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# TRINOMIAL CUBE

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. The Trinomial Cube travels in its own color-matched box, which is part of the process. An amazon.com link: Montessori Trinomial Cube .

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. Three green rectangular shapes are sized A x A x B. The brown ones are two sets of three, one set being A x A x C, the other A x B x B. There are six of the red ones sized A x B x C. The orange ones are also two sets of three, one set being B x B x C, the other A x C x C. Last, the three yellow ones are B x C x 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. Measured up, over and back, this block is (C x B) x A. The one below is (A x B) x C, which of course equals (C x B) x A, as it’s the same block. Duh. 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. Consider the top surface of this arrangement. It’s a square that’s (A + B) on a side. Hence its area is the square of (A + B). It’s clear geometrically as well that this is the same as A2 + 2AB + B2.

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. Here’s a cube that’s (A + B) on a side, with volume (A + B) cubed.  Below is its geometric expansion, A3 + 3A2B + 3AB2 + B3. 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