**From factors to exponents**

We encounter factors of numbers in several situations of life. Parents use them when they try to divide things equally among children. You use them when you try to divide the class into small groups to do some activities. For instance, if, in your class there were 35 students, you might divide them into small groups so that each group has the same number of students. But what if there were 37 students? Would you be able to divide them into equal groups?

**Understanding Factors**

This is a real example that helps in understanding factors, and children need such examples to help them when they are learning factors. Here is a little story you can use:

In a faraway village a group of children decided to explore their neighbourhood forest. As they wandered around the forest, they found a box hidden in a hole in a huge tamarind tree. They took the box out and found that it contained some rare coins. They quickly counted the coins and found that there were 45 of them, which they divided among themselves equally.

Now ask your students to guess how many children there were in the group. Of course, you will get

different answers, as there are several possibilities.

(i) There could have been 3 children and each child could have got 15 coins.

(ii) There could have been 5 children and each child could have got 9 coins.

(iii) There could have been 9 children and each child could have got 5 coins.

(iv) There could have been 15 children and each child could have got 3 coins.

(v) Or there could have been 45 children and each child could have got 1 coin.

Ask them if there are any more possibilities. There are none, right, so point this out. Also tell them the

reason why this is so: it is because the numbers 1, 3, 5, 9, 15, and 45 are the only numbers that divide

45. These are called the divisors or the factors of 45.

**Factorizing numbers**

Now draw attention to the number 45 again. Highlight the first possibility, 45=3 x15. Point out that 15 itself can be broken into factors and written down as 3x5. So we can write 45=3x3x5. In other words, the only factors of 3 are 1 and 3 and similarly for 5, its only factors are 1 and 5. Natural numbers that have exactly two divisors, namely 1 and itself, are called prime numbers and natural numbers that are the product of at least two prime numbers are called composite numbers.

Point out that in the above example we wrote 45 as a product of prime numbers. This is called factorizing 45 into a product of prime numbers. We can do this for any number. For example,

100=2x50=2x2x25=2x2x5x5

60=2x30=2x2x15=2x2x3x5

81=9x9=3x3x3x3

210=2x105=2x3x35=2x3x5x7

Get the children to try and factorize all the numbers from 2 to 100.

**Factors and multiples**

Factorizing is one thing, but one can also do the reverse: one can build up new numbers by multiplying

prime numbers. Provide examples to the children:

2x3=6

2x2x3x5=60

3x3x5x7=315

Let them have some fun computing such combinations on their own. This is where you can also help

children understand the difference (and the relationship) between factors and multiples. From the

above examples, take the number 60 and ask the children for its factors, putting them down in a column

on the board (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 20, and 60). Now, in a second column put down the numbers

that are obtained by multiplying 60 with another number. So you get 120 (60x2), 180 (60x3), 240, 300,

and so on —explain that these numbers are multiples of 60. Let them try this with smaller numbers. E.g.:

24 (factors 1, 2, 3, 4, 6, 8, 12, and 24; multiples 24, 48, 72, 96, 120, and so on).

**Factors to exponents**

Now that they know how to build new numbers, get them to do this using the same factors each time, but in varying numbers. Let’s say they should use the prime numbers 2, 3, 5, and 7. What if they used each of these only once? They would get the number 210. Now ask them if they can guess (or calculate) what

2 x2x3x3x3x5x5x5x5x5x7x7x7x7x7x7x7 will be.

The answer is a big number having 11 digits: 11117830500. The prime numbers that are the factors of 11117830500 are 2, 3, 5, and 7—the same

numbers that are the prime factors of 210! The difference lies in how many times 2, 3, 5, and 7 appear in the factorization of these two numbers. In 210, each of them appears only once, whereas in 11117830500, the prime number 2 appears twice, 3 appears three times, and so on.

Put up the equation on the board and get the children to look carefully at the factorization:

11117830500=2x2x3x3x3x5x5x5x5x5x7x7x7x7x7x7x7

It takes up a lot of space, doesn’t it? More importantly, to find out how many times each prime number

appears in the factorization, we need to count them. Now if we were to write it as 11117830500=2^{2}x3^{3}x5^{5}x7^{7}, where the small number at the top right showed us how many times a number appears, wouldn’t that be a neat way of keeping track of the number of times a prime number appears and saving space too? Tell the children that we read 2^{2}, 3^{3}, etc., as 2 to the power of 2, 3 to the power of 3 and so on. So 11 to the power of 3 will be 11^{3}=11x11x11.

Get the children to go back with this new notation and rewrite the factorization of some numbers:

45= 3^{2}x5

100=2^{2}x5^{2}

60=2^{2}x3x5

81=3^{4}

Using the above examples, you can get children to understand that prime numbers are the building blocks from which other numbers are made using repeated multiplication. Therefore, they are very important in mathematics. The fact that almost every number can be factored as a product of prime numbers as we have done here was known to the Greeks even before 300 BC, i.e., 300 years before the birth of Jesus Christ, more than 2300 years ago! Even ancient Greek mathematicians like Pythagoras and Euclid were fascinated by prime numbers and the factorization of a composite number into its prime factors

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## A needy lesson plan

A needy lesson plan