This lesson might be previewed at the close of the previous day’s lesson by showing an image, such as one found on the Computer Animation overhead, and asking students to draw a simple image on a grid for homework. Student images should be fairly simple with 8–10 key points and without symmetry (so effects of reflection and rotation will be easier to detect).
Introduce the lesson with a classroom discussion of cartoon animation. Some possible question prompts are:
- How were the earliest cartoons animated by hand?
[The first cartoons were produced by drawing thousands of cels, which are transparent drawing sheets. Many would have the same background but with very slight changes to the main characters. When strung together in rapid succession, the illusion of motion was achieved.]
- How could computers take over some of the burden of animating by hand?
[The earliest introduction of technology into the animation process involved simplifying the background scene production. In the last twenty years, film studios developed techniques for shaping the characters to give 3D effects and transforming images to give the effect of motion. In effect, the computer has taken over the time-consuming work of creating thousands of incrementally-changed cels. In order to have computers do this work, artists and computer programmers had to analyze the components of motion and describe the components in a way the computer could work with. The transformation process will be explored in this lesson.]
- Can you name some computer-animated films?
[Answers will vary. Some computer-animated films are Toy Story, Antz, Finding Nemo, Cars, Kung Fu Panda, and Up.]
- What are the components of motion? How are they related to geometric transformations?
[The language of geometric transformations includes translation, rotation, reflection, and dilation. It provides a concise way to express the basic components of motion.]
Divide the class into groups of 3 or 4 students who will work together. The groups allow for more computation by the group as a whole. If you need to use larger groups, you might want to add additional transformation matrices to Question 2. Each group member should compute a different matrix product in Question 2. Give each group the Computer Animation activity sheet, several quarter sheets of graph paper.
Assign each group an image. All members of a single group should work on the same image. You may:
- Use the overhead projector to project a single image from the Computer Animation overhead for all groups.
- Print out images from the overhead and allow groups to choose one.
OR
- Allow student to pick one of the images created for homework to use with the activity. Check images before students begin working to ensure they meet the criteria previous stated.
Students should work together in their groups to complete Question 1 on the activity sheet, so that each student in the group has a clear idea of how the image can be represented in a matrix. Students will probably find it helpful to label the points on their image and the columns of their matrix with the same labels (A, B, C, etc.). Groups may want to work together, as well, on the first transformation in Question 2a, plotting the points on graph paper and connecting the dots in the same way as in the original image. Allow them to do so as long as enough transformations remain to allow each student to work on one individually. You can either allow them to choose their own transformation or assign them to students based on their person strengths. The Computer Animation answer key gives the solutions for the runner image only, but can be used as a guide as you help students.
Students may struggle with translating their matrix product back into an image. If they haven't already done so, you might encourage them to label the columns in their matrix and label the points in their image. Students can use the original skeleton as a guide, much like a connect-the-dots drawing.
Encourage students to write out the product carefully on scratch paper so that they can be certain about their computations. Alternatively, you might allow students to use a graphing calculator or CAS to compute the matrix products. Using graphing calculators will allow more time to discuss results in a class where students have previously demonstrated proficiency with matrix multiplication.
Question 2c involves fractions, which some students will find intimidating. They may prefer to use decimals, or even ignore the denominators (using 3 and ±4 instead of 3/5 and ±4/5), replacing it at the end of their calculation. Allow them to work with the numbers in whatever form they are most comfortable.
Question 2d results in a reflection over the line y = x. You may need to offer students some coaching, as some will think the transformation is a 90° counter-clockwise rotation. You might point out, in the student image, that the orientation of the image has been reversed, so that the image must be a reflection rather than a rotation). If students' misunderstandings persist, assign them to compute and graph their images with the transformation matrix 
, which does result in the 90° counter-clockwise rotation.
Once all the products have been computed and images graphed for Question 2, groups should discuss what kind of geometric transformation resulted, and then write geometric descriptions under their products on the activity sheet. Note that while the matrices and images in the answer key are only for the runner image, the geometric descriptions will be the same for all images. For example, the result for Question 2a is a reflection over the y-axis, regardless of the image used.
When groups have finished their discussion of Question have them go on to answer and discuss Questions 3 through 5 in their groups, as well, allowing time for other groups to complete their discussions of Question 2. Once all groups have finished with Question 2, have different groups present their geometric descriptions of the transformations resulting from each product. The class as a whole may offer suggestions for improved language.
After concluding the presentations, allow groups to continue working until they finish the activity sheet. Questions 6 and 7 provide an introduction to more advanced transformations, allowing students to see how translation can result from matrix multiplication. All students in the group should agree on the matrix representation in Question 6, but they should split the work and have each individual in the group complete a different product in Question 7.
As a summary discussion, invite students to share something they learned as a result of this lesson. Be sure to highlight Questions 3 and 5, since they address the way matrix multiplication works and introduce the function of the identity matrix in transformations, respectively.
, where I is the identity matrix and r is the scale factor]
undoes the transformation in Question 2c. It is not possible to undo the collapse-to-a-point or flatten-to-a-line transformations. This question previews the idea of inverse matrices.]
to obtain counter-clockwise rotations by angle a, and might experiment with transformations of the form 
in order to rotate around points other than the origin.
will transform the point (1,0) onto (a,b), and the point (0,1) onto (c,d).
