Weaving Inverse Operations, Multiples & Frieze Patterns

It's been a super exciting few days in mathematical weaving land here at our house.  This weekend I figured out some ways to facilitate basic paper weaving and grid exploration for the youngers, riffing off Patrick Honner's Moebius Noodles guest post.   Last night I did a little more searching for how others have managed the logistics of paper weaving and found this fabulous example of warp management which, in turn, inspired some incredibly productive inquiry on my part. Ultimately, all this will turn into something I can do with my seven year old but, for now, let me show you what I did!

Here's how I started the morning, with multiple colors of paper sliced down to 1/2" strips.  Previously, I had tried my hand at weaving with 1" and 3/4" strips, but both were a bit too chunky for my tastes (although they're perfect for the young ones).  My first attempt at the new approach started with gluing the top of each strip down onto a piece of paper, with a space between each consecutive strip of paper.

Below is some experimentation with keeping the weft (horizontal) snug, in yellow, and a little looser, in green.  I love the look, but for a child's mathematical inquiry, I think it's better to keep both the warp and the weft snug, so the final design is as mathematically accurate as possible.

Since I now had some flexibility with the number of vertical strips I started wondering if the number would affect the ultimate design.  Curious about working with threes I started with a warp of nine strips. At this point I was just playing around to find a design I liked.  There's a basic reflection from top to bottom and left to right in each design.

Here is another multiple of three, a six-strip warp -- a frieze pattern, I think!  The design is made possible because of the two-color warp.

It's also where I started of thinking about inverse operations.  Weaving technique requires the use of some combination of overs and unders.  Row 1, from right to left: [3-over, 1-under, 1-over, 1-under].  Row 2 & 3: both the inverse of Row 1, but with different colors.  Row 4: repeating Row 1, but in a different color.  Then repeat!  Even now it seems like magic.  I can't believe I figured this one out.

Why do inverse operations matter?  From what I've read, and the number work I've done with my daughter, I know you can't really fully understand addition until you also understand subtraction.  Same for multiplication and division.  Add/subtract and multiply/divide are each two sides of same process.  It seems simple to our adult brains but it can be a very hard concept for a child to grasp fully.  I'm thinking that a focus on creating a weaving algorithm and it's inverse might really be a supportive numeracy effort. 

Here's another multiple of 3 warp:

After this I tried a warp of four strips.  Each woven 'unit' is made up of three horizontal yellow strips.  I love how the red and the yellow are rotations of each other from left to right, and reflections of each other from top to bottom. 

Also, notice that the second unit is the inverse of the first.  Instead of having the first yellow strip go 1 under, 3 over, the second unit starts 3 over, 1 under.  Interestingly, I usually weave from right to left, but when faced with an inverse, I found myself weaving from left to right.  I did it every time.  It was quite fascinating to watch myself in this process, which is why I suspect this would be really great for kids.  There's so much to learn and understand as you work to create a visually pleasing design. 

Here is a multiple of four, an 8-strip warp.  In this case, it's a basic over/under weave, but the two colored warp gives it some real variety.  I love this one.

Another four multiple, this time putting together the previous two ideas: the weaving algorithm of the first and the two colored warp of the second.  I wasn't trimming the edges on my designs at this point, which I think affects how your eye sees the patterns.

Now I was curious to see what it would look like all one color.  I think using green in both warp and weft brings out the structure in the weaving in a whole new way.  I like the green one untrimmed.

And finally, onto multiples of five.  This one is nice...

But this one is by far my favorite!

The first row is double strips that go [1 under, 2 over, 1 under, 1 over] and the second row is its inverse.  So simple yet very effective.  A ten strip warp is interesting because it's comprised of two, five-strip units.  A nice juxtaposition of odd and even.

It was such a fun day!  I think I understand the symmetries and the inverse weaving at this point.  I'm not sure how or if building a warp out of multiples affects things.  If you have any insights or see anything in my post that is mathematically shaky please feel free to correct me.  And, if I can clarify anything, please do let me know if you have questions.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

Mathematical Weaving, Part 1: Young Children & Grid Games

Patrick Honner's Moebius Noodles guest post about mathematical weaving has been in the back of my head for over a month and a half.  Mathematical weaving employs one of my favorite making materials - colored paper! - and I thought it would be fun to try with my seven year old.  I know the rudiments of weaving, but I wasn't sure how to get started, so yesterday I played around to try and figure out a few things.  It was actually sort of challenging, but I landed on some solutions and new grid games, so I thought I'd share.

I'm not done with my exploration, but what I have discovered so far is a perfect little unit for young children.  I am imagining that the weaving and the games would be completed in an enjoyable collaboration between adult and child over the course of a day or two.

I started by experimenting with loose 1" strips of paper as the warp (vertical strips) but soon found that much too unwieldy for even my adult hands.  The pieces were not connected to each other so they slipped all over the place and I had to use a lot of tape to keep the weft (horizontal strips) connected to the warp, which wasn't ideal.  So, I searched for some advice on how others have made paper weavings.  A quick Google search and I found this video (which is cool, but still too persnickety for the young ones) and this video which, although the cutting is somewhat haphazard, led me to a solution for how to weave paper without tape...

I first decided that a 3/4" width for vertical and horizontal strips made a more pleasing final product to my eyes than 1".  To make the vertical strips I folded a piece of paper in half and used my paper cutter to cut 3/4" strips from folded edge to about 3/4" away from the open edges closest to me.  Essentially, I was creating a paper warp that was still basically one piece of paper.

As you can see, below, the horizontal strips weave in very nicely and don't need any glue or tape to keep them in place if you focus on pushing them gently, but snugly, downward.  For the young ones, at least, a basic over/under/over/under weave is challenging enough.  Using two (or more?) horizontal colors creates visual interest and perhaps even a conversation about the patterns you see: alternating colors both vertically, horizontally and diagonally.  You can also make a connection to odd and even numbers.  Yellow squares in the design show up 2nd, 4th, 6th... places.  Green squares are 1st, 3rd, 5th...

The minute I finished the piece I thought - A GRID!  It's a grid!  The Moebius Noodles blog is very inspirational and a great source of grid games (my favorite so far is Mr. Potato Head is Good at Math) and I always have grids at the back of my mind these days because of them!  Here are some of the ideas I came up with using a newly woven paper math and one of my favorite math manipulatives -- pennies!

Adult: Oh look!  There are three different colors of squares in our woven grid.  I've got some pennies -- I wonder if we could make a square by putting pennies down on only one of the colors?

Adult: That does look like a square. Let's count and see if there are the same number of little squares (yellow, blue, yellow, blue...) that make up each side?  There are!  How many little squares are there on each side?

Adult: But, wait! Look what happens when I push a corner penny in toward the center!  Yep, it lands on a green square!  Let's do it with the rest of the corners and see what we get.  Oh, lovely.  A rhombus.

Adult:  The corners on the rhombus are on the yellow squares.  I wonder what would happen if we pushed them one square toward the middle?  Ooooh, look!  We have another square.  Is it bigger or smaller than our first square?  Each side on our first square was six little squares long.  This square has sides that are...three little squares long.  Cool.

Another exploration:

Adult: Here's a little story about a tiny X who wanted to get bigger.  Can you help him figure out how to help the X get bigger?

Or, how about the tale of some square numbers who also wanted to get bigger?  What little kid doesn't want to grow up?

And, here's my favorite.  It's a 'let's make a rule' kind of game.  The first penny goes in the bottom left hand corner, and you start counting from there.  The first rule here (pennies) was two over, one up.  Each time you repeat the rule, you start counting from the last token on the grid.

You're probably wondering about the buttons?  Well, that's a different rule: one over, one up.  Isn't it cool how they overlap, but not always?  Kids can make up their own rules after a little modeling or you can challenge them to guess a rule you made up and keep it going. 

And then, of course, the final thing would be to leave the pennies and the paper grid mat out to explore at leisure. 

I have some more questions about how to facilitate Patrick Honner's activity with slightly older children (first and second grade-ish).  One of my thoughts is that there is a basic algorithm for weaving that is a combination of overs and ups.  The design in the picture at the top of this post starts on the first line (weaving right to left) as 'two over, one under'.  The next line is different: 'one over, two under' and then the next two lines are actually the inverse of the first two.  Since my seven year old is already a fairly competent weaver, I think giving her some examples of how different combinations of over/under interact with each other would be a good place to start.  I'm also curious whether my daughter would be interested in the mathematical modeling at this stage in the game.  She's still a do-first, map-it-second (maybe) kind of gal.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

Prelude: Spiders Who Spin Fancy Webs

Here's an lesson in the making that I'm really excited about!  Last week, when I was working hard to figure out the basics of stars and their relationship to math, I ran across a really helpful description of how to make and notate stars.  One of the images the author uses is that of spiders spinning webs, one line at a time, between or across points.

Part of the reason I'm looking into stars is because I think there's real educational potential in this kind of inquiry for all ages, including young children. Most of the material I've run across seems to target fifth graders and above.  But, after experiencing how helpful the spider analogy was for me I thought, why not tell my seven year old a story about a spider who wanted to weave a really fancy web and have her spin some webs of her own?  The kid is pretty excited to be a spider in the near future, especially since the project has morphed into a lot of hammering and tons of embroidery floss.

Today I was in the middle of creating the framework for each web (six through ten points) when my daughter ran to get her rubber bands out of the math basket and started make designs (hers on the left, mine on the right).  Even thought it wasn't planned, I thought this was a really cool way to explore the circular structures and posts/points before moving into a more formal activity.











Stay tuned for more developments, but in the mean time, I've got a new page on Facebook. I'd love it if you'd pop over for a visit. Check it out here!

Plant the Star Seeds, Watch them Grow

Oh, what a wonderful day!  The star seeds I've planted have started to sprout.  That is to say that my seven year old is now officially along for the ride!

This morning for our 'show and tell' time she shared her 125 year old spelling book and her newest doll.  I decided I should share something too, so I pulled out my sketchbook from the past week.  I showed her my process of discovering different kinds of stars (the whole story here) including the all-in-one composite stars I made with 8, 10 and 12 points. 

"That's art!" she exclaimed, and was also very clear to point out that some of my other sketches (not colored in) were most definitely 'not art'.  This is an example of the kind of stars she looked at:

Later in the morning she asked me if we could draw together.  I said, "Sure, as long as I can do math stuff."  She agreed and I pulled out my sketch book, and she got to drawing.  Much to my surprise I saw a star emerge on her page.  I didn't think about taking pictures because I was pretty absorbed in figuring out how to make some 10 stars and sometimes pulling out the camera makes her grumpy.



















As she is wont to do, she talked her way through her drawing.  I wish I had paid better attention, but I did notice she started with one star and then divided it on the inside and then colored in each section. She colored in the white space around the rays which created a circle.

Around the circle she drew five rays, colored them in between each ray, and on the outside to make new points/rays.  Eventually it became a triangle.

At some point I realized she was doing her own star proof, trying to figure out from memory the basic structure of the stars I had shown her earlier in the morning.  In the video below I ask her to tell me about her drawing.  I probably interrupt too much, but it's obvious to me that she is working hard to understand how stars are built.  When I listen back to it I also heard a lot of mathematical terminology in her explanation.



I really just wanted a chance to hear what she thought about what she drew.  But, I think it is also a perfect snapshot of a child in the process of creating mathematical meaning for herself. Her star is and isn't like mine or even mathematically correct. But, it is her thinking process and her own little starry path of inquiry, which I find really exciting.  I wonder what will happen next?!?

Found Math

It's been over five months since I pretended I was Tana Hoban, the photographer who found math everywhere with her camera and turned the pictures into wonderful books for children.  Since then our eyes have been WIDE open, finding math just about everywhere we look. The more math we see, the better we become at finding it; the more math we find, the better we are at understanding it.

As my six (now seven) year old and I traveled around town this spring we found lots of shapes and patterns, parallel and intersecting lines, even spirals.  It's been your basic geometry kind of math, but we've had some incredible conversations about what we see and find. 

Lately, though, our math eyes have become remarkably more advanced.  For example, my daughter saw a tetrahedron in ropes staked into the ground, steadying a young tree.   I started the spiral inquiry, but she's the one that started seeing them everywhere we went, even places we go to regularly.  She still notices spirals all the time.  Recently, she found math in the most prosaic of circumstances...a moment of recursion in the restroom mirrors (one on each wall) at a local grocery store.   I guess math is everywhere!

As for me, my eyes have very recently been opened to stars.  I would have never recognized this particular star for what it is without the last week of exploration and inquiry under my belt.  There are actually at least four different kinds of stars in this picture.

And, here's an 8 star I also found today.

The stars and the rest of the photos are from our trip to the zoo and botanical gardens.  My daughter found some flowers that had dropped to the ground and shouted over to me, "Mama, look!  These have five petals!  A Fibonacci number!"

These circles were near the carousel at the zoo.  I love it!  And, that reminds me that, although it was impossible to get a picture of it, the carousel platform was round on the outside, but actually created out of twelve trapezoidal sections leaving an interesting hole in the middle -- a dodecagon!  Geez, I was really impressed with myself for seeing that one.