## Tuesday, August 28, 2012

### 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!

## Monday, August 27, 2012

### 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!

## Thursday, August 23, 2012

### 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!

## Wednesday, August 22, 2012

### 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?!?

## Monday, August 20, 2012

### 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.

## Sunday, August 19, 2012

### Seeing Stars!

It's been a week since I first viewed this picture of a set of 12 stars made by Paul Salomon of the fabulous Math Munch blog...

...and stars have been the best possible obsession since then.

First, I went to the star making applet Paul suggested and made an activity for my seven year old to explore.  We found that stars are great for factoring, visualizing groupings (multiplication), and for observing patterns of both numbers and shapes.

That night I looked into the 12 stars a little further. I constructed as many different 12 stars as I could (there are six, I figured out five) using a pencil, a compass and a straight edge.  (And some colored pencils, obviously.)  Read more about it here.

From there, I had enough information for us to try this:

It doesn't look like a lot, but a hexagon is the first step in constructing a dodecagon, at least the way I do it.  And, the kid did it all herself; there was lots of learning in that simple shape.

The next day we got our own set of stars in the mail!  I ordered a custom-made set from Paul (click there if you're interested in your own set!) and, the first thing after opening the box and sorting through them, the kid stacks them up and exclaims: "They're related!"  Excellent.

Luckily, my husband took some time off from work this week and I was free to take my lovely, sparkly stars and myself to a coffee shop where I spent a couple hours just playing around.  It really helped to have a physical model in front of me and I soon made some interesting observations on paper.  This time I traced the dodecagon centers that were part of Paul's set and built outward.

Thus started my journey of exploring stars by extending the outside edges of a polygon.  By moving outward I was able to find all six 12 stars in one colorful composite star.  Awesome!

Paul wondered if other sets of stars could be included in the all-in-one category.  It was nice to have a question to investigate.  I was a bit stuck on how to make other kinds of stars so this morning I re-watched Vi Hart's star doodling video before heading out to the Saturday market.  The kid was immediately entranced with the whole thing.

Before we could leave the house she just had to draw a few stars, in green, below.  She wanted to show me how she could make a five star and an asterisk star.

She then grabbed a pen and some paper and started drawing individual points in a circle and the resulting asterisk. See that blob on the right side of the picture, below?  She talked while she drew it:

"...all the points are connecting together....through the center....this one across from that one, this one across from that one..."

I love that a simple asterisk inspired all that reflection -- to my adult eye asterisks have almost no merit, but for a kid to notice a structure that I took for granted?  Well, that means we're on our way!

After our market time, I got to go out by myself again and do some more investigation.  I didn't have any nice shapes to trace so I spent some time figuring out how to make ten equal points in a circle.  I was using a protractor, but my first attempt landed me back at a dodecagon.  I tried again and ended up with an octagon, which I figured was good enough.

For the 8 star I extended the edges far enough so I could make two full sets of 8 stars.  It seems that those edges extend infinitely, which means you could keep building infinite numbers of larger and larger sets of 8 stars.  Paul called this 'infinite descent."

Sure enough, the 8 stars could be all-in-one, as well as the 10 stars and I'm betting any stars made from a polygon with an even number of sides would be all-in-one as well. I also noticed that the total number of stars made from any even-sided polygon is half the number of sides.  So, n/2?  (My attempt at notation -- the '/' is the best I could do for 'dividing'.)

Anyhow, I had trouble seeing all five stars in the 10 star series in the picture above, so I went online and found my way to a really fabulous explanation for drawing stars including a description of how to notate them which was super helpful.

As you can see, it transformed my investigation, and improved my ability to notate and describe stars.

Isn't it interesting the difference between stars made inside the decagon and the stars made on the outside?  When have time again I think I'll do each one separately and compare to the inside all-in-one version.

As you can imagine, I have so many more questions than answers after this week of inquiry including how odd number stars work.  I won't be able to keep up the same pace, but I feel like I have enough now to support my daughter's explorations.  I was really pleased, actually, at the thinking she did this week.  What could we do next....?

How about stringing rope around tent stakes in the yard and pretending we're spiders spinning different kinds of webs?  Extending the floor tape hexagon into a dodecagon?  Watching Vi Hart's video after all that and seeing how much more we understand?  Sound like enough for now!

p.s.  I was thinking how far both I and my daughter have come mathematically in the last year and decided to revisit my August 2011 posts.  Here's one from exactly a year ago, August 19, 2011 which I called Spontaneous Math / Math All Around.  Such a sweet journey it's been so far.

## Thursday, August 16, 2012

### Starting Stars: Hexagon/Dodecagon Edition

I was recently inspired by Paul Salomon's thinking about stars and mathematics.  From our conversations I am excited about trying to understand stars better myself, and also about what they might have to offer in terms of elementary math explorations with my seven year old daughter.

Paul mentioned to me that stars have connections not only to geometry, but also to number theory and group theory (a branch of modern abstract algebra).  The visual interest provided by different kinds of stars is really what pulled me and my daughter in, with both of us wondering aloud: What's going on there?

I started my inquiry last night by constructing a bunch of different 12 stars.  Sure I could have gone straight to the cool star applet Paul recommended, but I somehow found myself drawn to the classic geometer tools: pencil, straightedge and compass.  I wasn't sure how far I'd get, but I wanted to at least understand how a dodecagon (twelve sided polygon) was constructed so my kid and I could make one using floor tape in the morning.

I used an illustration from the book Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music and Cosmology as a reference.  I started with a circle 2" in diameter and then made six more circles with their centers evenly spaced around the circumference of the first circle.  From there it was pretty easy to connect the dots to make a hexagon, and use the outer circles to create the points/vertices of the dodecagon....

...and that's as far as the book illustration got me.

Then, looking at a photo of twelve pointed stars Paul had designed and cut out of plexiglass (I ordered my own set, it's coming soon!) I tried drafting my own versions.  Using the basic framework for the dodecagon I made two stars -- one made of two hexagons, the other out of three squares. It looks pretty straight forward, but at one point I really struggled with how to create the points of both stars -- you can see erroneous red lines in both pictures.

And here are two more stars, below; one made from four triangles, and another from a six line asterisk.  There was one more that I couldn't figure out, but I was still pretty satisfied with my efforts.  My lingering confusion is the system and language Paul uses to classify stars; his notation looks vaguely like Cartesian coordinates.  I've read over his explanations to me, but I really wish he were across a table from me illustrating it in real life. What I notice in the picture above, for example, is that the green side of a square intersects with two points, one on either side of the red and orange points.  I see it but I don't know how to talk about it yet.

I did all that last night.  This morning I really wanted to see if we could at least tape down a hexagon on the floor.  Despite the fact that many of my posts seem like my kid and I are in some sort of harmonious learning nirvana, she is actually sometimes quite a resistant learner which was the case today -- she really wanted nothing to do with me and my math.

However, I went ahead and started measuring out 12" edges for the floor hexagon. I had one already measured and cut but had to leave the room for some reason, and when I came back I found her doing this:

She was measuring, taping and cutting as she went and was doing an excellent job at eyeing the angles.  Her fourth angle was a little too big, so I pulled in a hexagon from our pattern block set to show her how to use it to make each interior angle the same.  Resistance flared again, but I had the immutable laws of geometry on my side.  "Well," said I, "it won't really be a hexagon unless all the inside angles are the same."  Silence.

I sat quietly by after that and at some point I noticed she had taken the yardstick to make sure all the sides were the same length.  I heard her mutter something about "...these two are thirteen inches..."  and then I saw she had decided to trim them.  Her hexagon is below -- pretty good! Then she made sure to let me know she was making a square around it so no one would touch it.  It was hers, not the math mommy's.  She even went so far to install a burglar alarm in the form of percussion instruments so she would know if someone crossed over the line.

Given the mood around here today, I'm pretty happy with what got accomplished.  She did a bunch of independently initiated measuring (length and angles), and applied her understanding of equal sides in an analysis of the hexagon she created. And all of that pretty much without me. Later, when we were out and about, she applied the activity to other areas of her life.  One moment found her running up the zig-zaggy library ramp to the exit and exclaiming: "All you have to do is move your feet at an angle and you can run really fast up the ramp!"  In a restroom she saw that the sink's drain were made of small circular holes that formed a hexagon, and congratulated herself that she wasn't calling it an octagon anymore.

I think I'll let it sit for a while, maybe long enough to order some more floor tape in a huge amount of colors.  Perhaps a pile of new tape and some scissors left lying around will be enough to inspire a more collegial attitude around here!

## Tuesday, August 14, 2012

### Stars, Factoring & Patterns

I made a new game!  Well, maybe it's really more like an activity, a really fun activity where stars, factoring, combinations and geometry are all rolled up into one very beautiful package.  My seven year old and I both learned a lot in this first round of what I hope will be a very fruitful inquiry into stars and their use in elementary math learning.

This whole activity is thanks to some recent interactions I've had with Paul Salomon of Math Munch and Lost in Recursion.  A couple days ago, Paul posted a photo of some amazing stars he designed and manufactured himself, using a laser cutter to and 1/4" plexiglass.  Aren't they cool?

I showed them to my daughter and although we both thought they were super cool, we also wondered what in the heck was going on there?  I noticed that the bottom row would fit into the centers of the top row, and I counted twelve sides and twelve rays/points in each star, but other than that I couldn't figure it out.  Paul was nice enough to explain it to me:

"The top row is every possible 12-pointed star. The bottom row is the cutout dodecagon that fits in the middle. Starting on the right we have a dodecagon (every line goes over one corner); then on the next one 2 hexagons (lines go over two corners); then 3 squares (lines go over 3); then 4 triangles (lines go over 4); then a single-pieced 12-pointed star (go over 5); then a a 6-line asterisk (lines go over 6).  Make sense?"

It did make sense, but only in a fuzzy sort of way.  I was still incredibly curious about what kind of math this was.  I thought I saw some geometry (shapes, right?) but I suspected it was more than that.  Here's what Paul told me:

"Geometry yes, but it also has connections to number theory. If you do this with 13 points, for example, every star comes in one piece (nothing like 3 squares), and that's because 13 is prime! It's a cool question to work on. If I have 10 points and I go over 4 points, how many pieces will I end up with?  Images like this also come up in group theory, a branch of modern abstract algebra.

Paul's patient explanation included a link to Vi Hart's video on doodling stars which was really helpful and a very cool star applet for playing around with different permutations (or is that combinations?) of points and lines. As always, I started thinking about how my daughter and I could explore these ideas together.  I have my own inquiry separate from hers, but it always seems to come back to one question: How can I use this new information with a seven year old in a way that is mathematically meaningful?

Tah Dah!

I used the star applet Paul recommended to make a 3 star into 6, then 9, then 12 etc. and then put them side by side on the same sheet of paper.  My thought was that it might be an interesting visual way to explore multiplication and groups (for example, 1 group of 3, 2 groups of 3, etc.) as well as part to whole.  Basically I had no specific ideas about how we were going to explore the sheet until we started, but I think, for the first try, it worked out rather well.  Here's how things went down this morning:

Me: We're going to do something with the number three.  There are three of one thing that we're going to add to another three and see what it looks like then.  Sort of like multiplication.

Kid: I already know how to do that.

Me:  I know, but this is a new and different way to think about it all plus we can use your new colored pencils!  So, let's look at this first triangle.  What can you find three of?

She quickly found each point/corner, which she spontaneously marked with blue colored pencil, a move that completely influenced the way I guided the lesson from that point on.  I've used geometry words with her in the past and I asked if she remembered the word for what we were calling a corner.  She didn't so I had her write down the word vertex.  Then I wondered if she could  find something else there was three of in the triangle. She took her orange pencil and marked the sides, and then I had her write down the word 'edge'.  I thought we were done, but she suddenly found the interior angles and marked them with a pencil.  Awesome.

On the second star I said, "Now we're going to see what happens when two of the same triangles are put together.  What do you notice?"  She started counting individual points and found there were six.  To clarify the 'groups of three' I asked her to find one triangle to trace -- it took a minute but she finally figured it out and, after that she quickly found the second triangle which she traced out in blue.

Me: So, since there are now two triangles, we have two groups of three.  How much is that all together?

She wrote down the number six inside the star using both the blue and orange which I was pretty happy to see.  I would have never even thought to prompt her in that direction, but it was a clear indication that she understood the number was a combination of two different numbers.  As you'll notice, she continued this practice all the way through the 18 pointed star.

Me: So, the next star is three groups of three.  Let's outline each triangle.  [This was challenging for her visually, but a good kind of challenge.]

Then it was time to move on to the next row.  Instead of tracing each individual triangle I suggested putting the same colored dot on each of the three points of any given triangle.  The first time she did this it took some concentration.  The second time she did a star this way it was no problem and she also started to notice that the colors went around the star in a pattern but, all of a sudden, she got suspicious...

Kid: Hey, wait a minute, this one is the same as the other one!

She started making marks while she counted the points and, sure enough, it was the same star twice.  It felt like the perfect imperfection for this lesson.  I always love it when kids discover anomalies or mistakes.  It means they're really paying attention.

By the time we got to the 18 star she exclaimed: "All these stars are making my head hurt!" but I encouraged her to persevere.  After she found the first triangle (with the pink dots) I asked her if she knew enough now to predict the placement of each consecutive color.  She put down the orange and right away saw that, if you go in the same direction (in this case counter clockwise) three dots of a new color always go counter clockwise to the previous color.

Even though I had said we could be done after the 18 star, we did get to the 21 star and it's good we did because I got to make another interesting mistake.  For this last star I wanted to show her a different color pattern I had noticed.  Since the previous star (18) needed six colors to highlight each individual triangle, I had her pick one more color for a total of seven.  Then I asked her to see what would happen if she just put down one color at a time in a sequence until she had used all seven colors. All was going smoothly -- she put down seven colored dots on the points starting with light blue, and I drew two little lines to show where the seven started and ended.  Then she repeated the same pattern (this time she went clockwise, I think) two more times.

It was at that point that I realized something was amiss; that we had visually shown three groups of seven instead of seven groups of three which was one of the main goals of this lesson.  A minor point, but one made much more obvious using the colors, and a result I am still puzzling over.

Here's the whole sheet where we left off:

I think my pictures tell a pretty good story, but in the moment there was some major flow happening.  The colors are not just beautiful visual additions to the designs but also really effective in illustrating the structure, combinations and multiples within the stars. Overall, I'm pretty proud of myself for setting up and guiding this little exploration, but I would love (love!) to hear your feedback on this activity and any ideas you have on what we could do next.

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delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.