My story begins with an unexpected adventure in pain. Appendicitis sidewhacked my life last week, but that’s not the story. It’s just the setting. During my recovery, I spent a lot of time in the smaller room of my hospital suite. I noticed this semi-random pattern in the floor tile, which made me wonder:
Did they choose the pattern to keep their customers from getting bored while they were … occupied?
Is the randomness real? Or can I find a line of symmetry or a set of tiles that repeat?
If I take pictures from enough different angles, could I transfer the whole floor to graph paper for further study?
And if the nurse finds me doing this, will she send me to a different ward of the hospital? Do hospitals have psychiatric wards, or is that only in the movies?
What is the biggest chunk of squares I could “break out” from this pattern that would create the illusion of a regular, repeating tessellation?
I gave up on the graph paper idea (for now) and printed the pictures to play with. By my definition, “broken” pattern chunks need to be contiguous along the sides of the tiles, like pentominoes. Also, the edge of the chunk must be a clean break along the mortar lines. The piece can zigzag all over the place, but it isn’t allowed to come back and touch itself anywhere, even at a corner. No holes allowed.
I’m counting the plain squares as the unit and each of the smaller rectangles as a half square. So far, the biggest chunk of repeating tiles I’ve managed to break out is 283 squares.
What Math Stories Will You Tell?
Have you and your children created any mathematical stories this year? I’d love to hear them! Please share your links in the comments section below.
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Kitten and I have been working through the lessons, and she loves it!
We’re skimming through pre-algebra in our regular lessons, but she has enjoyed playing around with simple algebra since she was in kindergarten. She has a strong track record of thinking her way through math problems, and earlier this year she invented her own method for solving systems of equations with two unknowns. I would guess her background is approximately equal to an above-average algebra 1 student near the end of the first semester.
After few lessons of Tanton’s course, she proved — within the limits of experimental error — that a catenary (the curve formed by a hanging chain) cannot be described by a quadratic equation. Last Friday, she easily solved the following equations:
and (though it took a bit more thought):
We’ve spent less than half an hour a day on the course, as a supplement to our AoPS Pre-Algebra textbook. We watch each video together, pausing occasionally so she can try her hand at an equation before listening to Tanton’s explanation. Then (usually the next day) she reads the lesson and does the exercises on her own. So far, she hasn’t needed the answers in the Companion Guide to Quadratics, but she did use the “Dots on a Circle” activity — and knowing that she has the answers available helps her feel more independent.
After teaching co-op math classes for several years, I’ve become known as the local math maven. Upon meeting one of my children, fellow homeschoolers often say, “Oh, you’re Denise’s son/daughter? You must be really good at math.”
The kids do their best to smile politely — and not to roll their eyes until the other person has turned away.
I hear similar comments after teaching a math workshop: “Wow, your kids must love math!” But my children are individuals, each with his or her own interests. A couple of them enjoy an occasional geometry or logic puzzle, but they never voluntarily sit down to slog through a math workbook page.
In fact, one daughter expressed the depth of her youthful perfectionist angst by scribbling all over the cover of her Miquon math workbook:
“I hate math! Hate, hate, hate-hate-HATE MATH!!!”
Translation: “If I can’t do it flawlessly the first time, then I don’t want to do it at all.”
Banished is a captivating fantasy story with a well-thought-out plot that would be a credit to any writer. But it is especially remarkable coming from a thirteen-year-old student who has been homeschooled all her life.
However, be forewarned. When you reach the final page and find the words, “Not the End…,” you will cry, “Oh! No!”
I for one feel as if I simply can’t wait to read the next installment to find out what happens to Chris and his friends. It’s that good!
We continue with our counting lessons — and once again, Kitten proves that she doesn’t think the same way I do. In fact, her solution is so elegant that I think she could have a future as a mathematician. After all, every aspiring novelist needs a day job, right?
If only I could get her to give up the idea that she hates math…
Permutations with Complications
How many of the possible distinct arrangements of 1-6 have 1 to the left of 2?