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# Montessori Encyclopedia: Binomial & Trinomial Cubes

The binomial and trinomial cubes are two of the sensorial materials in the Montessori classroom, although they also represent mathematical concepts.

### What are the Binomial & Trinomial Cubes?

As with all Montessori materials, abstract concepts are first introduced to the child through the handling of concrete materials.  The attractive binomial cube puzzle is a concrete representation of the mathematical formula of (a + b)³ = (a + b) (a + b) (a + b) = a³ + 3a²b + 3ab² + b³.  Similarly, the trinomial cube represents the mathematical formula of (a + b + c)³.

Of course, at the ages of 3-6, when children first use these materials, the emphasis is not on learning a formula but rather on refining the visual sense of discrimination (shape, color, pattern) while indirectly introducing the child to the concrete form of the formula. Other indirect purposes include the development of a logical-mathematical mind, critical thinking, refining observation, developing sense perception and developing the imagination.

First introduced as mathematical materials in the elementary years, the binomial and trinomial cubes are some of the few materials that are used throughout all levels of Montessori, from Children’s House all the way up to high school!

### What Do the Binomial & Trinomial Cubes Consist Of?

The binomial and trinomial cubes are made up of red, yellow, blue and black cubes and rectangular prisms that fit neatly together into a wooden box.  There is only one correct way to assemble the puzzle in order for it to fit back in the box, and so it is self-correcting in nature.   Self-correcting materials are beneficial to the child because they don’t require adult guidance.

As children handle the pieces of the puzzle over and over and learn to fit them together correctly, they can readily link together the concrete and abstract forms when they enter their elementary years!

Dr. Maria Montessori originally designed these materials for elementary age students, but when she saw how much the Children’s House students in enjoyed fitting it together as a puzzle, included it in their environment as well!

### Dr. Maria Montessori on the Binomial & Trinomial Cubes

Dr. Maria Montessori said of the binomial and trinomial cubes:

“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.” (Maria Montessori, in The Discovery of the Child)