Wow. When I saw this, I was so excited! Although it's been awhile since I worked with a matrix, or matrices, I thought I quickly understood what was happening. As I explored further, I realized, I needed a refresher. Plus, what an amazing way to illustrate the changes. This "Snow White" has enough detail that we can clearly see the x and y transformations. I love it and I've barely even begun to explore it.
I'm disappointed to report that the Algebra II courses in my school do not teach matrices. Why is this a surprise to me? I never saw a matrix until I set foot in College Algebra my first day of college. Oops, that was a major mistake! I know now that even the briefest exposure would have been invaluable. That being said, as the standards make a transition to understanding properties of shapes and proofs of theorems through transformations, this type of analysis and thinking cannot be ignored. I hope that by creating and applying this to mini lessons, lessons, and assessments, my colleagues will see the value and return to teaching matrices.
This image was obtained from this site.
Grade Level: High School
Course: Algebra II (Most likely)
Standards: N-VM.7, N-VM.10, N-VM.12, A-REI.9,
SMP: MP1, MP2, MP3, MP4, MP6, MP7, MP8
Skills: Properties of Matrices, Matrix Identities, Transformations of matrices, linear transformations
How to use this as a mad minute:
You have 60 seconds. Choose one transformation and explain why the image is accurate.
How to use this as a warm up:
You could ask the students to consider one of the following:
1. What does the identity matrix represent?
2. What would the identity Snow White look like? How do you know?
3. Explain the Wicked Queen matrix.
4. Personally, I think this would be more accurate, and more appropriate, if it were the Alice in Wonderland Matrix. Why?
5. Can you think of another transformation? Draw the matrix and the resulting "Snow White" image.
How to use this as a mini-lesson:
20 minutes, assuming kids DO know about matrices and the basics of transformations.
0:00--This Snow White image caught my eye. Take 2 minutes to read, study, explore, and think about it. Make notes, list questions, draw pictures, etc.
2:00--Ok, anyone want to share immediate thoughts and reactions? (Hopefully the kids will share some of these thoughts, but if not, try to lead them in these general directions. Remember that if they go in a direction that you don't expect, but is valuable, take that path and abandon mine!)
4:00--Let's talk about the identity. Why do you think the artist never drew the identity? Do you think you could? Take a minute to do your best. You may want to draw another one next to it to show the difference. For example, draw Scaley and then draw the identity.
6:00--Can we talk for a minute about why the artist used the name Scaley? Any guesses why? (If kids don't see the connection between "y" and the "y" value and the "y" axis, don't push it. You'll come back to it. If they do, ask them to check and see if their theory works for all of the examples as you discuss and review.)
6:00--What does the identity represent in any matrix? How does that relate to any geometric figure?
7:00--What's the difference between Scaley and Scalex? Do you think someone unfamiliar with those could figure it out by looking at the two images?
9:00--What about the difference between Reflecty and Reflectx?
10:00--So, let's revisit the question about Scaley vs. Scalex and Reflecty vs. Reflectx. Why those names? Are they clear and obvious, now? Did your hypothesis work out?
11:00--Let's look at the rotation example. If the matrix is named "A", the element A1,2 is -1. Why did that cause the rotation shown?
12:00--What would happen if you made A2,1 = -1? What kind of rotation would you get? Do you know?
14:00--What would happen if both elements were -1? Can you predict?
15:00--What do you think the Wicked Queen matrix does? Explain.
17:00--Take the next 3 minutes to draw a simple, yet distinct figure. Notice that Snow White has a bow on the left side of her head, and a distinct top and bottom half. Try to create at least 3 transformations by drawing the matrix first and then drawing the resulting figure.
How to use this as a full lesson?
Honestly, when I saw this, as much as I loved it, I really wondered why it wasn't Alice, from Alice in Wonderland, instead. After all, who better to use for stretching and shrinking?
I would do the mini lesson above, but then show the two movie clips below. One is 6 minutes long and one is 1 minute long. (The first is from my personal favorite live action version from 1985! Please let me know if these video clips expire or the links no longer work!)
I would challenge the students to create, while watching, a quick timeline of the transformations. For example:
Normal size--drink potion--become small
Small--eat cake--become larger than normal
etc.
Then, have them get in groups and try to create matrices that would create the transformations. They need to be able to back them up. Give them 10 to 15 minutes.
After this, ask them to share their results. (Poster? Write it on the board? Hold up their dry erase board?) They have to convince you that their matrix is better than another group's. (It might be more accurate, it might better account for her change in size in both directions, etc.)
Let there be a debate. Let kids really argue for why theirs is the best. You can decide if they should get time to revise their answers and try again, or if they have to stick with their first answers. (Revisions would encourage kids to learn, assess, and fix!)
From there, I would choose a simple cartoon figure and ask students to perform the given transformations. (Sample worksheet here.) I selected Kirby because he doesn't require much artistic strength to create simple transformations. Also note that the worksheet I made doesn't ask students to draw. If I had better software to create matrices, I might have included problems like that, but I didn't. Some of your more creative students may better show their understanding by drawing various Kirby images to match given matrix transformations.
How to use this as an assessment?
You could use the worksheet above as a quiz (though something requiring artistic ability is not very fair!)
You could also use the video activity as an assessment tool instead of a team project. You would need to create a rubric and more specific directions!
Please feel free to use any of these ideas and modify them to meet your needs. However, please acknowledge the original source of the items and my own lesson outlines. ©NatalieRSprigg 2013
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