Lesson 10

The purpose of this warm-up is to elicit the idea that some shapes can be described as transformations of other shapes, which will be useful when students specify sequences of rigid transformations that take one figure onto another in the next activities. While students may notice and wonder many things about these images, the important discussion point is that rigid transformations take sides to sides of the same length and angles to angles of the same measure.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, and point out contradicting information.

If sequences of rigid transformations or corresponding measurements do not come up during the conversation, ask students to discuss this idea. Reinforce that because the sizes, shapes, and angles did not change from Figure \(S\) to Figure \(M\), that transformation is called a rigid transformation. But the transformation from Figure \(S\) to Figure \(D\) is not a rigid transformation because the size changed.

If the difference between \(A\) and \(A'\) does not come up during the conversation, ask students to discuss this idea and tell them that \(A'\) is pronounced “\(A\) prime.” Explain that \(ABCD\) is called the original figure and \(A'B'C'D'\) is called the image of the transformation.

The purpose of this activity is to activate students’ prior knowledge of rigid transformations. Students also see that the image of a polygon is determined by the images of each of its vertices. Students build toward the concept that transformations are functions that take points as inputs and produce points as outputs so that distances and angles are preserved.

Some students may have trouble reflecting on the isometric grid. Ask these students to use tracing paper to fold across the line of reflection to find the image.

If students are stuck as to how to translate when \(w\) isn't connected to a vertex of the figure, remind students that \(w\) is telling them the direction and the distance, but the location doesn't matter.

The important idea for discussion is that rigid transformations preserve distances and angles. Display a student’s work for all to see and ask:

• For the reflection of \(S\) across line \(y\), how do the side lengths of \(S\) compare to the corresponding side lengths in its image? (The lengths are equal.)
• What is the measures of the angle in the upper left corner of \(S\)? How does this compare to the corresponding angle measure in any of the images of \(S\)? (This angle and all of its images measure 120 degrees.)

The purpose of this activity is to observe that the order of the transformations in a sequence of transformations can have an effect on the image.

Monitor for students who define a sequence that works even when the order is reversed to compare to students whose sequences don’t work when the order is reversed during the discussion.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Reversing the sequence means using the same steps, but step 2 becomes step 1. For example, 1) reflect across line \(\ell\) then 2) translate left 2 units would become 1) translate left 2 units then 2) reflect across line \(\ell\).

Select previously identified students to share their responses.

Highlight that reversing the steps in a sequence of transformations sometimes results in the same transformation, and sometimes it results in a different transformation.