Division Memorization Exercises


a. Introduction and List of Materials

The memorization of division is the synthesis of the four operations. For this reason the child must precede this work with a great deal of work with the other operations, especially multiplication. It is very important that the child know multiplication really well before going on to division.
The child has encountered division before via many materials and regarding many cases: distributive and group division, division with a 1 or 2 digit divisor. In order to go on, the child must memorize certain combinations in division.

…Division Bead Board: the numerals 1-9 across the top on a green background represent the divisors; the numerals 1-9 down
the left side represent the quotients; the 1 in the green circle in the upper left indicates that the numbers 1-9 below it represent
units; 81 holes.
…Orange box containing 9 green skittles
…Orange box containing 81 green beads
…Booklet of Combinations (36 pages)
…Box of Multiplication Combinations (same combinations as are found in the booklet)
…Box of orange tiles for bingo game
…Charts I, II (for control)



b. Initial Presentation

Preparation for Presentation
Before-hand, the teacher prepares pads of problems which consist of 81 forms. On each form, in the spaces at the top, is written a number, which is the constant dividend for that form. The forms are arranged on the strip in order according to the dividends, starting with 81, ending with 1. Under the forms on which a combination with remainders appears, the ordinal number of the form is written in red. On each form are written all of the combinations possible to perform on the bead board, beginning always with 9 (except, of course, when the dividend is less than 9) All of the combinations which have a remainder of zero are underlined in red.

In this box are 81 beads which we will distribute to these 9 skittles. The nine skittles are placed along the top green strip of numbers 1-9. These 81 beads are the dividend, so we write 81 under the column ‘dividend.’ Then write the sign for division. The beads must be divided among 9 skittles; nine is our divisor, so we write 9 under ‘divisor.’ Now give out the beads in rows until all the beads are given out equally.
Each skittle received nine beads, (note the 9 at the end of the row), so we write 9 under ‘quotient.’ The last column is for the remainder, but here we have no remainder, so we write zero in that column. Whenever we have a combination that has no remainder, it is very important for our work, so we underline ‘it’ in red.
Let’s try 81 ÷ 8. Remove one skittle and the extra beads to their respective boxes. What is the quotient and the remainder? 81 ÷ 8=9 r 9 In this game there are two rules to be followed:
1) The quotient should never be greater than 9.
2) The remainder may never be equal or greater than the divisor.
Therefore, we cannot have this combination because the remainder is too big.
Go on to 80. Change to a new form. Remove one bead and place it back in a box. Start with 80 ÷ 9=. Write the combination on the form, distribute the beads and count the remainder. 80 ÷ 9 = 8 r 8
Try 80 ÷ 8. Remove one skittle, redistribute the beads, and count the remainder.
80 ÷ 8 = 9 r 8. This cannot be used because the remainder is too big, being equal to the divisor. Erase or cross out this combination.
Go on this way until 72 so that the children see another page on which combinations can be underlined. At this point bring out the prepared roll of forms. On this strip we can see all the combinations that have zero remainder. All of the forms that have at least one combination with remainder zero, have been reproduced into booklet form for your work.

Aim: to understand how the combination booklet was formed



c. Division Booklets

…Division Bead Board
…box with beads
…box of skittles
…Combination booklet
…Chart I

The child chooses a form in the booklet, i.e. 7 Since 7 is the dividend, count out 7 beads into the box cover. The first combination is 7 ÷ 7 =, so 7 skittles are put out; the 7 beads are divided among them. Each skittle receives one; 1 is the quotient. There is no remainder. 7 ÷ 7 = 1 r 0
The next combination is 7 ÷ 6 =. Only six skittles are needed. Each skittle receives one and the remaining bead is placed in the bottom row, or in a box cover. 7 ÷ 6 = 1 r 1. Continue until the form is completed.

Control of error: Chart I. Find the dividend along the top and the divisor in red at the left. Go down and across to find the quotient where the fingers met. If there is a remainder, the box will be empty, thus move along the row to the right until a box is full. There you will find the quotient. To check the remainder subtract the dividend at the top of that column, from your original dividend. The difference is your remainder.

Note: When presenting the chart to the children, we identify the prime numbers as well, since they are shaded in red. 7,5,3,2 and 1 are special numbers because they only have as divisors, themselves and one. 7 can only be divided by 7 and by 1, and so on. These special numbers are called prime numbers.



a. Chart I

…box of loose combination cards, only those that have a remainder of zero
…Chart I (as a control)

In this box are only the even division combinations: those having a remainder of zero. The child fishes for a combination, reads it and copies it into his notebook. On the chart he finds the dividend at the top and the divisor on the left side. The place where the two fingers meet is where the answer is found. He writes the quotient to complete the equation.
Later the child can do the combinations in his head, write down the quotient and use Chart 1 only as a control.



b. Bingo Game (Chart II)

…box of tiles
…box of combination cards
…Chart II and Chart I (for control)

Note: Much practice should have preceded these exercises.

Exercise A:
Spread out the tiles face up. The child fishes for a combination, writes it down including the quotient, and finds the corresponding tile. The tile is placed on Chart II appropriately.

Exercise B. 
With all of the tiles in the box, the child fishes for a tile. He thinks of a combination that would yield that quotient and writes the equation in his book, i.e. 8 = 56 ÷ 7. The tile is placed on Chart II appropriately.

Exercise C. 
All the tiles are stacked as usual (this time forming a parallelopiped) The child chooses a stack, and one at a time he thinks of all the possible combinations that will yield that quotient, writes them down and places the tiles on Chart II appropriately. This continues until all the tiles of the stack which was chosen, are used. The child uses Chart I to check if he found all of the possible combinations and if they were placed correctly.

Note: For many children the aim of this work can be to fill up the entire board.

Group Game 1. 
The teacher or a child leading a group of children draws a combination and reads it. One child responds.

Group Game 2. 
The teacher draws a tile and reads the quotient. One child may try to give all of the possible combinations, or each child in turn may give one until all of the possibilities are exhausted.

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