RSS

Tag Archives: Quick Poll

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.)

0 Screen Shot 2017-02-09 at 5.02.00 PM.png

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.

1 02-09-2017 Image001.jpg

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.

2 Screenshot 2017-02-06 09.12.47.png

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

3 Angles in Circles 1.gif

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

4 Angles in Circles 2.gif

5 Screenshot 2017-02-06 09.13.47.png

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.

5_1 02-09-2017 Image002.jpg

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

Changing the numbers purposefully helped them see.

6 Screenshot 2017-02-06 09.18.51.png

I sent one more poll before we talked about why.

7 Screenshot 2017-02-06 09.25.26.png

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?

8 Screen Shot 2017-02-09 at 5.30.24 PM.png

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

9 IMG_1847.JPG

I saw a chord drawn that made a triangle visible.

10 IMG_1846.JPG

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

11 Screenshot 2017-02-06 09.34.15.png

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.

12 Screenshot 2017-02-06 09.42.22.png

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

 
Leave a comment

Posted by on February 9, 2017 in Circles, Geometry

 

Tags: , , , , , ,

Hinge Questions: Dilations

Students noticed and noted.

1 Dilations 4.gif

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”.

3.5 Screen Shot 2016-12-19 at 2.49.57 PM.png

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.

4 Screenshot 2016-11-11 09.24.27.png

Next we looked at this question.

5 Screenshot 2016-11-15 09.14.34.png

Students worked on paper first.

Then some explored with technology.

6 Dilations 5.gif

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.

7 Screenshot 2016-11-15 09.19.05.png

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

 
3 Comments

Posted by on December 20, 2016 in Coordinate Geometry, Dilations, Geometry

 

Tags: , , , , , ,

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.

 

Dilations 1.gif

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?

Dilations 2.gif

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

Dilations 3.gif

 

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

Screenshot 2016-11-11 08.58.34.png

Screenshot 2016-11-11 09.04.53.png

Here’s what NA noticed and noted.

image_123923953

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

Screenshot 2016-11-11 09.09.47.png

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?

Screenshot 2016-11-11 09.12.11.pngScreen Shot 2016-12-19 at 2.21.33 PM.pngScreen Shot 2016-12-19 at 2.28.17 PM.png

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

 

 
Leave a comment

Posted by on December 19, 2016 in Dilations, Geometry

 

Tags: , , , , , , , ,

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.

1-screenshot-2016-09-21-10-44-25

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.

2-screenshot-2016-09-21-10-44-43

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 …

 
7 Comments

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

 

Tags: , , , ,

Introduction to Curve Sketching, Part 1

Learning Intentions:

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.

  1. A student who knew which was which based on the power rule, which she learned during the last unit.
  2. 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.
  3. 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.

  1. 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.
  2. 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.
  3. 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).

14 Screen Shot 2016-09-24 at 6.53.43 PM.png

 

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

17 Screenshot 2016-09-23 13.09.24.png

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

18 Screen Shot 2016-09-24 at 6.59.06 PM.png

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?

19 Derivative Analysis 1.gif

19 Derivative Analysis 2.gif

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

19 Derivative Analysis 3.gif

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

19 Derivative Analysis 4.gif

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

19 Derivative Analysis 5.gif

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.

21 Derivative Grapher.gif

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?

22 09-24-2016 Image006.jpg

23 Screenshot 2016-09-23 14.00.39.png

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?

24 Screenshot 2016-09-23 14.02.36.png

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

25 Screenshot 2016-09-23 13.56.00.png

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

26 Screenshot 2016-09-23 14.03.01.png

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

 

 
1 Comment

Posted by on October 17, 2016 in Applications of Differentiation, Calculus

 

Tags: , , , , ,

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].

1 Rectangle 1.gif

2 Rectangle 2.gif

We practiced both, but we focused on describing.

3-screenshot-2016-08-29-09-03-264-screenshot-2016-08-29-09-03-34

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?]

5 Screenshot 2016-08-29 09.14.58.png

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

6-screenshot-2016-08-29-09-03-507-screenshot-2016-08-29-09-03-58

What about mapping a regular pentagon onto itself?

8 09-21-2016 Image008.jpg

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?

9 Screenshot 2016-08-30 17.08.44.png

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?

10 Screenshot 2016-08-31 09.10.36.png

What can you do other than a single rotation?

11 Screenshot 2016-08-30 17.09.12.png

12 Screenshot 2016-08-30 17.09.04.png

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

13 Screenshot 2016-08-31 09.10.59.png

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?

14-screenshot-2016-08-31-10-01-39

15-screenshot-2016-08-31-10-01-46

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

 
Leave a comment

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

 

Tags: , , , , ,

MP5: The Traveling Point

How do you give students the opportunity to practice “I can use appropriate tools strategically”?
MP5

When we have a new type of problem to think about, I am learning to have students give their best guess of the solution first. I’ve written about The Traveling Point before.

1 Screen Shot 2016-08-18 at 6.13.30 PM.png

Students sketched the path of point A. How far does A travel?

Students used paper and polydrons, their hands and string.

2 Screen Shot 2016-08-18 at 6.24.31 PM.png

3 IMG_0946.JPG

I sent a poll to find out what they were thinking about the distance traveled.

4 Screen Shot 2016-08-18 at 6.23.31 PM.png

Students then interacted with dynamic geometry software. Does seeing the figure dynamically move help you better see the path?

Traveling Point 1.gif

5 Screen Shot 2016-08-18 at 6.23.44 PM.png

Does seeing the path help you calculate how far A travels?

Traveling Point 2.gif

6 Screen Shot 2016-08-18 at 6.23.53 PM.png

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …


And the journey for my own learning continues. Thanks to Howard for correcting me. The second two moves do not travel a distance of 6, but the length of the circumference of the quarter circle.

Screen Shot 2016-08-24 at 10.38.58 AM.png

One student figured that out by the time the bell rang.

I look forward to redeeming this lesson this year, as the journey continues …

 
8 Comments

Posted by on August 23, 2016 in Geometric Measure & Dimension, Geometry

 

Tags: , , , , ,