The Montessori Decanomial – BR is making the geometric decanomial.

One of the anticipated lessons in the Montessori Elementary is the presentation of the decanomial.  It is a giant square (literally and figuratively) and is laid out in beads bars.  This is most often accomplished by third year students with legions of adoring first years lending hands with the beads.

The presentations for this work fall into three categories: numeric, geometric, and algebraic.  Binomials and trinomials are familiar to the child that is third year (nine). As far as the attraction in the mind of the child, doing the decanomial is similar to doing giant multiplication or division lessons.

One of my old students laying out the bead bars. 1 red bead = 1 x 1, two red beads = 1 x 2, one green bead = 2 x 1, etc

This layout can take days of precision.  It requires using a ruler to keep the lines straight.  It is important to point out that, as we go, the diagonal is comprised of the squares of 1 through 10. We discuss this and decide that we could replace these squares with the actual squares from the bead cabinet.

Our next visit with the work will have us discuss the commutative law.  We look at the two red beads and the one green bead bar.  Could we make the work safer and more stable if we replaced the two red beads with one green bar?  It says the same thing.  OK.  Can we do that with others?  The great exchange begins – 4 x 3 is the same as 3 x 4, 9 x 10 is equivalent to 10 x 9.

The child needs to spend some time thinking and observing this now.  He will be encouraged to write observations (at least five) in his notebook.

The next visit we discuss the look of the work and his observations.

“We have the square of one.

“Look at something I notice.  The two green bars (2 x1 and 1 x 2) would make a second square of 2.

“Are there any other ways to replace the bars and make more squares?”  All the bars are regrouped to make squares of the varying colors.

At the next visit, we look at the squares and wonder how many of each there are.  We also realize that the two squares of two make the cube of two. Wonder about three and so on. Upon completing the squares to cubes, the cubes are stacked into a step tower.  When we are done, a procession of children heads down to the 3 to 6 room to fetch the pink tower and compare the tower of numerical squares to the beloved early lesson of the pink tower.

These observations must be written in the child’s note book.

We will work later with the cubes moving backwards to the squares and discuss a formula and rule.  “The square of a decanomial in which the ten terms equal the first ten natural numbers equals the sum of the cubes of these numbers.”

I will challenge the rule and see if there is a way for the child to make a universal statement. We will do some problems.

Now, we arrive at the point of discussion for BR – the Geometric Decanomial.

We began with all the squares angling down the kitchen floor. He then filled in all the rectangles.

At our next visit I bring out envelopes labeled 0 to 9. Inside the envelopes are paper representations of the bead bars.  Beginning with the 0 envelope, we layout the squares in a diagonal.  The rectangles from the envelopes 1 -9 are then placed in their spots.  We discuss that there are 10 squares and 90 rectangles. We discuss that the rectangles above the squares are congruent to the rectangles below the squares.

This is made from paper that is painted blue(ish).

Then he is ready to discuss it Algebraically.



Filed under BR, Mathematics, Montessori, Pythagorean theorem

4 responses to “The Montessori Decanomial – BR is making the geometric decanomial.

  1. Anita Pishko

    I am having a bit of a problem finding the algebraic presentation and the language used for this lesson. Can you point me in the right direction?

    • EV

      Is this what you need?

      Materials: The geometric decanomial from the above lesson, Small pieces of paper, Regular sized paper, Pencil, Red pencil

      Presentation: What do you know about codes? How do we make codes? How do they work?
      Wait for it – this will be a rather rambling conversation.
      We are going to make a code to refer to each section of the decanomial square. Instead of numbers we are going to use letters.

      Get out the piece of paper and make a chart. Using the regular pencil, ask the students to identify the numbers that are used in the decanomial.

      then go back and put = signs next to each number.

      What letters do you think would be the most logical to label the numbers?
      That’s correct – since a is the first letter in the alphabet let’s use a for 1 and b for 2

      Place the letters next to the numbers – Use the red pen.


      With the chart in front of you, begin with the square of one – one times one – so we need to say what? Yes, a x a. Write that on one of the little tickets. Then flip the ticket over and write “a” squared. Place the ticket on the over the “1” squared.

      Move to the two side sections – 1×2 and 2×1. Look meaningfully at the chart while you “puzzle” over the way to write it. axb and bxa and place them over the sections.

      Do the algebraic layout the same way you label the geometric layout.

      You may want to say your hand is getting tired and there are other ways to write multiplication when using letters – one is with a dot and the other is by just leaving out any sign.

      Allow the child to help until the child is doing the entire process.


      • The elements of the decanomial need not be in numeric order – a does not necessarily = 1, b does not necessarily = 2… The numbers could be in any order, as could the alphabetic layout – although in algebra, the order is usually a, b, c…

  2. This is such a fabulous, well-written piece. I appreciate your clarity in describing this sequence of lessons. I really enjoyed it and found it helpful for my assignment.

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