1001 Leaders: Exploring the spiral multiplication table
Today, our guest is Joey Grether, an unbridled idealist, parent, and teacher. His journey to math play with kids started with designing tools and devices. Have you ever wondered how Montessori designed all the activities with her beads, or how Cuisenaire came up with the base ten rods? Here is Joey’s story about a colorful multiplication wheel that helps children and their adults explore factorization and primes. You can see how the ideas for student activities come up as a grown-up is playing, exploring, and designing.
In our 1001 Circles series, we feature math circles stories from the point of view of a circle leader, who acts as an invisible tour guide. In 1001 Leaders, the companion series to our 1001 Circles, we put the spotlight on the leaders themselves. What got them started and what keeps them going? What are their math dreams and worries? If you lead a math circle, an engineering club, or an informal playgroup, we would like to hear your story or interview you. Write moby@moebiusnoodles.com to talk about your adventures.
Updates:
- The sites linked here still exist, though the products aren’t commercially available. We hope you can make something similar from pictures.
- The Tesla connection was a hoax. You can see some discussion in the comments. Since the hoax was relatively benign, we are keeping the original write-up here as a memento/archive.
Hi there. I am a high school substitute teacher, graphic designer, and parent, with a degree in Cultural Anthropology. I have an 11 year old and a 3 year old. When the 3 year old was a little younger, I decided to work on a project that would combine numbers and letters. I sketched and struggled for a few weeks. One day I was impatiently waiting for something, looking at the clock as I was dividing up the letters for my project. I realized that if I combined the most used triads, ABC and XYZ, I could fit the rest of the letters in pairs around the clock. Once I did that, I knew how to fit other ideas into the circle: months, degrees, seasons, moon cycles, elements, the color wheel, and the music scale. I called the device The Learning Flower.
We have a lot of magnetic toys and puzzles around the house, so I made the Learning Flower into a vinyl magnet project. You can take apart the petals, and put them together again in a round cookie sheet. My 3 year old, Indara, would play with it on and off. Observing her and listening to other parents’ feedback, I made a simpler device with just colors, numbers, and letters. Indara seems to understand that one a bit better, because its middle disk supplies a picture for her to model. Here is a photo of another toddler playing with it, from a review at Tog&Glam blog.
I was showing the Learning Flower to friends. Some asked me where the math was, and I was kinda thrown back, thinking – math? This tool got 8 overlapping sets of symbols! Why add more? But then I tried anyway. It led me to develop a more complex device, adding days of the week, the periodic elements, and two more layers of numbers. As I played with the clock positions on my screen, patterns slowly began to stick out to me. First I noticed that 15 and 27 were in the 3 o’clock position, then that other numbers there were multiples of 3 too. Hmm… four positions (3, 6, 9, 12) contained multiples of 3, looking like a square. I started tracing other multiples. Then I saw multiples of 4 forming a triangle in positions 4, 8, and 12. Multiples of 2 make a hexagon.
I sketched out the first 12 dozen numbers in a 12-position spiral, using the same positions of the clock. I then tried tracing the multiples of 2, 3 and 4. I noticed there were numbers that just were not part of the first few multiples. I worked on the diagram for several more days, fascinated by the simple patterns. I underlined the numbers that were not a part of the 2, 3, and 4 multiples systems. It turned out that they were primes. The rest of the numbers were prime! And as crazy as it sounds, this realization of how primes are distributed around the 12 positions hit me on 12/12/12! Were all numbers in these positions prime? Mostly. I checked prime lists online to be sure, and all were, except 25, 35, 49, 55, 65, 121… that is, the multiples of 5, 7, 11, and 13. The patterns began to feel more comprehensive.
Since then, I have been developing the device, trying to understand it, sharing it, and presenting it. I started calling it The Number Spiral, but people confused it with the Ulam Prime spirals. On different versions of posters, I call it Number Patterns, 12x Spiral, Web of Compositability, The Symmetry of Divisibility, and Grethers Spiral. Which name do you like the most? My pick and the name of my site is 12x Spiral, because it implies the design.
Why do numbers work like that? As I played with the 12x Spiral, I have developed a theory. For me, and for students using the 12x Spiral, answering that question is the most useful aspect of the device. It leads to figuring out where numbers are placed in each of the 12 positions, by combing the divisibility rules. Every number has a place, and if we can figure out that place, we can figure out what the number’s divisors are. Here are just some of the patterns students can explore while trying to answer that one Why question!
The theory is that numbers are self-organized around the smallest, most highly composite number, 12. The number 12 and many of its multiples (24, 36, 48, 60, etc.) are HCNs: highly composite numbers (with lots of divisors), which are extremely useful for measuring and proportions. Why are there 12 eggs in a carton, 12 inches in a foot, 12 months in a year, 24 hours in a day, 360 degrees in a circle, 60 seconds in minute? Because highly composite numbers can be divided evenly in many ways. For each 12 numbers in one ring around the spiral, there are 4 candidates to be prime. I started calling the 4 positions where these candidates reside the prime fields: positions 1, 5, 7, and 11. When two prime numbers are separated by just one composite number, such as 5 and 7, 17 and 19, or 29 and 31, we call them twin primes.
Here is another way to look at the prime fields. Recall the hexagon, square, and triangle that are the shapes of 2x, 3x, and 4x. The prime fields sit next to the vertical line between the positions of 6 and 12 o’clock, which is the shape of 6x. In other words, all numbers in the prime fields are either 1 higher or 1 lower than a multiple of 6. We also understand this as the equation 6x +/- 1 that, for natural x’s, works for all primes other than 2 and 3 (though not all results of the equation are prime). I found this equation online while I was researching prime number theory.
I like to draw these spirals in chalk or make wall murals. I tried this activity first with my kids, then as math workshops with their friends. I just start drawing a spiral, and color-coding the 12 positions. I make the spirals large, because kids love to run to clock positions or particular numbers when they seek patterns. As the spiral grows, students begin to see where numbers and colors belong. They jump in to help color and draw, making the mural bigger and bigger. You can also use found materials. Once we raked rocks in a dirt lot into a spiral. We invited kids to walk to its center, and to lay out rocks in a triangle shape (like bowling pins) to show more number patterns. Once you start drawing or building a spiral project, it can inspire many math and art extensions.
Now that we are looking at number relationships and tracing the web of multiples, we can explore more complex patterns. For example, let us define a way for numbers to intersect one another. Say, 55 is a product of just two numbers, 5 and 11. Looking at the spiral, mark the multiples of 5, that is, 5, 10, 15, 20, 25… Connect the numbers with lines. They will form a beautiful star. Now follow the multiples of 11, that is, 11, 22, 33, 44, 55… They form a spiral. The star and the spiral intersect at 55 and then at 110. Try it with other primes: observe how the stars and spirals of the numbers 2, 3, 5, 7, 11, and 13 intersect one another. The stars and spirals relate to spirolaterals and Waldorf multiplication stars, but are also different, so you can explore them separately.
Here is another bridge to number theory, made visible by the 12x Spiral. When only two prime numbers are multiplied, or a prime is squared, we get semi-primes such as 35=5×7 or 49=7×7. The first non-prime exemption in the prime field is 25, which is a semi-prime, because it’s 5 squared. Want to know a neat pattern about primes squared? Find the first few on the 12x Spiral: 4, 9, 25, 49, 121. Starting from 5 squared, they all fall into the 1 o’clock position! In other words, all primes squared (besides 2 and 3) equal 12x + 1. Is it always true? How can you make sure without checking all the infinitely many primes?
Parents and math circle leaders who want to have a casual, easy math experience can start with checking the patterns of multiples. If you want to work with kids, draw the 12x Spiral with them! Or you can print the activity page I have created. Young kids can color, and older students can name each number’s divisors. The 12x Spiral poster is more of a reference, a way to invite math into the homes of kids.
Follow the multiples of 2, 3, and 4. See how their patterns become predictable. Then explore 8x, 9x, and 10x. For a more advanced task, seek links between patterns. For example, 5 and 17 are in the same position, next to one another. Are the patterns 5x and 17x linked? Let us find out! 5×2 and 17×2 are again in the same position, 2 rings apart. Look at 5×3 and 17×3. The same position once more, this time 3 rings apart. Ha. Awesome!
You can find patterns like this through simple, quick sketches. I would like people to get comfortable drawing this model, sharing it, thinking about it, and so, helping develop it! It doesn’t matter if our work is perfect, what matters is if we learn from doing it.
Up to now, we looked at positions to find numbers. The inverse game is more challenging: pick random numbers, and dial them in to the correct position on the spiral. The kids will eventually discover the strategy of using the divisibility rules (see below). For example, your number is 180. It is even (divisible by 2). Cut in half, and the result is still even (180/2=90): this means 180 is divisible by 4. The sum of digits 1 + 8 + 0 = 9, so 180 is divisible by 3. Since it’s divisible by 3 and 4, it’s also divisible by 12, and so it is in the 12 o’clock position. Do it for your birth date, your address, the current year 2014, whatever. Once you become more familiar with placing numbers into the web of compositibility, you will know their divisors at a glance.
Here are the rules of divisibility useful for these games. Knowing these rules, any number can be placed into one of the 12 positions on the spiral.
- If the last digit is even, the number is divisible by 2.
- Multiples of 3 have the sum of the digits divisible by 3.
- Halve a multiple of 4, and the result is even.
- If a number is divisible by 3 and 4, it is also divisible by 12.
The more you can understand how numbers relate, the easier it is to process them in your head. Say, you need 187 inches of 2×4 for a wall, but the store only sells it by the foot. How many inches do you need to add? How many feet will you need to buy? If you can visualize where the numbers belong, you know 187 is on the ring that is starts at 180 and ends at 192 in the 12 o’clock position. So you’ll need 192 inches, which divided by 12 gives you 16 feet. Many things use 12x as their systems of measure. If you go on a scavenger hunt, you will find examples such as 72 pixels per inch image resolution, or 12 packs of soda with 12 ounces each.
I have recently figured out how to make a simpler game. Print out the 12x Spiral poster for a visual reference, or draw the spiral using the first 36 numbers. Now take a pair of dice, roll them, multiply the numbers, and see how the result fits into the spiral. For example, if you roll 4 and 5, put the first die on the number 4 on the spiral. Now skip-count around the spiral by fours, as many times as the second die shows – in this case, five times: 4, 8, 12, 16, 20. There you are! 4 times 5 is 20. Another example: you got 3 and 6. Put the first die on the number 3. Skip-count from it around the spiral by threes, six times: 3, 6, 9, 12, 15, 18.
I’m sure there are many more ways to use the 12x Spiral, but I can’t figure them all out just by myself! Want to join? For more exploration, prompt students to look for patterns, try to draw some patterns of their own, or see what they notice about individual numbers. On that note, I am still developing this project and noticing new patterns. I am open to any and all ideas. Your and your children’s feedback will be greatly appreciated!
I believe the national math standard is to have 5 year olds know the first 10 numbers. What if I could get 5 year olds to know their number patterns (multiples) all the way up to 12×12? I think I can, but I need some support. The project is all about improving math literacy, and inspiring people to quit saying, “I’m no good at math, or art, or whatever.” Personally, I would love to teach kids that have rejected or are rejected by schools. It seems kind of ridiculous to me, but so far I have had a hard time getting people to understand and appreciate this project. Even educators seem hesitant to introduce the 12x Spiral. I think it’s because this is so new, while curricula are planned out months in advance and handed down from within the bureaucracy.
I would love to travel the world teaching any student of any age the web of compositability – with chalk murals, wall murals, or rocks even. I think this model is more useful than any multiplication table based purely on memorization. The 12x Spiral combines the multiplication table and the Sieve of Eratosthenes, allowing students to see how all numbers work together as patterns. It could resurrect many people’s interest in numbers, patterns, and math. If that’s too much to ask the universe at this time, I would just like more students to try it out. Maybe I could get a curriculum developing grant, or just sell a few posters to get young minds to explore the patterns. My ultimate lifelong dream is to run a 24/7 cultural recreation center.
Posted in A Math Circle Journey, Make
Shapes Out Of Shapes
What do you see?
People see shapes everywhere. Our minds deal with any visual experience by forming meaningful patterns, using a system called object recognition. Most of the time, we are not aware of this process. But you can make a game out of recognizing and making shapes and patterns.
Take this squash, for example. It doesn’t look much like squash, does it? It looks more like a duck. Because our minds analyze shapes automatically, we see all sorts of shapes where we don’t even expect them to be: a vegetable becomes an animal.
We recognize a shape by association with a shape we’ve already seen. You can invite your kids to play recognition games with potatoes and squash at the grocery store, or clouds and tree bark at the park. Or if your kid is a night-time person, people have been looking for shapes in the stars for thousands of years! At home, make random doodles, paint blots, or clay blobs and ask, “What Do You See?” Share what you see, when you are the one to catch a shape: “The stem of the squash is perfect for duck’s beak, because both are flattened cones.” Talking helps everybody, kids and adults, to grow their math eyes.
A shape is just a shape until…
But object recognition is only the first step! Take the game to the next level by adding your own details to the image your mind just found. Several teachers call this game “A Shape Is Just A Shape Until…”
From A Shape Is Just A Shape PDF book, by the kindergarteners at Stony Point Elementary School
From Bishop’s Blackboard 1st grade
This game gives kids more freedom to tweak and customize shapes than “What Do You See?” does. A shape is just a shape until you add more shapes. Then that shape can be anything!
Integrate in style
The two games I’ve described so far are easy and meditative, because human minds automatically recognize objects. Even newborns are good at that. But building totally new objects out of simple parts is much more challenging. Instead of seeing an existing shape, you need to make a new shape in your mind, as you build or draw it. Since people enjoy challenges, many cultures around the world have games and craft traditions for integrating shapes out of shapes.
From “Tangram” by Wanda Copier
Have you ever played with tangrams? The Chinese game of tangrams is about making shapes out of seven pieces called tans, without the pieces overlapping. Some shapes are abstract, but others are images of animals, birds, houses, and so on. The traditional collections of shapes are like galleries of impressionist miniatures.
In one version of the tangram game, you are given silhouettes of shapes. You have to reverse-engineer them to figure out how to make the same shape out of the tans. Some shapes are easy to make, others can take hours to figure out.
Another tangram game is to build your own silhouettes out of the seven tans. It doesn’t sound complex, but there are hundreds of different shapes that can be constructed with tans, and thousands of variations. Of course, you don’t have to play by these rules all the time. See what kind of shapes your kids can invent with more than seven pieces, or with overlaps.
If you or your kid want to look at more complex shapes out of shapes, try drawing celtic knotwork. Even the most complex knotwork is made of simple shapes repeating throughout the pattern.
If you or your kid are having a bit of trouble replicating knotwork, try using a grid. Grids help with keeping neat the lines, the angles, and the topology of the knots. Thinky Things has a database of instructions for learning how to draw knotwork.
As with tangrams, you can design abstract knots or flowers, animals, and people. One thing is consistent though: no matter the shape of the final product, each finished knot was built out of smaller shapes!
There are many paths to a single destination. How do you make a bird? Sometimes it’s as simple as walking through the supermarket and seeing a tomato that looks like a duck. Or you can go back in time through hundreds of years, to ancient China or Celtic Europe, and make birds out of tangrams or knotwork. Or, if you don’t want to go to the trouble of building a time machine, modern tools such as LEGO and Minecraft are all about making shapes out of shapes.
While learning about shapes out of shapes, your kid will be practicing object recognition, pattern-making, analysis, and synthesis, all of which are used by mathematicians, engineers, and countless other professions. Every time you see a 3D animation, walk across a bridge, or explore an aviary, you are seeing the product of a mind making shapes out of other shapes.
Posted in Make
Jobs for kids: here, now, and in sci-fi futures
In a Berkeley study, 4-year-olds outperformed adults at problem-solving that required unexpected moves. What if each engineer played with a couple of young kids to work though tough design challenges? #JobsForKids: Scout of Unlikely Possibilities.
A designer dad invited his 5-year-old, and an illustrator mom her 4-year-old, to pass the work back and forth. Deeply original stories, poems, and drawings emerged, reaching worldwide renown. What if art, design, and architecture studios had kid artists on premises? #JobsForKids: Surrealist Artist.
In our upcoming book, Problem-Solving for the Young, we describe the complementary roles of kids and adults in harmonious mixed-age math circles. Here is the table:
| Adults | Children | |
| Ideas | Write ideas down, sort and organize sets of examples, articulate knowledge | Generate diverse, creative, novel, unexpected ideas |
| Mathematics | Maintain consistency of patterns, extend patterns with new examples | Open up and maintain free play, break patterns to create new patterns |
| Process | Organize the process, manage time and tasks, maintain group well-being, nurture | Sense poor management practices, quickly react to dangers (“the canary”), invoke empathy and joy |
| Applications | Connect ideas to many life experiences and examples | Connect ideas to unexpected examples, look at familiar things from new angles |
| Aesthetics | Appreciate order and systems | Appreciate beauty and adventure |
Has your child ever surprised you with a clever solution to a home improvement dilemma?
Is your toddler better at conflict resolution than your whole HR department?
Can you see the future where kids fully participate in advanced, professional, math-rich endeavors?
Send us examples for our new #JobsForKids collection!
Email moby@moebiusnoodles.com or leave a comment.
Posted in Grow
Playing With Math: Newsletter July 1, 2014
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Hi, I am Moby Snoodles, and this is news about Natural Math.
Send me your questions, comments, and stories of math adventures at moby@moebiusnoodles.com
Playing With Math
We at Natural Math are at it again: producing and publishing an innovative, playful, Creative Commons book, crowdsourced and crowdfunded. Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is edited by our Sue VanHattum, with more than fifty people contributing stories, art, puzzles, and lots of math love. You can read more details and contribute to the campaign at IncitED.
This is a much anticipated book, about five years in the making. That’s why there are a lot of excited reviews, and more than half of the target amount collected in just the first week of the campaign. I would like to share quotes from a few of my favorite reviews:
To know Sue is to know that she loves teaching and learning mathematics, and she loves writing, therefore this book — this work of consummate love — has to happen. Playing With Math is really a collection of love stories because the authors, including yours truly, want to share something we’re pretty crazy about. It’s the stuff we do beyond the regular school day — we play with math after hours, at the dinner table, on a napkin at the coffee shop, with our own child or with a neighbor’s child, at a family picnic, with our in-laws whom we don’t even like. – Fawn Nguyen, math teacher
“Math-play adventures.” What an inviting phrase. What a promise, what a fundamentally different, and desperately needed vision of what we could be teaching when we teach math! If play is important to you, if learning, if your children are important, this book could restore your hope, or at least reassure you that there are alternatives, effective, tested, and meaningfully playful alternatives, alternatives that you and your children’s teachers can put into place, immediately. – Bernard De Koven, researcher and author
Each chapter of Playing With Math is written by someone not very different from you: parents who have formed kids’ math clubs, homeschoolers who foster math enthusiasm, and teachers who use math in unexpected ways. The book includes puzzles, games, and other activities as well as a wealth of online resources. It also offers something more vital. The real-life experiences shared by the book’s 30 authors enlarge our vision of the role math can play in our lives, one that’s joyfully creative as well as purposeful. – Laura Grace Weldon, author and GeekMom editor
1001 circles: your math adventures
Bill M. from Chicago emailed about trying out different symmetry activities with his twins: “My two 4-year old girls did not really dig the “symmetry miming exercise” [from the Moebius Noodles book – Moby], but I found something that they enjoyed: folding up paper towels (by squares, by triangles) and dipping each corner into colored liquid. We used water and food coloring like tie-dye. The result was colorful and symmetrical. We could play with symmetry across the various axes and try folding into various shapes to see how it would, literally, unfold. A tip: use lots of food coloring to make rich colors. ”
I love to see kids laughing out loud because they are happy about their math! Check out the cool 3D mesh climbing structure in the background, for body-scale math adventures.
Stephen Taylor is an enthusiastic dad who happens to be an engineer and loves math. Read his guest post for our blog, sharing detailed know-how about running Zometool workshops. Want to try leading your first group event? A one-time, free-building workshop for your children’s friends or classmates is an excellent format. Stephen writes: “I chose a workshop format of “discovery learning” in which just enough guidance is provided to the kids to stimulate their own thinking and creating. I wanted the setting to be very “real world”, like an architect’s studio or a scientist’s lab.”
Naveen posted a question to our Ask and Tell forum: “I had asked a open question to my 7 year old son and asked him to make a rectangle with 48 pieces of snapping cubes. He came up with lots of fun ways and really enjoyed it. Would anyone have suggestions for open challenges with the snap cubes?” Answer Naveen’s question, or see what ideas others shared.
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Talk to you soon! Moby Snoodles, aka Dr. Maria Droujkova
Posted in Newsletter
