Multiples, Prime Numbers, Factoring

When the child, by the aid of all this material, has had a chance to grasp the fundamental ideas relating to the four operations and has passed on to the execution of them in the abstract, he is ready to continue on the numerical processes which will lead to a more profound study preparatory to the more complex problems that await him in the secondary schools.

These studies are, however, a means of helping him to remember the things he already knows and to enlarge upon them. They come to him as a pastime, as an agreeable manner of thinking over either in school or at home the ideas which he already has gained.

One of the first exercises is that of continuing the multiplication of each number by the series of 1 to 10 which was begun by the exercises on the multiplication tables. This should be done in the abstract: that is, without recourse to the material. Let us, however, set some limit—we will stop when each product has reached 100. In order that these series of exercises may each be in one column the first exercises will stop with 50 and another can be used for the numbers from 51 to 100.

The two following tables (A and B) are the result. These are prepared in this manner in our material so that the child may compare his work with them.

To read over a column of the results of each number is to learn them by heart, and it impresses upon the child's memory the series of multiples of each number from 1 to 100.

With these tables a child can perform many interesting exercises. He has sheets of long narrow paper. On the left are written the series of numbers from 1 to 50 and from 51 to 100. He compares the numbers on these sheets with the same numbers in the tables, series by series, and writes down the different factors which he thus finds; for example, 6 = 2 × 3; 8 = 2 × 4; 10 = 2 × 5. Then finding the same number in the second column and the other columns his result will read, 6 = 2 × 3 = 3 × 2; 18 = 2 × 9 = 3 × 6 = 6 × 3 = 9 × 2.

In this comparison the child will find that some numbers cannot be resolved into factors and their line is blank. By this means he gets his first intuition of prime numbers (Tables C and D).

When the child has filled in this work from 1 to 50 and from 51 to 100 and has reduced the numbers to factors and prime numbers he may pass on to some exercises with the beads.

The children now meditate, using the material, on the results that they have obtained by comparing these tables. Let us consider, for example, 6 = 2 × 3 = 3 × 2. The child takes six beads, and first makes two groups of three beads and then three groups of two.

And so on for each number he chooses. For example:

The child will try in every way to make other combinations and he will try also to divide the prime numbers into factors.

This intelligent and pleasing game makes clear to the child the "divisibility" of numbers. The work that he does in getting these factors by multiplication is really a way of dividing the numbers. For example, he has divided 18 into 2 equal groups, 9 equal groups, 6 equal groups, and 3 equal groups. Previously he has divided 6 into 2 equal groups and then into 3 equal groups. Therefore when it is a question of multiplying the two factors there is no difference in the result whether he multiplies 2 by 3 or 3 by 2; for the inverted order of the factors does not change the product. But in division the object is to arrange the number in equal parts and any modification in this equal distribution of objects changes the character of the grouping. Each separate combination is a different way of dividing the number.

The idea of division is made very clear to the child's mind: 6 ÷ 3 = 2, means that the 6 can be divided into three groups, each of which has two units or objects; and 6 ÷ 2 = 3, means that the 6 also can be divided into but two equal groups, each group made up of three units or objects.

The relations between multiplication and division are very evident since we started with 6 = 3 × 2; 6 = 2 × 3. This brings out the fact that multiplication may be used to prove division; and it prepares the child to understand the practical steps taken in division. Then some day when he has to do an example in long division, he will find no difficulty with the mental calculation required to determine whether the dividend, or a part of it, is divisible by the divisor. This is not the usual preparation for division, though memorizing the multiplication table is indeed used as a preparation for multiplication.

From the above exercises (Table D) others might be derived involving further analysis of the same numbers. For example, one of the possible factor groups for the number 40 is 2 × 20. But 20 = 2 × 10; and 10 = 2 × 5. Bringing together the smaller figures into which the larger numbers have been broken, we get 40 = 2 × 2 × 2 × 5; in other words 40 = 2³ × 5.

This is the result for 60:

For these two numbers we get accordingly the prime factors: 2³ × 5; and 2² × 3 × 5. What then have the two larger numbers, 40 and 60 in common? The 2² is included in the 2³; the series therefore may be written: 2² × 2 × 5; and 2² × 3 × 5. The common element (the greatest common divisor) is 2² × 5 = 20. The proof consists in dividing 60 and 40 by 20, something which will not be possible for any number higher than 20.

We have test sheets where the numbers from 1 to 100 are arranged in rows of 10, forming a square. Here the child's exercise consists in underlining, in different squares, the multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10. The numbers so underlined stand out like a design in such a way that the child easily can study and compare the tables. For instance, in the square where he underlines the multiples of 2 all the even numbers in the vertical columns are marked; in the multiple of 4 we have the same linear grouping—a vertical line—but the numbers marked are alternate numbers; in 6 the same vertical grouping continues, but one number is marked and two are skipped; and again in the multiples of 8 the same design is repeated with the difference that every fourth number is underlined. On the square marked off for the multiples of 3 the numbers marked form oblique lines running from right to left and all the numbers in these oblique lines are underlined. In the multiples of 6 the design is the same but only the alternating numbers are underlined. The 6 therefore, partakes of the type of the 2 and of the 3; and both of these are indeed its factors.