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The Mysterious Trinomial Cube
Today, we’re going to highlight on one of our students’ favorite works; the trinomial cube. There’s real beauty behind the intricate details of this mathematical work, details that are hard to notice at first glance.
The direct aim of this work is rather obvious; to build a puzzle in a box. The indirect aim, an introduction to algebra and preparation for the formula a+b+c³. 
The child approaches the box with curiosity. It’s elegant details can spark the interest of anyone. Children are inherently captivated by mathematical materials. The colors, the difference is sizes and heights…”Why these specific colors?”, “Why do they fit together like that?”…
While there are many hidden lessons in this puzzle, the child is actively working towards building a concrete foundation for the abstract nature of the formulation, which they will later expand upon. The puzzle helps the child grasp the concept of a+b+c³ at such a young age, rather than simply memorizing the formula.
Through the child’s discovery, they come to unlock the intricate layers of the trinomial cube.
To begin, the guide carefully opens the box, and lays out all of the pieces, organizing them in different manners. With little, to no words, she places the cubes upon the lid, pairing the colors to one another. Each layer is placed in the box upon completion. The child is then invited to do the work on their own. There are many variations of the trinomial cube that call upon the child’s creativity, imagination and critical thinking.
“These small objects fascinate a child. He must first of all group them according to their color, then arrange them in various ways, making up a kind of little story, in which the three cubes are three kings, each one having a retinue identical to that of the other two, the guards being dressed in black. Many effects can be obtained through the use of this material…when playing with this material, a child forms a visual image of the arrangement of the objects and can thus remember their quantity and order. The sense impressions received from these objects furnish material for the mind. No object is so attractive for four-year-old children. Later on, by calling the kinds a, b, c, and writing the names of the separate pieces according to their dependence upon their own king, five-year-old children, and certainly six-year-olds, can store up in their minds the algebraic formula for the cube of a trinomial without looking at the material, since they have fixed in their visual memory the disposition of the various objects. This gives some idea of the possibilities that can be attained in practice.” (Montessori, The Discovery of the Child)
To see the lesson in full, click on the link below:
Parent Resource: the Trinomial Cube
There is so much to discover using the trinomial cube; a favorite work in our primary community. This blog article, courtesy of the Baan Dek Blog, unfolds all of the intricate details involved in the trinomial cube, and the true beauty behind this awesome mathematical work. Definitely a “must read”!
THE TRINOMIAL CUBE
“There are so many layers to Montessori.
No where is this more apparent than in the simplicity and elegance of the trinomial cube. While the primary aim may seem rather obvious, to build a little puzzle in a box, the secondary aim, an introduction to algebra and preparation for the proof of the formula (a+b+c)3, just blew us away.
The mystery of the material summons our attention. We approach the box with curiosity. We are enchanted by what it promises. We are allured by what might be inside. “What is inside?”, we ask ourselves. With our fingertips, we start to explore.
As we unfold the wooden box and engage, an entirely new world slowly reveals itself. The colors seem to call out to us. Black, red, blue and then yellow. “Why these colors?”, we wonder. Little by little, or more literally, piece by piece, we begin to see more.
“There seems to be an order here.” “Why these shapes?” “What’s going on here?” Then, as if naturally, we begin to form a rough idea of the concept behind the work. “I think I understand what’s happening.” “I can figure this out.”
…While children are inherently captivated by the mathematical precision of this seemingly innocuous puzzle, noting the different heights of the prisms, they are actively working to build a concrete foundation for the abstract nature of the formulation, which they will later expand upon.
…With Montessori, children are constantly working to create the foundations they will need to explore abstractions on their own terms. While many of us encountered this auspicious formula at some point, (a+b+c)3, very few of us had the opportunity to understand and grasp the concept, in such a visceral, hands-on type of way. What’s so key in this process is that it is by discovery alone that children come to unlock the intricate layers.
“ Whereas most of us had to memorize mathematical formulas, eventually solving the problems we faced, children who grow up with Montessori, will understand the problem, before they are faced with the formula. ”
In many respects, they’ll be able to see what it is they are solving…”
Healthy Beginnings Montessori 