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July 27 2020

Nim

Now that schools have closed, many of us are finding ourselves suddenly homeschooling. So I’ve decided to continue Math Monday as an online program in Zoom, every Monday at noon. This will be a program for parents and kids to attend together, and it will include activities for all ages, grades K-12. For kids this will be fun education; for parents this is games you can make part of your family math culture.

I’m going to host a small private experimental version of Math Monday tomorrow March 16 at noon. Just click on this link at noon to join. And if you can attend, please RSVP so I have an idea of how many are coming. All you need is a computer, tablet or laptop running Zoom. The rest we’ll handle from there.

This is short notice, so I expect many of you can’t make it, but if you happen to be available and are looking for something fun to do with your kids, I’d be glad to include you. And feel free to pass on this notice to other families that you think might be interested.

Here’s briefly how it will work.

I’ll open with a story that introduces a math game that you can play at home with things around the house.
I’ll play a demonstration game with a volunteer — these are games you can play remotely with someone else using nothing but Zoom. I’ll include a couple variations that let you dial up the difficulty of the game for older kids or for those who want more challenge.
Everyone plays a brief game at home. Then we all come back together and discuss.
I’ll end by suggesting followup activities and challenges you can continue to do at home, and providing additional resources for following up.

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July 27 2020

Achi

Math Monday Recap: Achi

Here are the followup resources from yesterday’s Math Monday.

SLIDES. Here’s are the slides in Google slide format. Feel free to use or copy them.

REPLAY. Here’s the video replay of this week’s session

Next Math Monday: String Figures

In next week’s Math Monday we will be making mathematical string figures.

To prepare, cut a 5-foot length of string (6 feet for adults) and tie the ends together to make a loop, for each person attending.

If you are not already on our mailing list, you can sign up here.

Resources

ACHI AND RELATED GAMES

Achi. Here are the rules, along with the rules for Tant Fant — same board, but each side starts with three pieces lined up in a row.

Tapatan. Same rules, but each side starts with just 3 pieces.

9 Holes. Simpler game with simpler board. Good for very young kids.

9 Men’s Morris. Bigger board, more pieces, more rules. A classic, but I find this game quickly devolves into a “back and forth shuttle” rut.

Morabaraba. Popular in South Africa, Botswana and Lesotho. I like this game better than 9 Men’s Morris, because it forbids the “back and forth shuttle” move.

OTHER TIC TAC TOE VARIATIONS

Tic Tac Toe and other three-in-a-row games is a book by ethnomathematician Claudia Zaslavsky (she writes about how math is practiced differently in different cultures) gives the history and rules for many tic tac toe variations, including Achi and Tapatan. Out of print, but I bought a good used copy through Amazon and it was worth it — a gold mine for teachers who want to include cultural history in their math instruction.

Gobblet Gobblers published by Blue Orange is a wacky superbly original board game in which larger pieces can gobble (replace) other pieces. Requires completely new sorts of strategy.

Otrio, by Marbles the Brain Store is an elegant abstract presentation of 3d tic tac toe as a wooden board game.

Ultimate Tic Tac Toe is a mindbending recursive version of Tic Tac Toe — each of the nine squares contains a smaller 3×3 tic tac toe board, and to win a larger square you must win the game in the smaller board. Here are videos of the rules, and a winning strategy. And this site lets you play against the computer, or with a friend across the internet.

Multi, by Synapse School math educator Federico Chialvo, uses rules similar to Ultimate Tic Tac Toe, but adds an ingenious educational component that teaches kids to think about factoring and multiples.

OTHER ONLINE MATH RESOURCES

James Tanton has compiled a master list of online math events and resources here. As soon as we’re ready to scale up, this will be a great place to publicize ourselves.

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July 27 2020

Hand Shapes

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July 27 2020

Counting Patterns

Last Math Monday: Counting Patterns

This last Monday we played games that involve making and counting patterns, using pennies and other objects around the house. It sounds simple, but it’s an important and richly creative activity that many teachers repeat daily as a warmup (look up “counting collections” for details). And it’s a fun thing to do with young kids — there are so many things around the house to count! The key is to not just count things one at a time, but to group things so they are easier to count — counting by 2s or 5s makes it go faster, and is called “skip counting”.

Here’s the slideshow from the event, including instructions for games you can play at home.
And here’s a video replay of the event.

Activity: Counting and Grouping

Here’s the basic exercise, so you can play along at home.

Get a bowl of 20 to 40 identical small objects, such as pennies, beans, checkers, or buttons.
Put a bunch of the objects on a sheet of paper, placemat, or tray.
Estimate the number of objects, and write it down.
Count the exact number of objects, and write it down. How close were you? The more you play this the better you’ll get at estimating.
Now try dividing your objects into some number of equal piles. Start by seeing if you can divide your objects into 2 equal piles. If that doesn’t work, try dividing the objects into 3 equal piles. Keep trying until you find a number that works.
If you CAN’T find a number that works, then congratulations, your number is PRIME, which is the name for a number that can’t be divided evenly into smaller groups. For instance, 7, 11, 13, and 20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 are prime numbers (this last example is the largest prime number found without the use of a computer).
If you CAN find a number that works, then congratulations, you number is COMPOSITE, meaning it can be composed of smaller numbers multiplied together. For instance, 10 is a composite number because it can be divided into 2 equal groups of 5 (10=2×5).
And if you CAN divide your number even into smaller groups, trying then dividing each of your smaller groups into yet smaller groups. For instance, 12 is 2x2x3, and 100 is 2x2x5x5, which can be visualized as:

Links and resources

Last Math Monday we watched these movies:

The Black Diamond Precision Drill team (12 athletes marching in recombining patterns): https://youtu.be/thmn21wkFGY
Watch the mesmerizing Factor Conga (counting from 1 to infinity one dot at a time) at: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Wolves and Sheep

Last Math Monday: Wolves & Sheep

This last Monday guest host Spencer Bowen, who leads the San Francisco math circles, led us in a classic mathematical puzzle called Wolves & Sheep. Solving this puzzle is well within the reach of all children (and even adults), and requires a willingness to keep trying things until you find a solution.

Here’s the slidesh[ow from the event, including instructions for games you can play at home.
And here’s a video replay of the event.

Make up your own Variations

I independently discovered essentially the same puzzle many years ago when I was writing puzzles for Discover Magazine. I called it “Queens at Peace”, and I framed it as a chess problem — place 10 black and 10 white queens on a chessboard so queens of opposite colors don’t attack each other. See the attached puzzle from Discover for details.

As a puzzle designer, I often invent new puzzles by taking an existing puzzle and changing one of the rules. The puzzle I started with was the classic “8 queens problem”, which is to place 8 queens on a chessboard so no two queens attack each other.

I encourage you to make up your own variations on the Wolves & Sheep puzzle. Keep at it, and it’s quite likely you’ll come up with a question that no one else has ever asked.

Start by considering what happens on larger (or smaller) boards. You can also try other types of pieces. How many black and white rooks can you place on a chessboard so rooks of opposite colors don’t attack each other? How about knights?

You can also vary a puzzle by keeping the rules the same, but changing the story. What characters could you use instead of wolves & sheep?

Links and resources

The wolf/sheep illusion on the first slide is from this video on YouTube, which contains a dozen pictures that can be seen in two different ways.

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July 27 2020

The Math: Topology

The Math: topology

Mobius strips are studied in an area of math called topology, sometimes called “rubber sheet geometry”, which studies properties of strings and surfaces that are allowed to bend and stretch, but not break or pass through itself. Mathematicians joke that a topologist can’t tell the difference between a donut and a coffee cup, because as this picture shows, one can be reshaped to make the other:

(This picture is from a rather silly movie by the wonderfully crazy mathematician/artist Henry Segerman, who loves to 3d print his ideas.)

Topology is only a little more than a century old, and includes knot theory, which classifies the different ways that a loop of string (or several loops) can be knotted. Here’s a fascinating video by my friend Carlo Sequin at UC Berkeley explaining the basics of knot theory.

Movies and Instructions for making your own shapes

Here’s Vi Hart’s fabulous video about Möbius strip made of Fruit by the Foot. She’s a popular YouTuber who yes makes super popular and entertaining movies about math crossed with art and music, plus a good deal of rapid fire irreverence.
Karl Schaffer shared the handout he uses with students for exploring what happens when you cut up THREE loops all joined together.
Here are instructions for making a Möbius strip by adding a single cut to a square. Artist/sculptor Max bill made a beautiful metal sculpture in this shape, as well as making many other sculptures equivalent. I showed everyone how to do this last Monday.
And here are my instructions for cutting and folding a Möbius strip made of just six squares and four triangles. It doesn’t look like a Möbius strip, but trace the edge and you’ll see that its one edge is in the shape of a square.
Jeanne Lazzarini, who attends Math Monday regularly, sent me this: Have you heard of BIG Architects Group? Founded by Bjarke Ingels from Copenhagen? Ask me about him some day…Bjarke’s innovative buildings make me think of stretching the boundaries of mathematical artistry and he’s gained world-wide acclaim for his buildings! —> Be sure to check out his möbius building — https://inhabitat.com/big-unveil-massive-mobius-strip-library-for-kazakhstan/big-diag02

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July 27 2020

Last Math Monday: Mobius Strips

Last Math Monday: Möbius Strips

Last Monday we got cut up and taped strips of paper to make Möbius strips — one of the stranger and more surprising objects studied by mathematicians. It was invented by mathematician August Ferdinand Möbius in 1858.

We traced a line around a Möbius strip and found that it only has one side (“surface”), unlike an ordinary sheet of paper. We tried cutting one in half and found that it fell apart into one piece (instead of two), and cutting it in half again created to loops that were woven together. And we found that if you tape two (regular) loops together at a right angle and cut both in half, you get…a square frame. Weird.

Here’s the slideshow from the event, including instructions for games you can play at home.
And here’s a video replay of the event.

Video replay

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July 27 2020

The Math: Types of Transformations

The Math: types of transformations

When you repeat your name, there are four ways you can change it. These are called geometric transformations. You can:

Translate it — slide it to a new spot without changing its angle. Like this: RR

Rotate it — turn it to a new angle. Like this: NZ

Reflect it — flip it over so the letters go the wrong way, like in a mirror. Like this: db

Scale it (also called Dilate it) — make it bigger or smaller. Like this: oO

Here’s a word pattern that uses all four geometric transformations. Can you read what it says? Can you see how the word is transformed? Hard question: how many copies of the word TREE are here?

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July 27 2020

Word Patterns

Last Math Monday: Word Patterns

Last Monday we drew different patterns by repeating our first name.

Here’s the slideshow from the event.
And here’s a video replay of the event.

And here’s a pattern I made with the name CHARLYE. Notice that this pattern contains two different pinwheels, one with R turning into L, and another with C turning into A. I had to make the Y and E narrow so they wouldn’t collide with the upside down CHARLYE coming at it from the other direction.

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