Weaving your way through mathematics

Today’s guest contributor to Moebius Noodles is Patrick Honner, an award-winning high school teacher and a passionate math enthusiast. My favorite part of Patrick’s beautiful blog is the Math Appreciation section, because you can adapt most ideas there for young kids. When I was four or five, I spent a lot of hours weaving with paper. I believe it helped me fall in love with mathematics.

Before starting on Patrick’s game, check out his TED talk on creativity and mathematics! – MariaD


Weaving is a fun and creative way to explore real mathematical ideas. Simple “mat” weaving offers a way to experience basic concepts in geometry and number theory, while encouraging the development of representation and modeling techniques– fundamental mathematical skills.

With some colored construction paper cut into long, thin strips, and some glue or tape, you can get weaving right away! Here are a few introductory activities. More examples and ideas can be found at my website: http://www.MrHonner.com/weaving/

A good place to start is the checkerboard pattern. It is simple, intuitive, and helpful in developing facility with the basic techniques of weaving. Start with two sets of strips of different color; align all of one color horizontally and all of the other color vertically. Now, a simple alternating over-under weave will create the checkerboard.

Green-Purple Weave

A more challenging activity with just two colors, each aligned horizontal and vertically, is to weave a tiling of the plane.

L-shape Tiling Weave

This activity definitely requires some planning. Once a type of “tile” is chosen, the weaver must figure out what kind of weaving pattern will produce the desired tilling of the plane. The orange-and-black weave above uses a “short L”-shaped tile and an alternating, 1-over / 2-under pattern. The orange-and-purple weave uses a “long L”-shaped tile and a similar 1-over / 3-under pattern.

Here’s where modeling and representation come into the process. With a blank grid, one can plan out the weave ahead of time, hopefully figuring out what kind of weaving pattern will produce the desired mat. A standard modeling approach can be used, or the weaver can develop their own representation—in both cases, the important mathematical skill of modeling is being developed. Here are some examples of different approaches to modeling various weaves.

Modeling Various Weaves

Through trial (and error!), the weaver can refine their modeling process and their plans to produce the desired weave.

Once the basic techniques of simple two-color weaving are mastered, more interesting and challenging projects can be undertaken. Using more than one color in an alignment (horizontal or vertical) opens up new patterns, as does using more than two colors. More challenging patterns, tilings, and inversions can be attempted. Here are some examples.

Challenging Weave Patterns

A fun mathematical follow-up to introductory weaving is to consider the question “Which kinds of patterns are weavable?” For example, the following two mats weren’t really woven—some pieces were cut out and taped over other pieces. An interesting and highly mathematical question is, “Would it be possible to produce these mats through weaving alone?

Draw 2 Weave

With some basic supplies and a few simple techniques, significant mathematical ideas can be explored through weaving. And once you’ve mastered the basics, you can start investigating circular weavings, hat and basket weaving, and even try your hands at mathematical knitting!

For more ideas, visit www.MrHonner.com

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Posted in Make

Newsletter July 15, 2012

Subscribe | Read online

Welcome to adventurous math for the playground crowd! I am Moby Snoodles, and I love to receive mail at moby@moebiusnoodles.com

Moby Snoodles composes the newsletter

Paul Salomon from Math Munch, a weekly digest of the mathematical internet, writes:

 The Moebius Noodles newsletter was fantastic! I’m expecting my first child in November, a girl. Seeing how well you understand math, I am really excited to have Moebius Noodles on my side as I try to help this girl love life. Thanks so much!

Happy news and glad words, Paul! I bet you will have some good baby math games to share very soon, too!

Book news

One of my favorite aspects of making the book is the transformation of ideas. It all starts from experience design. Then we play with kids, and the designs change – a lot! The ideas become alive. Next, we express these lively ideas in text – and texts are, in turn, represented as pictures. Illustrations, just like texts, go through many drafts and changes. Here is an example from the chapter about iterations of function machines.

November 1998

The cover of the Humanistic Mathematics Network Journal, featuring iteration of a function machine that halves numbers or objects, from the article on young math by Maria Droujkova.

1998 Humanistic Mathematics cover

January 2003

Children intent on working with a function machine that doubles the number of toy frogs.

Function Machine Frogs

July 2012

Two drafts of the “blueprint” page from the “Moebius Noodles” chapter about iterations. We changed from counters to cookies because you can – at least theoretically – break cookies into smaller and smaller pieces, forever.

Cookie Iteration

Our blogs

Kids Consult is an amazing consulting company founded by three girls, who are just a couple of years older than the target audience of the “Moebius Noodles” book. They review mathematics games, puzzles, books and other materials. We asked them to review three chapters from the “Moebius Noodles” book. Check out the full review on our blog!

 Function Machine

  • We love the illustration at the top of this page. It really adds reality to the concept.
  • Even though this is the most “mathy” of the set, it’s also the most interesting. This is probably because it has no pre-prescribed instructions. Even though the others do need directions and examples, try not to get too constrictive – some parents might think that these are the only variations for the activities. Be sure to mention that those are only possibilities.

Yelena and her son had a chat about signs and symbols – a central young math topic that is likely to play a large role in the next book we are preparing to write. Here is a part of their conversation; read the rest at our blog. I love how Yelena casually weaves a math idea of combining symbols into her son’s play.

Son: Mama, when my tree house is finished, I’m going to invite all my friends and put a big sign “no girls allowed”

 

Mom: How would you make a sign like that?

 

Son: Easy, I’ll just make a big red circle with a thick line across like that (draws in the air) and there will be a girl on it, like on bathroom doors.

 

Mom: Ok, but what if your cousin A comes to visit? Can she play in your tree house? (My son loves playing with his oldest cousin)

 

Son: (after some thinking about it) Sure!

 

Mom: But then you need to make a different sign. What would it look like?

 

Son: (after some more thinking) Ok, I’ll just put her picture next to the other sign. It has to be a smiling picture.

Lapware

Moving Fractal Illusion

Are math snakes in the picture moving or are they not?! Most people will see the motion if they keep focusing on different parts of the picture. Show this gem to your kids, for a beautiful “math minute” together. This mathemagical optical illusion, shared by our friend and supporter Science Jim, is an example of a fractal – that is, a self-similar picture. Self-similarity is deeply meaningful for young kids. It is one of the key topics in the “Moebius Noodles” book.

 

Talk to you again July 30th!

Moby Snoodles, aka Dr. Maria Droujkova

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Posted in Newsletter

Kids Consult reviews three chapters from “Moebius Noodles”!

Kids Consult is an amazing consulting company founded by three girls, who are just a couple of years older than the target audience of the “Moebius Noodles” book. They review mathematics games, puzzles, books and other materials. Here is their review of three chapters from the book, and some of the changes that resulted.

Kids Consult:

Function Machine

  • We love the illustration at the top of this page. It really adds reality to the concept.
  • Even though this is the most “mathy” of the set, it’s also the most interesting. This is probably because it has no pre-prescribed instructions. Even though the others do need directions and examples, try not to get too constrictive – some parents might think that these are the only variations for the activities. Be sure to mention that those are only possibilities.
  • Great idea with the actual building of the machines, but it would be good to put some sample photos up of actual built ones.
  • The sentence is way too long, which makes the content very hard to understand clearly and completely: “Use qualitative functions, for example, a machine that adds a sticker to each toy thrown into it by a baby or a machine that finds its mommy for each baby animal.”
  • Well done. This is amazing!

Double Doodle Zoo

  • This is a classic activity, and a favorite with kids. Good choice.
  • However, there may be one downside to this: they might get bored, and shun the activity. So, on this activity especially, though this applies to all, have more and more interesting variations. For instance, suggest that the kids try to make a chain with 2 or 3 different types of figures on it, for example.
  • Excellent job accommodating all the age groups involved.
  • Overall, a job very well done!

Snowflake Symmetry Demo

  • Move the snowflake making instructions closer to the beginning of the document. For kids and parents who start right in at the top, this does not help them have pretty snowflakes, but little messes and frustration instead.
  • Please explain how this relates to symmetry. Some kids may not realize this, and if their parents cannot explain to them what’s going on, you end up with some problems.
  • How does someone study times tables with snowflakes? Please explain.
  • Could this be combined with the previous chapter? It’s very similar…
  • It is great how you explore the possibilities of working with snowflakes. All in all, this is a great thing to try at home.

Question

Is this meant to be for parents to read and kids to look at pictures, or is it for just parents or kids to read on their own? If this is only for parents, then you don’t need as many illustrations, do you?
Special Snowflake Layout

MariaD:

Dear Kids Consult reviewers,

Thank you for the detailed and thorough review. It is very useful in our work. I will recommend your services to colleagues!

About Double Doodle, you wrote: “So, on this activity especially, though this applies to all, have more and more interesting variations” – we have variations inside light bulbs, which symbolize bright ideas (page 2) and the “blueprint” area that shows a few more ideas at the end (page 4). Retrospectively, I see that these ideas don’t necessarily take math to the next level of complexity, like some other chapters. Is it why you thought they were less interesting? Good catch; we will be checking all chapters for that now.

As you have seen, we don’t usually have visual instructions upfront. The layout of all chapters is:
Page 1 – inspiring big picture
Page 2 – short text description, mindmap of keywords, bright ideas
Page 3 – three ages and a picture representing them
Page 4 – blueprints for more ideas

Do you think we should move the basic visual instructions to Page 2 in the case of snowflakes?

The next iteration of the long sentence: “Use qualitative functions, that is, machines that work without numbers. How about a machine (good for babies and toddlers) that adds a sticker to each toy thrown into it? Or a machine that finds mommies for baby animals?”

Kids Consult:
Thank you. We are glad that the reviews were useful.

About the interesting variations, we were just making sure that parents would not think that those are the only possible ones that work.

We think that you should move the visual instructions to page 2 for the snowflakes.

So, is this for parents, kids, or both, to read and look at pictures?

MariaD:
We will play with the chapter layout to see how it works with instructions on page 2.

The idea is that parents will read the book, but kids will be nearby, looking at pictures and maybe discussing plans for games with parents. I picture a parent reading it, with a kid in the lap playing with something else (like a smartphone game) and looking at pictures occasionally.

Many parents are VERY SCARED of math. Pictures help them – they really do. Pictures are friendly and accessible. This was the goal here – therapy, if you will. People who aren’t scared of math will appreciate 3D modeling in the pictures, I think, so they won’t mind that being a picture book either. At least that’s the goal.

 

 

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Posted in Grow

Discovering Signs and Symbols

Lately my 5-year old has been very interested in signs – road signs, signs at the entrances to parks, museums, office buildings, etc. Which led to some really interesting conversations about how rules (and ideas in general) can be represented as symbols.

Unlike written words or letters (which are symbols as well), well-designed signs are much more intuitive and easier for pre-readers and early readers to interpret independently. By the way, have you ever noticed how many of the signs we encounter are the ones that prohibit something rather than inform or encourage? I never did until my son pointed it out saying “see, this sign says no smoking, this – no drinking, this – no music, this – no guns. Signs are for saying “no” to things.”

So back to the conversation that we, my son (S) and I (M for Mama) had a few days ago:

S: Mama, when my tree house is finished, I’m going to invite all my friends and put a big sign “no girls allowed”

M: How would you make a sign like that?

S: Easy, I’ll just make a big red circle with a thick line across like that (draws in the air) and there will be a girl on it, like on bathroom doors.

M: Ok, but what if your cousin A comes to visit? Can she play in your tree house? (My son loves playing with his oldest cousin)

S: (after some thinking about it) Sure!

M: But then you need to make a different sign. What would it look like?

S: (after some more thinking) Ok, I’ll just put her picture next to the other sign. It has to be a smiling picture.

M: What if (names of a couple of girls he knows well from playdates) want to come play? Will you let them into the treehouse?

S: (after even more thinking) Yes. All girls I know can come and play. Only girls who are strangers can’t come. And if they are not very little.

M: Ok, but then you have to change the sign.

S: (sounding a bit weary) I dunno. Put more pictures. (runs away)

We had a few more conversations about signs that were similar to this one. My son would come up with a very broad rule and a sign for it. I would then suggest scenarios that did not fit the rule and he’d adjust the rule. And we’d try to figure out how to create a sign that would accurately reflect the new rule.

Since all these conversations were completely “on the fly”, usually while walking or right after reading a bedtime story. Which, I figured out, is not the best time since we don’t get to put any of the sign designs on paper.

But now I’m thinking what kind of a sign-making game can I put together (something that wouldn’t take too long). Any suggestions? Please share!

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Posted in Grow