## A binomial Cube is a classic and unique Montessori sensorial puzzle designed to help develop a child’s visual perception of three-dimensional patterns, sizing, grading, and color-matching; while indirectly preparing a child for later algebra and cube root work.

In Montessori sensorial work, the binomial cube consists of a wooden box with a cover and two adjacent hinged sides, containing eight wooden blocks (prisms) that fit together to form a cube with the same pattern on each side and in the middle.

*The prerequisites for this work are the early sensorial works, like the pink tower, the brown stair, the knobbed cylinders, and the color tabs.*

**The** **Binomial Theorem**

What is a **binomial**?

It is an algebraic expression of the sum or difference of the *two* terms.

(a + b )

It is possible to square or cube a binomial. For example, to square a binomial like a + b, you multiply the binomial by itself. (a + b) ** ^{2 }**= (a + b) (a + b) = a

**+ ab + ab + b**

^{2 }**= a**

^{2 }**+ 2ab + b**

^{2 }

^{2 }To cube a binomial, you ** (a+b) ^{3}**. Thus, based on the

**Binomial Theorem**, the cube has a visual representation of the algebraic formula (a+b) (a+b) (a+b) or:

**(a+b)**=a

^{3 }**+ 3 ab + b**

^{3}

^{3}However, the Binomial cube is a **sensorial activity** since the child is not really calculating algebraic formulas but rather conceptualizes an abstract math concept in a tangible sensorial way.

A binomial cube is suggested for ages **three to six**, but you can definitely try introducing it earlier. When an activity is presented prematurely, the child will let you know, and you will simply have to wait till a later time. However, when an activity is presented late, the “sensitive period” had ended, and the window of opportunity and excitement is missed. Thus, the challenge is no longer there, and lack of stimulation would render this activity archaic. However, when presented early, a child might surprise you and show great interest, perseverance, concentration, and motivation to concur the challenge!

So, with Adrian, I started presenting the Binomial cube when he was around two years old, and now, at 29 months, he seems very comfortable with it.

## Exploring the Binomial Theorem at 2.5 YR

### Binomial Cube | Montessori Sensorial Work Presentation

First, the child would take all the cubes out of the wooden box and group/sort them by color on a table or a mat to help with the sense of order. Then, starting with the largest single-color cube (all-**red**), the child would complete the first level by placing the adjoining cubes so that the same colors “touch.” You may guide the child by saying: “*This is red, and this is black; let’s find a block/prism that also has red and black: red touches red and black touches black.”* Place a finger on top of two prisms to show the child that the blocks are the same height. Once the first level is completed, the child will move it into a wooden box. Similarly, the child would proceed with the second single-color cube (all-**blue**) in like manner. *The control of error*: the child runs his or her finger across the top of the cubes to see if all the cubes are the same height. (Adrian would do it a lot in the beginning, but at two and a half, he can visually discern when one of the cubes is taller or shorter, and he would self-correct.)

## Binomial Cube | Montessori Sensorial Step -By Step Instructions

The child would first group and sort all the cubes by color.

Starting with the all-**red **cube, a child will place the adjoining/neighbor cubes so that the same colors ‘touch,’

* Moving the completed level into a wooden box.Next, starting with an all-blue cube, proceed similarly as with the all-red – filling in the “neighbors”.*

Binomial cube is a great fine-motor activity, which prepares a child for later work with maths, especially algebra; aids eye-hand coordination, promotes concentration, and prepares a child for writing. Can we wish for more?

### Things to Know About Binomial Cube | Montessori Sensorial Work

*On the last note:* a Binomial cube is a very self-corrective puzzle, even if a child builds it incorrectly. As long as a child seems to be intrigued and not frustrated, don’t correct right away, rather let the child try to figure it out. Rather, ask questions: “*Does red touch red?*” and help only if the child seems frustrated.

## Extensions of a Binomial Cube | Montessori Sensorial Lesson:

*Extension 1*: build the cube outside the box; then show the child that all the cube’s faces have the same pattern on them. Turn the cube around to show the child that the back faces are the same too. Place both hands around the cube and lift it – to show the child that the bottom also has the same pattern on it.*Extension 2*: Show the child that the 6 inside faces have the same pattern by splitting the cube three ways: horizontally, vertically, and back to front. (This will expose each face with each split.)*Extension 3*: build the cube outside the box, splitting the layers so that the child can see the pattern on the box top on all sides of the cube. Next, you can build one layer next to the other layer. Finally, you can build the cube in the box without the use of sight.

##### Have you ever introduced a Binomial Cube? Leave a comment!

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## 1 Comment

Hi there, I’m a teacher and K06 tutor very interested in learning how to incorporate this into my teaching but don’t yet understand. Do you have any resources that may help? Thank you! Nicole, NH