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Hopewell Geometry – Right Triangle

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

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And they are given a diagram with the layout of some Hopewell earthworks.

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The second question is for students to explain whether or not the shaded triangle is a right triangle.

With which student explanation would you start in a class discussion?

How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

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Student B

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Student C

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Student D

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Student E

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Student F

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Student G

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Student H

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Student I

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Student J

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Student K

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Student L

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And so the journey continues …

 
2 Comments

Posted by on June 18, 2013 in Geometry, Right Triangles

 

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Hopewell Geometry – Similarity

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

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And they are given a diagram with the layout of some Hopewell earthworks.

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The first question is for students to explain which triangle is similar to Triangle 1. Some student responses are below. With which student explanation would you start in a class discussion? How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

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Student B

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Student C

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Student D

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Student E

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Student F

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Student G

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Student H

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Student I

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Student J

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Student K

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Student L

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Student M

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Student N

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Student O

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Student P

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I have to say that the most thrilling responses are those that justify the similarity of the triangles through dilations and scale factor. Teaching CCSS-M Geometry this year has forced me to change how we talk about similarity and congruence. And while we still discuss similarity postulates such as AA~, SSS~, SAS~ and congruence postulates such as SSS, SAS, ASA, our focus has been on talking about congruence of figures through rigid motions – and similarity of figures through a dilation and if needed, rigid motions. I will be even more comfortable having the transformational geometry-congruence-similarity discussions next year than I was this year. And I will be even more comfortable with sequencing the student work in our classroom discussion so that students can make connections between the different ways to justify similarity of figures.

And so I look forward in great anticipation as the journey continues …

 
4 Comments

Posted by on June 18, 2013 in Dilations, Geometry, Right Triangles

 

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Sides of a Rectangle

I used the MAP Lesson Finding Equations of Parallel and Perpendicular Lines as a start to our unit on Coordinate Geometry.

One of the interesting tasks was for students to determine the equations of three other sides of a rectangle given that one side had the equation y=2x+3. I would have never thought to leave off the x- and y-axes had I created the question, but it brought about some interesting responses from my students for us to discuss.

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We ran out of time to discuss the problem during the first lesson (imagine that!), so I asked for students to consider the problem outside of class. During the second class, I collected their work, but we did not go over it. The following are some of the responses that I got.

1. Screen Shot 2013-03-25 at 5.56.52 AM     2. Screen Shot 2013-03-25 at 5.57.01 AM

3. Screen Shot 2013-03-25 at 5.57.10 AM

4. Screen Shot 2013-03-25 at 5.57.18 AM     5. Screen Shot 2013-03-25 at 5.57.24 AM

6. Screen Shot 2013-03-25 at 5.57.34 AM

7. Screen Shot 2013-03-25 at 5.57.42 AM

I was very surprised by the progression of responses. It would have never occurred to me that students would have suggested the same line for the two opposite sides of a parallelogram. Several students remembered the slope criteria for perpendicular lines from their algebra class, but, again, more than one used the same line for the other pair of opposite sides of the rectangle.

I kind of liked the generalizing that several students did for the y-intercept of the equations of the sides, but they were not totally prepared for the practice of “reason abstractly and quantitatively”. Did they intend for their values of b or r (see above) to be equal? They used the same constant to represent the values.

The day after I looked at the student work, we revisited the problem as a class. We started with a Graphs page. Only a few of the students that I have this year used TI-Nspire handhelds last year in algebra, so their experience with graphing on TI-Nspire was limited. In fact, most had not thought of graphing the equations as an option. They  graphed the given equation. I showed them the first picture above. Could the same equation work for the opposite side? Some of them had to graph the equation again to realize that it wouldn’t work; others knew immediately.

Students determined a second equation for the opposite side. I used the Class Capture feature of TI-Nspire Navigator to monitor students’ progress – I am able to set Class Capture to refresh every 30 seconds so that I can walk around and discuss the problem with individuals or groups who need help.

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As students continued to work, I was pleased to see them troubleshooting their own work. They entered the practice of “attend to precision” on their own – I didn’t have to tell them that lines with slopes of 2 and -2 are not perpendicular.

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I didn’t have to tell them that y=-½+3 is the equation of a horizontal line instead of an oblique line. The technology provided them the opportunity to figure that out for themselves.

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Even if I ask a similar question on a no calculator assessment later, I want my students learning and making sense of the mathematics with the technology.

And speaking of a future assessment, I’ve been trying to think of a good question to follow up on this task. What if I give them the equation of one side of a square and ask for the rest?

Or the equation of one side of an isosceles trapezoid and ask for the rest? If I remember correctly, what happens with the slopes of the two legs is interesting.

At least I’ll never run out of problems as the journey continues…

 
1 Comment

Posted by on March 25, 2013 in Coordinate Geometry, Geometry, Polygons

 

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Zoom In

We started out our unit on rigid motions with a thinking routine called “Zoom In”, described in Making Thinking Visible. I showed part of a piece of fabric and asked students to write down what they saw, keeping in mind that we were in a mathematics class. Then I uncovered more of the picture and gave students time to write down what they saw. Eventually I showed them the entire piece of fabric and asked them to discuss what they saw in their groups. I had no idea if students would see reflections, rotations, and translations (and in fact, some saw flips, turns, and slides), but they did. It actually worked…they saw transformations…the perfect lead-in to our unit on rigid motions. And so we began to think about how we could show that one of the images was congruent to another image on the fabric. We began to develop the idea that two images are congruent if there is a rigid motion that maps one image onto the other.

This quote by Nobel Prize winner Albert Szent-Gyorgyi hangs on one of the walls of my geometry classroom: “Discovery consists of looking at the same thing as everyone else and thinking something different”. While we were all looking at the same piece of fabric, we didn’t all see the same transformations. We opened each other’s eyes to many types of transformations as we shared with each other and listened to one another.

We moved next to a MARS (Mathematics Assessment Resource Service) MAP (Mathematics Assessment Project) Classroom Challenge, Representing and Combining Transformations. This classroom challenge had some good ideas for talking about transformations, and it turned out to be a great way to start the unit. Students were given a coordinate plane with an L-shape drawn in (pre-image) and a cut-out L-shape (image) to transform as directed.

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We began with having them translate the L-shape, then moved to some reflections (about the x-axis, the y-axis, the line x=-2, the line y=x), and then moved to some rotations (90 degrees clockwise about the origin, 180 degrees counterclockwise about the origin). Physically moving the L-shape helped students differently than drawing the transformation or recognizing the transformation – even distinguishing between a reflection about the x-axis and about the y-axis. We asked students why they had transformed their shape as they did, getting them to start using the language of transformations: “The image is the same distance from the x-axis as the pre-image.”

Then we showed students a transformation and asked them to describe it in a Quick Poll. At this point, we had them enter into the CCSS Mathematical Practice of “attending to precision”.

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Notice that the class decided not to give “a translation” credit. They wanted students to be more specific about the type of translation.

There was a matching game in the MARS lesson that we didn’t get to. We should still be able to use it at some point in the unit. During this lesson, though, we eased the hurry syndrome by giving students time to understand the big picture of transformations before we dig deeper, spending a day on each type of transformation, making conjectures about what each type of transformation “buys us” mathematically.

And so, the journey continues…

 
2 Comments

Posted by on September 3, 2012 in Geometry, Rigid Motions

 

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