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Building Montessori Trinomial Cube at 31 months

Trinomial Cube (buy here) a classic Montessori sensorial material, is generally introduced after the Binomial Cube.  Once the child had mastered the Binomial cube, the transition to a Trinomial cube is rather quick since the method of completion remains the same (just more cubes and prisms to work with). The trinomial cube is a wooden box with a cover and two adjacent hinged sides, containing 27 wooden blocks that fit together to form a cube with the same pattern on each side and in the middle. The cube has a visual representation of the algebraic trinomial formula (a+b+ c) (a+b+c) (a+b+c) or: (a+b+c) 3. Both Binomial and Trinomial cubes are based on a Square of Pythagoras, which is a visual representation of the multiplication tables, 1-10, and the square of sums (binomial formula, trinomial formula, etc.)  

First, the child would take all the cubes out of the wooden box and group and sort them by color on a table or a mat. Then, starting with the largest single-color cube (all-red), the child would complete the first level ("the first floor of the Red King") by placing the adjoining cubes so that the same colors "touch". Once the first level is completed, the child would move it into a wooden box.  Similarly, the child would proceed with the second single-color cube (all-blue), completing "the second floor of the Blue King". Finally, the child would complete "the third floor of the Yellow King" in the similar manner starting with the smallest (all-yellow). (Read Maria Montessori's Story of the Three Kings below.)  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. 

building a Trinomial cube at 31 months from Anya on Vimeo.

Trinomial cube, in addition to aiding fine-motor and eye-hand coordination, also helps develop child’s visual perception of three-dimensional patterns, as well as prepares the child for later work with maths, especially algebra. As a result of this activity, the child learns dimension, size-gradation and primary colors.

DWELLING INTO DETAILS  and the Story of the Three Kings:

The following excerpt is from Montessori Materialized Abstractions, by Daniel Jutras Ph.D. The Canadian Montessori Teacher Education Institute.

TRINOMIAL CUBE Age: 3 ½ and up

Materials: the smallest and the largest cubes of the Pink tower; and the box containing:

  •  1 red cube, 1 blue cube, 1 yellow cube;
  • 3 thick and 3 thin red and black prisms;
  • 3 thick and 3 thin blue and black prisms;
  • 3 tall and 3 short yellow and black prisms,
  • 6 all black prisms;
  • a felt with a glued pattern simulating a river.

Presentation (from the Maria Montessori's story in The Discovery of the Child, Ballantine Books p. 278)

  1. Invite the child and show him where the trinomial cube is kept. Invite to work at a table or on a mat.
  2. Have the child carry the box to the table or to the mat and set it so the hinges of the box are in front of you and invite the child to unroll the felt.
  3. Invite to unroll the special felt, take the lid off, place it on the table so he can see the pattern.
  4. Begin by telling the child: "Do you remember the little castle we built before with two kings? Now this is the story of three kings. Would you like to hear the story?"
  5. Tell the story and invite to perform as the story goes on and on.
  6. Invite to take out all the blocks one by one by using the 3-finger grip and placing them at random on the top part of the felt.
  7. Invite to isolate the red cube, the blue cube and the yellow cube. "Do you remember how each king could organize his entourage?"  The child understands the term "entourage" because he has already built the binomial cube.
  8. Invite the child to place by sizes all the red prisms under the red cube, all the blue prisms under the blue cube and all the yellow prisms under the yellow cube. There should be three rows, side by side. (It is simple sorting of prisms by color.)
  9. Then you say: "Each king has two bodyguards dressed in black!" showing the black prisms. Invite to place each prism at either side of the red, blue and yellow cubes.
  10. As the child has already built the binomial cube, so he has a good idea how to build this trinomial cube. The directress is there to present the "story of the Three Kings and the King of Peace". When the story is over and the child is done, invite to marvel his work.
  11. If he wants to do it again, invite to disassemble it and place the cubes and prisms at random on the felt and repeat the building of the castle according to the "story".
  12. When finished, he lifts the two sides up from the box and places the lid back on the box (or he places the box-lid on the blocks forming the trinomial) and rolls the special felt.
  13. Invite to return the box back on the shelf.

Language: The story of the three kings (Ref.: The Discovery of the Child, Ballantine Books p. 278)

Direct Aim:

  • Visual discrimination of size and color
  • Sensorial construction of a cube

Indirect Aim:

  • Development of a logical-mathematical mind
  • Preparation for algebra
  • To form judgment, to reason, to initiate comparisons and to decide
  • To build a foundation for intellectual activity
  • To refine his power of observation and to educate the eye to distinguish differences in dimension
  • Develop sense-perception: child forms a visual image of the arrangement of the blocks and can thus remembers their quantity and order
  • To develop imagination

Point of Interest:

  • The "Montessori story" of the three Kings, their respective entourage and the King of Peace and his personal ambassador.
  • The pattern on the lid or bottom of the box
  • The colours and shapes of the cubes and prisms

Control of Error:

  • If the cube is not formed and the QpatternS is not seen from any side of the cube: when completed, the pattern of the trinomial cube repeats itself 18 times: 6 times in each of the external faces, and 2 times in each of the internal sections (the cube can be separated into two parts, twice in each dimension, or into two parts, one each dimension).
  • Noise/banging

Activities before: Binomial cube, sorting by colour, cylinder blocks

Activities after: Geometric cabinet, Decanomial square, geometric solids

Note: The 27 blocks per se represent the elements of this algebraic formula: (a+b+c)³ = (a+b+c) x (a+b+c) x (a+b+c), that is: a³ + 3a²b + 3ab²+ b³ + 3a²c + 6abc + 3b²c + 3c²a + 3c²b + c³

The child is of course not introduced to the formula. He is only interested in the colour and sizes of the blocks (cubes and prisms) and how he can build a big cube.

Story of the Three Kings

Once upon a time, there were four kings living separately with their royal suite in four respective kingdoms: three on one side of a river and one on the other side. The three kings living on the North side of the river in three different castles were the Red King; the Blue King and the Yellow King. Each of the three kings had an entourage of two bodyguards dressed in black and six attendants. On the South side of the river lived the King of Peace in a pink tower with his own suite.

One day, the smallest attendant of the King of Peace was named ambassador and sent to make a courtesy visit to all three kingdoms on the North side of the river. During his visit, the ambassador perceived that each king and his entourage were unhappy. Apparently, the red castle was too hot; the blue castle was too cold and the yellow castle was too small. The ambassador came back to the pink tower and informed the King of Peace of the state of affairs of the three kingdoms. In order to resolve the situation, the King of Peace in his kindness, prepared a plan to be presented to the three kings. If the three kings were to accept to live with their entourage in the same castle, but on a separate floor, in security, there will be harmony and happiness in each royal kingdom. The King of Peace sent his ambassador back to cross the river and present this plan to the three kings. At the approach of the ambassador, the Red King commanded all his attendants and bodyguards to be in position to receive the ambassador. His bodyguards and attendants realigned themselves.

The ambassador presented the King's plan to come to live in a brand new three stories castle. The Red King's immediate answer was: "No!" pretending he might be annoyed by the proximity of the two other kings: the Blue King and the Yellow King. He then said to the ambassador: "I prefer to stay in a castle which is too hot!" The ambassador then went to the Blue King to make the King of Peace's proposition. The Blue King commanded all his attendants and bodyguards to be in a position to receive the ambassador. Unfortunately, the king answered: "No! I prefer to live in a castle which is too cold than to share a castle with two royal neighbors!" Finally, the ambassador went to see the Yellow King. In order to receive the ambassador with dignity, the Yellow King commanded all his attendants and bodyguards to be in a position to receive the King of PeaceGs ambassador. On behalf of his king, the ambassador proposed the Yellow King to lead his suite to dwell in a brand new castle and promised him that he would not be annoyed by any neighbor. Unluckily, the ambassador received the same negative answer.

Discouraged, the ambassador went back to inform the King of Peace of this unanimous refusal. The King of Peace met all his advisers in the tall pink tower, and they arrived to a clever solution: each kingdom would live independently on a separate floor conditionally that each other kingdom has a spy who will reside permanently on each other floor of the brand new castle.

The King of Peace sent back his ambassador to the three kings to present this innovative proposal. The Red King received the ambassador and agreed to the deal. Protected by both bodyguards (all dressed in black who are next to other king's spies) and four of his attendants (2 children & 2 servants [chef & a maid]/thinner),  he led his royal suite leaving behind two of his own attendants chosen to become spies. The Red King chose to reside in the farthest corner of the first floor of the new castle. As the deal was that each king has to accept to be located close to, but not next to two special spies, two spies joined his suite: one from the blue kingdom and one from the yellow kingdom. When everybody was installed, the ambassador then went to visit the Blue King in his cold castle and invited him to move in high security with his suite on the second floor of the new castle, but not exactly over the Red King, but in the center of the room. The Blue King accepted the compromise of being convivial with two unlike spies, issued from each royal red or yellow kingdom. The Blue King established his suite on the second floor after settling in the center of the room. Then, the ambassador went to see the Yellow King and did the same claim: "Would you accept to move and stay in tight security on the 3rd floor? You will not be staying over the Blue King nor the Red King." The Yellow King who was living in a tight castle accepted all the conditions of the King of Peace's arrangement. His entire royal suite accompanied him with the two remaining spies after he has chosen for himself a far corner of the third floor of the new castle.

From then on, the Red King, the Blue King and the Yellow King lived separately in security, secluded on each floor, protected by their own bodyguards and attendants, but sharing convivially a spy from each other royal kingdom. The King of Peace's plan brought harmony and happiness forever. Therefore, the King of Peace's ambassador returned to his kingdom and positioned himself as an observer, till the next mission, at the very top of the famous Pink Tower.

The End

Please, let me know if you liked the story. 

The Story Of The King Of Peace is an inspiring story that explains the relationship between the Pink Tower and the Trinomial Cube. Even though the child already knows these materials from their primary years, this story will entice the elementary child to learn about the mathematical relations between these materials.

  • 1st floor: Biggest & Hottest Red King: has 2 body-guards all black, 2 children & 2 servants. And, according to the agreement,  Blue & Yellow Spy
  • 2nd floor = Blue King that is too cold (in the middle - does not want to be on top of Red King): has 2 body guards as big & strong as the King; 2 children (have blue in them); & 2 servants (all black). And, according to the agreement,  Red & Yellow Spy
  • 3rd Floor= Yellow Smallest King: has 2 body-guards of the same size, 2 children (bigger-grown up) & 2 all black servants.  And, according to the agreement,  Red & Blue Spy.


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