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:

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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:

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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

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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).

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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.

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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?

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About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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4 Responses to Fun with pentacubes

  1. Diana says:

    I would show it to them and I think they would find it interesting to see what they missed. I would stay on this topic a bit longer – so much extension work here. Like pick one of the 4 pieced cube structures (like the one that looks like the letter L) – if you drew it in 2D only and you showed them the 2D picture and said – that you covered the whole thing in paint – how many squares did you paint? See if they can visually recall what the 4 piece 3D cube looks like… Or if you had a bigger cube that you could fit exactly around one of those pentacubes – how many more cubes would they need to fill the bigger cube up completely…

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  2. Pingback: Holiday Math and More: Math Teachers at Play #114 – Denise Gaskins' Let's Play Math

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