let students explore necessary and sufficient conditions for the equations in a system with one, no, or many solution(s). We used a question from the activity so that students could solidify their thinking about the relationship between the equations for different solutions.

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We sent the questions as Quick Polls so that we could formatively assess their thinking and correct misconceptions.

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We like using this type of Quick Poll (y=, Include a Graph Preview) while students are learning. The poll graphs what students type so that they can check their results and modify the equation when they do not get the expected result.

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(Note: on the second day, we asked again for students to create a second equation to make a system with one, no, and many solution(s), but the given equation is in standard form.)


On Day 2 of Systems, the learning goals are slightly different.

Level 4: I can solve a system of linear equations in more than one way to verify my solution.

Level 3: I can create a system of linear equations to solve a problem.

Level 2: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We started with a task from Dan Meyer’s Makeover series, Internet Plans. Part of this makeover was a complete overhaul of the context. Students write down a number between 1 and 25. They view this flyer (from Frank Nochese).


And then they decide which gym membership to choose, using the number they wrote down as the number of months they plan to work out.


When I saw the C’s and B’s on the number line on the blog post, I actually thought that the image was beautiful – a clear indication that one plan is better than the other two for a while, and then another plan is better than the other two. This image absolutely leads to the question of which one becomes better when – and thinking about why plan A is never the better deal.

Our students started calculating. Several asked what $1 down meant before they could get a calculation. Then they shared their work on our class number line.

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We were dumbfounded at the results. Really. It was hard to know where to start. We were prepared to help student develop the question with correct calculations. We hadn’t thought about what we would do when students couldn’t figure out which plan was the better deal for a specific, given number of months.

What would you do next if this is how your students answered?


We started with the lone C at 4 months, hoping someone would claim it and share his calculations. He did: I divided $199 by 12 months and then multiplied by 4 to get the cost for 4 months.


Another student said he couldn’t do that because you had to pay $199 for 12 months no matter how many months you actually used it.


Another student said that A was the better deal because they were planning to get every other month free by working out more than 24 days each month. We discussed how realistic it is to work out at a gym 24 days a month.


We did eventually look at a graphical representation of each plan. And talked about what assumptions had been made using these graphs.

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Can you tell which graph is which plan?

Can you write the equations of the lines?

In the first class we never made it to actually calculating the point(s) of intersection (class was shorter because of a pep rally), but other classes did calculate the point(s) of intersection.

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It’s not a waste of time to think about Plan C costing $199 for 12 months.

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But then we really should have created a piecewise function to represent the cost for 13-24 months as well.

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The number lines weren’t much better in subsequent classes.


So maybe the students got to practice model with mathematics, just a little, even though we have a long way to go before students will be able to say, “I can create a system of linear equations to solve a problem”.

The teachers were definitely reminded of how much work we have to do this year, as the journey continues …


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SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 2)

We want every learner in our care to be able to say

I can reason abstractly and quantitatively.


But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?


A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?


In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?


Level 4:

I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations. 

Level 3:

I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.  


I can identify and connect the units involved using an equation, graph, or table.


I can attend to and document the meaning of quantities throughout the problem-solving process.


I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion. 

Level 2:

I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:

I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.


What evidence of contextualizing and decontextualizing do you see in the work below?

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[Cross-posted on Experiments in Learning by Doing]



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SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

We want every learner in our care to be able to say

I can reason abstractly and quantitatively.


I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively. (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)


Level 4:

I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:

I can reason abstractly and quantitatively.

Level 2:

I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:

I can define variables and constants in a problem situation and connect the appropriate units to each.


You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively. Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway. How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.


[Cross-posted on Experiments in Learning by Doing]


Posted by on September 28, 2014 in Standards for Mathematical Practice


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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 …

1 Comment

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


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