We began a foray into abstract algebra last week in Fundamental Understanding of Mathematics XLIX, with a look at a toy mathematical system that used a four color spinner. We defined how to read the spinner to get a single color to indicate it's position, and an operation that started at one color, rotated the spinner to another color, and had a color as a result. We discovered that the rotate operation was not commutative, that is, Red rotate Blue wasn't the same as Blue rotate Red.
The basic spinner looks like this:
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50a.png)
If you'd like to make one, take a half sheet of 8 1/2 by 11 inch paper (standard US letter size), fold along the dotted lines, color as shown, then fold in half with the colors on the outside, then fold along the diagonals, still keeping the colors on the outside, and tuck the flaps in after the last fold. Assembly hint: start in the middle.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50b.png)
Play around with it, it's easier to make than it seems. Alternate plan, cut a square out of an index card, draw the diagonals, and color. Report from the field: students who make spinners from folded paper tend to not lose their spinners, and seem more engaged in the process of using them.
We read the spinners by the color on top.
It's fairly simple to demonstrate that the rotation operation isn't commutative. Simply pick two different colors, figure out the rotation result, then switch them, and get a different result from the rotation operation.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50c.png)
In the first case, Blue is rotated to the Red position and Green winds up on top. In the second case, Red is rotated to the Blue position, and Yellow winds up on top, clearly a different color.
There are some rotate operations, however, where we can reverse the order: for example, B r B = B, and if we switch the order and put the first Blue after the rotate, and the second Blue before, we get B r B = B, the same thing. Any time we rotate to the same color we began with, we get that color. Investigating the actual movement of the spinner reveals that we either rotate it all the way around, or don't move it at all, since it's already in the right place. We draw a parallel with the identity property of addition, a + 0 = a. Not rotating the spinner is the equivalent of adding zero.
Now we add another operation. This time, we are going to flip the spinner, turn it over onto its other side. Some brief experimentation will reveal that it's not enough to just say "flip the spinner," because there are a number of different ways to flip it. This is an interesting group project, because it leads to a discussion of which is the "right" way to do a flip operation. Before fisticuffs break out, we note there are a number of ways to flip the spinner, and suggest it would be interesting to find out how many different ways there are.
It turns out there are four unique ways:
top to bottom,
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50d.png)
left to right,
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50e.png)
around the top left to bottom right diagonal,
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50f.png)
and around the top right to bottom left diagonal.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM50g.png)
There are a number of ways to describe them, though. A top to bottom flip could also be a bottom to top flip, for example. The change in position (which is what the flip accomplishes) is the same in both cases, so they are really the "same" flip.
Once these have been discovered, it's a good time to introduce the idea of notation. Trying to have a discussion about "Flipping the Spinner along the diagonal running from the top left corner to the bottom right corner" would get very tedious, very quickly. It can also be used to introduce the idea of arbitrariness and consistency in notation, by allowing groups to come up with their own notation. They may collaborate between groups to come up with common names for the different types of flips, or each group may come up with different names, which the class resolves with a class wide conference, where each group presents a justification for the names they chose, and the class comes to a consensus about which names to adopt so they can talk to each other about their discoveries.
We can't have that discussion in the diary (although it might be done in the comments) but, for the sake of moving forward, I'll assume the issue was hashed out, and the consensus is:
Vertical Flip or Fv
Horizontal Flip or Fh
Right Flip or Fr
and
Left Flip or Fl
I think I will wrap this diary up at this point, and try an experiment: homework.
Make a spinner, try the flip operations starting with each color, organize your results in a table, and present your findings in the comments.
Have fun in the comments.