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Sensorial Decanomial

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Squares and rectangles in colors corresponding to the bead stair, representing the factors of the decanomial square (Square of Pythagoras):

(a)^2 = a^2
(a + b)^2 = a^2 + 2ab + b^2
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
…and so on, through the square of the decanomial.

A wooden board, or a black/gray felt underlay.


  1. Place the wooden board on the rug, so that the two wooden edges show the right angle in the top left corner. 
  2. Remove lid and place it underneath the box. 
  3. Take out the square of each color from largest to smallest.  
  4. Place them in sequence concentrically.  
  5. Disassemble the squares, keep them out. 
  6. Take the small red square and place it in the upper left hand corner of the board. 
  7. Align the green square so its top left corner touches the red squares bottom right corner. 
  8. With the other two green pieces, place them adjacent to the green square, forming a larger square. 
  9. Repeat this process until all the squares and rectangles have been placed on the board. Go from smallest to largest, one color at a time. 
  10. Admire the completed board.  
  11. Show the child how to return the squares and rectangles to the box, leaving the squares on top.  


Exercise 1, Presentation

The child performs the presentation.

Exercise 2, Removing a Square
  1. With sensorial decanomial all laid out, remove a square from its place. 
  2. Look at the gap, place the square on the rug and remove its corresponding rectangles. 
  3. Slide the rectangles into the new gap towards the direction of the top left corner. 
  4. The rectangles will fit into the gap, except for the rectangle with the corresponding width of the removed square. 
  5. Remove the rectangle of the same width from each color, then push the rectangles towards the upper left hand corner.  
Exercise 3, How Many Squares Can be Made?
  1.  Explain to the child that we are going to see how many squares can be made. 
  2. Start with the red square and put it into the top left corner of the rug.  
  3. Remove all the green pieces, and place the green square below the red square.  
  4. Using the other two green pieces, form a square and place it next to the green square.  
  5. Remove all the pink pieces, and place the pink square below the green square.  
  6. Use the remaining pink pieces to make squares, place them next to the pink square.  
  7. Remove all the yellow pieces, and place the yellow square under the pink square. 
  8. Use the remaining yellow pieces to make squares, place them next to the yellow square. 
  9. Repeat this process until all squares of all colors have been made. 
Exercise 4, How to Make a Larger Square
  1. Take out a square from the box, place it on the rug. 
  2. Take out a smaller square and touch the top left corner of the smaller square to the bottom right corner of the larger square.  
  3. Find corresponding rectangles to fill in the gaps, forming a binomial square.  
  4. Superimpose a larger square on top of the binomial to verify it’s a square. 
  5. Invite the child to explore. 



Control of Error

Visual disharmony

Direct Purpose

Building Sqaures

Indirect Purpose

Preparation for mathematics.


4 1/2, 5 for later exercises.


Just help the child get started with the material, but don’t build the whole thing yourself.

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