Composition of Transformations

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SWBAT draw transformed figures on graph paper and specify the sequence that transformed the figure.

Big Idea

Students will perform a variety of transformations on a single object and describe how the object was transformed.

Do Now

5 minutes

In the Do Now, students are given a graph with several triangles plotted on it. They are asked to describe specific transformations that map one triangle onto another. This activity requires skills learned in previous lessons. In the previous lessons, students looked at specific transformations in isolation. For this activity, students must distinguish the different transformations on the same grid. It leads to the lesson activity where students will explore compositions of transformations. After about 4 minutes, I go over the answers to ensure students are familiar with the different transformations. 


15 minutes

We begin the Mini-Lesson looking at question 1 from the Do Now.  Most students have identified the single transformation as a reflection over the x-axis.  I ask, “Are there any other transformations that can be performed on triangle C to result in triangle B?” We discuss the idea of performing more than one transformation on a figure to get a specific image.  This is called a composition. 

I hand out the worksheet, “Compositions of Transformations Mini-Lesson” and we discuss the notation for compositions. On the worksheet, I included a pictorial example using a picture of Alice to connect to previous lessons where we investigated transformations in the context of Alice’s Adventures in Wonderland, by Lewis Carroll. Students need to be able to precisely use the notation for compositions (MP6). The most important thing for students to remember about the notation for compositions is the order in which the transformations are performed. The right-hand transformation is performed first and the left-hand transformation is performed second.

We work as a class to practice three examples of compositions on the same triangle before students work independently. 


15 minutes

In the first part of the Activity, students describe the sequence of transformations that maps one triangle onto another. This is the same diagram as the one in the Do Now; however, students are looking at different preimages and images. Each triangle can be mapped using a composition of transformations.

This part of the activity is naturally differentiated since there is more may be more than one correct sequence and the answers can be written different ways depending on the level of the students. Most students are able to write the sequence using composition notation. Some students write out the sequence in words. As long as the students understand the order the transformations need to be performed, I accept either the notation or the written description.

In the second part of the activity, students are given figures to plot and compositions to perform.  Although I have included a grid on the paper, some students find it confusing to have so many shapes plotted on their graphs. These students can answer each question on a separate graph using paper from their notebook.

After about 12 minutes, I ask for volunteers to show their work on the document camera.


10 minutes

Think-Pair-Share:  Students take time to think about the following scenario:

A composition is commutative if the transformations can be performed in any order and result in the same image. Which compositions of transformations, generally or specifically, are commutative?

After 2 minutes of thinking, they share their answer with the person next to them.  We then discuss the answer as a whole class. 

I give students a homework assignment to come up with three specific examples of commutative compositions.