a. The Whole Product

Materials:
…The Horizontal Bead Frame, which lies flat on the table.
It is less sensorial in that hierarchic colors and spaces between the classes have been eliminated.
…box of 4 series of gray cards on which 1-9 is written in black (to serve as multiplier)
…strips of white paper on which multiplicand will be written.

(note: the black lines are drawn on the board beneath the wires; they will indicate where to begin the multiplication when multiplying by units, tens, hundreds or thousands.)

Presentation:
All of the previous operations can be done with this material, but we will do the most interesting: multiplication with a two-digit multiplier.
Write down a problem, 6542 x 36 = and show the child how to set up this problem. Place a white strip over the zeros and secure it with a rubber band or tape. Write the multiplicand on the strip so that the digits correspond to the correct wires. Find among the gray cards the digits needed to form the multiplier. Place the 6 over the lowest green dot which represents the units, and the 3 over the blue dot for the tens. The beads should be at the top to start.
Begin multiplying 2 x 6 = 12, bring down 2 units and 1 ten. 4 x 6 = 24 tens – bring down 4 tens and 2 hundreds and so on. After the multiplicand has been multiplied by the units we can turn over the card ‘6’.
In order to multiply by the 3 tens, 30, we must move the multiplicand to the left one space to let one red zero show. This is just like multiplying the number by 10.
The black line indicates that we start with the row of tens. Continue multiplying, making changes as necessary. In the end we read the product and record it.

b. Partial Products

Age: 7-8 years

Materials:
…box of 4 series of gray cards on which 1-9 is written in black
…strips of white paper

The procedure followed here is exactly the same, except that when the child has finished with one multiplier he turns over the card, reads the partial product, writes it and clears the frame before beginning with the next multiplier. In the end he adds abstractly to total the partial products.

c. Carrying Mentally

Age: 8 years

Materials:
…box of 4 series of gray cards on which 1-9 is written in black
…strips of white paper

The child sets up the multiplication problem on the frame.

 2443 x 46

3 x 6 = 18 move down 8 units, remember one ten in your head.

5 x 6 = 30…+1 = 31 move down 1 ten, remember 3 hundreds

4 x 6 = 24…+3 = 27 hundreds-move down 7 hundreds, etc.

Record the partial product and clear the frame before beginning multiplication by the tens.

Note: The work done with this frame is on a higher level of abstraction than the work with the hierarchic frames. In both activities the tens, hundreds and thousands of the multiplier were reduced by a power of 10, while the multiplicand increased by a power of 10. The same work was done in two different ways.
At the end of this work the child should understand that when he starts multiplying with a new digit of the multiplier, he must move over one hierarchy. The partial products must start from the same hierarchy as the corresponding digit of the multiplier.
This activity forms the basis for an understanding of the function of multiplication with a multiplier of two or more digits, and a preparation for abstract solution. The child doing this activity will be stimulated to invent his own problems.