Beth Menzie AMSTERDAM HIGH SCHOOL, AMSTERDAM, NY
Geometry : Unit #4 - Transformations : Lesson #3

Composition of Transformations

Objective: SWBAT recognize composition notation and perform compositions of transformations.
Standards: HSG-CO.A.5 HSG-CO.D.12 MP1 MP2 MP3 MP5 MP7
Subject(s): Math
60 minutes
1 Do Now: Construction - 15 minutes

For the Do Now, I ask my students to construct lines of reflection using a compass and a straightedge (MP 5). 

In the first problem, they choose a point and its image and construct the perpendicular bisector.  (This is a construction that they have been doing since Unit 1).

For the second problem, students can again construct a perpendicular bisector or, if they realize that the line of reflection is y=x, can construct an angle bisector, using the x and y axes as the sides of their angle. We will discuss the possible approaches when it appears everyone has finished.

2 Discussion of the Homework - 15 minutes

The homework assignment was to complete the last four problems on Name that Transformation.  These problems are interesting and typically lead to lots of great discussion (MP 3).  

In Number 7 one transformation is a reflection over the line x=1.  The students have practiced finding equations of vertical and horizontal lines in previous sections, so I expect most of them will see this. Other possibilities include a translation and a reflection over the point (1,3).

Number 8 tends to be difficult for my students, until we look at the mapping of points: (4,-2), for example, can be seen as mapping to (-2,4).   Because they have learned and worked with the rule for a reflection over y=x, most are able to conclude by looking at the coordinates that this is a reflection over y=x.  As I stated in previous sections, I take the time to ask questions about this line y=x:

  • What are some points on this line? 
  • What is its slope? 
  • What does this line do to the quadrants in which it lies? 

This problem can also be seen as a clockwise rotation of 270 degrees, or, a counterclockwise rotation of 90 degrees, both centered at the origin.

Number 9 is a great problem for discussion in that four different transformations are possible:

  1. A reflection over the origin
  2. A rotation of 180 degrees
  3. A translation
  4. A dilation of factor -1.  

The dilation of -1 will be important in the next activity, the Compositions Project.

Number 10 is a reflection that the students have not previously seen. The line of reflection is y=-x. Generally, a number of my students are able to come up with this answer, based on their work for problem Number 8 and during the Coordinate Geometry unit.  This transformation can also be seen as a rotation of 90 degrees counterclockwise, or 270 degrees counterclockwise.

3 Introduction to Composition: A Project - 45 minutes

The Compositions Project provides the opportunity for students to develop a deeper understanding of the different types of transformations, to reason, and to pay really close attention to the structure of their diagrams (MP 1, MP 2, and MP 7). The An Introduction to the Project video provides a little more detail about my goals for and my activity during this project.

A listing of materials for each pair of students, in each stage of the project:

  • Part 1: directions to Part 1, half a sheet of construction paper, approx. 6 in. square of graph paper, template, scissors, glue, dice
  • Part 2: directions to Part 2,  "1, 2, 3" sheet (plus glue and construction paper from Part 1)
  • Part 3: directions to Part 3, a "composition" paper, a whole piece of construction paper, glue

I have included some sample images, to more clearly illustrate the final product.

An "Aha" Moment for Me! Deeper discovery about dilations.
Complex Tasks
4 Practice with Compositions (and Homework) - 10 minutes

When students finish with their Compositions Project, they can begin working in their groups on the worksheet entitled Composition of Transformations Worksheet.  Those problems that they do not finish in class will be their homework for this evening.

5 Ticket Out the Door - 5 minutes

The Ticket Out the Door is intended to provide assessment data for me.  Specifically, do all of the students understand the concept of composition and composition notation? Also, we have finished about half of this unit at this point, so I would like to find out those topics with which they are feeling comfortable and those which are confusing for the students. 

I have found that, rather than simply giving students compositions to perform, the students find it a lot more interesting to create their own compositions.  They often compete to see who can find the simplest or the most complex, and this serves to enhance their engagement in the subject.