I ask students to work in pairs Translations Practice. In this task, students make connections between the coordinates, graphs, and verbal and symbolic representations of shapes under translation. Working together encourages risk-taking. I also think it promotes individual understanding and accountability for learning. Pairs of students perform translations and see that while translations change the position of a figure, the pre-image and image are congruent because corresponding angles and sides are congruent. Giving students tracing paper in this activity (and throughout the unit) helps reinforce observations that translations, reflections, and rotations produce congruent figures.
To keep the learning on track I have pairs of students check in with me after each section. As they check in, I look over their work and ask questions like “how do you see the translation in the graph and why does it make sense with your ordered pair rules?” or “how do you know the figures are congruent?” which require them to justify their answers (MP3). When pairs have sufficiently demonstrated their understanding to me, I ask them to work on reflection and translation practice.
In this practice, students also discover reflecting a figure over the x-axis and then the y-axis is equivalent to rotating the figure 180o about the origin—again, tracing paper can really help students to understand this equivalence. What is important is that this discovery highlights different transformations or combinations of transformations producing the same result, a understanding which is pivotal to students’ ability to engage in algebraic manipulation and symbolic sense making (MP1).
When pairs have sufficiently demonstrated their understanding to me, I ask them to work on Transformations Page 1, which gives them opportunities to make these transformations concrete. In this practice, students also discover that reflecting a figure over the x-axis and then the y-axis is equivalent to them rotating the figure 180o about the origin—again, tracing paper can really help students to understand this equivalence—which highlights the idea that different transformations or combinations of transformations can produce the same result, an understanding that is pivotal to students’ ability to engage in algebraic manipulation and symbolic sense making, which is critical to their success in future math classes.
As typical for all of our lessons, we debrief the lesson’s main ideas in our Notetakers at the end of the lesson. Since students have now played around with translation, reflection, and rotation, it is an opportune time to discuss what stays the same and what changes under translation, reflection, and rotation and to organize our ideas in a table.