As for the other operations, we examine the special cases using as a starting point the combination that is most familiar. A chart will be constructed as follows:
0- Calculate the quotient
72 ÷ 9 = ? ( 72 divided by 9 gives me what number?)
1- Calculate the Divisor
72 ÷ ? = 8 ( 72 divided by what number gives me 8?)
2- Calculate the Dividend
? ÷ 9 = 8 ( What number when divided by 9 gives me 8?)
3- Inverse of Case Zero—Calculate the Quotient
? = 72 ÷ 9 ( what number will we obtain by dividing 72 by 9)?
4- Inverse of The First—Case, Calculate the Divisor
8 = 72 ÷ ? ( We obtain 8 as a quotient when dividing 72 by what number?)
5- Inverse of the Second Case —Calculate the Dividend
8 = ? ÷ 9 ( We obtain 8 as a quotient when dividing what number by 9?)
6- Calculate the Divisor and the Dividend
8 = ? ÷ ? ( We obtain 8 by dividing a certain number by another number. What is the first number and the second number?)
Note: Here we also see the relationship between multiplication and division. In cases 2 and 5 the child must multiply to find the dividend.
SEARCH FOR QUOTIENTS
…bingo tiles for multiplication
Have the child find one bingo tile to match all the dividends along the top of Chart II. These are placed in a box cover or something.
The child fishes for a tile, i.e., 24. Let’s try to find all the quotients with zero remainders that can be made with this dividend. Start with 24 ÷ 9 =. It won’t work so leave it blank, and go on24 ÷ 8 = 3 and so on with the child giving the correct quotients. At the end erase those that would not yield zero remainders, thus leaving space to correspond with Chart I. Notice how the column of quotients matches the column under 24 on the chart.
These are the combinations I wanted, because now we can do 3 x 8 = 24. Write this to the right of 24 ÷ 8 = 3. 24 was my dividend: now it is my product. Go on in the same way for the other combinations making a second column.
The child will realize that if 24 ÷ 8 = 3, then 24 ÷ 3 = 8. It is a sort of game where the numbers change positions.
…to find quotients with zero remainders
…to realize the relationship between multiplication and division memorization of division
Indirect Aim: indirect preparation for the divisibility of numbers
Materials: Chart I
Recall the child’s attention to the numbers in pink on the chart, which were called prime numbers, and which can be divided only by themselves and 1. These are very important numbers because they form all of the other numbers. We can see that this is true. (refer to the chart) 1, 2, and 3 are prime numbers. 4 is not a prime number, but is made of 2 x 2, and 2 is a prime number. Go on to 6 which is not prime. It is made up of 3 x 2; 3 and 2 are prime numbers.
If you try to decompose any number, you will find that it is made up of prime numbers. Invite the child to choose one of the dividends, and think of one combination: 24 = 3 x 8. 3 is a prime number, but not 8, 8 is made up of 2 x 4. 2 is a prime number, but not 4, 4 is made up of 2 x 2. 2 is a prime number.
Try another combination of 24 to check.
24 = 6 x 4 neither 6 nor four is prime
6 = 2 x 3 both are prime
4 = 2 x 2 both are prime
Direct Aim: to realize the importance of prime numbers
…to prepare for divisibility of numbers, LCM-least common multiple GCD-greatest common divisor and
…reduction of fractions to lowest terms
…(for all of division) to memorize the combinations necessary for division.
…to stimulate an interest that will help him to use the experiences acquired previously.