The operations of addition, subtraction and multiplication can be performed with the large bead frame. But since the beads are connected to the wires, the beads frame cannot be used for division. The hierarchic material for division consists of loose beads.
Up to this point division has been done with the decimal system material to give the concept, including division with a 2-or 3-digit divisor, and group division. These concepts were reinforced with the stamp game. Following a research of the combinations necessary for memorization, where the quotient was limited to a maximum of 9. Division was dealt with indirectly in many of the multiplication activities. Using this material the dividend may have up to 7 digits and a divisor of 1-, 2- or 3-digits may be used.
Maria Montessori referred to this material “as an arithmetical pastime for the child.” This work clarifies the analytic procedure for the development of the operation. The fundamental difficulty of division is obtaining the digits of the quotient, recognizing their values and placing them in their proper hierarchical position.
At this level more importance is given to the quotient, that is, what each unit receives, and not so much to the quantity to be divided.
MATERIALS: RACKS & TUBES
…7 test tube racks: 3 white, 3 gray and 1 black
…each rack contains 10 test tubes
deposit each tube contains 10 loose beads
[These are the “deposit” from which quantities are drawn. The racks are white for the simple class, gray racks for thousands and 1 black rack of green beads for units of millions.]
7 bowls – 1 to correspond to each rack:
dividend the exterior of the bowl corresponds to the color of the rack the interior of the bowl corresponds to the color of the beads The dividend is formed in these bowls, just as was done with the stamp game.
3 bead boards: in hierarchical colors for units, tens, hundreds
For a 1-digit divisor the green board is used
divisor For a 2-digit divisor the green and blue boards are used
For a 3-digit divisor the green, blue and red boards are used.
[As in memorization, the distribution is done on the boards.]
Box with three compartments containing nine skittles of each of the three hierarchic colors that represent the divisor.
Also: A large tray to hold the racks while they are not in use.
SMALL DIVISION (1-DIGIT DIVISOR, 4-DIGIT DIVIDEND)
Isolate the racks that are needed to form the dividend. Place the other racks on the tray. Pour the quantities into the respective bowls. Place the green skittles on the green board for the divisor.
Begin distributing “bringing down” the units of thousands, that is moving the rack and bowl closer to the board. After distributing the units of thousands, record the first digit of the quotient in its hierarchical color, reading the number at the left side of the board.
Remove the beads from the board and place them back into the tubes. There is one units of thousand bead remaining in the bowl, which can’t be distributed as is. Change it for 10 hundreds (pour the hundreds into the hundreds bowl). Having finished with the units of thousands, place the rack and the bowl out of the way on the tray.
Continue in the same way for hundreds, tens and units.
Note: Here also the operation is reduced to the level of memorization.
Recall the problem 81 ÷ 8 = which couldn’t be done before. This does not mean that it couldn’t be done, just that it couldn’t be done with that material. Try it using this new material.
It is important to emphasize that every time a hierarchy is considered, a digit must be placed in the quotient. If there are not enough beads to distribute, we must still record a zero. This is where the child could easily make a mistake.
Set up the problem as before and begin distributing the units of thousands. Record the first digit of the quotient using the hierarchical color green. There are no beads in the bowl, so the remainder is zero. Write the remainder under the 9. Put the thousands away.
Bring down the hundreds, that is, move the bowl closer to the board and write the 2 next to the 0. Since we can’t distribute these two beads, write the next digit in the quotient and write the remainder. Remove the beads.
The two hundreds must be changed for tens. Put away the hundreds. Bring down the tens. Now we must distribute 21 tens. Continue in this way.
Note: It is important to work through a problem such as 1275 ÷ 3 = to demonstrate the grouping of the first two digits. In a case such as this we do not record a zero in the quotient for the first hierarchy
3rd Level-Group Division
Note: The child will never reach abstraction using the distributive division technique.
Recall the concept of group division. Take out 15 loose golden beads. Invite the child to find out how many groups of three can be formed.
Relate word problems to demonstrate the difference between distributive division and group division:
1. I have 12 pencils which I must give to 6 children. How many pencils will each child get? 2. What kind of division did we do? distributive division.
2. I have 25¢. I want to buy pencils costing 5¢ each. How many pencils may I buy? 5. This time I had to think of how many groups of 5 are in 25. This is group division.
Note: The Difference here is mostly in language, for this is an important step in the development toward abstraction. At this level the child incorporates the other operations in a conscious way. The quotient is no longer written in the hierarchical colors, because by this time, the concept should be firm in the child’s mind.
Set up the materials as before. We must see how many times this group of 5 (indicate the skittles) is contained in this 7 (the units of thousands). Distribute the beads. Record the quotient. One group of 5, that is 5 x 1 = 5. Write 5 under the 7 and subtract. This is our remainder. Check to see if the number of beads in the bowl matches the difference.
Change the remaining two units of thousands to hundreds and bring down the hundreds.
Now we must find how many groups of 5 are contained in 26.
Age: from 6 – 7 years ( at 7 years old, the child should reach abstraction)
Note: When the child has reached abstraction of division, he has actually reached abstraction for all the operations, since division involves all of the operations. Before progressing to Big Division (having a divisor of more than one digit) the child should have reached abstraction with small division, that is, without the materials.
BIG DIVISION (TWO OR MORE DIGITS IN THE DIVISOR)
Recall with the child the presentation of the concept of division with a two-digit divisor, using the decimal system materials and the arm ribbons. Introduce the bead board for tens. Recall that each blue skittle represents 10 units. If I give 10 to the blue skittle, what must I give to the green skittle? After this concept is already recalled, begin division.
Set up the material as before. Bring down the tens and units of the thousands, one for each board. Distribute the beads: one 10,000 for the tens; one thousand for the units. The first digit of the quotient is 1, but 1 what? The result is what one unit receives, so it is one thousand. Record the digit in color.
Remove the beads from the board. Change the 10,000 to ten units of thousands and out away the ten thousands. Move the rack and bowl of units of thousands to the left, to the tens board. Bring down the hundreds. Distribute.
When the bowl of the lesser of the two hierarchies being considered is emptied, continue changing and distributing. However, when the bowl of the greater hierarchy is emptied, we must stop, record the digit of the quotient, and move on to another hierarchy.
3rd level-Group Division with a two digit divisor
Recall the meaning of group division in the same way as before.
Set up the problem as usual. Bring down the first two hierarchies. How many times is 24 contained in 88? First, we must find how many times 2 is contained in 8. Distribute the beads 4 times. Now we must see if 4 is contained in 8, 4 times also. Distribute the beads. It doesn’t work. Take off one row of beads from the board and place them in the bowl. Change a thousand bead to 10 hundreds and distribute.
Since we want to make groups of 24, there must be the same number of groups of 2, as there are groups of 4. Thus 3 groups of 24 were made. 3 what? Refer to what one unit received – 3 hundreds.
Multiply 3 x 24, carrying mentally and recording the product beneath 88 in the dividend. (The number that we had to carry in the small multiplication, corresponds to the number of changes that were made while distributing.) Subtract; the difference should match the quantity that remained in the bowls. Remove the beads. Change. Bring down a new hierarchy and continue as before.
LONG DIVISION WITH A THREE DIGIT DIVISOR
Note: If the child has reached abstraction in division with a 2-digit divisor, he will encounter very little difficulty here because the mechanics of the operation are the same. Thus, the material will be used less by the child. The material is used for the presentation to be certain that the child has understood the concept. At this level, group division is used immediately.
Recall the activity done with the decimal system material and arm ribbons. Note the difference that the centurion received over the decurion and the unit. Present the materials.
The procedure follows the pattern set down previously, now using 3 bead boards, and bringing down 3 hierarchies at a time. Remember that the first digit tells what all of the others must receive: How many times is 2 contained in 5? 2 We must see if 3 is contained in 6 2 times and if 4 is contained in 4 2 times.
DIVISION WITH ZERO IN THE DIVISOR
51,252 ÷ 207 =
Recall the similar case in the stamp game where a counter took the place of zero for the skittles. Here the board without any skittles reminds us of the zero. Move down the three hierarchies- one for each board. The hierarchy by the empty board reminds us of what the tens would receive if there were any. Distribute as before using group division.
19,293 ÷ 370 = 676 ÷ 300 =
Make the child conscious of what the units would have received, if there were any , in order to determine the value of the digit of the quotient. Hierarchic colors can be used for recording the quotient.
Dividend: 70,569 ÷ 229 =
Place the dividend in the bowls, leaving one empty. Do not put it back on the tray since it will be needed for making changes. Bring down the hierarchies as usual, ignoring the fact that one bowl is empty; that hierarchy corresponds to one of the digits in the divisor.
Age: 9 years (by 9 1/2 the child should reach abstraction)
…mastery of long division
…knowledge of the reasons for every aspect of the procedure