MoMath and MegaMenger

Yesterday we visited the Museum of Mathematics in NYC to help out with their MegaMenger build.   The boys had a blast!

Menger Boys
This was our 3rd visit to the Museum and I’m sure there will be many more.  One of the fun attractions is this tricycle with square wheels (sorry for the poor quality of this video):

The MegaMenger project is an incredible project in which people from all over the world are working together to build a giant Menger Sponge out of business cards.  The website for the project is here:

The boys were actually so excited about participating in this project yesterday that we started the day today building a level 2 Menger sponge out of snap cubes.  Although I enjoyed the project, too, I wouldn’t have described my excitement as “build a new level 2 Menger Sponge at 5:30 am the next day excited,” but hey, I’ll take it:

Snap Cube Menger

With that new morning build, there was really  no doubt at all what our Family Math project for the day would be 🙂  We began by simply reviewing our trip to MoMath and some basics about the Menger Sponge.  The specific topics for the day are going to be volume and surface area.  For all but the last movie the questions will revolve around Menger Sponges of ever increasing sizes, like the one being build in the Mega Menger project:

Having touched on the volume of the Menger Sponge in the last movie, we now dive into the volume calculation in more detail.  What I liked here is that each kid had a different way of calculating the volume.  So fun to see the different approaches to counting here!  I showed a third way, too, that has  a sort of surprising twist.  The Level 2 figure we talk about at the end is the shape that we constructed out of the snap cubes that is pictured above:

The surface area calculation is only slightly more tricky.  As with the approach to volume, both boys had different approaches to counting the surface area of the Level 1 Menger Sponge.  It turned out that my younger son’s method was actually the same as mine, so I didn’t add a third counting method here.    Taking through my older son’s direct counting method and my younger son’s method of counting the overlaps was really enjoyable.    We finished by wondering which of these two methods was easier to generalize to the higher level sponges.

Next we attempted to calculate the surface area of the level 2 sponge.  The level 2 sponge is the one that we made out of snap cubes this morning.  Our contribution to the MoMath Mega Menger build amounted to the construction of two of these sponges out of business cards.  The construction from folding business cards took a bit longer than the one from snap cubes, though the business card construction was at 2:00 in the afternoon and was followed by BBQ at Blue Smoke in Manhattan, so maybe I should call it a draw 🙂

The math in this video is the most difficult to follow in this project, but hopefully we work through it slowly enough.  To calculate the surface area of the Level 2 sponge we use the method my younger son suggested in the last video.  We first assume that the surface area is 20 times the surface area of one of the Level 1 sponges (since it takes 20 level 1’s to make a level 2) and then subtract out the surface area that vanishes when two sides touch.  We break down this calculation into two pieces.  The first part is for the middle pieces that touch two other Level 1 sponges, and the second part is for the corner pieces that touch three.  After a 3 minute calculation, we arrive at the surface area of the Level 2 sponge:

Finally – the punch line!  I thought that ending the project with the lengthy calculation above would kill the excitement we had going this morning, so I went in a different direction for the last movie.  Instead of building ever larger sponges, what happens if we start with a sponge of a fixed size and make a Menger Sponge by cutting holes of ever decreasing size in it?    Even thinking about this question may seem strange, but the result is both fun and a little bit perplexing.  Luckily to answer it we can use the numbers we’ve already calculated in the previous videos – we just need to adjust the scale of the sponges.  Adjusting that scale is an interesting lesson all by itself, btw!    After spending a minute or two talking about what tripling the side length of a cube does to a cube’s volume and surface area, we look at the volume and surface area of a the different levels of Menger Sponges with a fixed edge length.  The result is a neat surprise.

So, despite the super early start (!!), we had a really fun morning.  I’m happy that we had a chance to help out the MoMath team with their Mega Menger build.  Hopefully many other kids around the world will get to help out with this project – it is such a great opportunity to hold an amazing math project in your hand.  Exploring the math behind the Menger Sponge seems like a project that lots of kids would love.

Also, if you’ve made it this far and happen to be in the NYC area, head over to MoMath today (Sunday October 26, 2014) to help them finish the build!  And now having finished this morning’s project and written up this blog post by 8:30 am, it is time to take a nap!! ha ha.


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