RSS

Tag Archives: Navigator

Performance Assessment – Rigid Motions

One of the tasks that we gave our students at the end of our unit on Rigid Motions came from Illustrative Mathematics: The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected. I took the task and made it dynamic in a TI-Nspire document. Instead of creating the line of reflection for two static triangles, they had to construct it so that it still worked when we moved one of the vertices of the original triangle. They were also required to show some measurements on their screen that justified their work.

On the next page, △ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

Many students were successful on this task. Most of them constructed the perpendicular bisector of one of the segments with endpoints that were pre-image point and its image.

Then next task that we asked them to do was the following, which I first heard about from an instructor at the University Lab School in Honolulu.

On the next page, you are given segment AB. Construct a regular hexagon ABCDEF with segment AB as one of its sides.
-You may not use any Shapes tools.
-You many not use any Measurement tools.
When you are finished, we will use Measurement tools to justify your construction.

After proposing the task to the students, I made sure they knew what we meant by regular hexagon. Then I let them think and work for a little while before the class discussed a few questions. Someone wanted to know what one of the measures of the angles of the regular hexagon was. So I drew a triangle and asked for the sum of the measures. We ended up having a mini-lesson – and didn’t even get to the point of generalizing the sum of the measures of the triangle (that will come later) – just enough for them to have what they needed to make their hexagon using transformations.

 

Most students rotated the sides all of the way around to create their hexagon.

A few students rotated segment AB twice and then reflected the segments to get the rest of the hexagon.

 

All of the students learned something about hexagons that they had not previously considered.

And then one more…

I love the angles of rotation that the students used to rotate the pentagon onto itself…angles that are easy to use because of technology. My students are entering into the practice of use appropriate tools strategically.

And so the journey continues….

 
1 Comment

Posted by on September 26, 2012 in Geometry, Rigid Motions

 

Tags: , , ,

What’s My Rule?

In What’s My Rule, students move point Z and observe how W follows. Z is mapped to W according to some rule that the students are trying to determine.

Image

There are all sorts of great questions to ask with an activity like this. After moving Z for a few seconds, I ask “Can Z and W ever coincide?” And then, of course, someone has to tell the class what coincide means in terms of points.

If I move Z to Quadrant I, then where will W go?

If I want W to be in Quadrant II, then where must I put Z?

I sent a Quick Poll to get student responses for the rule.

Image

ImageImage

ImageImage

Students entered the mathematical practice of attending to precision. Students entered into the mathematical practice of constructing viable arguments and critiquing the reasoning of others. Students entered into the mathematical practice of reasoning abstractly and quantitatively.

We went back to the TNS document and showed why Z was not always the image of W reflected about the line y=-x.

Image

We also showed what happens when we require Z to be on the function y=x^3 (I’ll eventually learn how to do LaTex…) and origin symmetry.

Image

There are 7 transformations for students to explore in What’s My Rule. I could write an entry about each one. But instead, I’ll just share one more snippet.

Image

In this problem, Z gets mapped to W through a translation with the rule (x,y)→(x,y-4). One question I asked was where Z would be if W was in Quadrant IV. Obviously, Z can be in Quadrant I. Can it be anywhere else? Someone suggested that Z could be in Quadrant IV. So Z can be in Quadrant I or Quadrant IV. Can it be anywhere else? After a while (a long while…I had to wait, easing my hurry syndrome), someone suggested that it could be on the positive side of the x-axis.

And so, the journey continues….

 
4 Comments

Posted by on September 22, 2012 in Coordinate Geometry, Geometry, Rigid Motions

 

Tags: , , , ,

Mapping an Image onto Itself

CCSS G-CO 3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We are incorporating this learning objective into our unit on Rigid Motions. And we are still learning how best to ask questions to get students thinking about how to carry an image onto itself.

We started with a rectangle and asked students to perform a reflection that would map the image onto itself.

A few students tried to “eyeball” where they could draw a line of reflection. But I was excited at how many used the Construction tools to construct a line of reflection.

Some chose to construct the perpendicular bisector of a side of the rectangle. Some constructed the midpoints of more than one side of the rectangle and then used the segment tool to connect. Some constructed the midpoint of one side of the rectangle and then constructed a perpendicular to the side through its midpoint.

One nice feature of the TI-Nspire CX is color. I made sure that the rectangle had a colored border so that when a student reflects the image onto itself, the result will default back to a black border, instead of the blue border of pre-image. For those without CX models, students can hover over the border of the rectangle and see that the reflection worked when they see the object selection “polygon” on top of the rectangle.

After reflecting the rectangle onto itself, we grabbed movable vertices of the rectangle to make sure that the reflection “stuck”. Those students who drew instead of constructed a line of symmetry for the rectangle soon saw two rectangles instead of only the one. So they undid their work and tried again constructing a line of symmetry.

I gave students a hexagon next…This time most didn’t have to actually construct any lines of symmetry to come up with a solution.

So how could we take this to the next level? Students know that reflecting an image about a line of symmetry will carry the image onto itself.

In the next lesson, we asked students to create an image that would map onto itself when reflected about a given line. We started with the line y=x.

I was surprised at how many students thought to create a circle with its center on the line. I would not have thought of a circle myself since we have mostly been reflecting, translating, and rotating polygons.

Others created triangles, regular pentagons, regular hexagons, squares. We got to see everyone’s work using the Class Capture feature of TI-Nspire Navigator. And we got to talk about the properties of the shapes (will any triangle work?) and the coordinates of the vertices.

Rotations are next…We will see what happens when we begin rotating an image onto itself. Or creating an image that can be rotated onto itself.

And so the journey continues….

 
1 Comment

Posted by on September 12, 2012 in Geometry, Rigid Motions

 

Tags: , , , ,

Defining Congruence

Defining congruence using CCSS is not the easiest concept that I have ever taught to a group of students…and I am not convinced that I have it all right. My students have always come in to their high school geometry course thinking of two congruent figures as those with the “same size and shape”. So instead, we are asked to define two objects as congruent when there is a rigid motion that maps one object onto the other. I’ve mulled over how to introduce this to students (in the back of my mind, most of the summer), and I decided to use a document from Geometry Nspired to begin.

Transformations: Translations actually begins with the pre-image and image on top of each other, directing students to move point C to begin the translation. I decided instead to give the students the document with the triangle already translated and ask students how we know the two images are congruent.

Image

Even though we can see that perimeters, areas, and some corresponding side lengths are equal, we decided that the two triangles were congruent because we could grab and move C’ to translate triangle A’B’C’ onto ABC.

Image

Triangles ABC and A’B’C’ are congruent because there is a rigid motion that maps one onto the other.

After exploring what happens to coordinates during a translation and introducing more sophisticated notation than “left 2, up 3” to talk about translations, I asked students to justify that one triangle was a translation of another:

Many students noted that they missed the x-y coordinate plane. How could they show that this was a translation without being able to talk about “x units right and y units down”? Several students began to measure sides, angles, perimeter, and area of the triangle. But just because our previous experience with those measurements tells us the triangles are congruent doesn’t mean one has to be a translation of the other.

Some began to think about the mapping of A to A’, B to B’, and C to C’.

Is it enough to show that segments AA’, BB’, and CC’ are congruent? Is there some other relationship between the lines that contain those segments?

One student noticed that the lines that contain the segments are parallel to each other. How would we show that they are parallel? Students remembered that we could use slopes to show parallel lines…but how else could we show that the lines are parallel? Another student said something about the distance between the lines. So we talked about what “distance between two lines” means.

And so, the journey continues…

 
1 Comment

Posted by on September 7, 2012 in Geometry, Rigid Motions

 

Tags: , , , ,

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.

Image

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”.

ImageImage

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

 

Tags: , , , , ,

A Locus of Points

(Part 1 – A Locus of Points) As we were finishing up our unit on constructions and special segments in triangles last week, I wanted to introduce the idea of a locus of points. But I wanted to do it through exploration. So we started with the following question:

Image

Students were familiar with “equidistant” but they were not familiar with “locus of points”. I was proud that none of them got caught up with the terminology so much that they couldn’t try something. This is a fairly new question type in TI-Nspire Navigator. Students drop a point on the graph. If the point they initially drop isn’t what they want to answer, then they can grab it and move it where they want before submitting their response. I was expecting the majority of the students to submit the midpoint. And they did. But even then, the picture that we got was still beautiful:

Image

Some students remembered that any point that lies of the perpendicular bisector of the segment with the given points as endpoints would be equidistant from the two endpoints. And then we were able to go in as a class and connect their submitted points with the equation of the line, y=1.

The next question I asked was for an equation:

Image

So they moved away from the visual of dropping a point to writing an equation, and most students were successful.

Image

Looking back, I realize that I need to add “in a plane” to both of my questions. But that is one reason I am blogging this year. Reflecting on the lessons that I am presenting to my students will make me a better teacher.

(Part 2 – The Summative Assessment) Students took a summative assessment on our first unit in class at the end of the week. We had studied triangles during the unit, but we had a question on the assessment that extended the content of the unit to quadrilaterals.

Image

I am proud of my students for their work. It may seem simple, but ultimately I want them to learn how to problem solve while they are taking their tests…to not be intimidated by questions they haven’t exactly seen before in class. They are off to a good start.

And so the journey continues…

 
2 Comments

Posted by on August 26, 2012 in Geometry, Tools of Geometry

 

Tags: , , ,

Aside

Why “Easing the Hurry Syndrome”?

A Google search of “technology speeds up life” results in about 173 million results in less than half of a second. What I find in my classroom, however, is that using technology actually slows down the pace.

In the midst of my usual rush to cover all of the required standards, when we use TI-Nspire (a dynamic, interactive platform) to explore a difficult concept, the questions that students ask slow us down. The platform that we use encourages students to ask “what happens if…” questions. The platform that we use allows the students to find answers to those questions interactively so that the teacher is not the only expert in the classroom.

In the midst of my usual rush to cover all of the required standards, when I use TI-Nspire Navigator to send students a Quick Poll to check for understanding, their responses sometimes let me know that we need to spend a little longer “attending to precision”. Seeing that their response doesn’t get marked correct up on the board in front of the class is not the same thing as the teacher writing the correct answer on a whiteboard and having students compare to what they have in their head or paper as their answer to the given problem. And so we look at their responses, letting the students determine whether or not the answers are correct or incorrect, letting students determine what error was made to produce an incorrect answer, “critiquing the reasoning of others”.

Image

Most importantly, using TI-Nspire Navigator gives every student a voice – even the most quiet. I no longer have to rely on their polite nods to determine whether they are “getting it”. Using technology eases the hurry syndrome, forcing me to pay attention to the questions students have and allowing me to assess their progress in a timely manner.

This year, I am trying to document our journey to create a CCSS Geometry course – and we are all trying to ease the hurry syndrome, which we are already finding difficult. For example, we had planned to do the compass and straightedge constructions for perpendicular bisector and angle bisector, along with letting students explore the relationship of the centroid of a triangle to each median on Wednesday of last week. What we found, however, is that whether they were supposed to or not, our students didn’t really know how to use a compass, much less know what things were equal in the construction when they completed it. If we wanted the students to do the constructions well, we were going to have to move medians to the next class period.

And so the journey to ease the hurry syndrome continues…

The Title

 

Tags: , , ,