Binomial cube is a classic Montessori sensorial puzzle, designed to help develop childโs visual perception of three-dimensional patterns, sizing, grading, and color-matching; while indirectly preparing a child for later algebra and cube root work.

The binomial cube is 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. Based on a Binomial Theorem, the cube has a visual representation of the algebraic formula (a+b) (a+b) (a+b) or: (a+b) 3. However, Binomial cube is a sensorial activity since the child is not really calculating algebraic formula, but rather conceptualizes an abstract math concept in a tangible sensorial way. The prerequisites for this work are the early sensorial works, like the pink tower, the brown stair, the knobbed cylinders, and the color tabs.

Generally, you would introduce Montessori Sensorial Materials in the following order starting at around two years old:

knobbed cylinders, pink tower, brown/broad stair, red rods, color tablets, knobless cylinders, botany cabinet, geometric solids, sound cylinders and bells, touch boards/fabrics/tablets; thermic bottles/tablets, smelling jars, baric tablets, and finally the Binomial cube.

Binomial cube is suggested for ages three to six, but I personally would rather be early, than late. 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, the window of opportunity and excitement is missed, the challenge is no longer there and lack of stimulation would render this activity archaic. So, with Adrian, I started presenting the Binomial cube since he was two years old, and now, at 29 months, he seems very comfortable with it.

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.)

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

Starting with 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 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?

https://player.vimeo.com/video/174915990

On a last note: Binomial cube is a very self-corrective puzzle: even if the child builds it incorrectly, but seems to be intrigued and not frustrated, donโt correct right a way, let the child try to figured it out. Rather, ask questions: "Does red touches red?" and help only if the child seems frustrated.

p.s. Julia built her first Binomial cube at four years old, but I think it was not because of inability to complete the activity earlier, but because of lack of opportunity to do so. I was introduced to Montessori method of education when she was three and a half, and I was four months pregnant with Adrian. By the time I had researched and purchased all the materials, few months had gone by. Would have she been able to complete the Binomial cube earlier? Absolutely! Much earlier? Probably, yes. But there is no way to tell now. So, try to anticipate and be "early" with your child – and he or she might just surprise you and show great interest, perseverance, concentration and motivation to concur the challenge!

I would love to hear your thoughts – how early do you think Binomial cube should be introduced at?

Extensions of a Lesson:

1. Extension 1: build the cube outside the box; then show the child that all the cubeโ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.
2. 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.)
3. 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.