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A Surfer's Tessellation  Middle School Problem of the Week
Posted:
May 2, 2001 10:49 PM


Hi
In case you didn't notice, the Piedmont High 1st Period Geometry class is going to be mentoring this week's Middle School Problem of the Week. http://mathforum.com/midpow/solutions/solution.ehtml?puzzle=114
I'm almost caught up with submissions from last week so I've had a little more time than usual to think. Some might say that could be dangerous. :)
Did you know that the students in the Geometry class designed the problem and chose the title. I know that the tiles on the floor have to "tessellate" so I thought their title was very clever.
But ever since I finished finding the cost of the tiles, I've been digging deeper. What else is there in this problem? Here's what I've come up with so far:
1. Once the floor is completely tiled, will it be a regular tessellation, or a semiregular tessellation. or something else?
2. The tiling pattern fits "exactly" on the given floor. Is there a smaller, but similar, shape that would also fit the pattern exactly?
3. How many different floors could you design that would accommodate this exact pattern? Make them similar, make them different, just have fun with it!
If you want to create a similar problem for Elementary students, using the given tiling pattern... why not try one of these:
1. Make a chart that has # Green tiles (8") #Red tiles (4") # Green tiles (4") Total area 1 2 2 128 2 4 4 256 3 6 6
Let them look for patterns. Can they take it all the way to an algebraic representation? If they can state that there are twice as many red tiles as green tiles they are starting to make the connections that are needed. x 2x 2x 128x
I hope I got that right.
2. After making the chart, look at which of the rows will make rectangles. Wait, they'll all make rectangles, so that's not exactly what I mean to say. I want to know which rows will make rectangles, keeping the original tile pattern. Can you prove to me that we can continue to build rectangles as we add one more and one more tile pattern? Here's where we could use some square tiles in two sizes.
By the way, I don't know all the answers to these questions. I'm not sure that any of them make sense. But I'd love to have a discussion about some of them.
Do you have any other ideas you'd like to share? Judy



