Transformations, as a topic, lends itself well to hands-on activities and intuitive understanding. However, there are some basic definitions, terminology and conventions that students need to know. I prefer to teach these in context by repeated exposure, but it is also important to give students reference materials they can refer to when they get stuck. For that reason, I provide Notes on Transformations
Even though I'm giving students notes, I want it to feel like a learning experience, not a note-copying exercise. My style in delivering the notes is to sandwich the times when students are actually writing something down with verbal lead-ins on the front end and visual reinforcement on the back end. For example when I am about to introduce prime notation my lead in would go something like, "When we have an image and a preimage on the same coordinate plane, it helps to have a way of knowing which is which. We also need to keep track of which vertex on the image corresponds to which vertex on the preimage. That's why mathematicians have agreed on prime notation as a convention." Then I'd give the notes on prime notation. After the students have copied that section of the notes, I show an example of a transformation on Sketchpad, which automatically uses the prime notation for transformed figures. The notes are just a reference point, an anchor, a "link" if you will, but the memory of the things I've said, done, and shown is hopefully what will "open up" when students "click on" the "link".
The objective in this section is to get students actively thinking about what they have just finished taking notes on. I use the Check for Understanding: Transformations resource as the medium for this section, but the most important thing for me during this section is student engagement. I think of the PowerPoint resource as a conversation piece, not as a presentation.
This section is very open-ended. The very first slide, for example, shows a picture of a transformation and merely asks, "What does this say?" To make sure that all students are engaged, I call on random non-volunteers after a think-pair. I accept all factual answers. Some students want to be funny and say things like, "The rectangle is yellow". I would respond, "Great observation that it's a rectangle, and it is, in fact, yellow...can anyone else make a significant factual statement based on the picture?" If the conversation stalls, I refer students to their notes, "Look at your notes and see what vocabulary and concepts you can relate to this picture." Before we move on from the slide, I make sure that we've talked about prime notation, preimage vs. image, type of transformation, arrow notation, angle of rotation (follow-up to determining that it is a rotation).
There are some strategies I use throughout this section, which I'll talk about next.
1. Mandating that students communicate complete thoughts using complete sentences with academic vocabulary. Here's an example. In response to the first slide, a student says "It's Rotation." I would redirect them saying, "I know ABCD underwent a rotation because..." Typically, the student will try to pick the sentence up from where I left off, but I insist that they start back at the beginning of the sentence, "You say 'I know ABCD underwent a rotation because..."
2. Stirring up student conversation. Here's an example. On the first slide, a students says "I know That rectangle ABCD underwent a translation because it slid upward and to the left." "Thank you," I would say, "[To the class] Raise your hand if you agree with the claim that ABCD underwent a translation." Then "Raise your hand if you want to dispute the claim." Suppose a student raises her hand and says, "I disagree with the claim. I think the figure was rotated because if you look at where A', B', C' and D' ended up, you can see that the figure had to turn around" Then I would come back to original student and ask, does Jane Doe's argument make you reconsider or would you like to maintain your original position?" Then I might re-poll the class.
3. Poll the class. Here's an example. On the second round of slides (with the non-rectangular figures). I might start the conversation by taking a class poll to get first opinions on the type of transformation that has taken place. To do this, I would work with the class to develop some physical gestures to represent translations, reflections and rotations. For example, translation might be a palm facing downward and sliding along an imaginary line. A reflection might start with palm facing upward and then flipping it as if putting a top bun on a hamburger. A rotation might be represented by holding an imaginary steering wheel and turning it. This wakes students up, reinforces the concepts, and injects some fun into the situation. So everybody ready, on 3, 1-2-3...show me what type of transformation HIJKL underwent. And then I'd follow up by stirring up student conversation.
4. Ask Probing Questions. For example, a student identifies a transformation as a rotation, reflection, or translation. I would ask questions like: "Being as precise as you can be, what is the angle of rotation?" or "Would you say the line of reflection has a positive or negative slope? Positive or negative y-intercept?" or "Can you determine the translation rule? translation vector?"
This section of the lesson takes place in the computer lab. I use an activity called Introducing Transformations which comes from Exploring Geometry with the Geometer's Sketchpad Version 5 from Key Curriculum Press.
In the activity students learn to:
1. Mark vectors and translate figures.
2. Determine that translations are isometries
3. Mark a center of rotation
4. Rotate a figure using dynamic center and angle of rotation
5. Determine that rotations are isometries
6. Mark a line of reflection (mirror line) and reflect figures
7. Determine that reflections are isometries
8. Make generalizations about translations, rotations, and reflections.
See Introducing Transformations Demo to get an idea of what students will be doing in the activity.
The purpose of this cooperative activity is for students to take inventory of everything they have learned during the lesson, make sense of it, and represent it on a poster. I create groups of 3 to 4 students each. I give each group four pieces of square origami paper (different colors), scissors, markers, and a large sheet of gridded easel pad paper.
1. Cut out an asymmetrical figure from one of the square sheets.
2. Create a poster that shows this figure after translation, rotation, and reflection.
Be sure to include all relevant vocabulary and concepts. Be as precise as possible.
As students are working, the first issue that comes up is the issue of the images being congruent to the preimage and how to make that happen. Other issues are forgetting to label the poster with words like image, preimage, translation, reflection, rotation. Some students forget to use prime notation.
I try not to tell students what they need to ask. Instead I try to ask open-ended questions that push them in the right direction. For example, if a student has forgotten to include the prime notation, I would ask "How could your poster be explicit about which vertices on the image correspond to which vertices on the preimage?"