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Category Archives: Rigid Motions

The Magic Octagon – Dan’s, Andrew’s, and mine

I had saved Andrew’s post in my folder for a recent lesson, which was about Dan’s video.

We paused halfway in, and students decided where it would be. They answered a Quick Poll to let me know, and by the time they had all answered, some had changed their minds.

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We quickly looked at the responses, and they decided using time would be easier to decipher than some of the other descriptions.

I sent a second poll. I waited for everyone to answer, even the ones who wanted to take their time thinking about it.

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And then we continued to watch.

We paused for the last question, they discussed with their team, and then we finished watching.

Good conversation. But we didn’t get to the sequel proposed by one of Andrew’s students: If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too? Why or why not?

So I emailed that question to my students.

  • Yes, the two points move like opposite hands on a clock moving closer to each other and overlapping at 5:00. At about 11:00 they would overlap again. Otherwise, there is no overlap.
  • They would be at 5:00. This is because when he flips the magic octagon, the back arrow also flips, causing the new time to be 3:00 instead of 9:00. This means that if you were to find a line of reflection, you could flip the octagon on that line and the arrow would always land right where the previous one did. If this was on transparent paper, you can see that if one arrow points to 5:00, then the other one would be pointing at 7:00. But if you were to flip the octagon on the reflection line which intersects 12:00 and 6:00, then you would continuously get 5:00 because of the reflection.

As I got the responses from students, I realized that I wished I had asked a different question. While I did include why or why not, and it was obvious from the responses that students didn’t just answer yes or no, I wish I had asked “At what time(s), if any, are the front side and back side arrows at the same time?”

I am reminded of something I can no longer find that I read in a book. A group of teachers observed a “master” teacher for a lesson and then went back to their own classrooms to teach the lesson. The teachers asked the same questions that the master teacher asked; however, the lessons didn’t go as hoped. The teachers were not asking questions based on what was happening in their own classrooms; they were asking questions based on what had happened in the other classroom.

I love reading blog posts and learning from so many mathematics educators. They give me ideas that I wouldn’t have on my own. In fact, as my classroom moved toward more asking and less telling, I used to say that my most important work happened before the lesson, collaborating with other teachers and deciding what questions to ask. I’ve decided otherwise, though. My most important work happens in the moment, not just asking, but also listening. And then, if needed, adjusting what I planned to ask next based on the responses of the students in my care. And so the journey will always continue …

 
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Posted by on November 15, 2016 in Geometry, Rigid Motions

 

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MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

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This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

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Then she rotated the parallelogram 180˚ about that point.

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Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

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How else could you carry a parallelogram onto itself?

 
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Posted by on September 22, 2016 in Geometry, Rigid Motions

 

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MP6 – Mapping a Figure Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

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We practiced both, but we focused on describing.

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I asked the student who listed several steps to share his work.

  1. rotate rectangle 180˚ about point A
  2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
  3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

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What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

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What about mapping a regular pentagon onto itself?

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Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

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This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

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What can you do other than a single rotation?

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This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

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The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

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It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 
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Posted by on September 21, 2016 in Geometry, Rigid Motions

 

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Reflected Triangles + #ShowYourWork

I’ve used the Illustrative Mathematics task Reflected Triangles for several years now. This year students practiced “Show Your Work” (from Jill Gough) along with coming up with a correct solution.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem

Which of the directions can you follow exactly to construct the line of reflection? Are there any that need clarification?

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Students wrote first and then constructed their lines of reflection. I used the Class Capture feature of TI-Nspire Navigator to watch. We’ve spent longer on this task in the past, but this year, I only had about 5 minutes for a whole class discussion. Whose work would you select for the whole class to see?

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We started with Kaelon’s construction.

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How is Holly’s construction different? Can you tell what Holly did?

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How is Phillip’s different? Can you tell what he did?

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We are well on our way towards learning how to “describe or illustrate a solution in a way that the reader understands without talking to me”, as the journey continues …

 
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Posted by on September 8, 2015 in Angles & Triangles, Geometry, Rigid Motions

 

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Carrying a Figure Onto Itself + #ShowYourWork

G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our content learning goal for the day: I can map a figure onto itself using transformations.

Our practice learning goal: I can attend to precision.

Combining those, we were working on: I can show my work.

Do your students know what you mean when you ask them to show your work?

Jill Gough has written a transformative leveled learning progression for showing your work. This was our first day in geometry this year to focus on it.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem.

For this task, our focus was on describing clearly the transformations that would carry a rectangle or equilateral triangle onto itself so that a partner could follow the steps.

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Which of the following is clear?

Reflect ABCD about a line through the middle of the rectangle.

Reflect ∆ABC about its center.

Rotate ∆ABC 60˚.

Reflect ABCD about the perpendicular bisector of segment AB.

Rotate ∆ABC 180˚ about point A.

Students set to work individually, paying attention to their language. I walked around to see what they were writing.

I noticed MR’s first, which said, Translate ∆ABC using vector AA. As I looked more closely, I realized that she was mapping the triangle on the left side of the page onto the triangle on the right side of the page, but even so, she had come up with a remarkably trivial solution, had she been mapping the triangle onto itself.

The next student that I saw had rotated ∆ABC 360˚ about point A.

And then the next student that I saw had dilated ∆ABC about point A using a scale factor of 1.

I decided at this point that perhaps a class discussion was in order to limit additional trivial solutions to this task. So we talked about transformations that will, of course, map the figure onto itself, such as rotating the image about one of its vertices 0˚ or 360˚, and also, really, are simple and not very interesting.

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And then I let them work some more. The idea was for them to write a transformation or sequence of transformations and have their partner try it, following their directions exactly. The partner helped revise the directions as needed if the directions didn’t work the first time.

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Instead of selecting particular students to share their work with the whole class, I asked students to write at least one set of their successful mappings in a shared Google Doc so that they could see multiple solutions to both the rectangle and the triangle.

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Thanks to the leveled learning progression, I think we are off to a good start practicing “show your work”, as the journey continues …

 
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Posted by on September 7, 2015 in Geometry, Rigid Motions

 

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The Center of Rotation

This is the first year we have tried Identifying Rotations from Illustrative Mathematics.

△ABC has been rotated about a point into the blue triangle. Construct the point about which the triangle was rotated. Justify your conclusion.

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This reminds me of the Reflected Triangles task, which we have used now for several years.

I got a glimpse of students working on the task using Class Capture. I watched them make sense of problems and persevere in solving them.

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We looked at all of the auxiliary lines that LJ made, trying to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Jarret’s work, who used technology to perform a rotation, going backwards to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Justin’s work, who rotated the given triangle about A to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Quinn’s work, who knew that if R is the center of rotation, then the measures of angles ARA’, BRB’, and CRC’ must be the same.

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Students took those conversations and continued their own work.

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The next day, Jared shared his diagram. What can you figure out about the relationship between the center of rotation, pre-image, and image looking at his diagram?

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In my last two posts, I’ve wondered what geometry looks like if we start our unit on Rigid Motions with tasks like these instead of ending the unit with tasks like these. Maybe we will see next year, as the #AskDontTell journey continues …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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The Line of Reflection

I’ve blogged about the Illustrative Mathematics task Reflected Triangles before. I really like that it asks students to determine the line of reflection given the pre-image and image instead of determining the image given the pre-image and line of reflection.

△ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

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How would you construct the line of reflection?

Work on your mathematical flexibility to come up with more than one way to construct the line of reflection.

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I used Class Capture to monitor student work and selected some students to share their approach with the whole class.

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Many students approached the task like Paris:

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What can we learn about the relationship between the pre-image, image, and line of reflection from the additional line in Max’s diagram?

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I’ve always used this task towards the end of our unit on Rigid Motions, but I am thinking about using it earlier in the unit next year. Maybe the task itself could be an #AskDontTell approach for students learning what we want them to learn about the relationship between the pre-image, image, and line of reflection. We will see next year, as the journey continues …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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Transforming a Segment

The task:

Given segment AB, construct a regular hexagon ABCDEF with segment AB as one of its sides.

-You may not use any Shapes tools.

-You many not use any Measurement tools.

When you are finished, we will use Measurement tools to justify your construction.

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This task is a good one for working on math flexibility. You can construct the hexagon one way? Great! Now find another way to construct the hexagon.

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I used Class Capture to monitor students working.

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Whose would you select for a whole class discussion?

 

This year, we started with someone who did rotations only.

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Then moved to someone who did rotations and reflections.

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Then moved to someone who had a number on their page that was different from everyone else.

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I’ve written about this task before, here and here. As I think about the students I will have next year, I wonder whether we could start our unit on Rigid Motions with this task, instead of ending the unit with it. We’ve been using our new standards for 3 years now in geometry, but it is really only next year that we will have students who had most of the new standards in middle school. I’m beginning to think about how that changes what we’ve been doing.

And so the journey continues, constantly making adjustments to meet the needs of the students we do have and not those we did have …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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The Diagonals of a Rectangle

CCSS-M.G-CO.C.11.Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

How can you use rigid motions to show that the diagonals of a rectangle are congruent?

I had a few students come in during zero block to work on proofs. (If you read my last post on the diagonals of an isosceles trapezoid, you’ll know why.)

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We were trying to show that AC=BD. Students used look for and make use of structure to compose the rectangle into two triangles. Which two triangles should we show congruent if we want to show that the diagonals are congruent?

∆ACD and ∆BDC.

One student showed the two triangles congruent by SAS. (Opposite sides of a rectangle are congruent, all angles in a rectangle are right and thus congruent, and CD=CD by reflexive.)

But a pair of girls wanted to use a rigid motion to show that the triangles were congruent.

First up: rotating.

Rotating which triangle?

∆ACD

About what point?

The center of the rectangle. Where the diagonals meet.

How many degrees?

90˚

(I’m not sure whether they really thought we should rotate by 90˚ or they chose 90˚ because we seem to rotate by 90˚ and 180˚ more than any other angle measure.)

We need paper. And scissors.

What is the image of ∆ACD when we rotate it 90˚ about the intersection of the diagonals?

Not ∆BDC.

What is the image of ∆ACD when we rotate it 180˚ about the intersection of the diagonals?

∆ABC

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How can we show that ∆ACD and ∆BDC are congruent?

A reflection?

About what line?

About the perpendicular bisector of segments AB and CD.

Are you sure?

Yes.

Once the triangles are congruent, then the corresponding parts are congruent, and so we can conclude that the diagonals are congruent.

And so the journey continues … with an apology to my former students for not using scissors more often.

 
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Posted by on December 2, 2014 in Geometry, Polygons, Rigid Motions

 

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The Diagonals of an Isosceles Trapezoid

It was the day before the test on Polygons, and so I thought that writing a proof and then giving feedback on another team’s proof might be helpful.

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Students worked alone for a few minutes, thinking about what was given and what could be implied. Then they worked with their team to talk about their ideas and to begin to plan a proof.

Some were off to a good start.

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Some were obviously practicing look for and make use of structure.

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Some were stuck.

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I talked to several groups, listening to their plan, asking a few questions to get them unstuck.

And then I got out colored paper on which to write the team proof.

 

The clock was ticking, but I thought that surely they would be able to trade proofs with another team for feedback within a few minutes.

 

I talked to another group. They were reflecting ∆ABC about line AC.

What will be the image of ∆ABC about line AC?

The answer? ∆ACD.

Of course that is wrong. It seems so obvious that ∆ABC is not congruent to ∆ACD. And I’m also wondering how that helps us prove that AC=BD, since BD isn’t in either of those triangles. But that’s where this team of students is. I now have the opportunity to support their productive struggle, or I can stop productive struggle in its tracks by giving them my explanation.

 

My choice? Scissors. And Paper. And more time.

What happens if you reflect ∆ABD about line AC?

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Oh! The triangles aren’t congruent.

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So are there triangles that are congruent that can get us to the diagonals?

∆ABC is congruent to ∆BAD.

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How do you know?

A reflection.

About what?

This pencil!

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So what is significant about the line that the pencil is making?

It’s a line of symmetry for the trapezoid.

It goes through the midpoints.

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(One of the team members was using dynamic geometry software to reflect ∆ABC in the midst of our conversation, but I don’t have pictures of her work.)

 

So the plan was for team to write their proofs on the colored paper and then trade with other teams for feedback. Great idea, right? So how do you proceed with 15 minutes left? Proceed as planned and let them give feedback with no whole class discussion? Or have a whole class discussion to connect student work? Because as it turned out, no two teams proved the diagonals congruent the same way. I chose the latter.

 

I asked the first team to share their work.

 

Their proof needs work. But they have a good idea.

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They proved ∆AMD≅∆BMC, which makes the corresponding sides congruent, so with substitution and Segment Addition Postulate, we can show that the diagonals are congruent.

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Next I asked the team to share who proved ∆ABC≅∆BAD using a reflection about the line that contains the midpoints of the bases. Their written proof needs work, too. But they had a good idea.

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Another team proved ∆ACD≅∆BDC.

 

Another team constructed the perpendicular bisectors of the bases. Since the bases are parallel, a line perpendicular to one will be perpendicular to the other. I’m not sure they got to a reason that the perpendicular bisectors have to be concurrent. They could have used ∆AZD≅∆BZC to show that. Instead, they used a point Z on both of the perpendicular bisectors (they know that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment) to reason that ∆AZB and ∆DZC are isosceles & then used Segment Addition Postulate and substitution to show that the diagonals are congruent. Not perfect. But a good start.

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NCTM’s Principles to Actions discussion on support productive struggle in learning mathematics says, “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to ‘rescue’ students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such ‘rescuing’ undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics.”

 

So while I didn’t rescue my students, we also never made it to an exemplary proof that the diagonals of an isosceles trapezoid are congruent. Did they learn something about make sense of problems and persevere in solve them? Sure. Is that enough?

 

Would it be helpful to lead off next year’s lesson with this student work? Or does that take away the productive struggle?

 

Is it just that we have to find a balance of productive struggle and what exemplary work looks like, which is easier in some lessons than others? If so, I failed at that balance during this lesson. Even so, the journey continues …

 
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Posted by on December 1, 2014 in Geometry, Polygons, Rigid Motions

 

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