Fun While it Lasted

The first group (sixth graders) arrived in my classroom after what appeared to be an intense recess. They were, all of them, either drenched in sweat or nursing some kind of injury.

I surveyed the carnage. Turns out half their class was absent this day as well.  I decided that dancing was not in the cards.

"Okay guys," I said, "come sit down near me. Let's talk about our options for today."  I outlined my plan. They had played around with the straws and pipe cleaners last week and loved it.  I had given them a chance to figure out for themselves how the materials worked and they loved the experimentation and play, even going so far as to exclaim "This is better than Xbox!"

Today, I wanted to challenge them.  I showed them a sheet with pretty good but not overly helpful illustrations of the Platonic solids.  I told them they could work individually or in teams - the goal was to see if, as a class, they could make at least one of each solid.  Most chose the octahedron, surprisingly. But by the end of class when the three-person team was finally, after a lot of muddling and helpful argument, finishing up their icosohedron, a bunch of other kids decided they wanted to make one too.

Since class was almost over at that point, it was a race down to the final possible minute.  Because once you make an icosohedron, you also have to spin it!

I love how certain aspects of this solid are made more obvious through...movement!

Later that day the fourth and fifth graders were more up for dancing but they also got a chance to work a second time with the straws and pipe cleaners.  Their particular challenge this time was to build something using an odd number of edges in their starting shape.  This essentially meant three (triangle) and five (pentagon) as the base shapes.  They grumbled. Some made squares anyway.  I reminded them of the challenge.  They grumbled some more but then...

Extra Fancy Math

Here are two brand new varieties of the Platonic solids, the extra-fancy beaded cube and tetrahedron, from the mind of my seven year old:

We had a nice mini-conversation about how many straws/edges she needed to build up from the triangular base to build the tetrahedron.

And, later in the day, the beaded tower with pyramid topper (and flag, which I couldn't get into the photo) was finally complete:

Proving, yet again, that children are like cats.  When you're engaged in an activity (like cutting straws for next week) and not paying them any attention, both a child and a cat will become intensely interested in doing what you're doing, or at least something related.  At least my daughter doesn't lie on top of my laptop's keyboard.

And the Award for Favorite Platonic Solid Goes to....

After constructing all the Platonic solids out of straws and pipe cleaners (some with my daughter's help) our favorite platonic solid is hands-down the icosahedron.  I made this one with one green pentagon end, and one yellow pentagon end.  It 's strong and sturdy, spins well, like a top, and is fascinating to look at from every angle!

Here's another view:

In a past post, I observed that illustrations of these solids are rarely to scale.  I wondered how a icosahedron would compare to the unwieldy dodecahedron I made.  Here's a family portrait of all the platonic solids, all constructed with six inch edges.  See what you think:

Yup!  Looks like a family, to me (Papa solid, Mama solid, sister solids, baby solid, lol)!  But, clearly, the icosahedron, despite more faces and the same number of edges, is smaller than the dodecahedron.  For what it's worth, I guess I have my answer about the scale issue. 

Hmmm...I wonder what I should try next?  And, here's a question: if the kid's not involved in the making, would my obsession be considered parental neglect, do you think, or could I just call it 'enriching her learning environment'?   I'm thinking about one of these (source), something truncated or stellated...

Solids and an Observation on Scale

We've been exploring platonic solids around our house lately.  We've made them using toothpicks and marshmallows.  We've made them with straws and pipe cleaners.  We're even in the process of building a house using straw and pipe cleaner cubes as the building blocks.

Both myself and the kid are learning a lot through the making process, but also from the the fact that we've filled our living environment with the structures.  Because they're so open, there are all sorts of relationships within the solids themselves that we observe by simply walking through a room on the way to somewhere else.  This is one more great example of environment being an effective teaching and learning tool.

The kid is definitely interested in all this, but me?  I have to say that I'm a bit obsessed.  Questions keep coming up, and they are loud ones that are demanding answers.  Luckily, an office supply store near us had bags of 100 brightly colored straws on sale for $1.00 a bag, so we've got enough supplies to go on for a while.

Here is the dodecahedron I made from 6" straws and some pipe cleaners.  Twelve pentagonal faces, 30 edges and 20 vertices.  (I know because I counted, multiple times.  Sometimes the things that seem the easiest on paper are actually the hardest to figure out in real life.)

And here it is compared to three other platonic solids: a cube, a tetrahedron, and an octahedron, also made with 6" straws and some pipe cleaners.

Wowsa. Not only is the dodecahedron much bigger, it's also really obvious that its not strong enough (using these materials) to hold its shape.  

Here is what happened when I cut the straws in half and made another:

 

It supports itself much better and feels and looks, well, more solid than the first.  Here it is in relation to the other solids: 

Interesting!  Remember, three of the solids shown are made with six inch edges and the dodecahedron is made with three inch edges.

Have you ever noticed that when you see illustrations of platonic solids that they all appear to be about the same size?  Here's an example (source):

Or, this (source):

I never would have observed this if I hadn't built a 3D model.  Score another point for hands-on learning! 

I wonder what will happen when I try the last platonic solid, the icosahedron?  Even though it has eight more faces than the dodecahedron the faces are triangles.  I'm thinking that even with using 6" straws the end result won't be as cumbersome. Wish me luck!

Straws, Solids and a Possible Parthenon

I'm trying to brush her hair.  We've got places to go, people to see. She darts over to the straw/pipe cleaner tetrahedron and brings it back to me.

"Mama!"

"What?"

"This tetrahedron does NOT have equal sides."

"What do you mean?"

"Well...[positioning the tetrahedron so that one triangular face is toward her, and pointing through to show the two other triangular faces coming together in the back]...you see these sides are two together, but this one...[losing steam]"

"Those sides are called faces. I think what you're showing me is that there are two faces that look like they're opposite each other, and that there's a third in front that has no match?"

"Yes."

"Well, 'equal faces' or 'equal sides' does not mean there is always a match.  What it means is that each face is the same shape and the same size. [Note to self: How should I really be describing the sides of regular polyhedra?]  For a tetrahedron that means that each face is a triangle shape.  Let's look at the cube...all its faces are squares of the same size, that's what equal means in this case.  It doesn't matter whether there is a match for each side...

"Here, look. Let's count the number of faces of each solid you made with marshmallows and toothpicks...  [counting together, four...six...eight]  It doesn't matter how many faces there are, just that they're all the same.  Hey!  I know, wanna try making an octahedron out of straws to go with our tetrahedron and this cube?

"Mama!"

"What?"

"Let's make a model of the Parthenon out of straws and pipe cleaners!"

"Uh, okay...maybe we should finish building these solids first?  So then we'd have some building blocks to work with?"

"It'll be math and history at the same time!!  That'll be really fun!!"

"You're absolutely right! [rummaging]  Hey, here's that book I got at the library book sale, on buildings of the ancient world.  Wanna see?  Here's the Parthenon."

[Kid, looking at the book, flipping through the pages.]  "Or we could build a castle.  But what would we do for the round towers?"

Never did get her hair brushed.  Didn't really get to building the Parthenon either, but we did build the octahedron and get out the door, eventually. 

This is the first time we've tried making structures with straws and pipe cleaners.  The beauty of platonic solids made with a unit length of 6" or so is that because they're not really, well, 'solid', you easily look through them to the other side which is what prompted the whole exchange, above.  You can also play around with them more easily, experimenting with how they fit together and observing how the relate to each other.   Just as exciting is having them around the house with us, keeping us company.  We've been admiring the toothpick/marshmallow sculptures and solids from last week as well.   

After making all these wonderful structures, it's also been wonderful to share our space with, well, space.  And, I am reminded over and over, that just because I look at something doesn't mean I understand it.  Even for me, as an adult, physically constructing these solids is bringing to light a whole new understanding of structure, relation and order.  As always, this is an amazing (and fun!) learning journey for both of myself and my daughter.   

Marshmallow Math: Solids & Sculpture

Given my daughter's resistance to formal lessons, it really was the perfect storm.  A supervised math activity of new concepts thinly disguised as 'something to do while mama makes dinner'?  Yes, please!

It came about as a sort of semi-premeditated accident: It was time to get dinner ready and there was a lot of prep work to it.  I had the mini-marshmallows and toothpicks at the ready, as well as buy-in from the kid who had seen this post of the amazing, huge, Serpinski-esqe giant tetrahedron built by the kids at Almost Unschoolers the day before.

I laid out some newspapers on the kitchen floor and poured the marshmallows into one bowl, the toothpicks into another.  Completely off the cuff I said, "Here's how you can make a square base, and then build up from there."  I built about one half of a cube and then said, "I really want to make tetrahedrons, but I don't have time now.  See what you can do."

The kid jumped in enthusiastically.  She started by making one cube, no problem. She added on another cube, then two more to make a four-cube base.  She built it up a level, and up a level again. 

"I'm an artist!  I'm making a sculpture!  Let's put these piece under glass, like in a museum. It's marshmallow art."  Despite this enthusiastic chatter she was actually a little worried that the kids at Almost Unschoolers would be mad at her for copying their work. This led to some conversation about copying vs. learning the basics so you have the skills to express your own personal vision, and then, thankfully, our attention was diverted. 

The tower/sculpture of cube units started to lean. 

These marshmallows are tricky!  The kids at Almost Unschoolers made it look easy, but it's not!  There are all sorts of structural problem solving challenges to overcome. I suggested turning her 'sculpture' on its side (pictured below) so it would be a little more sturdy.  We counted how many cube units made the structure.

When she was done, thinking toward my goal of tetrahedrons, I casually mentioned she might try a triangle base.  After some experimentation she started making this:

It has an interesting twisting quality to it and is my personal favorite of all her 'marshmallow art' pieces.

By this time, dinner was in the oven and I had time to experiment myself.   I made a tetrahedron, then attempted a larger one using four tetrahedral units.  It was a pretty strong structure, but only after I figured out that if you're going to connect the tetrahedral unit patterns, you need to share the same marshmallow where the vertices make contact.  (An aside: Spell check said that is was 'vertexes' instead of 'vertices'.  What do you think?)

She worked on another sculpture and then...lo!  The girl created a tetrahedron of her own, then played around and discovered, on her own, how to create an octahedron!

The picture below shows all of today's creations. The 'sculptures' which she created first, are in the back row.  The geometric/platonic solids, created at the end, are in the front.  What's interesting to me, in terms of my daughter's learning process, is that it is more apparent than ever that she really wants to explore and discover new things on her own first and then, when she's had her fill, she comes more easily and willingly toward 'the point' of the lesson. 

Whatever you call this afternoon's explorations it was a fun, full hour of inquiry.  It was also the first time, except for origami, that I've willingly pushed us into 3D math.  I'm learning right along with her, and it's an amazing journey.