Playing with the Etch-A-Sketch to experiment with linear functions, circles and sinusoidal functions

 The Etch-A-Sketch toy has been around since the 1960's, at a time when lots of people were first getting TVs. The TVs of that era had horizontal and vertical hold knobs to stabilize the picture -- so the Etch-A-Sketch has knobs that control the horizontal and vertical movements of an internal mechanical cursor. No electricity required, and the image is erased by shaking the screen (something like sand inside erases the image.) Some people have figured out how to do amazing drawings with this simple drawing tool! We will aim to use it to understand some aspects of algebraic geometry in a more visceral way, with a fun retro toy.

You can access a virtual Etch-A-Sketch here, by the way!

Here are some experiential ways to play with an introduction to linear functions (and then circles and sinusoidal functions) with students working in groups of 3 with an Etch-A-Sketch!

Instructions:

1. Form a group of 3: one person operating the horizontal (H) knob, one the vertical (V) knob, and one person as timekeeper.
   
   
2. Start by playing a bit with drawing on the Etch-A-Sketch. Most people try to draw a house with a slanting roof. Notice how you have to use both the H and V knobs together to get a diagonal line (or anything that's not strictly horizontal or vertical!)
   
   
3. Use a permanent marker to mark a dash as the home position for the H and V knobs, so that you'll be able to tell when you've done a full turn, a half turn, etc.
   
   
4. Now have just one person operate the H knob, and a different person the V knob, and cover the screen with a piece of paper so that you can't see what you are drawing till it is done. You'll need the help of the time keeper to keep a steady beat by clapping or counting or otherwise directing the actions of the H and V people. Try the following:

• (a) Turn the H and V knobs the same distance, in the same direction, at the same rate. What have you drawn? (Take a photo of it, and of each successive drawing).
• (b) Turn the H and V knobs the same distance, in the opposite direction, at the same rate. What have you drawn, and why?
• (c) Turn the V knob two turns for every one turn of the H knob, in the same direction (and then in the opposite direction). What do you predict will be the result, and why? Did the result match your prediction?
• (d) Turn the H knob two turns for every one turn of the V knob, in the same direction (and then in the opposite direction). What do you predict will be the result, and why? Did the result match your prediction?

    5. Let's do some translation from "turn the knob" instructions to linear equations:

• (a) V is the same as H -->
  V = H --->
  ??
• (b) V is equivalent but opposite to H -->  ??
• (c) V is equivalent but twice as much as H --> ??
• (d) H is equivalent but twice as much as V --> ??
  
  Any surprises or revelations? How was it to do this with the screen covered with a piece of paper (as opposed to what it would be like looking at the screen while you draw)? What difference did it make to your mathematical noticing or awareness to have two people separately operate the H and V knobs, rather than doing it all yourself?

And now: CIRCLES!

1. Again with one person on the H and one on the V knobs, the timekeeper coordinating, and a piece of paper covering the screen, think about how you could draw a circle -- and then try doing it several ways! Keep a record of your results with photos, and caption them if you can with the strategy you tried.
   
   What strategies worked best for drawing circles? Was it easy or hard -- and why?
   What does this tell you about what a circle is? Does it shift your sense of the definition of a circle in any way? 
   
   
2. As an interesting extension: Now try drawing a sine (sinusoidal) curve, also with one person per knob, the screen covered with a piece of paper, the timekeeper keeping everyone on track.
   
   What did you notice about what the person on the H knob and V knobs did differently?
   What does this tell you about what a sinusoidal curve is?
   Did it connect with the way you drew a circle in any way?
   Did this experience connect at all with ideas from differential geometry?
   
   Once everyone has tried this, we will share and discuss what we noticed and discovered about these shapes using this fun toy!