Fun with pentacubes

Today we started a unit on geometry with the second graders.  The first activity had them explore how many different structures can be made out of 4 snap cubes.  They very quickly came up with the five tetrominoes:

After a bit more tinkering, they declared that those were all the possibilities, but I told them to keep trying.  It was at least a few minutes before the leap into the third dimension was made.  But once they found one, they quickly found the remaining 2 as well, for a total of 8:

 

They even immediately saw the non-equivalence of the top right and the bottom second from the left (I was kind of hoping that there would be some disagreement and discussion about it 🙂 ).

Since this planned part of my lesson went by so quickly, I decided to take a leap into uncharted territory and see what the students could do with 5 cubes.  I had never tried enumerating all the pentacubes myself and didn’t know how many there are (I knew that there are 12 pentominoes).   However, I think that this fact made the exploration process even more exciting for the students.

By far the most interesting part for me and for them was comparing the different 3D structures that they were building and determining whether they are equivalent.

This structure

was built multiple times and each time we were comparing two of them, one of the students would say that they’re different and the another would rotate them to be in the same orientation, thus convincing the first one that they are the same (and the two roles kept switching!).

By the end of class, the students were able to come up with 26 different structures that we all collectively agreed on being different (including all 12 pentominoes).

After class, I decided to look up how many different pentacubes that are altogether (I admit, I was too lazy / didn’t have enough time to do a thorough check myself).  It turns out, there are 29, so the students found almost all of them!  I was impressed.  But even more so, I was thrilled by how excited they were by each new one that they found.

And now the question is how to keep that excitement going tomorrow.  Should I tell them that there are 29 and have them try find the remaining 3?  Should I just show them the remaining 3?  Should I switch to a new topic entirely?