My son's board game :^)

My son, we'll call him R, goes to Wildcat Community FreeSchool, where the kids get to play most of the day, and learn lots of good stuff that way. Wildcat has lots of voluntary, free form classes, like art, Spanish, music, and other impromptu things, and there are also semi-mandatory classes in language arts and math. (If the parent says they're taking care of these subjects at home, the child doesn't have to go.)

R has been uncomfortable in crowds for the past year or two, and has hated 'having to' go to class. So he didn't. We do plenty of reading at home, and I'm all about math. But the classes are pretty fun, and he finally started going last week. (I don't know that it will last.) I'm thrilled. I trust that he'll learn just fine without it, but he's very social, and I think he'll be happier going to class.

On Thursday, the teacher (go Sarah!) had the kids rolling dice and marking their results on a chart with maybe 8 boxes above each of the numbers 1 to 6. Doesn't sound particularly exciting, does it? But that's because you're not 7 years old. The kids were watching the numbers race to the top. They were fascinated.

Yesterday evening R was messing around with the dice, and suddenly said, "I can do the game Sarah showed us!" And he made his own chart and raced the numbers. (Meanwhile I'm sitting near him, reading all the blogs I follow.) When he got tired ot that, he made his own board game:

We played it together today, and I loved it! Yes, it's utterly simple. But it was amusing what happened sometimes. The image doesn't show up very well, so I'm going to bore some of you with the details: Space 0 says 'Start', 1 is back 1 (indicated by arrow), 2 is back 2, 3 is forward 3, 4 is lose a turn (frowny face), 5 is back 2, 6 and 7 have no special actions, 8 is back 6, 9 to 12 - no actions, 13 is back 7, and spot 14 (not visible) is the goal. Landing on spot 8 takes you back 6, and then back 2 more to the start. Landing on spot 5 takes you back 2 then forward 3 to spot 6. It was a topsy turvy game.

This is what I want for my son - enough exposure to math that he has a glimpse of its power, and enough freedom to play with math however he wants. Yeay!

Richmond Math Salon and Base Three

I've been doing this for a year and a half now, and it's finally beginning to gel. There are some anchor families who come most of the time. There are new families coming. I'm starting to be able to describe how important it feels to me to work with parents and kids together.

Most of the folks coming to the math salon are homeschoolers. It's even more important for them than for others that the parents enjoy math. But, really, every kid, whatever sort of schooling they experience, will have a richer experience of math if their parents enjoy it.

But most parents don't. Most people over 6 or 7 or 8 don't. (You know, after they've had enough arithmetic lessons to think it's either boring or senseless.)

So if they can bring their kids to a math salon or a math festival or... and work on something together, then the parents can relearn how to have fun with math.

Today I had a sign up on my front door:

Enter at your own risk!
This doorway takes you
Through a spacetime warp,
Onto the planet of Triplay.
Are you ready?!

When people come in, there are games like Blink!, Set, Rush Hour, and Quarto to help them settle in. And Lori, the mom in one of the families that has come often, was there to assist me, since I was expecting a larger than usual crowd, with younger kids. (Although I've had months where tons of people are going to come, and then half of them can't make it.) Lori's daughter Audrey helped me too, sharpening pencils and playing a game with her mom, so mom would be ready to explain it to other parents.

We did this for close on an hour, I'd guess. Then I gathered all 20 of us in my small living room (except the girl who was sleeping in a bedroom). I started us off by talking about the people on Triplay having only 3 fingers on a hand - we all held up our hands, either with two fingers folded down, or fingers grouped in the Star Trek (live long and prosper) way.

We began discussing how the people of Triplay (Triplatians?) count: one, two, hand, hand and one, hand and two, two hands. We were all holding up our hands while we counted and talked about it. When we got to three hands, we needed a new word. We voted on whether our word would be handhand or handred. We voted for handhand - the cuter one. But then we got to four hands, which is handhand and hand. Too confusing! So I talked about how our language evolved over time to handred.

We kept counting, decided that 3 handreds was a handsend. (We didn't write it down at first. When we were writing these down later, it was interesting to think about how they might be spelled. It felt like a linguistics exploration too.)

The kids started drawing people from this planet, and some of the adults drew too. After a while of that, we began to design the Triplatian monetary system, and came up with names for the coins. I had a bunch of plastic chips available to use as coins, but I'm not sure how much use they got. We came up with our own names for the 3 cent piece (trickel), the 9 cent piece (handrime) and the 27 cent piece (hansellar). I'm not sure what the 81 cent piece was named.

A few of the adults began working on how numbers are written on Triplay - 1, 2, 10, 11, 12, 20, 21, 22, 100, and 101 are the first ten numbers, if you use our place value system and their base (three). They did that with no suggestion from me, so I was all the more excited to see the work they were doing. Two of the people working on that had mentioned having traumatic childhood experiences at home with math, so you know that diving in like that took some courage. (Isn't it amazing sometimes how our efforts to parent well push us to grow in ways we never expected?)

As usual, the adult conversation eventually meandered from our activity over to what coop classes they've found, and questions about books. I pulled out at least a dozen books, and we talked about how they do things in China (read Liping Ma's book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, for the good stuff, but we also talked about how pressured kids are there, as in many countries).

This was the best math salon I've done yet*. Other sessions there've lots of people, and things seemed to go well, but this time it felt like we really had a great balance of individual and group exploration, and I was tickled that I've gotten better at working with younger kids (drawing pictures, yeah!). Four people came later, so there were 24 people in all - the most that have ever come to the math salon.



===

*I've had lots of math salon sessions flop, mostly due to low attendance, but can't find the energy or courage after those to write. I'm so impressed by the math teachers who blog about what they're struggling with. I will try harder next fall, when I'm back to teaching, to tell you all about some of my bad days.

Sleep and Learning

I sleep about 9 hours on a good night. (Sometimes I wake up in the night, start thinking too much, and can't fall back to sleep.) My 7-year-old son sleeps 11 hours a night. I've always hated waking to an alarm, and haven't had to do that for years now. We go to bed between 8 and 9 pm, and if my son sleeps late, I let him. (His school is very unusual, and it's ok for him to arrive late.)

I'm happier - and I can teach better - when I've had enough sleep, so it's been a priority of mine for many years now. Recently, I've been hearing about how important sleep is for our health and our brains. The article I read just now (Snooze or Lose, at NY Magazine) gives even stronger evidence than I've seen before. This paragraph got me started writing this post:

In Edina, Minnesota, an affluent suburb of Minneapolis, the high school start time was changed from 7:25 a.m. to 8:30. The results were startling. In the year preceding the time change, math and verbal SAT scores for the top 10 percent of Edina’s students averaged 1288. A year later, the top 10 percent averaged 1500, an increase that couldn’t be attributed to any other variable.

A combined score of 1500 is in the top 1% of all test takers. 1288 is in the top 15 to 20%. This is a huge jump. The article gives detail about how sleep helps us learn too:

Dr. Matthew Walker of UC Berkeley explains that during sleep, the brain shifts what it learned that day to more efficient storage regions of the brain. Each stage of sleep plays a unique role in capturing memories. For example, studying a foreign language requires learning vocabulary, auditory memory of new sounds, and motor skills to correctly enunciate new words. The vocabulary is synthesized by the hippocampus early in the night during “slow-wave sleep,” a deep slumber without dreams. The motor skills of enunciation are processed during Stage 2 non-rem sleep, and the auditory memories are encoded across all stages. Memories that are emotionally laden get processed during R.E.M. sleep.

At the end is a companion article about how to get more sleep. The bit of advice I found most helpful is to limit exposure to tv or computer screens in the last hour or two before bed. I guess if I want to sleep more soundly, I'd better stop surfing a bit earlier.

Sweet dreams!


(Kudos to Rebecca Zook, who posted about this in her blog, Triangle Suitcase.)

Challenge: Write a Kids' Poem about Math

Back in September, Sean Nash wrote a post, over at his nashworld blog, about reading a book of Mother Goose to his 2-year-old daughter and not reading this one:
She's recognizing words, maybe he's lucky she let him skip it. I wonder what the rule of three was back then.

Reading the comments, including one from a dad who's encouraging his daughter to write science poems, got me thinking about what more we could do. So...

Here's the challenge: Write a little kids’ poem that’s as catchy as that nasty bit quoted above, and that tells of the beauty of math, or, that mentions math and challenge, both in a positive way.

If you start working on this, please let me know, even if you don't come up with something you like. I know I need to let this thought percolate for a while before I'll be able to come up with anything.

Joint Mathematics Meetings in SF Next Week

I'll be attending. If you'll be there and would like to meet up, please email me. I'm suevanhattum on hotmail.

I'm planning to buy a few books I've heard good things about:
Euler's Gem, by Dave Richeson, who blogs at Division by Zero,
The Calculus of Friendship, by Steven Strogatz (read Sam Shah's review),
and some of James Tanton's books.

Pythagorean Triples

I got interested in this problem last summer at the Math Circle Institute, held at Notre Dame. I have a bad memory and am having fun reconstructing what we figured out together there.


Introduction to Pythagorean Triples
The Pythagorean Theorem tells us that a right triangle with legs a and b and hypotenuse c will always have the relationship a²+b² = c². (Do you know how to prove that?) When all three sides are whole numbers, we have a Pythagorean triple. The most famous of these is 3²+4² = 5², often referred to in this context as (3,4,5). The 3-4-5 triangle was used in Egypt to help make perpendicular sides for their magnificent buildings. A loop of rope with 12 equally spaced knots (3+4+5 = 12) was pulled taut at knots 0, 3 and 7 to make a precise right angle.

If the three sides don't all have a factor in common, then they make a primitive Pythagorean triple (PPT). 6²+8² = 10² is not a PPT because all three sides have a factor of 2.


Starting to Explore
When I'm making up problems for students, I often want another Pythagorean triple, so I have one more sitting in my brain, 5²+12² = 13². Are there others? Are there infinitely many PPT's? How would we find more?

One approach to exploring these involves thinking about parity. (Parity refers to whether a number is odd or even.) I see that both of the PPT's above, (3,4,5) and (5,12,13), are odd+even = odd. Can we have odd+odd = even? What about even+even = even? If we have odd+even = odd, does the odd leg have to be the shorter one?

Do other questions occur to you?

Recommendation: stop reading and start playing as soon as you have a thought about how you might proceed.

I knew there were more PPT's, but couldn't remember any others. I wanted to find a few more, so I could see any obvious patterns. So I made a list of the first 25 perfect squares and looked for pairs that would add to equal another perfect square, or subtract to equal another one. I found (8,15,17) and (7,24,25). Well, that's one question answered: the odd leg does not have to be the shortest side. I see that all 4 hypotenuses are odd. So I want to address the question of whether odd+odd = even is possible.


Question1: Is odd+odd = even possible?
Here's how I start: Suppose we have two odd legs. We can let a = 2n+1 and b=2m+1. Then a²+b² = ... Can we come to a contradiction? [See the hint at the end for a bit more direction.]


Question2: If a is an odd number, can I find a PPT for it?
I notice that all 3 triples in which the odd leg is the short one include consecutive numbers for the other leg and the hypotenuse. Hmm. If a is the short leg, then b+c = a² in all 3 of those cases. Is that important? What if I write a²+b² = c² as c²-b² = a²? Oh! A square minus a square can factor. So I'd have (c-b)(c+b) = a². What does that get me? [I found a way to get a PPT for every odd number. Can you?]


Question3: Must the even side be a multiple of 4?
I wanted more triples, in case it would help me see more patterns. So I set up a spreadsheet with column a holding 1 through 100, row 1 holding 1 through 254 (first time I've ever used all available rows!). Column b and row 2 had the squares of these numbers. The rest of the spreadsheet showed a 0 if the square root of the sum of these squares was not a whole number, and otherwise showed the number. [Here's the formula in cell c3: =IF(INT(SQRT($B3+C$2))=SQRT($B3+C$2),SQRT($B3+C$2),0)]

For most multiples of 4, I found a PPT. And I didn't find any for the other even numbers. So I wanted to know whether the even side had to be a multiple of 4. If the even side is a, then b and c are odd, and c²-b², with b=2m+1 and c=2n+1, can be explored.


Questions 4 and 5: Multiples of 3, 4 and 5
This reminded me that I had read (whose blog was that on?) that in every PPT, 3 will be a factor of one side, 4 will be a factor of one side, and 5 will be a factor of one side. (As in (5,12,13), one side may contain more than one of these factors.) I realized I'd already proved it for 4. I started trying to prove it for 3.

I mentioned parity earlier. Even numbers can be expressed as a=2m, and odd numbers can be expressed as b=2n+1. Similarly, if we want to think about whether side c will be a multiple of 3, we can look at three cases: c=3m, c=3m+1, or c=3m+2. Using this, I started with the question of whether c would be a multiple of 3. (It wasn't in any of the triples I'd found.) If it is, then neither a nor b can be. (Why?) Once I solved that problem, I wanted to prove that one of the legs would be a multiple of 3. Suppose b is not a multiple of 3 and consider c²-b². There will be 4 cases, each of b and c can either be 3x+1 or 3x+2. What does this make a?

I tried to think about 5 in this way but got nowhere. I'm writing this blog post in hopes that explaining my thinking will help me get further on some of my dead ends.

I have a few other questions I haven't answered:

• Given a multiple of 4, how can I come up with a PPT?
• Can the same number show up in 3 different PPT's?
  

Let me know if you have fun playing with this. Maybe the directions your thinking takes will be different than mine...

Hint: c² is a multiple of 4. (Why?) What about a²+b²?

Math Teachers at Play #21

Welcome once again ...
to the Math Teachers At Play blog carnival. Puzzlers, riddlers, thinkers, doers, novices, experts, come one , come all!

[photo by santarosa]

First off, in honor of the number 21, is a puzzle, fresh from the oven.

The Numberland News runs personal ads. 21 was looking for a new friend and put an ad in.

Two-digit, semi-prime, triangular, Fibonacci number seeks same. I'm a binary palindrome, what about you?

Will 21 find a friend?


Elementary
Kendra at Aussie Pumpkin Patch has written about the estimation lesson she did with her sons, which they started by reading Counting on Frank. It sounds like fun!

What is your child's favorite small toy? Will it help them learn division? Ashley at HyperHomeschool headed for the Legos, and here's what happened.

A really good way to understand place value is to work with other number bases. A book I recently discovered, How to Count Like a Martian, by Glory St. John, tells a detective story in which the history of other number systems plays a starring role. The last few chapters discuss place-value-based systems. Want a more hands-on approach? Sol, at Wild About Math!, offers us some math magic with index cards, based on binary numbers.

Megan Wong has written some books she'd like to share with us: Math Power is Fun and Brain Power is Fun are part of her Mind Power series.


Algebra and Geometry
At Let's Play Math, Denise gives another algebra lesson in Pre-Algebra Problem Solving: 4th Grade. I'm thinking this would be a good first step even for adults just learning algebra. I've seen mention of bar diagrams plenty, but this is the best illustration of their use I've seen so far.

Maria Miller offers us 3 videos of her proofs of some basic geometric relationships in Angles in a parallelogram and a triangle.

John Golden at Math Hombre shares his geogebra sketches (available as webpages and geogebra files) at Net Results. Students can use these to create their own prisms and pyramids. Print the nets, fold them up and see the funky solids.

One of my favorite things about MTAP is discovering new blogs. Here's one: Guillermo presents a Tutorial on Geometer's Sketchpad.


Calculus and ...
Pat's Blog has Fun with Parabolas.

Dan at Mathrecreation offers us a curious population model, with directions for exploring the logistics functions in Fathom.


Our Favorite Proofs
Brent, at The Math Less Traveled, presents a proof that pi is irrational. He thinks calculus students should be able to follow it. I've treid to follow this proof in other places and not had the patience for it. So far, his explanation is right up my alley.

The Count, at Discrete Ideas, gives us Discretely Simple, on his two favorite proofs.


On Teaching
Simple things can make such a difference. “What’s a question that someone else might get wrong?” So simple, and such a good way to get students thinking. Here's JD's post on it.

Riley Lark, at Point of Inflection, offers us his index cards. Well, the students get the cards, and some quickie interaction.

Then they can review with Trashketball. Post by Dan Greene at The Exponential Curve.


News

• TI and Numb3rs
• DARPA's red balloon project
• TIMSS Advanced (from Math Education Research Blog)
• I wrote to Ivar Ekeland, author of The Cat in Numberland, and he wrote back! He says the publisher is "preparing a re-issue in 2010". Yeay! I think I'll be getting lots of copies of it next year around this time to give as presents.
  

The Holiday Connection
When mathematicians hear about the gifts "my true love gave to me" on the 12 days of Christmas, they start counting. How many gifts would that be altogether? Sol at Wild About Math! wrote this post a few years back. And John at The Endeavor wrote this post more recently. One of the commenters on John's post wondered: "Funny that the 12 days of Christmas turn out to have just short of one gift per day for a full year. A coincidence?"

Remember the Soma Cube? Rachel, at Minds in Bloom, gives directions for making it here, and thinks a home-made puzzle would make a great gift. If you'd like to do that for this holiday season you may not have time to wait for the cubes to be shipped. But if you can wait, the cubes are pretty inexpensive online: 100 1" plastic cubes on Amazon for about $15, 100 1" wooden cubes for about $10 here, or 1000 centimeter cubes for $25.

Between Maria and JD and a few others around here, I've started to think that creating puzzles (or authoring math, as Maria would say) is something I too can do. So last week I made a logic puzzle, Holiday Logic. I hope you'll enjoy it.




This edition of MTAP was composed in Richmond, California and Chicago, Illinois. It's coming out late in the day because leaving home and flying here yesterday, even though uneventful, did take up my whole day. May your holidays be peaceful. May peace spread exponentially from our hearts through our actions to the world around us.

Holiday Logic Puzzles

I'd like to share two puzzles here. One is by Mike Shenk, who has a site called puzzability, and is interviewed here. It's called Oh Deer!, and it's a killer. The other I created last week. I wanted a logic puzzle with a holiday theme for my math salon, and I wanted it to be much easier than Oh Deer! When I searched online, I got tired of seeing so much about buying presents. I also wanted to include other holidays besides xmas.

I've always loved doing these puzzles, but this is my first attempt at creating one. I believe it has a unique answer. Please let me know if I goofed.

I wish you all peace and joy during these winter holidays.

Holiday Logic
by Sue VanHattum

1. The Green girl’s favorite Christmas tradition is singing carols.
2. The Brown boy celebrates Kwanzaa indoors.
3. DJ and Jordon joined their friend in her candle lighting ceremony.
4. Layla and Amani joined their friend for his annual walk through the woods.
5. The Gold girl came to Jordon’s house to join his family in their feast.
6. The Fox family celebrate the Yuletide, and Amani comes to their party.
7. Amani couldn’t make it to the Gold family’s Hanukkah celebration.

[Edited for clarity.] Each child celebrates just one holiday with one special activity as a tradition in their family, though they do join in the fun with their friends this year. Your mission: Decide who celebrates each holiday, and what they do to celebrate.

Oh Deer! A logic problem by Mike Shenk
(first published in Games Magazine, December 1992)

Twas the night before Christmas, and at the North Pole
The last-minute planning was taking its toll.
As Santa was hastily making a scheme
For the placement of deer in his sleigh-pulling team,
The good Mrs. Claus was crocheting bright bows
To be worn by these reindeer (four bucks and four does).

The ribbons were colored in eight festive hues:
One ocher, one rose, one cerise, one chartreuse,
One maroon, one magenta, one white, and one blue.
(These ribbons helped Santa keep track of who's who.)
The deer pulled the toy-laden sleigh in four rows,
Arranged so no row held two bucks or does.

The order of pullers was changed year by year,
For Santa was thoroughly fair with his deer.
He summoned the elves and instructed them thus:
"Let's hitch up the reindeer with minimum fuss.
The bow on the buck behind Dasher is white,
While Blitzen, a doe, sees cerise to her right.

The blue bow is nearer my sleigh than is Dancer,
But nearer the front of my team than is Prancer.
The doe in chartreuse gets a front-of-team honor,
But not on the same side as Cupid or Donner.
Now Comet stands two spots ahead of the rose.
And three deer of four on the right side are does.

The cerise bow is worn two in back of maroon,
One of which is beside the bright ocher festoon.
Oh-Cupid's in front of a buck, by the way.
Well, that's how they line up for pulling my sleigh.
I trust that you elves, being clever, now know
Each reindeer's position and color of bow."

In no time each colorful ribbon was tied
And the team was hitched up for the transglobal ride.
Can you ascertain where each member fits in?
Who's Comet? Who's Cupid" Where's Donner? And Blitzen?
Who's Dasher? Who's Dancer" Where's Vixen? And Prancer?
With logical thought, you'll determine the answer
And write down the color and place for each deer.
Happy Christmas to all, and to all much good cheer!

Hannah, Divided, by Adele Griffin

Over the past year I've learned a new term: 2e, or twice exceptional, is a term used by advocates of kids who are exceptionally smart, along with having exceptional learning differences. (I'm paraphrasing Tiffani, who blogs beautifully at Child's Play about 2e issues. Another blog I've enjoyed on the subject is Life Among the Gifted.)

Hannah, Divided is the sweet story of a girl who would be designated 2e nowadays. Growing up during the depression on a farm, she's not too worried about her struggles with reading, and the comfort she takes in numbers is very personal. She doesn't much expect either her learning trouble or her gift to take her away from milking cows and sharing the chores with her family. But they do.

Her teacher, Miss Cascade, has prepared the one-room schoolhouse and all its students for a visit from a possible benefactor. On the day Mrs. Sweet arrives, she takes an interest in Hannah's math abilities and quizzes her after school. On the way home, Hannah is so fired up, she just has to run, and count.

Finally, she took this year, 1934, and divided it by two. Over and over, skipping the decimal point like a checkers piece until it stood at the front of the line.

Granddad McNaughton encouraged her mathematics. Sunday afternoons, they passed gleeful hours inventing games with figures and sums, making up riddles and puzzles to solve.

She's given a chance to go to Philadelphia to study math, and takes it. Leaving home is difficult for her, and she leans on her need to pace her room 32 times, get each item in exactly the right place, and tap her paper 32 times. As hard as it is, the math she's able to learn from her tutor at Ottley Friends' School makes all the hardship worthwhile.

As I read this simple story, I kept thinking of how moving it might be for my young friend Artemis, and other kids like Hannah. I hope some of you have a chance to enjoy it soon.

Math Teachers at Play coming up...

Submissions are due in a week, on Wednesday, December 16. Please send your favorite posts that haven't been in MTAP yet, new or old. Send your own; send links to other people's posts that you like; send math questions that have you puzzled.