I created a survey in our Canvas course for students to reflect on what they had learned in this unit.
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
I Can statements:
Level 1: I can identify and define transformations and composite transformations.
Level 2: I can perform transformations and composite transformations.
Level 3: I can determine the congruence of two figures using rigid motions.
Level 4: I can apply transformations and composite transformations to figures in the coordinate plane.
Level 4: I can map a figure onto itself using transformations.
Every student answered “strongly agree” or “agree” to each of the I can statements.
We focused on learning math using the Standards of Mathematical Practice. I asked students which practice they used most often during this unit.
55% of students said that they most used make sense of problems and persevere in solving them.
The students have some experience with transformations in middle school. Could they tell that we were deepening their understanding of rigid motions? Did any of the lessons feel like repeats of material they had previously learned?
- None of them were complete repeats because I learned about them more detailed than before.
- Before now, I had a very general knowledge of transformations, but actually putting this knowledge to practice was new.
- I had already learned most of the basics of transformations. I knew reflections were flips, translations were slides, and so on and so on. However, I’d never gone in depth with the mathematical reasoning behind these transformations and their causes and effects, which brought a whole new light on the subject.
- I believe that the basic ideas of 1A-E I knew but I still had room to grow on the knowledge these subjects.
- I felt that I already knew how to do translations and reflections, but in this unit, I noticed new things about them such as that a reflection changes the orientation of the pre-image and that a translation keeps the same orientation as the pre-image. I also knew about rotations, but I wasn’t great at actually preforming a rotation.
- I learned how to really think about math differently. I learned to double check my work- A LOT. I learned how to tell counterclockwise and clockwise apart. I learned translations and how they move, reflections and how they work, and rotations and how they twist. I learned about matrices and how complex they can be, although I’m still a little confused (I think I have it, but I’m not so sure). I learned the first parts of geometry.
- I have learned how to reflect a figure across the y=x line and the y=-x line.
- I learned how to map figures onto itself and reflect on lines other the the x and y.
- I learned some formulas to help with reflections. I also learned what directions a figure should be rotated if I’m only give the degrees to rotate.
- I learned how to see if shapes on a coordinate plane are congruent, and how to map something onto itself using rigid motions.
- A lot of things in this unit were repeats of what I have learned in the past, but there was so much more that I didn’t know about all the transformations. For one, I didn’t even think about the distance from each side to the line of reflection in the past. I just knew the simple things like the points were congruent. I’ve learned a lot and couldn’t type it all. People who don’t learn anything within a whole unit are crazy because you can always learn something even if it’s just a small little thing. I would say that I’ve decided not to give up on hard problems like I would in the past. I didn’t really learn that, but I think it’s important to geometry and, really, everyday life.
I wonder (maybe too often) whether what we are doing in class is important. The student reflections give me evidence that my students are recognizing new content and also connecting it to what they knew before our high school geometry class.
And so the journey continues…