RSS

Category Archives: Geometry

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: , , , ,

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.

1_1 Screen Shot 2016-11-14 at 8.30.14 AM.png

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.

2-screen-shot-2016-10-25-at-1-16-23-pm3-screen-shot-2016-10-25-at-1-16-38-pm4-screen-shot-2016-10-25-at-1-16-48-pm

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.

Screenshot 2016-01-27 09.07.23.png

Screenshot 2016-01-27 09.07.17.png

We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

5 Trapezoid.png

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 …

 
Leave a comment

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

 

Tags: , , , , , ,

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.

21 09-21-2016 Image009.jpg

22-screenshot-2016-08-31-10-01-1823-screenshot-2016-08-31-10-01-25

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.

24 Screenshot 2016-08-31 09.13.49.png

Then she rotated the parallelogram 180˚ about that point.

25 Screenshot 2016-08-31 09.13.54.png

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?

26 09-21-2016 Image001.jpg

How else could you carry a parallelogram onto itself?

 
Leave a comment

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

 

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: , , , , ,

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.

1 2015-03-16 09.28.06.jpg

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

2 2015-03-16 09.48.23.jpg

How could we arrange 2 square units into a square?

3 2015-03-16 09.41.51.jpg

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

4 Screen Shot 2015-03-20 at 2.14.53 PM.png

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

5 Screen Shot 2016-09-20 at 1.18.28 PM.png

6 Screen Shot 2016-09-20 at 1.39.10 PM.png

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

 
Leave a comment

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

 

Tags: ,

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?

7 Screen Shot 2015-02-06 at 9.37.21 AM.png

Could we use the intersection of the angle bisectors of an equilateral triangle inscribed in a circle to find the center of the circle?

8 Screen Shot 2015-02-06 at 9.39.58 AM.png

Could we use the perpendicular bisector of a chord of a circle to find the center of the circle?

9 Screen Shot 2015-02-06 at 9.41.51 AM.png

Could we use the intersection of the perpendicular bisectors of a pentagon circumscribed about a circle to find the center of the circle?

10 Screen Shot 2016-02-08 at 9.31.59 AM.png

Could we use the intersection of the perpendicular bisectors of several chords of a circle to find the center of the circle?

11 Screen Shot 2016-02-08 at 9.32.20 AM.png

Could we use a right triangle inscribed in a circle to find the center of the circle?

12 Screen Shot 2016-02-08 at 9.33.21 AM.png

And so the journey continues …

 
Leave a comment

Posted by on September 15, 2016 in Circles, Geometry

 

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: , , , , ,