Tag Archives: Class Capture

A Heuristic Approach to Angles in Circles

I am taking a qualitative research class right now, and my mind is full of lots of new-to-me words (many of which my spell checker doesn’t know, either): hermeneutics, phenomenology, ethnography, ethnomethodology, interpretivism, postpositivism, etc. One that has struck me is heuristic, the definition of which I can actually remember because I try to teach heuristically. (The word does not yet roll off of my tongue, but the definition, I get.)

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On Monday, our content was G-C.A Understand and apply theorems about circles

  1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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We started with a Quick Poll. I asked students for their best guess for the angle measure. I showed the results without displaying the correct answer, noting the lowest and highest guesses.

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Students moved to the technology. What happens to the angle measures as you move the points on the circle?

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They moved to the next page, which revealed more information. What happens to the angle measures as you move the points on the circle?

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I sent the poll again. There was one team who hadn’t answered yet, so I made a brief stop by their table. Last semester, I remember reading something about how a certain example might give students the eyes to see what you’re trying to get them to see. So we moved the points around to look something like this.

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If you have 49 and 43, how can you get 46?

Changing the numbers purposefully helped them see.

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I sent one more poll before we talked about why.

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So we gave our best guess, and then we used technology to explore. Students practiced MP8 I can look for and express regularity in repeated reasoning as they noticed what stayed the same and what changed with an angle whose vertex is in the center of the circle. They generalized the result. But we hadn’t yet discussed why that happens.

Students practice MP7 I can look for and make use of structure. By now they know our mantra for MP7: What can you make visible that isn’t yet pictured?

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I saw a line constructed parallel to the given line, which made alternate interior angles visible.

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I saw a chord drawn that made a triangle visible.

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I asked students to write down everything they knew about the angles in this diagram.

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They made suggestions about what we know. They didn’t say the relationships exactly like I would. I wrote them down anyway. They didn’t recognize the exterior angle of the triangle and so ending up proving the Exterior Angle Theorem again off to the side. I wrote it down anyway.

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And so the journey continues, always trying to enable my students to discover or learn something for themselves (and sometimes succeeding) …

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Posted by on February 9, 2017 in Circles, Geometry


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Notice & Note: Dilations

How do you give your students the opportunity to practice MP8: I can look for and express regularity in repeated reasoning?

SMP8 #LL2LU Gough-Wilson

We started our dilations unit practicing MP8, noticing and noting.


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What would you want students to notice and note?

How do students learn what is important to notice and note?

An important consideration when learning with self-explanation is to look at the quality of the explanation itself. What are the students saying or writing? Are they just regurgitating bits of text or making connections to underlying principles? Do the explanations contain predictions about what is going to happen, try to go beyond the given instruction or do they just superficially gloss over what is already there? Students who make principle-based, anticipative, or inference-containing explanations benefit the most from self-explaining. If students seem to be failing to make good explanations, one can try to give prompts with more assistance. In practice, this will likely take iteration by the instructor to figure out what combination of content, activity and prompt provides the most benefit to students. (Chiu & Chi, 2014, p. 99)

We had a brief discussion about what might be important to notice and note. We’ve also been working on predictions, thinking about what you expect to happen before trying it with technology:

What happens when the center of dilation is on the figure, outside the figure, and inside the figure?

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What happens when the scale factor is greater than 1? Equal to 1? Between 0 and 1? Less than 0?

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I observed, walking around the room and using Class Capture, selecting conversations for our whole class discussion.

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Here’s what NA noticed and noted.


We looked at Hannah’s Rectangle, from NCSM’s Congruence and Similarity PD Module. Students had a straightedge and piece of tracing paper.

Which rectangles are similar to rectangle a? Explain the method you used to decide.Hannahs Rectangle.png

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What would you do next? Would you show the correct responses? Or not?

Would you start with an incorrect answer? or a correct answer?

Would you regroup students based on their responses?

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I started with a student who didn’t select G and then one who did. Then I asked a student who selected C to share why he chose C and didn’t choose F. We ended by watching Randy’s explanation on the module video.

And so the journey continues, always wondering what comes next (and sometimes wondering what should have come first) …

Chiu, J.L, & Chi, M.T.H. (2014). Supporting self-explanation in the classroom. In V. A. Benassi, C. E. Overson, & C. M. Hakala (Eds.). Applying science of learning in education: Infusing psychological science into the curriculum. Retrieved from the Society for the Teaching of Psychology web site:


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Posted by on December 19, 2016 in Dilations, Geometry


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MP5 – The Center of the Circle

How do you give students the opportunity to practice “I can use appropriate tools strategically”?


How would your students find the center of a circle?

Every year, I am amazed at the connections students make between properties of circles that we have explored and what the center of the circle has to do with those properties.

We started on paper.

Some students moved their thoughts to technology.

Whose work would you select for an individual and/or whole class discussion?

Could we use the tangents to a circle from a point to find the center of the circle?

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Could we use the intersection of the angle bisectors of an equilateral triangle inscribed in a circle to find the center of the circle?

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Could we use the perpendicular bisector of a chord of a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of a pentagon circumscribed about a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of several chords of a circle to find the center of the circle?

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Could we use a right triangle inscribed in a circle to find the center of the circle?

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And so the journey continues …

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Posted by on September 15, 2016 in Circles, Geometry


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Notice and Note: Dilations

Are you familiar with Notice & Note: Strategies for Close Reading? Here’s a link to Heinemann’s Notice & Note learning community, and here’s a sample PDF. I wonder whether our Standards for Mathematical Practice are similar to the Notice and Note literary signposts.

It’s not enough to just read a text. We want students to read for understanding and comprehension. The literary signposts help students with close reading of a literary text.

Similarly, it’s not enough to just explore math with dynamic graphs and geometry. We want students to explore for understanding and comprehension. The math practices help students learn how to interact with a mathematical problem or concept … and what to notice.


Last week, we explored dilations.

What do you need for a dilation?

A figure, a point (which we’ll call the center of dilation), and a number (which we’ll call the scale factor)

We used our dynamic geometry software to perform a dilation.

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About what things might you be curious as you explore dilations?

(I thought of Kristin when I used the word curious.)

What happens when the center of dilation is inside the pre-image?

What happens when the center of dilation is on the pre-image? (on a side, on a vertex)

What happens when the scale factor is between 0 and 1?
What happens when the scale factor is negative?

How do the corresponding side lengths in the pre-image and image relate to each other?


I asked students to practice look for and express regularity in repeated reasoning as they explored the dilation. Do you know what it means to look for and express regularity in repeated reasoning?

Find a pattern.

Yes. Figure out what changes and what stays the same as you take a dynamic action on the dilation. Begin to make some generalizations about what you notice.

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And don’t just notice, but actually note what you’re thinking.

The room got quiet as students noticed and noted their observations about dilations. I monitored student work both using Class Capture and walking around to see what students were noting.

(I promise I’ve tried to make it clear to students that dilation has 3 syllables and not 4 … but we do live in the South.)

Eventually, they shared some of their findings with their team, and then I selected a few to note their observations for the whole class.

BB showed us what happened when he perfomed a dilation with a scale factor of -1. He had noted that it was the same as rotating the pre-image 180˚ about the center of dilation.

SA talked with us about when the dilation would be a reduction. She had decided it wasn’t enough to say a scale factor less than 1 or a fractional scale factor but that we needed to say a scale factor between 0 and 1 or between -1 and 0.

FK showed us that when she drew a line connecting a pre-image point and its image, the line also contained the center of dilation.

PS noted that when the scale factor was 2, the length of the segment from the center of dilation to a pre-image point equaled the length of the segment from the pre-image point to its image.

When the scale factor was 3, the length of the segment from the center of dilation to a pre-image point equaled one-half the length of the segment from the pre-image point to its image.


We next determined a dilation and set of rigid motions would show that the two figures are similar.

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Translate ∆DET using vector EY.

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Rotate ∆D’E’T’ about Y using angle D’YA.

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Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.

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Then we looked at dilations in the coordinate plane. I knew that my students had some experience with this from middle school, and so I sent a Quick Poll to see what they remembered.

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Due to the success on the first question, I changed it up a bit with the second question.

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But I wonder now whether I should have started with the second question. If they could do the second question, doesn’t that tell me they can also do the first?

I’ve rearranged the polls to try that the next time I teach dilations.


We ended the lesson with a triangle that had been dilated. Where is the center of dilation?

And so the journey continues, with hope that noticing & noting will make a difference in what students learn and remember …


Posted by on November 17, 2015 in Dilations, 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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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 …


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


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