Binomial Cube and Trinomial Cube- Montessori

The Montessori Binomial Cube and Trinomial cube are part of the sensorial curriculum area. The material is comprised of 8 wooden blocks which, when put together correctly, create a cube. Two of Montessori’s sensorial materials, the binomial and trinomial cubes, visually introduce the child to the concept of cubes of binomials and trinomials, while providing a fun challenge.

The binomial cube, when first introduced to the child, is presented as a challenging, three-dimensional puzzle. The cube is made up of a number of colored blocks, which fit together in a specific way. Assembling it uses a child’s fine-motor skills and requires the ability to discriminate between the blocks based on multiple characteristics. Unlike Montessori’s iconic pink tower, for example, the binomial cube does not isolate only one quality. Some blocks have one color, others have two. Some blocks are cubes, while others are rectangular prisms. While the pink tower blocks vary only in size, the binomial cube’s blocks vary in color, size, and shape! This makes the binomial cube a more complex sensorial material, and it requires organized thinking to master. The binomial cube’s big brother, the trinomial cube, is a child’s more complex next step.

The binomial cube is a concrete representation of the cube of a simple binomial equation: (a+b)3. The eight wooden blocks represent the expansion of this equation. The top of the box is painted to show what the assembled cube looks like from the top. Each block represents a component of the equation a3+ 3a2b + 3b2a + b3.
The indirect purpose of the binomial and trinomial cubes in the Children’s House becomes the direct purpose when the material is re-introduced in the elementary classrooms. Elementary-aged children use the cube to develop an understanding of the expanded equation (a+b)3 = a3 + 3a2b + 3b2a + b3.

After a child has mastered building the binomial or trinomial cube inside of the box, he may then try building it outside of the box, or building each layer separately in order to observe similarities in patterns. Over time, the child’s familiarity with the cube’s physical aspects will lead to an internalized understanding of the abstract concepts the cube represents.