...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.