Wednesday, May 13, 2015

Squaring and Cubing: Decanomial

Me Too is on track to finishing everything I had planned for him this year right on schedule.  He has been working so productively during our work sessions he had time to receive some math presentations before Kal-El has.  Kal-El will get to this soon, but in the meantime Me Too is the first to tackle the decanomial sequence in the "Introduction to Squares and Cubes" (Cultivating Dharma) or "Squares and Cubes of Numbers" (KotU) section of the albums.   MRD has this in its own chapter called "The Decanomial."

There are many "steps" or "levels" or layers to this work and each of the three albums I referenced do these things in a slightly different way and slightly different order.  I read them all, which was handy because everybody had different pictures, and did what made the most theoretical sense to me.  For another perspective, my friend Abbie has posted some of her son's work with the decanomial and in this post she links to a video series that shows a sequence of work for the decanomial.  The sequence they use is a little different than all of the albums I referenced as well.

One major difference between MRD and most of the others is that MRD has the child set up the decanomial from scratch several times, each time a different way.  In reality, in a classroom that is probably how it has to work.  Even if several other kids didn't also need the decanomial that day, it is unlikely that it would survive set up on the floor for five days like ours did (we did this on a Wednesday, Thursday and Monday).  If your child doesn't know their times tables and needs the extra practice, that might be the way to go.  My kids know theirs and would have gone bonkers setting this thing up five times.  We followed the sequence in the KotU album which has you set up the square once and then transform it several times.

First Me Too built the decanomial vertically one factor at a time.  1 taken 1 time, 1 taken 2 times, 1 taken 3 times, etc.,  Then, 2 taken 1 time, two taken two times, two taken three times.  This is the same as MRD step one.  Oddly, MRD has the child set up the decanomial the next time in the same way, except horizontally (1 taken 1 time, 1 taken 2 times...). The red strip winds up along the top instead of down the left side.  As this is a square that seems silly.  If I wanted to have my child build the decanomial horizontally I think I would have them build it so it looks the same in the end, but is constructed using a different sequence of facts.  I would have them build the first horizontal row 1 taken 1 time, 2 taken 1 time, 3 taken 1 time,  taken 1 time, etc.,

After the square was built.

Next Me Too found all of the existing squares (1x1, 2x2, 3x3, etc.,) and replaced them with an actual bead square from the bead cabinet.  We did all of this the first day.

Next we transformed the square using the commutative property.  1 taken 2 times is two and 2 taken one time is also two.  So, we replace the two "ones" with a single 2-bar and then continued such replacements throughout the decanomial.  That ended our second day.

On our third day he combined groups of bars to find additional squares.  For example, he combined 9x1 and 9x8, 9x2 and 9x7, 9x3 and 9x6, 9x4 and 9x5 to make additional nine squares.  I noticed in the EdVid video the guide combines any bars she wanted breaking up groupings in the process.  For example, grabbing two 9-bars from the 5x9 and combining them with the 7x9 group to make a square.  I think that is pedagogically bad.  Combine existing rectangles with another existing rectangle to make a square.

This, by the way, is one of the few times in the elementary sequence when I have not had enough bead bars.  I was short eleven 9-bars even after borrowing every one from the negative snake game.  I threw our bead square on the photo copier and we cut out the rectangles we needed.  I was also short bead bars when we did some of the multiples work, particularly when showing multiples of two-digit numbers.  We borrowed squares to make that work.  I don't think I'll buy another decanomial box, but I'm putting it out there.

After we made all of the squares Me Too stacked them on the diagonal.

"false" squares

Next Me Too exchanged all of the "false" squares for the real ones from the bead cabinet.

And of course he stacked them.  That ended our third day.  We could continue on to the paper decanomial(s).  However, those are on the KotU scope and sequence for year two and Me Too has numeration work that is more pressing.  He has finished multiples, but I would like him to ideally cover factors before beginning the geometric decanomial.  Now Kal-El is taking a turn with the decanomial beads.  He does need to do the paper decanomial(s) this year.  I'll see what Me Too is up to when Kal-El is finished.  If you are doing the MRD sequence you can now do everything we just did but BACKWARDS.  Start with the cube, exchange each cube for squares, lay out the squares as they were before, exchange each square for the corresponding bars, etc.,  Again, I'll see what Me Too is up to.

I will switch to the MRD album to complete the decanomial sequence at this point.  Much of the potential work with the paper decanomial(s) is in an extensions list in the KotU albums. There is nothing wrong with that, but the MRD happens to have a sequence for the extensions and full presentations with pictures so I might try that.  The MRD album has pictures of EVERY step for most of what we did above.  I mean pages of laying out the decanomial each time with one picture per row.  This was handy when transforming the square using the commutative property.  KotU could have used a few more pictures of that transformation.  MRD somehow manage to NOT have a picture of the critical transformation where you combine the bead rectangles to make squares.  That was the one time I really wanted a picture.  KotU had one.  Yea!  In COLOR.  The MRD pictures are also in black and white which is not ideal for this work.  It took several re-readings to determine that their first and second layouts were technically the same except for stripe direction.  Sigh.

I have to make our paper decanomials now (equations on graph paper, algebraic equations on graph paper, products on white and yellow paper).  I am going to store these, or some of these, in our old sensorial decanomial box (Square of Pythagoras box) instead of in envelopes.   One question I have is whether I should make these on graph paper. OR, because we no longer need our sensorial decanomial, I could use my label maker and use black ink on clear labels and put the equations on the actual pieces of our sensorial decanomial.  I could do the numeric on one side and the algebraic on the other.  The benefit of the graph paper is that you can see the number.  However, Montessori kids are so attuned to the color coding that they can "see" the number on those too.  Anyone have an opinion?


  1. I know I responded elsewhere, so just to share here for your readers :) I personally like the idea you've suggested, but also see the benefit to moving on in the path to abstraction - removing the color cue and providing the visual of the "graph". Or do both ways. What I need to do is look ahead into the algebra album to see what foundation is being laid here... it seems to connect, but I can't verify that at this second.

  2. Great Blog,Very informative.Thank you for sharing,Good to see kids having fun as well as learning.Keep up the good work.
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  3. Wow! I can't wait to do that with the kids! Looks fun. Thanks for sharing!

  4. PS - KotU has been updated to reflect the input given here ;) The backwards work was always present, but perhaps not emphasized? And the other ideas ;)

    1. Jessica, you mean the backward sequence in MRD, as well as pictures for the individual steps? Thank you! I was so torn over the idea of purchasing MRD and KOTU (financially not sound for me, but may have to do it anyway), but this makes things so much easier. :)

      MBT, I especially really enjoy your thought processes and sequenced, step-by-step tutorials in these types of posts, as it helps clarify how things are approached in the different albums. You're doing an amazing job with your children and this blog. Please keep it up!