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
- 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.
- 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
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