Thinking in Threes

My seven year old daughter and I are really quite a team.  One of us will share an idea or an observation and all of a sudden the other of us will be inspired into action.  For example, she found a triangle in this clover last week and I immediately knew what we could do with it.



And, although she sometimes eschews direct participation in projects I think up, my kid is generally always around as I'm making something.  In this particular case, I chatted with her about what I was doing while I glued and pasted, and she made a lot of observations, which is good enough for me.

Isn't it cool?!?

It's a (dried) clover Sierpinski triangle!  We picked clovers, flattened and dried them in a sketchbook, and finally found some time (and a glue stick) to paste them down.  I don't know about my kid, but I am really enjoying all the different types of Sierpinksi triangles we've made over the last few weeks: out of candies, with our straight edge and ruler, with our colored pencils, with money, and now this!

It's fun to find math, wherever we go and it's even more fun to make math out of the things we find. 

A Whole Lotta' Triangles Goin' On, and On, and On...

We've been on a triangle kick around here, of a Sierpinski nature.  When we made this fractal with candies, a question entered my mind -- after you divide a triangle by connecting the midpoints of each side, is the Sierpinski triangle defined by the three outside triangles or the one center triangle?

My answer right now is: probably both.

When we built a Sierpinski triangle with colored candies, two different ways, my seven-year-old really did not see the structure.  She saw lots of other things, but not that.

So, I pulled out a sheet with the triangles already drawn in and we colored it one recent morning.  The first, largest middle triangle we left blank.  I asked her to pick a color and color in the next-largest triangles, of which there were three.  And then the next largest (pink), purple, yellow, orange...

"The triangles get smaller...It's so pretty!" she gasped.



















What we both noticed is that that center triangle then gets surrounded by three more, smaller triangles, and the pattern repeats. 

This afternoon, the kid was eying the change jar... "Mama, I have a GREAT idea!"

"What is it?"

"Let's do some money math!"

"But you are really good counting money, why do we need to do that?"

"Well, it'll be good for both of us.  I get the money, and you get the math."

"Hmmm...."

"We'll do triangles with the money!"

"Well, okay, but I've not made up my mind about what we'll do with the money once we count it..."

Kid emits a heavy sigh: "Okay..."

First we made a passable Sierpinski triangle with pennies and counted them up. 27!  It was great she wanted to do it because although the pattern seems easy to identify, constructing one this way is still a bit challenging and we both still need some practice.

(And my mind is reeling with a question I can barely pose, let alone answer -- the numbers 3, 9, 27 etc. usually refer to the number of triangles in the fractal, but it seems that when we build it out of candies or coins, the numbers of individual pieces share the same pattern, but in a different way...)

The quarters came next, both versions a great opportunity to talk about each triangle (3 coins=small, 9 coins=medium and 27 coins=large) needing three triangles to complete the next larger triangle. It's a similar observation to the one we had when we were coloring, but sort of reversed.  Instead of getting smaller by threes, this one is getting bigger by threes.  Hmmmm....

Here's what I know right now:

Things are not always as straightforward as they seem, especially fractals. Having to teach something, anything, to a child always reaffirms this point for me.

Coin currency is a way awesome math manipulative, on so many levels.

I've got two more Sierpinski triangle activities up my sleeve, one of which is about growing the pattern (but not with chocolates or candies), the other about dividing it into smaller parts (but not on paper).  No hints on these, though!

Also, If you have any photos to share of money math you're doing with kids, or Sierpinski triangles, or math you find around the house or when you're out and about, please don't hesitate to post a photo at my new Math in Your Feet Facebook page!  We'd love to see what kind of math-y things you've been doing or making!

Weaving Geometric African Motifs: Part 2

I've got paper weaving on the brain.  Everywhere I go I see new designs I want to try out, but I'm still working through an incredible source of inspiration, the Woven and Constructed exhibit that happened in conjunction with the recent Lotus World Music Festival.  You can read more about my first design here.

My second design, also inspired by an African textile, was much less difficult to plan and execute but, in a way, I found the simplicity and balance to be much more pleasing.  Here is the big picture of this Kasai "Velvet" Prestige Raffia Panel from the Democratic Republic of the Congo:

And here is the part I decided to reproduce in the paper weaving realm -- it's actually something called a raffia flat stitch on raffia cloth:

I was also inspired by the squares in this design.  It's country cloth made by the Mende or Vai people of Sierra Leone.  This one is actually a piece of weaving, with a cotton warp and weft and indigo dye:

I love the idea of a square turned on its corner, so I played around with a bunch of different options:

Sweetly Sierpinski

Day One
Me:  I'm going to make a triangle out of three candies.  Can you copy it? Good! Now I'm going to make this triangle bigger by adding two more triangles.  Let's see if you can make the same as me.

Kid:  Look, Mama!  This side is red, yellow, red, yellow.  And the other side is yellow, orange, yellow, orange, and the third side is...

Me:  Great!  So let's see if we can make another triangle just like our first two, except with different patterns. [Kid starts right in...]

[I've been thinking lately about the importance of modeling inquiry, especially in the math we do.  I want her to not just identify patterns, which she's good at, but also consider those patterns malleable to the whims of her own curiosity.  Also, if I ask a question that leads to my desired result it's generally a lot more successful than giving her a direction.]

Me: What would happen if we added on two more triangles the same size but with different patterns?

[With these sized candies it was easier to build the fractal structure if we made each 3-candy triangle out of three of the same color.  I also had to point out that the right and left sides of the top triangle had to be extended diagonally to make this work, which was a bit challenging for her to see and do.]

Kid:  Oh look!  There's a triangle in the middle!

Me:  Let's take this big one apart and see how many different colored candies we used. These three columns each have six candies and the orange row has how many...?  How many candies is that all together?

Me: I wonder if we could make another big triangle using the same candies, but different color patterns?

Day Two
I have been waiting for months and months to use this sheet I found here.  It's meant for older kids, I think, but we adapted it just fine for the candy approach.

A Seven Year Old's "Math Thought"

Kid, scrambling for some paper and a pencil: "Mama!  Hold on a second, I have a math thought!"

Silly me, I just thought she was trying to avoid having her hair brushed... It took her three tries, but on the third try she had it just right.

Me: "Tell me about it..."

Kid: "This is a square, and if you draw two triangles you get a rhombus."

"Scrar" = square
"Robis" = rhombus
"T" = triangle

Me: "That's really cool!  How did you come up with that?"

Turns out it was this funny little geometric-solid-ish toy I bought her for $6 at the local children's science museum. This is what she saw:

Weaving Geometric African Motifs, Part 1

When I found out about a new exhibit of African textiles and baskets I knew I had to go.

I was in the middle of the first flurry of exploring mathematical paper weaving inspired by Patrick Honner's guest post over at Moebius Noodles.  At the time, I was having fun experimenting with warps made from various combinations of multiples as well as the different patterns created by a weaving algorithm and its inverse, and using Fibonacci numbers to create interesting designs.  So, a chance to see traditional, often geometric, weaving designs up close and personal was definitely something I did not want to miss.

The Woven & Constructed exhibit was fabulous.  It was something else entirely to see these incredible, LARGE pieces hung with care, side by side, ceiling to floor. Photos are one thing, but these pieces had a presence that is hard to describe, but felt by everyone in the gallery.

Over the next few weeks I will be working on woven paper designs inspired by the motifs in some of my favorite pieces from this exhibit.  Here is my first design which is, I realized after the fact, inspired by a design that is pieced rather than woven.  This may have been why it was so hard to weave!

Here is a close up view of the picture above, a pieced design from the Democratic Republic of the Congo:

Here is a sketch in my book, as I tried to figure it out (and a sneak peek at some other ideas):

And here I worked to graph it out and figure out the basic pattern.  The weave pattern was 4 and an inverse of half (basically), and the warp was built on multiples of 4.

And here is what the woven piece looks like: 

Oh my gosh, it was SO hard to keep track of the six individual lines of weaving; I had to start over a couple times.  Working with slightly different paper also slowed me down a little. I had decided to use 1/4" strips instead of 1/2" so I could get more iterations of the design in the piece.  I liked the thinner strips but will probably like them better cut from colored copy paper rather than this high-end construction paper used here.

I am really enjoying this dual artistic/mathematical inquiry through paper weaving.  It's easy and satisfying to have an idea and finish a piece in a short amount of time.  And, every time I finish one piece I have questions that lead me right to the next, but different, piece. It's an exhilarating feeling, honestly.  This is exactly the kind of work students and I do in Math in Your Feet and I feel certain that this kind of double inquiry can be similarly structured for paper weaving, even as early as first grade.

What I find fascinating about this little project is that I used a different set of skills today than in previous weavings.  Compared to my previous efforts, there was a lot more analysis and a lot less "I wonder what would happen if...?" kind of thinking.  If I had to choose, I'd probably pick the "I wonder.." approach, but I can appreciate how reproducing other people's designs can inform both my design process and my mathematical understanding.  It was certainly a good challenge to move a design from the inspiration piece to the paper version. 

More to come!

p.s. Join us at the Math in Your Feet Facebook page where I share fun stuff like the math my daughter and I find all around us as well as interesting links to things I find around the web relating to elementary math learning...

Joint Ventures / Spinning Stars

I've been over in weaving land for a while (young children and grids here, inverse operations and multiples here, and Fibonacci here), but recently came back around to our star adventure.  As with most things, my seven year old learns best in an environment of exploration and self-directed learning, so I've been careful to present this star inquiry primarily as just that...exploration.  What's amazing to me is that although she and I are exploring different things about stars, we seem to be on parallel tracks, moving forward together.

A few weeks ago I built the skeletons for the "spiders who spin fancy webs" out of plywood and copper brads, but things were left there for a while.  Yesterday I decided I was ready to pull out all the beautiful embroidery floss we acquired for this project and try spinning some star webs. 

Below are some of my experiments with six, seven and eight pointed stars.  I do love how the different colors look but it was way too futzy for me to get the effect I wanted; mostly it was hard to keep the tension steady between posts.  I quickly determined that rubber bands work best in this activity, and that is now my official recommendation for when you try this at your house or classroom.

The best part of trying to 'spin' these webs was that I saw the six pointed star next to the seven and eight pointed stars. That got me wondering: What it would look like if I drew the stars using the same color for each variety of star within the same circle?  Would I see anything new in the stars, especially compared to each other?

When you read about how to draw stars you often see the words "over two points" or "over three points."  If the number of points = n that means that for me (n, 1) was red, (n, 2) was purple, (n, 3) was green, etc. And, all that means is that you use the starting point as 0, and if you go directly to the next point, that's (n, 1).  If you skip the 1 and go directly to the second point, that's (n, 2), etc.

Here are the six and seven pointed stars (three colors, green is 'over three' points).  I loved the different flavors of even-ness, balance and symmetry between the even and odd numbered stars. 

The eight and nine pointed stars (four colors, blue is 'over four' points):

 And the ten and eleven pointed stars (five colors, yellow is 'over five' points):

 

What thrilled me was that my daughter had a similar but completely separate line of inquiry happening around the same time.  She started her 'spinning' when she saw me doing it, but her drawings (which she made the day after the spinning) actually happened before I had a chance to explore the ideas I shared above:

If math is about asking questions and exploring different ways to answer and understand the questions you pose, then I'd say we've been doing some math.

The absolutely fascinating thing for me is that through this kind of visual inquiry I am not only learning a lot about stars, I also appear to be finding my way to thinking and reasoning numerically.  As I figured how to divide the circle into fifths, sixths, sevenths, etc. with my compass and protractor, I started wondering if there was some kind of pattern that would emerge if I compared those measurements.  In a 5-pointed star, for example, the points are 72 degrees apart and a 6-pointed star is comprised of points that are 60 degrees apart.  I kept an informal chart and didn't come up with anything that seemed newsworthy, but it was very interesting to observe myself open up to this line of questioning.  As someone who is slowly but surely remediating herself in math, I am thrilled that this visual design approach is helping me begin to see numbers as interesting, useful and, perhaps, even friendly!

p.s. As always, a huge thank you to Paul Salomon of Lost in Recursion and the Math Munch blog.  He teaches math at St. Anne's school in Brooklyn but I was lucky enough to benefit from his help and feedback over the summer months via the magic of the inter-webs.

p.p.s. Check out my new Math in Your Feet Facebook page!  I'd love to see you there!

Weaving Fibonacci

I'm still exploring how to use paper weaving specifically to engage elementary students in mathematical inquiry.

So far I've found that paper weaving has the potential to open up discussions of different kinds of symmetries, encourage exploration of inverse operations, and support inquiry into how number multiples can influence design.  That's what I have at the moment, but I'm sure there's more. 

The other night I asked myself, "How could I weave Fibonacci numbers?"  I picked three colors for a warp (vertical strips).  Green=2, Blue=3, and Purple=5.  That gave me a warp of ten.  I had already played around with weaving multiples of five...

...but this was a completely different process.  I wove with a weft (horizontal strips) using a simple over-under weave, but used the same color/number pattern as the warp.

Holy cow!  Do you see what happened?  I got square numbers!  Green=4, Blue=9, and Purple=25!  

It was late but I decided to try one more Fibonacci warp.  Instead of weaving 2, 3, 5 for the weft as well, I played around with a basic over-under weave with red, orange, yellow, orange, red... as the color pattern.  I love the little orange boxes with alternating red and yellow centers that resulted.

Looking at these two pieces again I have a bunch more ideas for playing around with warp, weft and weaving pattern in relation to 2, 3, 5.  But, I so love when artistic inquiry and mathematical inquiry merge like this, to such beautiful and interesting results.  It's what we do in Math in Your Feet and it's clear to me that a design approach is perfect for integrating math with other mediums as well.

p.s.  The Math in Your Feet Facebook page is developing into a fun place for me to share interesting bits of mathematical art and design.  I'd love to have your company over in that space as well!

Bubbly Serendipity: Geometry Edition

This morning the kid was experimenting with bubbles and paint, and found a way to make bubble prints...

...and this afternoon I was scrolling through Geometry Daily (a new minimal geometric composition each day) and found this (which can be found here)! 

Even the colors match -- I love it when stuff like this happens!  

p.s. Check out the new Math in Your Feet Facebook page where I'm 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!

Kitty Census Redux

Perhaps you'll remember the kitty census that happened at our house last fall.  I just re-read the post and nearly fell out of my chair.  Not only is it touching and hilarious (at least to me) it was a completely different activity last year than what happened this morning.

Last year the kid was a newish six.  This year she's a newish seven.

Last year it was my idea; that morning, the kid just wanted to take her cats to the vet.  When I suggested we could do both she agreed, but only so she could take care of her kitties.  She had no interest in data whatsoever.

This year it was all her idea.  I ventured a suggestion about how to sort and classify all the cats, but it was clearly her gig and so I backed off.  I would have preferred to sort by material (stuffed, ceramic, plastic, flesh and blood, etc.) and then sort into sub categories from there, but the kid had other plans, as you'll see:



It may look straight forward, but the sorting process was actually quite involved.  I mean, really, what's the difference between a multi-colored cat and a spotted one? How come the otherwise solid grey kitties with the white paws didn't get put in the multi-colored pile?  The point is, she had to figure out what the parameters were to make those groupings; any one of us could come up with multiple variations on the same theme and we'd all be right as long as we were being consistent with the rules.  I'd say that's real math right there.

One more difference: Last year she was always performing, but unselfconsciously.  For this year's census she was intent on presenting her process and results in a formal way, specifically on video.  She's the one who decided what to say.  As always, I am just the faithful assistant who just happens to ask a few clarifying questions from time to time.



Yep, you heard right.  Our cat Lucy and my daughter have been added to the final kitty census tally.  72 cats!  Good thing only two of them need to be fed. 

p.s. I've got a new Facebook page where I'm 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!