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Category Archives: Geometry

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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Squares on the Coordinate Grid

I’ve written before about Squares on the Coordinate Grid, an Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

How do you provide opportunities for your students to practice I can look for and make use of structure?

SMP7 #LL2LU Gough-Wilson

How do you draw a square with an area of 2 on the coordinate grid?

It helped some students to start by thinking about what 2 square units looks like, which was easier to see in a non-special rectangle.

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What’s true about the side length of a square with an area of 2?

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How could we arrange 2 square units into a square?

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How do you know the figure is a square? Is it enough for all four sides to be square root of 2?

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CC made his thinking visible by reflecting on his learning after class:

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“Now drawing the square root of two exactly on paper is nearly impossible unless you know how to use right triangles.”

 
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Posted by on September 21, 2016 in Coordinate Geometry, Geometry

 

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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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Assessing the Centroid of a Triangle

The centroid of a triangle is often called the balancing point of the triangle. It is the point at which the medians of the triangle intersect.

Students used technology to explore the relationship between the vertices of a triangle in the coordinate plane and the vertices of the centroid.

If your students knew the relationship between the vertices of a triangle and the vertices of the centroid, how would you expect them to answer the following question? (I included this question on an end of unit assessment.)

The vertices of a triangle are (a,b–c), (b,c–a), and (c,a–b). Prove that its centroid lies on the x-axis.

A few of my student responses are below.

What learning opportunities could I have provided in class to better prepare my students for this question without just giving them a similar problem?

And so the journey to provide meaningful learning episodes that prepare students to answer questions they haven’t seen before continues …

 
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Posted by on August 22, 2016 in Angles & Triangles, Geometry

 

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When Your Team Is Working Well Together

Have you seen Jill Gough’s blog post Strategic Teaming: leadership, voice, our hopes and dreams? Jill reminds us that strong teams both set norms for their work together and then self assess to ensure that they are functioning within their norms.

How do you provide your students the opportunity to set norms for the work that we have to do together?

I asked my students what it looks like when your team is working well together.

Here’s a wordle of their responses.

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I see communicating, cooperating, talking, participating, strategies, but what strikes me most from their suggestions is everyone.

Some lengthier responses from the students:

We are all talking about our strategies. Everyone considers all possibilities presented by the team. Everyone is contributing and listening to what each other has to say, respecting each other. We communicate reasons the answers may be correct or wrong. We will work together to figure out multiple solutions, or the one correct solution, or if there is no solution.

We’ve agreed to these norms.

Everyone …

Respects

Contributes

Listens

Questions

Collaborates

Communicates

Since I want to be transparent about formative assessment being for students as well as teachers, I showed them Popham’s levels of formative assessment.

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We are working well together when the whole class is using formative assessment (and not just the teacher). We want all students in our class to meet the learning goals. Not just the “smartest”; not just the fastest. This isn’t survival of the fittest where some can adapt and others will grow extinct. Everyone can learn. Everyone will learn.

The start of another school year has come and gone as the journey continues …


Popham, W. James. Transformative Assessment. Alexandria, VA: Association for Supervision and Curriculum Development, 2008. Print.

 
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Posted by on August 18, 2016 in Geometry, Student Reflection

 

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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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MP7: The Diagonal of an Isosceles Trapezoid

 

SMP7 #LL2LU Gough-Wilson

I’ve written about the diagonals of an isosceles trapezoid before.

When we practice “I can look for and make use of structure”, we practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We ask: “what you can you make visible that isn’t yet pictured?”

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We make mistakes; the first auxiliary line we draw isn’t always helpful.

Or sometimes we see more than is helpful to see all at one time.

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We persevere.

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Even with the same auxiliary lines, we don’t always see the same picture.

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We learn from each other.

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

 
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Posted by on August 16, 2016 in Angles & Triangles, Geometry, Polygons

 

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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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SMP7 – The Triangle Sum Theorem

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How do you provide opportunities for your students to look for and make use of structure? I’m finding that deliberate practice in looking for and making use of structure is making the practice a habit for my students.

SMP7 #LL2LU Gough-Wilson

We ask: “what you can you make visible that isn’t yet pictured?”

We practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We make mistakes; the first auxiliary line we draw isn’t always helpful.

We persevere.

We learn from each other.

 

Months ago, our goal was to prove the Triangle Sum Theorem.

We thought first about what we already knew … what we had already added to our deductive system.

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Then we practiced “I can look for and make use of structure”.

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

 
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Posted by on July 1, 2016 in Angles & Triangles, Geometry

 

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