In my previous post on Mathematical Curiosities (www.imath.us/math), I wrote that I would explain in more detail later. So, here are a few of the experiences I have had utilizing “Math Curiosities and Puzzlements” in my Free Form Fridays this year.
What in the World? – These are some of the best curiosities that I have used. This one was an article in The Atlantic that talked about caffeine related emergency room visits. Here is the graph that they included in the article:
I then simply asked the question: “What do you wonder?” The students took off from there … “Why would caffeine send you to the emergency room?”, “What kind of events are ‘caffeine related’?”, “How much caffeine would you have to have in order to be sent to the ER?” And they kept going.
Some students focused on the age ranges and why there were more or less ER visits. Some focused on the percent increase in ER visits among the age groups. Some focused on the amount of caffeine in different drinks or foods. And still others focused on the amount of caffeine that is considered safe or harmful. They spent 20 to 30 minutes researching and coming up with their conclusions that they then shared with the class.
Coloring Problem – Technically it is the “4 Color Theorem” (you can find it here: https://www.youtube.com/watch?v=ANY7X-_wpNs). However, that would give away what I wanted to happen with this curiosity. The idea is quite simple: you can color any figure so that no two colors are adjacent to each other. What is the least number of colors that you can use to color it in?
Here is the original figure I used:
I allowed a bit of exploration as they colored the original figure. My only question was, “What is the least number of colors needed to color in this shape according to the conditions given?”
Most students insisted it was 6 colors. Some used only 5. There was only one student who was actually able to figure out that it was 4 colors. After a bit of exploration, I showed them the video (link above). I then gave them a plethora of shapes to color (including maps) and they worked on using only 4 colors.
When it came time for the student choice portion of the day, many chose to continue coloring (Most students would love to simply draw and color during math class!). However, given the constraints, they were actually thinking mathematically and utilizing pattern recognition to do so!
Ultimate Tic-Tac-Toe – The students LOVED this one. Below you will see the tic-tac-toe board we used. The rules are very simple:
Basic rules for you:
- On each turn, you mark a square on a mini-board.
- When you get three in a row on a mini-board, you win that mini-board.
- When you get three mini-boards in a row, you win the game.
- However …
- You do not get to pick which of the 9 mini-boards to play on.
- Whichever square on the mini-board your opponent plays, you must play in the corresponding square on the larger board (and likewise whichever square you play on the mini-board will determine where your opponent has to play).
- Exception: if the mini-board is already won where you are supposed to play, you can pick any spot to play.
The students spent the rest of the class time playing against each other. I continually asked about a pattern or a strategy that would allow them to win. Some think they had it figured out, others were just enjoying the challenge of playing. But, it really was a time where they had to think strategically (and even mathematically) in order to do this. Weeks later, they still play this when they have extra time at the end of class.
One group decided to make the Super-Mega-Ultimate Tic-Tac-Toe. They used poster board and ended with boards within boards within boards. It is incredible!
There is much more to say here, but I think you get the idea … get students talking and thinking about math. Ask them to notice or wonder or think mathematically about most anything and then let them run with it. It is time well spent and they are learning far more than you could possibly imagine.