# Algebraic Materials

Binomial Cube

Material Description:

Wooden cubes and rectangular prisms painted in various colours; the box contains pieces forming a cube (a+b)3and the square of the binomial is painted on it’s lid. A felt mat.

## NEW CASA EMAIL COURSE

Sign up for our email course that will guide you through the Montessori approach to the Casa.

All proceeds will go towards maintining Montessori Commons and improving the quality of the content.

Presentation:

• Bring the box and felt mat to two chowkis, for the first presentation the keep the box on the chowki, for subsequent ones keep it on a mat on the floor
• Remove the lid and place it with the painted control showing
• Open both sides of the box and place each piece at random on the table
• Rebuild the cube layer by layer in the box replicating the control
• The first layer
• Placing the red cube in the far corner
• Put the red and black prisms against the correspondingly coloured faces of the red cube
• Take black and blue pieces and fit them into the spaces
• The second layer
• Place the red and black prism in the far corner, corresponding to the colours
• Place the identical black prisms at the side
• Place the blue cube in the final near space
• Dismantle the cube and invite the child to rebuild

Exercises:

1. As in presentation
2. Build the cube outside of the box, show that all sides and horizontal and vertical ‘slices’ of the cube replicate the control
3. Build the cube in two layers, the base starting with the red cube, the top layer beginning with the blue cube, superimpose the top layer on the base layer ensuring that the colours match the control

Language:

None

Criteria of Perfection (Control of Error):

• Exercise one; the box itself
• Exercise two and three; controlled y the child’s visual sense

Direct Aim:

• To build a cube

Indirect Aim:

• Preparation for Mathematics
• The cube of the Binomial
• The cub root
• Introduction to algebra, especially the formulae (a+b)3 at the Elementary Level

Age at Presentation:

Three and a half years and above

Footnote:

To indirectly prepare for algebra the Absorbent Mind will take in a non-verbal impression of the formulae (cube and cube root)

Trinomial Cube

Material Description:

Wooden cubes and rectangular prisms painted in various colours; the box contains pieces forming a cube (a+b+c)3 and the square of the binomial is painted on it’s lid. A felt mat.

Presentation:

• Bring the box and felt mat to two chowkis, for the first presentation the keep the box on the chowki, for subsequent ones keep it on a mat on the floor
• Remove the lid and place it with the painted control showing
• Open both sides of the box and place the prisms of matching colours with their corresponding red, blue, yellow cube and the black ones all together
• Indicate the cubes
• Rebuild the cube layer by layer in the box replicating the control
• Placing the red cube in the far corner and continue layer by layer matching the colour, size and shape of the prisms
• Dismantle the cube and invite the child to rebuild

Exercises:

1. As in presentation
2. Build the cube outside of the box, show that all sides and horizontal and vertical ‘slices’ of the cube replicate the control
3. Build the cube in three layers, the bas starting with the red cube, the middle layer beginning with the blue cube and the top layer beginning with the yellow.  Superimpose the middle layer on the base layer and the top onto the middle, ensuring that the colours match the control

Language:

None

Criteria of Perfection (Control of Error):

• Exercise one; the box itself
• Exercise two and three; controlled y the child’s visual sense

Direct Aim:

• To build a cube

Indirect Aim:

• Preparation for Mathematics
• The cube of the Trinomial
• The cubes root
• Introduction to algebra, especially the formulae (a+b+c)3 at the Elementary Level

Age at Presentation:

After the Binomial cube

Sensorial Decanomial

Material Description:

Squares and rectangles in colours corresponding the the beed 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, etc through the square of the decanomial

A black or grey felt mat large enough for the complete square or a tray

Presentation:

• Bring the box and felt mat or tray to a large working mat on the floor
• Introduction – Tower
• Remove all of the squares from the box and place them concentrically to form a pyramid
• Building the Decanomial Square
• Remove the red square and place it at the top left corner
• Remove the green square and place it diagonally (below and to the right)
• Remove the green rectangles and complete the square
• Remove the pink square and place it diagonally (below and to the right)
• Remove the pink rectangles and complete the square
• Continue, now working with the child, until all the squares and rectangles have been used, fitting them exactly
• Show the child how to put away the material; placing the large square in the box and the rectangles on top, in order of descending size
• Note: Give the amount of rows depending on the child’s ability and concentration

Exercises:

1. As in the presentation
2. After completing the square, show the child how to decrease it by removing any square and fitting the rows together (to do this the child will need to eliminate those rectangles which are the same size as the square which has been removed)
3. How many squares can be made?

Place the red square as in the presentation and put the green directly below with a small gap, remove the two rectangles and place side by side, to the right of the green square.  Do the same for the pink square and rectangles

(The child may notice that the number of squares increases progressively by one)

1. How many ways can a larger square be built?
• Choose a larger square and two smaller ones which will fit perfectly within it’s diagonal (e.g. the blue, pink and lilac squares)
• Explore the other possibilities with the same large square involving the child.

(The child may notice that two squares are sometimes insufficient and a third square is required)

Criteria of Perfection (Control of Error):

• The child’s ability to recognise visual disharmony

Direct Aim:

• To build a square
• To discriminate size, shape and colour

Indirect Aim:

• Indirect preparation for Mathematics especially finding the square of a number

Age at Presentation:

Four to five years