To introduce the idea of transformations to my students, I briefly review the definition of transformation and give examples of translation, reflection, and rotation in a whole-class discussion. Since students have a strong association with all of these words in their everyday life, it is important to give them opportunities to tinker around with their newfound understanding of what these terms mean mathematically. I ask students to work through several examples in their groups before we debrief answers as a whole class and discuss confusions, misunderstandings, and insights. In particular, I find that #6 provides an opportunity to have a rich discussion on the importance of flexible thinking when working with transformations.
I find it challenging to get students to really grasp the precise definitions of all of our transformation vocabulary because they use these words all the time in their everyday lives. “Reflection,” for example, can be tricky for students to wrap their heads around because they have a strong, intuitive sense of the term already. I print a description of the term “reflection” backwards to engage students in digging deeply into the definition of the term—students must use a mirror in order to decipher the description correctly and, then, as a group, define “reflection” in twelve words or less. We then debrief groups’ definitions as a whole class and transition into the next task.
In this task, students are given various sets of points that they must first reflect over the x-axis, then reflect over the y-axis. My goal is for groups of students to notice patterns for themselves, which they can describe with words and express symbolically as an ordered pair rule. A great extension for groups who may finish early is to ask them to come up with a rule for reflecting over the line y=x or the line y=-x.
As in many group tasks in math, there is a risk for one or two students to “take over” in a group—perhaps they see patterns quickly, perhaps their teammates perceive them as having high academic or social status. For this reason, it is important to have a structure that will support equitable participation and respect for others’ thinking within the group, which is why I run the task as a Participation Quiz.
As the Participation Quiz nears its end for all groups, I make sure to debrief the key points with the whole class on our Transformations Notes. While every group should have discovered patterns in the coordinates as they were reflected over the x- and y-axes, I make sure to have at least one group share out their discoveries with the rest of the class so they have a mathematical voice in the classroom and an opportunity to test their understandings and receive feedback from their peers.