# Tag Archives: transformations

## Transformations with Matrices

We finished up our unit on Rigid Motions by taking a brief look at transformations using matrices. At the beginning of class, we made sure each student could transform a given point in the coordinate plane and generalize the transformation.

Our students do not have any experience multiplying matrices. I didn’t want to get into a huge lesson on multiplying matrices, but I also wanted my students to realize that they had the background they needed to make sense of transforming triangles using matrices.

We used TI-Nspire technology to look for regularity in repeated reasoning. We started by multiplying the matrix that represented the vertices of our triangle (1,4), (4,–3), and (–2,5) by the matrix shown to observe how matrix multiplication works.

By what matrix should we multiply if we want the output to be the vertices of the triangle after it has been reflected about the x-axis?

It took NR only two tries before she got the correct transformation matrix. It took others more than two tries. But once students had the matrix for a reflection about the x-axis, it didn’t take long to get the matrix for a reflection about the y-axis.

And then reflections about the lines y=x and y=-x.

And then rotations 90 degrees and -90 degrees about the origin.

I honestly do not care that students remember the matrix that produces a certain transformation. I am not even sure that I should have spent any class time on this topic (although I have seen questions on some state assessments about this topic). What I care about is that students know that they can make sense of problems and persevere in solving them. I care that they know that they can figure out the matrix that produces a certain transformation instead of me giving them a list to memorize.

And so the journey to create students who can figure out mathematics instead of being told mathematics continues …

Posted by on September 28, 2013 in Coordinate Geometry, Geometry, Rigid Motions

## 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.

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.

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.

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.

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.

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….

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

## 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

## 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.

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.

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…

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

## 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.

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”.

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…