Binomial Cube
The binomial cube is a wooden box holding eight blocks: a red cube, a blue cube and six prisms with black sides. A child of about 3.5 to 4 takes it apart and builds it again by matching colors and sizes, with the pattern on the lid as a guide. Years later the same blocks show (a + b)³.
Sourced from the texts, not yet presented at home
| Typical age | 3.5 to 6 years |
|---|---|
| Area | Sensorial |
| Before this | |
| Materials | |
| Direct aim | To build a cube from its parts by matching color and size. |
| Control of error | The box and the colors. A block in the wrong place leaves a gap or won't go in, and a red face next to a black one shows the mismatch. The lid's pattern checks the top layer. |
| Leads to | |
| Time | About 10 minutes (our estimate). |
| Grouping | One child |
| Parent prep | None. The box lives on the shelf, closed. |
| Cost | Buy about $30 to $113. Make: better bought than made. |
| Other names | Cube of a binomial, Binomial cube box |
What the work teaches
Direct aim. To build a cube from its parts by matching color and size.
Indirect aims, the ones you will not see at once
- Seeing a small difference between two sizes that look nearly the same
- A whole made of parts that fit back together only one way
- Indirect preparation for algebra: the cube of a binomial, worked out with the same blocks in the elementary years
- A careful hand, lowering blocks into a box that leaves no room for error
Set it out before you invite the child
Eight wooden blocks in a hinged box. Alison's lists its box at 3.5 by 3.5 by 3 inches.
The pieces: one red cube, one smaller blue cube, three prisms with red and black faces and three with blue and black faces. Each red face is a square the size of the red cube's side, each blue face a square the size of the blue cube's, and every black face is a rectangle with one side of each.
The lid shows the top layer seen from above: a red square, a blue square and two black rectangles. Two sides of the box fold down on hinges so the layers show.
Montessori's 1917 version was for older children: a 6 cm cube, a 4 cm cube and six prisms that together make a 10 cm cube, kept in a cube-shaped box. Painting it as a puzzle for 3 to 6 year olds is the later tradition.
A table, with room in front of the box to lay out two layers of four blocks.
Presentation, step by step
- 1Carry the box to the table with both hands. Turn it so the lid's pattern faces the child, and lift off the lid.
- 2Open the hinged sides so both layers show.
- 3Take out the top layer one block at a time. Set each block on the table in the same position it had in the box.
- 4Take out the bottom layer the same way and lay it in front of the top layer.
- 5Rebuild the bottom layer in the box, then the top layer, matching each block by color and size. When two blocks look alike, hold them face to face to compare.
- 6Close the sides, put the lid on, and invite the child to take it apart and build it.
Details that hold the child's attention
Folding the sides down and seeing inside the cube.
A red-and-black prism next to a blue-and-black one: the same black sides, different ends.
Red faces meeting red, blue meeting blue, black meeting black.
The last block dropping into the only gap left.
How the child sees a mistake
The box and the colors. A block in the wrong place leaves a gap or won't go in, and a red face next to a black one shows the mismatch. The lid's pattern checks the top layer.
The words come after the hands
| Words | Shown with |
|---|---|
| cube | The red block, the blue block, and the whole thing when it's built. |
| prism | The six blocks with black sides. |
| layer | The four blocks on the bottom, and the four on top. |
| binomial | The adult word. The child doesn't need it until algebra. |
The three-period lesson, scripted
You need the red cube and one red-and-black prism.
| Period | What you do and say |
|---|---|
| 1. Naming | "This is a cube." "This is a prism." |
| 2. Recognition | "Give me the cube." "Give me the prism." |
| 3. Recall | Point to one and ask "What is this?" |
Our suggestion: only if the child asks what the blocks are called. In the primary years the cube is a puzzle, and nobody names (a + b)³ to a four-year-old.
Extensions and variations
- Building it outside the box, on the table, with the lid as the only guide
- Building it blindfolded, by touch
- Laying out all eight blocks by kind first: one red cube, three red prisms, three blue prisms, one blue cube
- The trinomial cube, 27 blocks in three colors, once the binomial cube is easy
- In the elementary years, (a + b)³ worked out with the blocks and written as a³ + 3a²b + 3ab² + b³
Variations
- Leaving the lid off, so the child builds the top layer without the pattern to copy
Make it or buy it
| Material | Make it | Buy, approx. |
|---|---|---|
| Binomial Cube | Buy instead | $30 to $113 |
Prices noted . Approximate US street prices for home-use versions; they vary by maker and change often, so recheck before buying.
Making it at home
Not worth making. Eight blocks in two sizes have to be cut and painted exactly, and a box with folding sides adds more work.
What to check when buying a set
Check that the box closes with every block in place and that the painted faces are clean. Budget sets do the primary-years job; the algebra use later needs no better set.
Printables for this age
This work needs only the material, so it has no printable of its own. The shelf plan below places it among the other works for the same age.
- Shelf plan for age 3PDF, 1 page, 250 KBEvery work for this age on one page, in order, with a column to date each presentation.
Every printable for one age, by email
The shelf plan and the lesson printables for your child's age in one email, then three short emails on the first lesson, what to make or buy, and what comes next.
Signs the child is ready, and signs it is done
Ready for the first presentation
- Builds the pink tower, the brown stair and the red rods without misplaced pieces.
- Lines up a block's corner with another by eye.
- Enjoys puzzles of a few pieces and finishes them.
- Carries a box level with two hands.
Ready for the next lesson
- Builds the cube in its box with no trial fits.
- Builds it on the table, outside the box.
- Sorts the eight blocks into their four kinds at a glance.
- Asks for a harder cube, which is the cue for the trinomial cube.
Fitting it into a home, not a classroom
| If you have | Do this |
|---|---|
| A tight budget | Buy it rather than make it. Eight blocks in two lengths that differ by little need exact cuts, and the cheapest set in our price table was under $30. |
| A toddler sibling | The small blue cube fits in a toddler's mouth. Use the cube at a table and count eight blocks back into the box. |
| An older child at home | A child of 9 or 10 can use the same box for (a + b)³. Montessori's 1917 book gives algebra to older children with a cube built the same way. |
| A child who would rather do puzzles than shelf work | Offer the cube as a puzzle at the family table. All it needs is a flat surface and a little quiet. |
When it does not go to plan
| When | Try this |
|---|---|
| The child can't tell where a prism goes. | Show the face-to-face check again. Two blocks held together show at once whether the faces match. |
| The blocks get tipped out in a heap. | Present again, laying each block out in the position it had in the box. That layout is what makes rebuilding possible. |
| Interest fades after one build. | Offer the blindfold version, or building outside the box, on another day. |
| The lid won't close. | One block is turned. Have the child look at the top layer against the lid's pattern. |
Hazards to check before each session
The blue cube is small enough for a toddler to swallow. Count eight blocks back in after every turn.
Check a secondhand set for chipped paint, and sand any rough edges before use.
Where other sequences and standards place it
We have not bought the paid homeschool curricula, so they are not listed here.
| Framework | Where it sits |
|---|---|
| This site's Sensorial sequence | Lesson 10 of 12 published, after Red Rods (Long Rods), before Trinomial Cube. Checked in: the scope and sequence |
| Montessori, The Montessori Elementary Material (1917) | Solid geometry, under the powers of numbers: a 10 cm cube built from a 6 cm cube, a 4 cm cube and six prisms, kept in a cube-shaped box. It's an elementary material in the book. Checked in: Part IV, Solid Geometry |
| Common Core, Kindergarten math | K.G.B.6: compose simple shapes to form larger shapes. The cube asks for it in three dimensions. Checked in: tasks.illustrativemathematics.org |
| Head Start Early Learning Outcomes Framework, preschool | P-MATH 10: the child explores the positions of objects in space. Checked in: www.in.gov |
Binomial Cube: questions parents ask
What is the binomial cube in Montessori?
A box of eight blocks that make a cube: a red cube, a blue cube and six black-sided prisms. Children of 3.5 to 4 use it as a building puzzle. Older children use the same pieces to see why (a + b)³ = a³ + 3a²b + 3ab² + b³.
What age is the binomial cube for?
Most albums give it at about 3.5 to 4, once the size materials are easy, and on this site it runs to 6. Montessori's 1917 book uses a binomial cube with elementary children studying the powers of numbers.
Does my child need to understand algebra to use it?
No. At 4 it's a puzzle about color and size, and the algebra waits for years. The point is that the hands have already built (a + b)³ many times before anyone writes it down.
Binomial or trinomial cube first?
Binomial. It has eight blocks in two colors; the trinomial has 27 in three. Move on once the binomial cube goes together without trial fits.
Sources cited by book and chapter
- Maria Montessori, The Montessori Elementary Material (1917)
Part IV, Solid Geometry
The cube of a binomial uses a cube with a 6 cm edge, a cube with a 4 cm edge, three prisms 4 by 4 by 6 cm and three prisms 6 by 6 by 4 cm, with which the 10 cm cube can be made.
These combinations are kept in special cube-shaped boxes into which the 10 cm cube fits exactly.
Cubes and prisms put together show that 2 cubed and 2 to the sixth have the form of a cube, the squared numbers the form of a square, and the others a line.
- Maria Montessori, Dr. Montessori's Own Handbook (1914)
Sensory Education
The order of sensory education is recognition of identities, then of contrasts, then discrimination between objects very similar to one another.
To concentrate the child's attention the sense is isolated, with silence in the room and the eyes blindfolded for exercises that do not concern sight.
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What to offer next
Once the child chooses this work unprompted and finishes it without help, the next lesson to prepare is Trinomial Cube.
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