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I FIRST encountered a Trinomial Cube—as opposed to its purely algebraic and possibly intimidating namesake, (A+B+C)3—when I lived in the Caribbean in the 1970s and daughters Suz and Beth went to the St. Thomas Montessori School. One day soon I’ll share bits about Maria Montessori and the Association Montessori Internationale she founded. This time, I’m going to dance back and forth between mathematics and early childhood education as encouraged by this artful collection of multi-colored blocks.
Contained in its color-coordinated box are 27 blocks. Three of them are cubes, the blue one the largest, the red one intermediate, the white one the smallest. The remaining 24 are rectangular shapes, not cubes though their sizes relate to those of the cubes.
Let’s call the blue cube size A, the red cube size B and white cube size C.
Notice that you can describe the others in terms of these three basic sizes, A, B and C.
Of course, we’re doing algebra here without calling it that (or being intimidated by the word). The Trinomial Cube is introduced at the Primary Montessori level (ages 3-6), where there’s little math per se. It’s purely color- and shape-matching, together with some very subtle guidance in handling the blocks: Kids are encouraged to pick up each block with thumb and forefinger, the remaining fingers folded under.
You’ll never see a Montessori 4-year-old grasping a pencil like a lollipop. Also, a kid’s preference for right- or left-handedness is respected.
Let’s visit the algebraic side. Multiplication is commutative, that is A x B = B x A. It’s associative, i.e., (A x B) x C = A x (B x C). And it’s distributive across addition: A x (B + C) = (A x B) + (A x C). We take these properties pretty much for granted, but it’s fun to relate them to our blocks.
Now, let’s get a bit fancier:
(A + B)2 = (A + B) x (A + B) = A2 + 2AB + B2. Furthermore, we can look at this expansion in terms of the blocks.
Moving up to the cube (A + B)3 raises us out of two dimensions. But the blocks are equally useful. Look at them below, and you’ll not be surprised by its expansion.
And, when you learn to put all the blocks back with colors matched to the box sides, you’re ready for the full trinomial,
(A + B + C)3 = A3 + 3A2B + 3AB2 + 3A2C + B3 + 6ABC + 3B2C +3AC2 + 3 BC2 + C3.
These trinomial terms are precisely the 27 blocks of the Montessori Trinomial Cube. ds
© Dennis Simanaitis, SimanaitisSays.com, 2012