Trinomial Cube
The trinomial cube is a hinged box of 27 painted blocks: a red cube, a blue cube, a smaller yellow cube, eighteen prisms with black sides and six prisms that are black all over. A child of about 4 to 4.5 who builds the binomial cube easily rebuilds this one in three layers, matching colors and sizes.
Sourced from the texts, not yet presented at home
| Typical age | 4 to 6 years |
|---|---|
| Area | Sensorial |
| Before this | |
| Materials | |
| Direct aim | To build a cube of 27 blocks from its parts by matching color and size, one layer at a time. |
| Control of error | The box and the colors. A block in the wrong place leaves a gap or stops the sides from closing, and an end of the wrong color shows at a glance. The lid's pattern checks the top layer. |
| Leads to | Later lessons in this line are in preparation. |
| Time | About 15 minutes the first time (our estimate). Twenty-seven blocks take a while. |
| Grouping | One child |
| Parent prep | None. The box lives on the shelf, closed. |
| Cost | Buy about $40 to $211. Make: better bought than made. |
| Other names | Cube of a trinomial, Trinomial cube box, (a + b + c)³ cube |
What the work teaches
Direct aim. To build a cube of 27 blocks from its parts by matching color and size, one layer at a time.
Indirect aims, the ones you will not see at once
- Sorting by two features at once, the color of the ends and the size of the block
- Holding a layer of nine blocks in mind while building the next one
- Indirect preparation for algebra: the cube of a trinomial, worked out with the same blocks in the elementary years
- Patience over a long build with a clear finish
Set it out before you invite the child
Twenty-seven wooden blocks in a hinged box. Alison's lists its box at 4.25 by 4.25 by 4 inches, with the sides hinged so it opens out flat.
Three cubes in three sizes: red the largest, blue next, yellow the smallest. Eighteen prisms have black sides and two square ends in red, blue or yellow. Six more prisms are black all over.
Every colored end matches the cube of the same color in size. That's the key the child works out: a red end belongs beside the red cube's size.
The cube goes together in three layers of nine. The lid shows the top layer seen from above.
The trinomial cube isn't in Montessori's books from 1912 to 1917. Her 1917 book describes a binomial cube for older children, and the painted trinomial puzzle for 4 year olds comes from the later albums.
A table with room in front of the box for three layers laid out side by side.
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 all three layers show.
- 3Take out the top layer one block at a time. Set each block on the table where it sat in the box, so the nine make a square.
- 4Lay the middle layer in front of the top one the same way, then the bottom layer in front of that.
- 5Rebuild the bottom layer in the box first. Start with its cube, then match each neighbor by the color of its ends and its size.
- 6Build the middle layer, then the top. When two black prisms look alike, hold each against its neighbor to check that the faces meet edge to edge.
- 7Check the top layer against the lid, close the sides and put the lid on.
- 8Invite the child to take it apart and build it.
Details that hold the child's attention
The new yellow cube, smaller than the blue.
The six all-black prisms, which have no colored end to go by.
Each layer of nine laid out on the table like a small floor plan.
The red and blue blocks in one corner, which together make a binomial cube inside the big one.
How the child sees a mistake
The box and the colors. A block in the wrong place leaves a gap or stops the sides from closing, and an end of the wrong color shows at a glance. The lid's pattern checks the top layer.
The words come after the hands
| Words | Shown with |
|---|---|
| cube | The red, blue and yellow blocks, and the whole thing when it's built. |
| prism | Every block that isn't a cube: the 18 with colored ends and the 6 all-black ones. |
| layer | Each set of nine blocks. |
| trinomial | The adult word, for three terms. The child doesn't need it until algebra. |
The three-period lesson, scripted
You need the yellow cube and one all-black prism.
| Period | What you do and say |
|---|---|
| 1. Naming | "This is a cube." "This is a prism." |
| 2. Recognition | "Give me the cube." "Find another prism that's black all over." |
| 3. Recall | Point to one and ask "What is this?" |
Our suggestion: only if the child asks. The names carry over from the binomial cube, and the building is the lesson.
Extensions and variations
- Building it on the table, outside the box, with the lid as the only guide
- Building it blindfolded, by touch, once it goes together in the box without trial fits
- Sorting all 27 blocks by kind first: three cubes, the prisms by end color, then the six black ones
- Finding the binomial cube inside: the red and blue blocks alone make an eight-block cube in one corner
- In the elementary years, (a + b + c)³ worked out with the blocks and written in full
Variations
- Leaving the lid off, so the top layer has no pattern to copy
- Building the bottom layer only, for a child who tires before 27 blocks
Make it or buy it
| Material | Make it | Buy, approx. |
|---|---|---|
| Trinomial Cube | Buy instead | $40 to $211 |
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. Twenty-seven blocks in three sizes have to be cut and painted exactly, and the box needs hinged sides.
What to check when buying a set
Check that the box closes with every block in place and that the colored ends are cleanly painted. The algebra use years 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 4PDF, 1 page, 248 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 binomial cube in its box and outside it, with no trial fits.
- Sorts objects by two features at once, such as color and size.
- Finishes a puzzle of 20 or more pieces.
- Asks for a harder cube.
Ready for the next lesson
- Builds all three layers in the box with no trial fits.
- Builds it on the table, outside the box.
- Sorts the 27 blocks into their kinds at a glance.
- Points out the binomial cube hidden in one corner.
Fitting it into a home, not a classroom
| If you have | Do this |
|---|---|
| A tight budget | Buy it; 27 blocks in three sizes need exact cuts. The cheapest set in our price table was $39.95. |
| A toddler sibling | The yellow cube is small enough to swallow. Use the cube at a table and count 27 blocks back in. |
| A child who stops halfway | Build one layer a day. Leave the finished layers in the box and the rest laid out on a tray on the shelf. |
| An older child at home | A child of 10 or 11 can use the same box to see (a + b + c)³. Keep it on the shelf after the primary years. |
When it does not go to plan
| When | Try this |
|---|---|
| The black prisms all look the same to the child. | Hold each one against the cube beside it. A prism belongs where its faces meet its neighbors edge to edge. |
| Blocks get tipped out in a heap. | Present again, laying every block out where it sat. With 27 blocks the layout matters more than with the binomial cube. |
| Frustration in the middle layer. | Put it away and offer the binomial cube for a week. Then try the trinomial again. |
| The lid won't close. | One block in the top layer is turned. Have the child compare the layer with the lid's pattern. |
Hazards to check before each session
The yellow cube and the smallest prisms are choking-size for a child under 3. Count 27 blocks back after every turn.
Check a secondhand set for chipped paint and splinters 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 12 of 12 published, after Binomial Cube. Checked in: the scope and sequence |
| Montessori, The Montessori Elementary Material (1917) | Only the binomial version appears, under the powers of numbers: a 10 cm cube built from a 6 cm cube, a 4 cm cube and six prisms. The trinomial cube extends the same idea to three terms. Checked in: Part IV, Solid Geometry |
| Montessori, The Montessori Method (1912) | Few strongly contrasting stimuli first, then many with finer differences. Two colors and eight blocks come before three colors and 27. Checked in: Chapter XII, Education of the Senses |
| Common Core, Kindergarten math | K.G.B.6: compose simple shapes to form larger shapes, here in three dimensions. Checked in: tasks.illustrativemathematics.org |
Trinomial Cube: questions parents ask
What is the trinomial cube in Montessori?
A box of 27 painted blocks that build a cube: red, blue and yellow cubes, 18 prisms with black sides and colored ends, and 6 prisms that are black all over. Children of about 4 build it as a puzzle. Years later the same blocks show (a + b + c)³.
What age is the trinomial cube for?
Most albums give it at about 4 to 4.5, after the binomial cube, and on this site it runs to 6. Sellers vary: Nienhuis lists 3 to 4 years, The Montessori Room 4.5 and up.
What's the difference between the binomial and trinomial cube?
The binomial cube has 8 blocks in two colors and builds in two layers. The trinomial has 27 blocks in three colors and builds in three layers. The red and blue blocks of the trinomial cube make a binomial cube of their own in one corner.
Is the trinomial cube in Montessori's own books?
Not in the books we checked, from 1912 to 1917. The 1917 Montessori Elementary Material describes only a binomial cube, for older children. The trinomial cube comes from the later albums.
Sources cited by book and chapter
- Maria Montessori, The Montessori Elementary Material (1917)
Part IV, Solid Geometry
For the cube of a binomial the material is 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.
- Maria Montessori, The Montessori Method (1912)
Chapter XII, Education of the Senses
One should proceed from few stimuli strongly contrasting to many stimuli in gradual differentiation, always more fine and imperceptible.
- Maria Montessori, Dr. Montessori's Own Handbook (1914)
Sensory Education
Sensory education proceeds from the recognition of identities and of contrasts to discrimination between objects very similar to one another.
It is well to isolate the sense, blindfolding the eyes for exercises that do not relate to sight, so the child concentrates on the stimulus.
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What to offer next
This is the last Sensorial lesson published so far. The ones that follow are in preparation; until then, the Sensorial lessons page lists the whole area in order.
- Geometric cabinet3 to 5 years
- Mystery bag3 to 5.5 years
- Binomial Cube3.5 to 6 years
- Geometric solids3.5 to 6 years
- Trinomial Cube4 to 6 years