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Category Archives: Circles

Blending Technology with Paper and Pencil

My geometry class is 1:1 this year; each student has her own MacBook Air. Students share responses to questions digitally in class using TI-Nspire Navigator for Networked Computers. Students explore mathematics using TI-Nspire dynamic graphs and geometry software. Students explore mathematics and share responses digitally using Demos Activity Builder. We use Canvas, an online learning management system, for assignments. We use Google Drive for sharing electronic documents with each other, and we use MathXL, online homework with built-in learning help, to practice mathematics. What place does pencil and paper have in my students’ learning and understanding of mathematics?

Even though many of the tasks that my students do for geometry take place digitally, I am convinced that pencil and paper plays an important role in how much mathematics my students not only learn but also remember. In a Wall Street Journal article, “Can Handwriting Make You Smarter?“, Robert Lee Hotz reports that students who take notes by hand usually outperform students who type notes when assessed more than one day after the class period. Students who type notes quickly type everything the professor says, but students who handwrite notes have to process the information while they are hearing it to select what is important to remember (Hotz 2016).

Hotz cites the work of Mueller and Oppenheimer published in Pyschological Science. Their research studies showed that students who took notes by hand performed better on conceptual questions than those who took notes on a laptop. Students performed about the same on factual questions. Their hypothesis for why is that students who take notes by hand choose which information is important to include in their notes, and so they are able to study “more efficiently” than those who are reviewing an entire typed lecture (Mueller and Oppenheimer 2014). Note: These studies are on college students; I have found little research on grade school students.

For several years now, my students and I have been learning how to learn mathematics using the Standards for Mathematical Practice. MP8, “look for and express regularity in repeated reasoning”, has pushed me to think about having students record what they see instead of just noticing and discussing it.

SMP8 #LL2LU Gough-Wilson

One of the ways that I’ve learned to talk about “look for and express regularity in repeated reasoning” is to ask students to notice what changes and what stays the same as we take a dynamic action on a geometric figure. Consider a recent learning episode from my classroom.

Students were told that our learning intention was “I can look for and express regularity in repeating reasoning”. The content was conceptual development of the equation of a circle in a coordinate plane using the Pythagorean Theorem. I did not share that specific content with students up front, however, because I wanted it to be revealed as the lesson progressed. I showed them a dynamic right triangle in the coordinate plane.

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What changes? What stays the same?

I could have let them simply discuss what they noticed. But instead I asked them to “Notice & Note”, using words, pictures, and numbers to write and sketch what they saw.

Then I asked them to share what they noticed with a partner and add to their own notes as desired.

 

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Our classroom discussion revealed that the equation of the circle formed by tracing point P was x2 + y2 = 52.

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Students continued to “Notice & Note” as they moved a circle around in the coordinate plane. What changes? What stays the same?

 

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As we moved P around in the coordinate plane, and then as they later moved the circle around in the coordinate plane students noted what they saw. Eventually, students generalized the center-radius form of an equation.

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Notice & Note, by Kylene Beers and Robert E. Probst, is a guide of signposts (strategies) for close reading of text. Students are taught signposts to notice while they are reading, and they are asked to stop reading and note what the signpost might imply. “Again & Again” is one signpost. Do you notice an event in the text that keeps happening again and again? Do you notice a phrase in the text that is repeated again and again? Stop reading, note it, and think about what that might mean (Beers and Probst 2013). How might we take advantage of the ways that students are learning to read text in their English Language Arts (ELA) classes to guide students in inquiry based exploration of mathematics?

In her online course, Sunni Brown, author of The Doodle Revolution, states that “Tracking content using imagery, color, word pictures and typography can change the way you understand information and also dramatically increase your level of knowledge and retention” (Brown 2016). How do we make tracking content using words, pictures, and numbers a reality in the 1:1 classroom? My experience is that it doesn’t happen without deliberate emphasis on its importance.

In Reading Nonfiction, Beers and Probst write “When students recognize that nonfiction ought to challenge us, ought to slow us down and make us think, then they’re more likely to become close readers” (Beers and Probst 2016). Our ELA counterparts are on to something. Effective classroom instruction is not just about creating learning episodes for our students to experience the mathematics using the Math Practices. Effective classroom instruction incorporates practices that will help students remember what they are learning longer than for the next test.

 

As I think about our district’s continued implementation of 1:1 technology, I am convinced that we need to pay attention to when we are asking, encouraging, and requiring students to use pencil and paper to create a record of what they are learning. I am interested in thinking more about how we might blend the use of dynamic graphs and geometry software with Notice & Note – using words, pictures, and numbers, along with color, so that students not only have a record of what they are learning but also have a better chance of remembering it later. And so, the journey continues …


References

Beers, G. Kylene, and Robert E. Probst. Notice & note: Strategies for close reading. Portsmouth: Heinemann, 2013. Print.

Beers, G. K., & Probst, R. E. (2016). Reading nonfiction: Notice & note stances, signposts, and strategies. Portsmouth: Heinemann.

Brown, S. (n.d.). Visual Note-Taking 101 / Personal Infodoodling™. Retrieved April 25, 2016, from http://sunnibrown.com/visualtraining

Hotz, Robert Lee. “Can handwriting make you smarter?” The Wall Street Journal. 04 Apr. 2016. Web. 25 Apr. 2016.

Mueller, P. A., and D. M. Oppenheimer. “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking.” Psychological Science 25.6 (2014): 1159-168. Web.

 
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Posted by on August 3, 2017 in Circles, Coordinate Geometry, Geometry

 

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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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MP6 – Defining Terms

Screenshot 2016-01-27 09.07.23.pngHow do you provide your students the opportunity to attend to precision?

1-screen-shot-2016-10-25-at-1-18-52-pmWriting 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.

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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.

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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.

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We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

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

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

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And so the #AskDontTell journey continues …

 
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Posted by on November 14, 2016 in Circles, Geometry, Polygons

 

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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”?

MP5

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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What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?

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As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis

 

Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?

 

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As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?

 

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Notice and Note: The Equation of a Circle

I wrote in detail last year about how our students practice I can look for and express regularity in repeated reasoning to make sense of the equation of a circle in the coordinate plane.

This year we took the time not only to notice what changes and what stays the same but also to note what changes and what stays the same.

Our ELA colleagues have been using Notice and Note as a strategy for close reading for a while now. How might we encourage our learners to Notice and Note across disciplines?

Students noticed and noted what stays the same and what changes as we moved point P.

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They made a conjecture about the path P follows, and then we traced point P.

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We connected their noticings about the Pythagorean Theorem to come up with the equation of the circle.

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Students moved a circle around in the coordinate plane to notice and note what happens with the location of the circle, size of the circle, and equation of the circle.

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And then most of them told me the equation of a circle with center (h,k) and radius r, along with giving us the opportunity to think about whether square of (x-h) is equivalent to the square of (h-x).

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And so the journey continues … with an emphasis on noticing and noting.

 

 

 
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Posted by on March 19, 2016 in Circles, Geometry

 

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The Circumference of a Cylinder

We talked about pi earlier this week in geometry, and we used Andrew Stadel’s water bottle question to start.

I’m not one to pull of the wager that Andrew used (unfortunately, my students will agree that I am a bit too serious for that), but we still had an interesting conversation.

Compare the circumference and height of the water bottle.

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Here’s what they estimated by themselves.

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Then they faced left if they thought height > circumference, straight if =, and right if height < circumference. (I saw Andrew lead this at CMC-South year before last … I certainly didn’t think of it myself.) They found someone who agreed with their answer, and practiced I can construct a viable argument and critique the reasoning of others.

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Next they found a second person who agreed, and practiced I can construct a viable argument and critique the reasoning of others again. (By this time, we decided it was easier to raise 1, 2, or 3 fingers based on answer choice rather than turn a certain direction as it was a challenge for some to see someone turned the same direction.) Finally, they found someone who disagreed, and practiced I can construct a viable argument and critique the reasoning of others.

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I sent the poll again.

It didn’t change much.

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So without discussion, I sent a poll with a bit more context … a cylindrical can holding 3 tennis balls. Would the can of tennis balls help them reason abstractly and quantitatively?

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Apparently not.

Here’s what they thought by themselves.

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And here’s what they thought after talking with someone else.

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The clock was ticking. I still wanted us to talk about pi. I asked someone who correctly answered to share her thinking with the rest of the class to convince them. And we used string to show that the water bottle circumference was, in fact, longer than its height.

I intended to follow up with this Quick Poll. But I was in a hurry and forgot. Maybe next year.

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You can find more number sense ideas from Andrew here.

I’ll look forward to hearing about how they play out in your classroom, as the journey continues …

 
 

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Ask, Don’t Tell

I was invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom.

Ask, Don’t Tell (Part 4): The Equation of a Circle

Ask, Don’t Tell (Part 3): Special Right Triangles

Ask, Don’t Tell (Part 2): Pythagorean Relationships

Ask, Don’t Tell (Part 1): Special Segments in Triangles

While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

“Ask Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask Don’t Tell” learning opportunities begin to activate students as owners of their learning.

I haven’t always provided “Ask Don’t Tell” learning opportunities for my students. My coworkers and I spend our common planning time thinking through questions that we can ask to bring out the mathematics. We plan learning episodes so that students can learn to ask questions as well. (Have you read Make Just One Change: Teach Students to Ask Their Own Questions?)

After the Special Right Triangles post, someone commented on NCTM’s fb page something like the following: “Really? You told students the relationships without any explanation?”

I have always used the Pythagorean Theorem to show why the relationship between the legs and hypotenuse in a 45˚-45˚-90˚ is what it is. But I think that’s different from “Ask Don’t Tell”.

I have been teaching high school for over 20 years. And yes. I really used to tell my geometry students the equation of the circle. I told them definitions for special segments in triangles along with drawing a diagram. I told them how to determine whether a triangle was right, acute, or obtuse. And I told them the relationships between the legs and hypotenuse for 45˚-45˚-90˚ and 30˚-60˚-90˚ triangles.

I’ve also been in meetings with teachers who have not thought about decomposing a square into 45˚-45˚-90˚ triangles or an equilateral triangle into 30˚-60˚-90˚ triangles to make sense of the relationships between side lengths.

You can see on the transparency from which I used to teach that I actually did go through an example where an equilateral triangle was decomposed into 30˚-60˚-90˚ triangles; even so, I failed to provide students the opportunity to look for and make use of structure.

special-right-triangles

Purposefully creating a learning opportunity so that the mathematics unfolds for students through questions, conjectures, and exploration is different from telling students the mathematics, even with an explanation for why.

As you reflect on your previous school year and plan for your upcoming school year, what #AskDontTell opportunities do and can you provide?

 

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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.

 
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Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry

 

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The Circumference of a Circle

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

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Create an argument for comparing the height and circumference of the bottle.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

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And sent another Quick Poll.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

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We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

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Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …

 

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