“Some of Maria Montessori treasures unveiled”
by Daniel Jutras Ph.D.
Saturday February 7th
PNWMA
[email protected]
www.montessori-institute.ca
The Canadian Montessori Teacher Education Institute
C. Sizes and Colours
THE BINOMIAL CUBE
Age:
3+
Materials:
1 red cube, 3 black and red prisms, 1 blue cube,
3 black and blue prisms (8 blocks). A box with a pattern on its lid and repeated
on the bottom of the box.
The 2 cubes and 6 prisms, which form a cube is the “materialized
abstraction” of the algebraic formula:
(a + b)3 = a3 + 3a2b + 3ab2 + b3
a = the dimension of the reds
b = the dimension of the blues
The child is not introduced to the formula. He is only interested in the
sensorial building the cube by colours and shapes. There is another
presentation that is more mathematical which leads to the algebraic
formula. This is for YOUR information and understanding.
Presentation # 1 (sensorial):
1. Invite a child to come and listen to a story of 2 Kings that live in a
special castle.
2. Show the child where the binomial cube is kept in the environment and
invite him to carry the box to the table.
3. Take the lid off and place it so you can see the pattern. (The top left
corner of lid should be placed so it touches the bottom right corner of
the box.)
4. Show the child how the sides of the box open or remove completely
the “box-lid”.
5. Show the child how to take out a block by using the three-finger grasp.
Take out all the blocks, one by one, placing them randomly on the
table. Invite the child to help take them out. Show the child that the
pattern on the lid is also seen on the bottom of the box.
6. Begin by telling the child a little “story” (Ref.: The Discovery of the
Child, p. 278) in which the two cubes represent “two kings”: a big red
and a smaller blue.
7. “There is a blue king and a red king and they live in a castle.” Invite
the child to find the red and blue cube, isolate the red and blue cube on
the table on top. Then continue: “The blue king has followers and the
red king has followers. Can you find the followers of the red king?
Can you find the followers of the blue king?”
8. The child places the red and black prisms under the red cube and the
blue and black prisms under the blue cube, making two rows side by
side. Tell the child that no follower is lower or higher than the King,
they must be the same.
9. Show the child where the red king lives (first floor). Place the red cube
in the left corner of the box (following the pattern glued on the bottom
of the box).
10. Show the child where one of the followers goes, beside the king (“red
on red”, black on black).
11. Now invite the child to place the next one.
12. You can tell him the blue and black block is the “spy” on that floor
where the red king lives. "The spy will report to the blue king".
13. If needed, invite the child to use the pattern designed of the lid or the
bottom of the box for assistance in building the cube.
14. Invite the child to build the second layer by showing the child where
the “blue king” lives and then following the story for the red spy.
15. The child will discover on his own through trial and error how to build
the sensorial binomial cube.
16. When the child is done, marvel at his work together.
17. If he wants to do it again, invite him to disassemble the cube and place
the pieces randomly on the table again.
18. If he likes, he may build the binomial on the pattern on the lid itself
instead of in the box. If he does it this way, you can show him the
pattern from all sides of the cube.
19. When he is finished, invite him to lift the two sides up to close the box
and place the lid back on the box (unless the lid is the box itself). Invite
him to return the box back to the shelf.
20. Invite him to use this activity anytime.
Presentation #2 (mathematical):
1. The teacher-guide says, “Just watch my hand.” Then removes the
blocks one at a time starting with the blue cube.
2. As you remove each cube and prism, arrange them according to the
same heights, then place your hand over the cubes, to show the child
each time the two different heights.
3. Invite the child to place his hand over the arranged cubes as well.
4. Then say, “Could you help me place the blocks back into the box?”
5. The last cube that you removed from the box (red cube) goes in first
when putting the blocks back, and the first cube (blue cube) you took
out, will go in last.
6. Invite the child to try the activity, “Would you like to take out the blocks
and place them as I just did?”
7. When he is finished, invite him to lift the two sides up to close the box
and place the lid back on the box (unless the lid is the box itself). Invite
him to return the box back on the shelf.
8. Invite him to use this activity this way whenever he would like.
Language:
The story of the two kings / "Just watch my hand.", and the invitation to try
Direct Aim:
Sensorial construction of a cube
Indirect Aim:
•
•
•
•
•
•
•
•
•
•
Development of a logical-mathematical mind
Preparation for algebra
Preparation for writing
To form judgment, to reason and decide
Builds a foundation for intellectual activity
To refine the child’s power of observation
Develop perception
To become a better explorer of the environment
Initiate comparisons
Visual discrimination of size and colour
Point of Interest:
Maria Montessori's story of the kings told in the book The Discovery of the Child
(Maria Montessori), the moment the last piece is placed in building the cube, the
colours and shapes of the cubes and prisms, and the pattern on the lid and
bottom of the box
Control of Error:
• The cube is improperly constructed
• The cube is formed in a way that the pattern cannot be seen from any
side of the cube
• Unable to place the cubes/prisms
• Knocking over the cube
3 Activities Before:
3 Activities After:
1. Superimposition of shapes
2. Colour tablets (box #1 and # 2)
3. Pink tower
1. Trinomial cube
2. Geometric solids
3. Pythagoras' square
Sensorial Presentation – Binomial Cube
Mathematical Presentation – Binomial Cube
a3 + 3a2b + 3ab2 + b3
The Canadian Montessori Teacher Education Institute
C. Sizes and Colours
TRINOMIAL CUBE
Age:
3½+
Materials:
1 red cube, 1 blue cube, 1 yellow cube, 3 large and 3 small red and black prisms,
3 thick and 3 thin blue and black prisms,3 tall and 3 short yellow and black
prisms, and 6 completely black prisms (27 blocks)
The cubes and prisms in the box, which form a cube, is the “materialized
abstraction” of the algebraic formula:
(a+b+c)3 = (a+b+c) x (a+b+c) x (a+b+c), that is:
2
2
2
2
2
2
a3 + 3a b + 3ab + b3 + 3a c + 3b c + 6abc + 3c a + 3c b + c3
The child is not introduced to the formulae. He is only interested in the
colour and sizes of the cubes and prisms and how he can build a big cube.
This is for YOUR information and understanding.
a = the dimension of the reds
b = the dimension of the blues
c = the dimensions of the yellows
Pattern of the trinomial cube for the control of error:
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)
Presentation #1 (sensorial):
1. Invite a child to come and listen to a story of 3 kings that lived in a
single beautiful castle.
2. Show the child where the trinomial cube is kept in the environment and
invite her to carry the box to the table. Indicate to the child to place the
box so the hinges of the box are in front of you.
3. Invite the child to take the lid off, place it on the table so you can see
the pattern. The top left corner of lid should be placed so it touches the
bottom right corner of the box. Invite her to put the sides of the box
down or remove the box-lid completely (she will know how to do this
from the previous activity).
4. Remind the child to take the blocks out using the three-finger grasp.
Invite her to take out all the blocks, one by one, placing them randomly
on the table. Show the child the pattern of the lid is repeated in the
bottom of the box.
5. Begin by telling the child “Do you remember the little castle we built,
well today we’re going to build a bigger castle.”
6. “Remember we had a red king and a blue king, well now we have a
yellow king as well.”
7. Invite the child to isolate the red cube, the blue cube, and the yellow
cube.
8. “Each of the kings has followers. Can you find the followers of the
kings?” The child understands the term “followers” because she has
done the binomial cube.
9. The child places 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. Remind the
child if necessary that all the followers have to be the same height as
the king, “Remember, the kings do not like it if any one in the kingdom
is taller then him, so make sure the body guards are the same height
as the king."
10. Then say: “Each king has two “bodyguards” showing him the black
prisms. Invite him to place each prism at either side of the red, blue,
and yellow cubes.
11. The child has already built the binomial cube so he knows the story
and can build that part of the trinomial.
12. Now by indicating to the pattern on the lid, or the bottom of the box
where a pattern is glued, ask if the child if he can continue to build it,
red on red, black on black and so on… You may show him where one
of the two bodyguards goes. You are there to assist him if he needs
any help.
13. Now he begins to build the second layer. After successfully completing
the second layer he goes on the third until the whole cube is
constructed and completed.
14. When the child is done, you will both marvel at his work.
15. If he wants to do it again, invite him to disassemble it and place the
cubes and prisms at random on the table.
16. He may build the trinomial on the pattern itself instead of in the box. If
he does it this way, show him the “pattern” from all sides of the cube.
17. When he is finished invite him to lift the two sides up from the box and
place the lid back on the box (or he places the box-lid on the blocks
forming the trinomial). He will then return the box back on the shelf.
18. Invite him to use this activity anytime.
Presentation #2:
1. The teacher-guide says, “Just watch my hand.” Remove the yellow
cube first and then the others, one at a time, in order.
2. As each block is removed, arrange them according to the same
heights, in rows (columns), and then place your hand over the blocks
to show the child the three different heights.
3. Invite the child to place her hand over the arranged cubes as well.
4. Then say, “Could you help me to place the pieces back into the box?”
5. The last cube (red cube) that is removed from the box will go in first
when putting the blocks back, and the first cube (yellow cube) taken
out, will go in last.
6. Invite the child to try the activity, “Could you take out the blocks and
place them as I just did?”
7. When she is finished, invite her to lift the two sides up to close the box
and place the lid back on the box (unless the lid is the box itself). Invite
her to return the box back on the shelf.
8. Invite her to use the activity this way anytime.
Language:
The story of the three kings (Ref.: The Discovery of the Child, p. 278) / "Just
watch my hand.", and the invitation to try
Direct Aim:
Sensorial construction of a cube
Indirect Aim:
•
•
•
•
•
•
•
•
•
•
•
Development of a logical-mathematical mind
Preparation for algebra
Preparation for writing
To form judgment, to reason and decide
Build a foundation for intellectual activity
To refine the child’s power of observation
Develop sense-perception
To become a better explorer of the environment
Initiate comparisons
To educate the eye to distinguish differences in dimension
Child forms a visual image of the arrangement of the blocks and can
thus remembers their quantity and order
• Visual discrimination of size and colour
Point of Interest:
The “Montessori story of the three kings and their respective retinue, the moment
at which the last piece is placed in building the cube, the pattern on the lid or
bottom of the box, and the colours and shapes of the cubes and prisms
Control of Error:
• If the cube is not formed and the “pattern” is not seen from all sides of
the cube
• Noise or banging
• Knocking over the cube
• Unable to place the cubes/prisms
• Unable to close the box after building the cube
3 Activities Before:
3 Activities After:
1. Binomial cube
2. Sorting by colour
3. Cylinder blocks
1. Geometric cabinet
2. Geometric solids
3. Constructive triangles
TRINOMIAL CUBE
Age: 3 ½ and up
Materials:
- The smallest and the largest cubes of the Pink tower;
- 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,;
- six 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.
4.
Invite to unroll the special felt, take the lid off, place it on the table so he can see the pattern.
3.
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?”
4.
Tell the story and invite to perform as the story goes on and on..
5.
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.
6.
Invite to isolate the red cube, the blue cube and the yellow cube. “You remember
how each king could organize his entourage, do you?” The child understands the term “
entourage” because he has already built the binomial cube.
7.
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.
8.
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.
9.
As the child has already built the binomial cube, so he has a good idea how to build this
trinomial cube. The directress/director 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.
10.
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”.
11.
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.
12.
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 colour
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 “pattern” 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 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
neighbours!” 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 position to receive the King of Peace’s 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 neighbour. 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 and four of his attendants, 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 centre of the room.
The Blue King accepted the compromise of being convivial with two unalike spies, issued from
each royal red or yellow kingdom. The Blue King established his suite on the second floor after
settling in the centre 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 for ever. 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
TRINOMIAL CUBES
(Template for cutting)
a³
a³
abc
b³
a²b a²b bc² b²c
b³
abc
c³
abc
a²b a²c abc b²c b²c abc
a²c a²c
3
a²c ab²
+
bc² bc² abc
+
+
+
ab² ab² abc
+
+
+
+
+
+
+
ac²
abc
ac² ac²
3
3
(a +b)³ =
6
+
(a + b + c)³ =
3
3
3