# Complete this sentence: Math is ______!

Yesterday I saw a super cool student project posted on twitter – click the link in the tweet to see the stop motion video:

and a few follow up pictures, too:

The student’s project gave me an idea for a fun activity that would tie together some ideas from arithmetic, algebra, and geometry.  We started by building a replica of the student’s cube out of our snap cube set and then talking through some of the ideas we’d be looking at today:

After the short introduction we moved to the living room to start in on the project.  I like to emphasize the problem solving strategy of looking at easier problems first.  Keeping with that spirit, to start in on the geometry we first looked at a line.

I’m not sure that I did a good job explaining the connection between the algebra and the geometry right away, but hopefully looking at this easier example helped get going with that connection.  Also, because my son mentioned the relationship between Pascal’s triangle and $(x + 1)^n$ in the last video, I decided to count the type of building block we use at each stage so that we return to that connection at the end:

Now we moved on to talking about turning a 3×3 square into a 5×5 square with the snap cube pieces we have handy.  My younger son picked up on a pattern was able to construct the 5×5 square pretty quickly.  He struggled with the algebra (which isn’t surprising, since we’ve not talked about algebra!) so we spent a bit of extra time on the connection to algebra in this movie.

I got so caught up in the algebra in the last video, that I forgot about the geometry.  Instead of moving on to the three dimensional case right away we returned to the 2 dimensional case to talk about dimensions and geometry.  The specific geometry that I wanted them to see here is how the number building blocks of each type we have at a given stage relates to the number of those blocks in the next stage.    Understanding how to count the number of building blocks at each stage helps understand the connection to Pascal’s triangle (although, I did not emphasize this connection at the end, unfortunately):

With all of this background we are now able to understand the student project that was posted on Twitter yesterday.  We look at (i) how this cube arises from sliding a square in the third dimension, (ii) how to count the number of different building blocks, and (iiI) how to write down an expression for $(x + 2)^3$ from our construction.  There is so much fun math hiding in this shape!

. . . and whoops – watching the movie just now it turns out that we did not actually write down the expression for $(x + 3)^3$  Dang!  Oh well, it comes from the numbers on the screen:  $(x + 2)^3 = x^3 + 6x^2 + 12x + 8$.

Now that we’ve understood a little bit about how to make 3 dimensional cubes from 2 dimensional cubes, we tried the next step – understanding how to make a 4 dimensional cube!  Everything works the same way as before, though it is now much harder to visualize so we have to let the math guide us..   The multiple connections that we’ve talked about in the prior stages help us understand the 4 dimensional shape even without being able to see it.  Amazing!

To wrap it all up we went back to the white board to see the connection between Pascal’s triangle and the number of pieces in each of our constructions.  I wrote the two triangles on the board side by side and my younger son noticed the connection between the two number triangles.  I was hoping one of them would see it, so I was super happy when he noticed the relationship!  We finished up just by talking about how cool all these connections are:

The connection between the algebra, geometry, and arithmetic demonstrated by this student project are really are amazing.  Seeing the project posted on Twitter yesterday helped me see a great way to explore these connections – that’s why I love looking at all of the math people share online!  Probably no better way to end the blog post than with what my younger son said at the end of the last video:

Me:  Complete this sentence:  Math is . . . . .

him:  crazy!