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.
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?
In a lazy, I-don’t-want-to-do-school mood, Princess Kitten was ready to stop after three math problems. We had gotten two of them correct, but the last one was counting the ways to paint a cube in black and white, and we forgot to count the solid-color options.
For my perfectionist daughter, one mistake was excuse enough to quit. She leaned her head against me as we sat together on the couch and said, “We’re done. Done, done, done.” If she could, she would have started purring — one of the most manipulative noises known to humankind. I’m a soft touch. Who can work on math when there’s a kitten to cuddle?
by tanjila ahmed via flickr
Still, I managed to squeeze in one more puzzle. I picked up my whiteboard marker and started writing:
Kitten complained that some math programs keep repeating the same kind of problems over and over, with bigger numbers: “They don’t get any harder, they just get longer. It’s boring!”
So we pulled out the Counting lessons in Competition Math for Middle School. [Highly recommended book!] Kitten doesn’t like to compete, but she enjoys learning new ideas, and Batterson’s book gives her plenty of those, well organized and clearly explained.
Pizzas at Mario’s come in three sizes, and you have your choice of 10 toppings to add to the pizza. You may order a pizza with any number of toppings (up to 10), including zero. How many choices of pizza are there at Mario’s?
[The book said 9 toppings, but I was skimming/paraphrasing aloud and misread.]
Our power finally came back on this afternoon after a couple of cold, dark nights.
We heated the house with candles and stove-top burners (propane). We found that if you get enough candles lit in a room, it really can make a difference, and the stove was able to keep the kitchen in the upper-50’s, even above 60 degrees at times, which was pretty comfortable. We dressed in lots of layers and wore gloves, drank a plenty of coffee and hot chocolate, listened to oldies on the battery-powered radio, and went to bed early each night. (Hooray for sleeping bags!)
It’s high time I got back on track with my Alexandria Jones posts, so I’ve been working hard on a short introduction to probability, to go along with The Birthday Surprise. Or, more honestly, I’ve been procrastinating on a short intro … well, anyway, here’s a little of what I’ve been reading around the interwebs lately.
If a girl and a half
can read a book and a half
in a day and a half,
then how many books can one girl read in the month of June?
Kitten reads voraciously, but she decided to skip our library’s summer reading program this year. The Border’s Double-Dog Dare Program was a lot less hassle and had a better prize: a free book! Of course, it didn’t take her all summer to finish 10 books.
I’m afraid that Math Club may have fallen victim to the economy, which is worse in our town than in the nation in general. Homeschooling families have tight budgets even in the best of times, and now they seem to be cutting back all non-essentials. I assumed that last semester’s students would return, but I should have asked for an RSVP.
Still, Kitten and I had a fun time together. We played four rounds of Tens Concentration, since I had spread out cards on the tables in the library meeting room before we realized that no one was coming. Had to pick up the cards one way or another, so we figured we might as well enjoy them! She won the first two rounds, which put her in a good mood for our lesson.
I had written “Prime numbers are like monkeys!” on the whiteboard, and Kitten asked me what that meant. That was all the encouragement I needed to launch into my planned lesson, despite the frustrating dearth of students. The idea is taken from Danica McKellar’s book Math Doesn’t Suck.
Niner (pronounced Neener), who takes the photos for my blog header — which reminds me, we’re about due for a new one of those… — has started a new blog. She calls it 19 & Still Alive, because “the world doesn’t end when you’re 16 if you don’t go to Prom or don’t get your driver’s license. (I never went to Prom, and I didn’t go through Driver’s ED until 17, but I’m alive, amazingly.)”
There won’t be any math there, or at least I don’t expect to see any; the blog will be mostly her rambling thoughts about whatever catches her interest. But she does have a wonderful graphite drawing based on the photo Masarwa man. On her blog, you can click the image to see him up close and personal. Wow!
Kitten strongly dislikes math when forced to do it on her own, so I am trying to get back into the habit of doing “Buddy Math” with her. We take turns working the problems in her workbook: mine, hers, mine, hers, and so on down the page. We work each problem out loud, explaining how we got the answer and checking each other as we go.
In a way, it is like Charlotte Mason-style narration applied to math, since my daughter has to process her thoughts in order to explain how she worked the problem, which fixes the math concepts more deeply in her mind.
No, it’s not math, but it looks like a great way to kick-start Princess Kitten’s long-neglected blog. She sat down at the computer and browsed the links to other kids’ posts for over an hour last night, occasionally laughing out loud. Then she opened her Dashboard and started to type a response to the green assignment.
I’ll have to let her know there’s a new post up today. Check it out:
Unfortunately, Homeschool Kids Write has disappeared from the web. The Wayback Machine link gives a taste of what the site was like, but It’s just not the same without the Mr. Linky connections to all the children’s writings.
Have you ever noticed how very different little girls are from little boys, in the way they play and in the way they think about things? Princess Kitten has been playing around with Backwards Math again, and my first thought was, “No boy would ever have done this with numbers.”
Math concepts: mental calculations, math vocabulary, and anything else you want to include Number of players: any number, but I think it works best with two players who alternate asking questions Equipment: imagination and, if necessary, scratch paper
Many years ago, I read a magazine article by mathematical music critic Edward Rothstein, wherein he described a game he invented for his daughter:
“What number am I? If you add me to myself, you get four.”
Rather than explaining the rules of the game, let me tell you a story…
My yard, my life! This was our front yard and driveway last weekend, as we rushed through our last-minute preparations for the County Fair. Niner, too old for 4-H, was determined to get her photos entered in the open competition before Saturday afternoon’s deadline. She rolled up her pants legs and waded through the drink, using her feet to feel out the edges of the driveway and marking the path with red-flagged fence posts so the Jeep could make it through.
Just in case you were wondering why there were no new posts this week.
It took 4+ years of practice and one solid hour of sweating — which included punching and kicking his dad as hard as he could (holding the karate pad is no easy job!) and the breaking of two boards — but it was worth it.
Last night, Chickenfoot received his black belt, being presented in the photo by his instructor, Sir John.
[Disclaimer: I gave in to blogger’s poetic license for a catchy title. To tell the truth, I am sure Chickenfoot wouldn’t consider himself a math geek at all. Math is far from being his favorite subject, even though he is good at it.]
Things are still hectic, but at least the phone company guy found the problem and got our “extended DSL” service working. “Extended DSL” is what you get when you live out in the boonies. No guarantees that it will be faster than the ancient modem, but at least it doesn’t tie up the phone line anymore.
And it is a bit faster, so I finally get to enjoy You Tube. If the video doesn’t display properly, you can find it at this link:
Even the child in question agrees with that. Chickenfoot is suffering from Peter Pan syndrome: “I don’t want to grow up!” This week, someone asked him, “You’re 17, aren’t you?” Not yet, but 13 has passed into memory…
I got tricky with the hidden present this time. The outside of the envelope changed insignificantly. It read:
I’m your last present.
Can you find me?
I’m hiding some place
That you cannot see…
But the real departure from normal came with the poetry inside. The poem was a success, in that he laughed at all the appropriate spots, but the hiding place turned out to be a little too good. Can you find the two clues in all this mess?
Our youngest child turns 9 today. Time shoots by at rocket speed, doesn’t it? Every year, the older kids complain that their baby sister isn’t allowed to grow up — they still feel like being nine years old themselves!
I have mentioned before our family tradition of the “hidden present.” [See this post and that.] Every year, one birthday present is hidden somewhere in the house, with the clue placed in an envelope to be opened after all the other gifts are unwrapped.