Mathematical Star Ornaments

I've finally succeeded in bringing my daughter (age 7.5) completely over to the mathematical stars camp thanks to this post at The Crafty Crow called "Woven Cookie Stars." 

My daughter is (in)famous for knowing her own mind and possessing very firm opinions about what interests her.  More power to her, I say, but it's required some flexibility on my part.  Over time I've been most successful in influencing her interests when I engage in my own colorful math related projects and 'just happen' to leave some scraps around for her to work with...if and when she so desires.

It was in this environment that I embarked upon our mathematical star inquiry this summer.  The details of this journey can be found in my Seeing Stars post, but I will quickly point to the picture below as evidence that although she often feigns disinterest in the 'math-y' things I do, her mind and heart are still permeable and receptive to the beauty and complexity of math, including stars.  Here's a picture from the child this summer showing her thinking about how stars are structured (explained by her in the post Plant the Star Seeds, Watch them Grow):



We really haven't done much with stars since late August.  I was stalled after a lackluster experiment with 'spinning' stars around nails; the whole thing was just too futzy.  But when I saw Crafty Crow's post I knew it was the answer I had been looking for.

I have lots of cardboard lying around.  I have scissors.  I have spice jar lids to trace.  I have embroidery floss.  My daughter saw my first efforts last night and immediately had to try it out herself.  She used up the only sparkly embroidery floss we had on hand, so today we went to the craft store and she chose her own thread -- sparkly gold and purple and some lovely shades of green. When I said I wanted to take a picture of her stars, this is how she arranged them, in all their glory:

Mathematically speaking, she's figured out a lot all on her own  Her stars are all variations of twelve-pointed stars (12 stars).  She started by wrapping thread around the circle, sort of randomly.  From there she very quickly figured out how to make her favorite star, the six pointed star she calls "the Jewish star" which is created using two overlapping equilateral triangles.  She made about twenty of those, and the last ten were completely uniform.  I encouraged her to see what would happen if she made two more triangles on top of the 6 star and lo!  There was a 12 star that echoed her work in the activity I developed for her, described in the post Stars, Factoring and Patterns.

I also encouraged her to try and make an asterisk star and wondered what it would look like if she used a combination of colors.  I really wish I had her aesthetic -- her stars are so gorgeous.

Here is what I was working on while she followed her bliss.  Initially I wanted to see how many 12 stars I could make which led me to play around with layering one color/version over another.  On the left are set of {12, 5} stars, one in each color.  The next column is a {12, 4} star combined with an 12 asterisk.  I think I like the one on top the best.  The final row (far right) is my least favorite but still pretty.  It's a {12, 3} star (see the squares?!) with a thick asterisk. (Here's more about notating stars.)

 
Every time I finished a new version or my daughter tried a new color to overlap the sparkly thread we ooooohed and ahhhhhhed.  It was a delightful morning, to say the least.  Just for fun, I experimented with a 16 star.  A little too busy for such a small space, but I do like the the one on the left.

And, although most of my time was spent trying to keep up with my daughter's demand for "more circles!" I still had time to experiment with a snowflake feel.  'Cause, you know half of twelve is six and six is a snowflake.  I want to come back to this soon.

So, by now, maybe you're itching to get started.  Remember, all the instructions are at The Crafty Crow.  And, remember, both my daughter and I got started by just playing around with the materials. Our first two or three tries each were quickly discarded but the process was incorporated into these darling beauties. Have fun!

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!

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.

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.

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!

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. 

________________

Malke Rosenfeld 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.