Purpose:
To show by this geometrical form of multiplication that the multiplier is never a constant as is the multiplicand.
It is only indicative of how many times a number is taken or a given quantity is repeated.
To show that a succession of lines creates a surface that is why it is called geometrical multiplication.
Preparation for square root and for factoring.
Preparation for division by helping the child to visualize the divisibility of numbers.
The geometrical formation is an indirect preparation for exercises that follow later in connection with geometry and algebra.

Materials:
Nine boxes marked 1 x, 2 x, 3 x, etc. each containing 55 bars of each number from 1 to 9.
A box with tens and colored bead bars.
A felt mat.

Control:
The teacher.

Age:
5.5 – 6.5 years

Presentation:
Bring the answer box, a yellow mat and a box marked ‘7x’ to a table or a mat.
Introduce the materials. Place one seven bar from the ‘7x’ box horizontally at the upper left of the mat (multiplicand).
Count the number of beads on the bar, i.e. 7.
Remove a seven bar from the answer box (product) and place it vertically below the multiplicand.
State the equation, 7 one time is 7.
Proceed in the same manner, increasing the multiplier by one each time up to 9 inclusively.
Count the multiplicand in the same manner as the snake game – counting to ten and placing a ten bar, any remaining beads are represented by the appropriate bead bar from the answer box.
When complete, go over the table verbally with the child.

Exercise:
As in the presentation, the child works through the tables 1 to 9.

Exercise 2:
Multiplying by 10: At a table, the child chooses a multiplicand, ie. 4. The child writes the multiplicand centerd on a page and takes out 10 four bars from the answer box and arranges them horizontally. The child counts as in the presentation, and lays out the product vertically. Say, “When we have 10 fours it’s the same as 4 tens or forty.” “When you multiply by 10 all you have to do is add a zero to the multiplicand.” Repeat with a few more examples. Then child may continue on his own.

Exercise 3:
Divisibility of product: Select a number, ie. 12. Find the number of ways to make 12. Begin with 1 and see if you can make 12 by counting, adding each bar on one at a time. The use of 1 works however at this point note that the multiplier should be less than 10. Continue in the same manner for 2 through 9. Leave out the combinations which make 12. Child may write down combinations.

EXERCISE 4:
Commutative Law: At a table, using the answer box layout place 7 five bars horizontally as before. Then 5 seven bars. The child counts the beads placing the answer vertically below. Compare the answers. They are the same. Turn the beads to compare. Review and write the equations. 7 x 5 = 5 x 7 Try a few more examples. Conclude that it does not matter which side of the multiplication sign the multiplicand and the multiplier are on – the answer is the same.

EXERCISE 5:
Making the decanomial Build the decanomial square (see the sensorial album).
Set out two mats. On a mat set out the supply of bead bars.
Use the other mat to build the decanomial square, in the same manner as above, using the bead bars.
When the decanomial is complete visually explore the layout.
Note the squares on the diagonal and have the child exchange them with the squares from the bead cabinet.

Note: the number of bars in the band is the number squared; the number of beads in the band is the number cubed.