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

- 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.

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.

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

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

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.

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

Changing the numbers purposefully helped them see.

I sent one more poll before we talked about **why**.

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?

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

I saw a chord drawn that made a triangle visible.

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

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.

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

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’.

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.

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’.

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.

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 …

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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 18^{th} National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

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.

Next we looked at this question.

Students worked on paper first.

Then some explored with technology.

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.

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

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

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?

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

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

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.

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?

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: http://teachpsych.org/ebooks/asle2014/index.php

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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.

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.

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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Writing sound definitions is a good practice for students, making all of us pay close attention to what something is and is not.

I’ve learned from Jessica Murk about Bongard Problems, which give students practice creating sound definitions based on what something is and is not.

What can you say about every figure on the left of the page that is not true about every figure on the right side of the page? (Bongard Problem #16)

Last year when I asked students to define circle, I found it hard to select and sequence the responses that would best contribute to a whole class discussion without taking too much class time.

I remember reading Dylan Wiliam’s suggestion in Embedding Formative Assessment (chapter 6, page 147) to have students give feedback to student responses that aren’t from their own class. I think it’s still helpful for students to spend time writing their own definition, and possibly trying to break a partner’s definition, but I wonder whether using some of last year’s responses to drive a whole class discussion this year might be helpful.

- a shape with no corners
- A circle is a shape that is equal distance from the center.
- a round shape whose angles add up to 360 degrees
- A circle is a two-dimensional shape, that has an infinite amount of lines of symmetry, and a total of 360 degrees.
- A 2-d figure where all the points from the center to the circumference are equidistant.

We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

Does how you define trapezoid depend on how you construct it?

Can you construct a dynamic quadrilateral with exactly one pair of parallel sides?

And so the #AskDontTell journey continues …

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We didn’t have time to answer that question during the first lesson of the unit, but we started with it during the second lesson.

Should the cubic start decreasing or increasing? How do you know?

[Yes, I do know that I can anonymize the names. However, by the time I thought of that, these two responses were no longer adjacent. It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time. ☺]

Then we looked at these.

They all have the same basic shape. Is one more right than the other?

Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.

I had the Second Derivative Grapher from Calculus Nspired queued during the previous class, but I decided it would be better for students to “do” instead of “discuss” for the last few minutes of class. We looked at it next to make the relationship between the second derivative and concavity of the original function more clear.

Students put all of this information together to begin to analyze a function given the graph of its derivative. The results of our formative assessments seem to indicate that students have a better understanding of the relationship between f-f’-f” than they have in the past. I’ll know for sure later today, though, after their summative assessment. And so the journey continues …

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Level 4:

I can use the graph of the derivative to sketch a graph of the original function.

**Level 3:**

**I can use the graph of the original function to deduce information about the first and second derivatives.**

**I can use the graph of the derivative to deduce information about the second derivative and the original function.**

**I can use the graph of the second derivative to deduce information about the first derivative and the original function.**

Level 2:

I can determine when a function is concave up or concave down and where it has points of inflection.

Level 1:

I can determine when a function is increasing or decreasing and where it has maxima and minima.

We were on the first day of a new unit. I included two questions on the opener to ensure students know what we mean by increasing/decreasing and concave up/concave down intervals. As expected students were familiar with increasing/decreasing and not so familiar with concave up/concave down.

Based on the results, we discussed what it means to be concave up and concave down. Someone asked how we would be able to tell for sure where the graph changes concavity, which we get to learn during the unit.

We started the lesson with a few Quick Polls for students to determine which graph was the derivative, given the graphs of a function and its derivative. The polls were based on Graphical Derivatives from Calculus Nspired. I sent the poll, asked students to answer individually, stopped the poll, asked students to explain their thinking to a partner. If needed, I sent the poll again to see whether they wanted to change their response after talking with their partner. I had 6 polls prepared. I sent 3.

I listened while students shared their thinking. I selected three conversations for the whole class.

- A student who knew which was which based on the power rule, which she learned during the last unit.
- A student who knew that the slope of the tangent line at the minimum of the parabola should be zero, which is the value of the line at z=0.
- A student who noticed that the line (derivative) was negative (below the x-axis) when the parabola was decreasing and positive (above the x-axis) when the parabola was increasing.

Again, as I listened to the pairs talking, I selected a few students to share their thinking with the whole class.

- The first student who shared used the maximum and minima to determine which had to be the derivative, since the derivative is zero at those x-values.
- The second student thought about what the slope of the tangent line would be at certain x-values and whether the y-values of the other function complied.
- A third student volunteered a fourth student to discuss her thinking: she noted that the graph of the function (b) changed concavity at the max/min of the derivative (b).

After students talked, I sent the poll again to see if anyone was convinced otherwise.

Two students briefly discussed how they used increasing/decreasing and concavity to determine the derivative.

Next we began to solidify what increasing/decreasing and concave up/concave down intervals look like using Derivative Analysis from Calculus Nspired.

I asked students to notice and note.

Where is the function increasing? Where is it decreasing?

What is the relationship between the slope of the tangent line and where the function is increasing and decreasing?

Where is the function concave up? Where is it concave down?

What does the tangent line have to do with where the function is concave up and concave down?

Can you look at a graph and estimate intervals of concavity?

I was able to see what students were noting on paper and hear what they were noting in our conversation, but I didn’t send any polls during this part of the lesson.

Next we looked at Derivative Grapher from Calculus Nspired.

We changed the graph to f(x)=cos(x). We already know the derivative is f’(x)=sin(x). What if we were only given the graph of the derivative? How could we use that graph to determine information about the original function?

I had more for us to discuss as a whole class, but I wanted to know what they had learned before the class ended. I used a Desmos Activity called Sketchy Derivatives to see what students had learned – given a function, sketch its derivative; and given a derivative, sketch an antiderivative. The original activity was from Michael Fenton. I modified it to go back and forth between sketching the derivative and antiderivative instead of doing all derivatives first and all antiderivatives second, and I added a few questions so that students could begin to clarify their thinking using words.

We spent the last minutes of class looking at an overlay of some of their sketches.

Could you figure out exactly where to sketch the horizontal line?

Most students have the vertex of the parabola near the right x-coordinate. Should the antiderivative be concave up or concave down?

Most students have the derivative crossing the x-axis near the correct location.

The bell rang. Another #lessonclose failure. But thankfully, there are do-overs as the journey continues …

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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.

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.

Then she rotated the parallelogram 180˚ about that point.

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?

How else could you carry a parallelogram onto itself?

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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].

We practiced both, but we focused on describing.

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

- rotate rectangle 180˚ about point A
- 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?]
- Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

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

What about mapping a regular pentagon onto itself?

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?

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?

What can you do other than a single rotation?

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

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?

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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