Category Archives: Dilations

5 Practices: Dilations

5 Practices for Orchestrating Productive Mathematics Discussions might be the book that has made me most think about and change my practice for the better in the past 10 years.

At the beginning of our second day on dilations, I asked students to work on this.

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Because of the 5 Practices, I pay attention differently when I walk around and monitor students working. I know that I looked for different student approaches before I read the book, but I didn’t consciously think about selecting and sequencing them for a whole class discussion. I often asked for volunteers. And then hoped that another student would volunteer when I asked who worked it differently [who had actually worked it differently and correctly].

I asked a few questions of students while I was monitoring them to clarify what they were doing and selected and sequenced a few to share. The student work above looks similar at first glance, but there are subtle differences in their thinking that make important connections about dilations.

TM shared first. She used slope to find the vertices of the image. She went down 1 and to the right 3 from C to X, and then because of the scale factor of 2 went down 1 and to the right 3 from X to get to X’. She went down 3 and to the right 2 to get from C to Z, and then went down 3 and to the right 2 from Z to get to Z’.

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JA shared next. He focused on the line that contains the center of dilation, image, and pre-image. He knew that X’ would lie on line CX and that Z’ would lie on line CZ.

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MB shared next. He also used slope, but a bit differently from TM. He noticed “down 1 and to the right 3” to get from C to X and so because of the scale factor of 2 then did “down 2 and to the right 6” from C to get to X. He noticed “down 3 and to the right 2” to get from C to Z and so then did “down 6 and to the right 4” to get from C to Z’.

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I had not seen additional methods while monitoring. This exercise didn’t take too long, and so I didn’t get around to everyone. [This is where Smith & Stein’s advice about keeping a clipboard to pay closer attention to whom you check in with and whom you call on helps so that you aren’t checking in with and calling on the same few every time you have a whole class discussion.] I hesitated before I asked, but I did then ask, “did anyone find X’Y’Z’ a different way?” [This is also where I am learning to trust my students to recognize when their method is different.] TC raised his hand. I treated C as the origin and used coordinates. He shared his work and showed that the coordinates of X (3, -1) transformed to X’ (6,-2) with a dilation about the origin for a scale factor of 2.

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And so the journey continues, thankful for friends like Gail Burrill [one of my voices] who recommend authors like Smith and Stein to help me think about and change my practice for the better, making me feel like a conductor rehearsing for a beautiful, exciting mathematics masterpiece …


Posted by on December 21, 2016 in Dilations, Geometry


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Hinge Questions: Dilations

Students noticed and noted.

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I wanted to be sure that they could answer a dilations question based on their observations. I had two questions premade in my set of Quick Polls. Which question would you ask?

In the past, I would have asked both questions without thinking.

I am learning, though, to think more about which questions I ask. If we only have time to ask a few questions, which questions are worth asking?

From slide 34 in Dylan Wiliam’s presentation at the SSAT 18th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

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I decided to send the second poll. I decided that if they get that one right, they can both dilate a point about the origin and pay attention to whether they are given the image or pre-image. If I had sent the second poll, I wouldn’t know whether they could both do and undo a dilation.

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Next we looked at this question.

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Students worked on paper first.

Then some explored with technology.

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What do you want your students to know about the relationships in the diagram?

What question would you ask to see whether they did?

I asked this question to see what my students were thinking.

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And so the journey to write and ask and share and revise hinge questions continues …


Posted by on December 20, 2016 in Coordinate Geometry, Dilations, 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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Rigor: Trig Ratios

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help our students build procedural fluency from conceptual understanding?

I wrote last year about using technology to develop conceptual understanding of Trig Ratios.

This year, we started the lesson a bit differently. I read a while back about Boat on the River, a 3-Act that Andrew Stadel had published and that Mary Bourassa had used to introduce right triangle trig, but I had never taken the time to look it up.

We watched Act 1 to begin the lesson.

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Students submitted what they noticed and wondered.

Then we thought about what information would be useful to know, along with thinking about what information would actually be reasonably attainable.

For example, many bridges are made to a certain standard height or have the clearance height painted on them. This one is no exception … the bridge height is given in the Act 2 information.

Students decided the length of the mast was attainable, too. And the angle at which the boat is leaning. Maybe there was a reading on the control panel.

So we ended up with something like this:

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Students’ experience with right triangles to this point had been the Pythagorean Theorem, similar right triangles/altitude drawn to hypotenuse, and special right triangles.

I told them we’d come back to the boat problem by the end of class.

Next, I asked students to draw a right triangle with a 40˚ angle and measure the side lengths. I collected their side lengths, again, telling them that we would use this information later in class. (I wish I had asked them to do this part the night before and submit via Google Form … maybe next year.)

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Students practiced I can look for and express regularity in repeated reasoning along with Notice and Note while first watching B move on the Geometry Nspired Trig Ratios activity and then observing what happened as I pressed the up arrow on the slider.

SMP8 #LL2LU Gough-Wilson

Eventually, we uncovered that the ratio of the opposite side to the hypotenuse of an acute angle in a right triangle is called the sine ratio. We connected that to triangle similarity as our content standard requires.


  1. Define trigonometric ratios and solve problems involving right triangles
  2. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

We checked the ratio of the opposite side to the hypotenuse for the right triangle they had drawn and measured. How close were they to 0.643? Students immediately noted that there was a problem with the ratios that were over 1 and talked about why.

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We continued practicing I can look for and express regularity in repeated reasoning along with Notice and Note to develop the cosine and tangent ratios.

And then we went back to Boat on the River. What are we trying to find? What ratio could we use? How would we know whether the boat made it?

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When I cued up the video for Act 3, the students were thrilled to find out they were actually going to get to see whether the boat made it. And at the end, they spontaneously clapped.

Someone asked me in a workshop recently how long a 3-Act takes. There are plenty on which we spend majority of a class period. Or even more than one class period. This one took less than 10 minutes of our lesson, but the payoff is worth more than our whole unit of Right Triangle Trigonometry. It gave us a way to develop the need for trig ratios that my students have just had to trust we need before. For these students, trig ratios don’t just solve right triangles; trig ratios can help with planning trips down the river. Coupled with the formative practice that students got during the next class, Boat on the River helped us balance rigor, one of the key shifts in mathematics called for by CCSS.

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And so the journey towards rigor continues … with thanks to Andrew for creating Boat on the River and Mary for blogging about her students’ experience with it and to my students for their enthusiasm about learning which made evident during this lesson through applause.

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Posted by on January 23, 2016 in Dilations, Geometry, Right Triangles


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Soccer Ball Inflation

We tried Soccer Ball Inflation again this year.

I haven’t found many opportunities during our first semester of geometry for students to engage in multiple steps of the modeling cycle. So I’m glad for the few problems that at least let students define the problem, decide what information is useful to know, and begin to formulate a model to describe relationships between what is important.

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We watched Nathan Kraft’s Soccer Ball Inflation video on 101 questions.

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Most students wanted to know how many pumps it would take to fill the other balls.

What information do you need to know to figure it out?

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This was the end of November. It’s not the last time I’ll ask my students what information they need to know to figure out the answer to a question, but it was the first. It takes practice figuring out what information is useful, especially when it has been given for so long. Most of what they wanted to know (except for the answer) isn’t very useful or even possible without complicated measurement tools.

So I asked, “What’s easy for us to know? What’s easy to measure?”

The radius.


Last year, I noted in my blog post that I gave them the circumferences (because that’s what Nathan included in Act 2, and I didn’t want to do any calculating). Dan called me out on this:

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I’m not the only one who’s been living inside the “ideal” math world for too long. So have my students.


I asked, ”Is it easy to measure the radius?”

Oh. I guess not. The circumference.


Okay – so the circumference. I gave them the circumferences of all three balls. They knew from the video that it had taken 9 pumps for the smaller ball.

And finally … What assumptions are we making here?


They worked. I watched.

I’ve learned not to be surprised at the faulty proportional reasoning that happens every single year.

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Most students said 14 pumps would fill the medium ball.

Why doesn’t that work?

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The few who had gotten it correct actually calculated the radii from circumferences, and then calculated volumes from the radii.

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No one recognized that the cube of the ratio of the circumferences would equal the ratio of the volumes.

And so the journey continues … trying to escape the “ideal” math world, one lesson at a time.

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Posted by on December 13, 2015 in Dilations, Geometry


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Student Misconceptions about the Similarity of Rectangles

Have you used any of NCSM’s Illustrating the Standards for Mathematical Practice modules? The modules are written to use as professional development with teachers. We have been using some of the Congruence and Similarity module with our students for several years now.


In the task Hannah’s Rectangle, students are asked which rectangles are similar to rectangle A. Each student had a copy of the rectangles, a piece of wax paper, and a straightedge. They really had a protractor, but I asked them not to use it to measure length.

I sent a Quick Poll to find out which rectangles are similar to A, and this is what I saw, with responses separated:

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And responses grouped together:

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Ten students selected the correct similar rectangles.

Seven students selected every rectangle as similar to rectangle A.


What would you do next?


I showed the responses separated, without correct answers marked.

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We looked at D. Is it similar to A? Why or why not?

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Then a student offered his misconception: I selected them all because I thought all rectangles were similar.

(I’ve used this task several times, and that misconception didn’t surface until this year. LJ wasn’t alone in his thinking … 7 students had selected all rectangles – it just hadn’t occurred to me why they had done so until he gave his reason.)

Are all rectangles similar?

With what shapes can we say, “All ___ are similar.”?

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Why aren’t A and F similar?

One student had already determined that C was similar to A. She eliminated F because it had the same base length as C.

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Another student dilated E about its center to get A, showing that the diagonals were collinear.

Another student dilated B about its top left vertex to get A, showing that the red lengths were equal.

I showed my students the video of Randy sharing his thinking with his class. Several students had used a similar method, but they didn’t use the same wording as Randy in explaining their thinking.

And so the journey continues … learning more every year about student misconceptions and grateful for those who write tasks to expose those misconceptions.

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Posted by on November 28, 2015 in Dilations, 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.

SMP8 #LL2LU Gough-Wilson.png

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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Practicing Formative Assessment: Hopewell Triangles

This year’s Mathematics Assessment Project Hopewell Triangles task on similarity and right triangles played out differently than previous years.

2013 – Right Triangle

2013 – Similarity

2014 – Misconceptions

In general, students didn’t have as many misconceptions as the prior year.

As students were working by themselves, I did see one misconception that I was sure to bring out in our whole class discussion.

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(Look at ∆D above.)

On another trip around the room, I saw this on his handheld, which helped him correct his own mistake for ∆D.

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I particularly enjoy seeing different ways that students explain why the triangles are similar (#3) and why or why not the triangle is a right triangle.

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I sent a TNS document with questions to collect student responses for some of the questions. When I collected it after students worked individually, we had a 70% success rate. (I’ve changed the Student Name Format to Student ID to keep student names concealed.)

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At that point, we changed to Team mode, and students talked with each other about their work and I told them that they could change answers in their TNS document as they discussed their work. Students are making it a practice to not mark an answer unless they have an explanation to go with the answer. When I collected their work the second time, I knew that our whole class discussion needed to start with the third question. (I also knew who needed to come in during zero block for extra support.)

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Also of note is that the first collection of question 3 had these results:

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But the final collected had these results:

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Why is ∆1~∆A?

Why isn’t ∆1 similar to ∆F or ∆E?

We finished the discussion by discussing a misconception that students had last year. Anna thinks that ∆2 is a 30-60-90 triangle. Do you agree? Why or why not?

And so the journey continues, using formative assessment to make instructional adjustments to meet the needs of the students who are currently in my care …

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Posted by on February 28, 2015 in Dilations, Geometry, Right Triangles


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Angles & Arcs

Circles: CCSS-M G-C.A Understand and apply theorems about circles

2. 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.

4. (+) Construct a tangent line from a point outside a given circle to the circle.

How intuitive is the relationship between an angle with a vertex inside the circle and the intercepted arcs of the angle and its vertical angle?

We started our lesson with a Quick Poll.

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About one-third of the students intuited the relationship.

Students interacted with the Geometry Nspired activity Secants, Tangents, and Arcs.

What happens as you move point A?

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What’s the least amount of information needed to calculate the angles and arcs? If I give you the measures of two the intercepted arcs, how can you determine the measure of an angle?

I sent the poll again. Do you want to keep your answer or change it based on what you observed with the dynamic geometry software?

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So why is the angle measure half the sum of its intercepted arcs?

We practiced look for and make use of structure. What do you see that isn’t pictured? What do we know so far about circles?

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Students thought about what auxiliary lines might be helpful for proving this relationship.

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And then we looked together at some of their ideas for proving the result. Which of these would be helpful for our proof?

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What about when the vertex of the angle is outside the circle?

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This time only one person got the correct answer. I initially thought that he had intuited the relationship, but after talking with him about how he got 70, I realized that wasn’t true.

We went back to the TNS document. What happens when you move point P? How can you determine the angle measure when given the two intercepted arcs?

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A Class Capture gave me evidence that most students were making observations and testing their conjectures.

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The bell rang as one student shared his conjecture: subtract the arcs, and divide by 2.

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Jill Gough challenges us to provide Ask-Don’t-Tell learning opportunities for our students. What Ask-Don’t-Tell learning opportunities are you already providing for your students? What new Ask-Don’t-Tell learning opportunities can you provide your students this week?

And so the #AskDontTell journey continues … one lesson at a time.


Posted by on February 24, 2015 in Angles & Triangles, Circles, Dilations, Geometry


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Pythagorean Theorem Proofs

CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

We’ve been teaching our CCSS Geometry course for three years now, and this is the first year that we have been able to spend more than a little class time on proofs of the Pythagorean Theorem. (Our students are coming to us knowing more mathematics than three years ago. Our students are coming to us more willing to take risks and use the Standards for Mathematical Practice than three years ago. We are making progress just in time for our legislators to decide that collaborating with other states to write standards and assessments was a bad idea.)

We started with the Mathematics Assessment Project formative assessment lesson (FAL) on Proofs of the Pythagorean Theorem. This FAL is one that includes student work. Students focus on SMP3: construct a viable argument and critique the reasoning of others.

As students practice look for and make use of structure, I asked them to share what they noticed and wondered.

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Then we looked specifically at a diagram drawn to scale, and students noted what they knew to be true (and why).

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As we started to examine the student work proofs so that students could critique the proofs, SC asked to go back to the previous page. I wonder what will happen if we reflect the outer right triangles about their hypotenuses into the center square.

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What do you think will happen?

The triangles will make a square.

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I think I’ve said before that technology slows me down in the classroom. Students notice and wonder more than they did before, and the technology gives us the chance to see what happens so that we can make sense of why it happens mathematically. I am not the only expert in the room. The student who gets mathematics without technology is not the only expert in the room. Our use of technology increases our confidence and lifts all in the room to experts. And so the journey continues …

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Posted by on February 1, 2015 in Dilations, Geometry, Right Triangles


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