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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?

Trapezoid.gif

And so the #AskDontTell journey continues …

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

 

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MP6 – Mapping a Parallelogram 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.

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.

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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.

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Then she rotated the parallelogram 180˚ about that point.

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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?

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How else could you carry a parallelogram onto itself?

 
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Posted by on September 22, 2016 in Geometry, Rigid Motions

 

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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].

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2 Rectangle 2.gif

We practiced both, but we focused on describing.

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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?]

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

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What about mapping a regular pentagon onto itself?

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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?

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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?

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What can you do other than a single rotation?

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

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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?

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

 

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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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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.

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

Students used paper and polydrons, their hands and string.

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

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

Traveling Point 1.gif

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

Traveling Point 2.gif

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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.

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

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

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

 

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MP8: The Centroid of a Triangle

We had been working on a unit on Coordinate Geometry.

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

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked them to “drop a point” at the centroid of the triangle. We looked at the responses on the graph first and then as a list of ordered pairs.

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What is significant about the coordinates of the centroid?

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between the coordinates of the vertices of a triangle and the coordinates of its centroid?

Some students needed to interact on a different grid setup to see a relationship.

Centroid_2.gif

After a few minutes, I sent another poll to find out what they figured out.

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And then we confirmed student conjectures as a whole class.

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

 

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MP8: The Medians of a Triangle

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

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

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I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.

Centroid_2.gif

I sent another poll.

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And then we confirmed student conjectures on the spreadsheet.

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

 
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Posted by on August 15, 2016 in Angles & Triangles, 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?

What_s_My_Rule.gif

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?

 

What_s_My_Rule_2.gif

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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Marine Ramp, Part 2

After talking about how Marine Ramp played out in the classroom with another geometry teacher, she decided to try it the next time our classes met, and I revisited it.

Our essential learning for the day content-wise was still:

Level 4: I can use trigonometric ratios to solve non-right triangles in applied problems.

Level 3: I can use trigonometric ratios to solve right triangles in applied problems.

Level 2: I can use trigonometric ratios to solve right triangles.

Level 1: I can define trigonometric ratios.

[Since I had recently read Suzanne’s post about #phonespockets and tried it during Part 1, I turned on the voice recording again. When I listened later, I noticed how l o n g some of the Quick Polls took. And I could also hear that students were talking about the math.)

We went back to the Boat Dock Generator to generate a new situation.

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I sent the poll, and the responses were perfect for conversation.

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About half used the sine ratio, using the requirement that the ramp angle can’t exceed 18˚. The other half used the Pythagorean Theorem, which neither meets the ramp angle requirement:

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Nor the floating dock:

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We generated one more.

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And had great success with the calculation.

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So I asked if they were ready to generalize their results.

And whether we were making any assumptions about the situation.

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As expected, they generalized with sin(18˚). And they didn’t really think we were making any assumptions. Except a few about the tides. And that the height was the shorter side in the right triangle.

Which was the perfect setup for the next randomly generated situation.

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(I would have kept regenerating until we got a similar situation had the random timing not worked out as perfectly as it did.)

While students worked on calculating the ramp length, I heard lots of evidence I can make sense of problems and preservere in solving them … lots of checking the reasonableness of answers. “No – you can’t do that.” “That won’t work.” “Those sides won’t make a triangle.” And I think it’s telling that no response came in that calculated with the sine ratio.

Here’s what I saw after 2 minutes:

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After 4 minutes:

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And after 5 minutes:

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So were you making any assumptions?

Yes!

And that assumption was … ?

We assumed you could always use sine. But this time, using the sine ratio didn’t work, so we used cosine.

We talked about the smallest answer.

How did you get 26.8?

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We did the Pythagorean Theorem and then rounded up to be sure the ramp would meet the floating dock. After looking at the Boat Dock Generator, most of the class decided they might not want to walk on that ramp.

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27.9 came from using the cosine ratio. Would you feel more confident on that ramp?

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Will the cosine ratio always work?

We ended wondering whether we could generalize what will always work, given the maximum ramp angle, the distance between low and high tides, and the distance from the dock to the floating ramp.

Next year, we will go farther into generalizing, which might look something like this.

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We started our Right Triangles Unit with Boat on the River, and we ended with Marine Ramp.  More than any other year, students had the opportunity to actually engage in many of the steps of the modeling cycle – a big change from how I used to “teach” right triangle trigonometry through computation only.

Modeling Cycle

So tag, you’re it now, and I’ll look forward to hearing about your experience with Marine Ramp as the journey continues …

 
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Posted by on February 19, 2016 in Geometry, Right Triangles

 

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