**Teaching Fractions to Young Students**

You own a bakery and the only thing you sell are loaves of bread. Not only that, but every loaf is identical. So when customers come into your shop they just say how many loaves they want..

{1,2,3,4,...}

One day a customer comes in and explains that they love your bread but your loaves are too big. They ask if you have smaller loaves. You don't. But then you have a creative idea. You take a knife and cut a loaf into 2 equal sized pieces. You sell one piece to the customer and they are happy.

But what did you just sell? It was not a loaf. It was something less. You chopped a loaf into 2 equal pieces and sold one of the pieces. You sold "one out of two", so you could represent that as 1/2.

This idea is popular with your customers. Pretty soon you are chopping your loaves into 5 equal pieces and selling customers 1, 2, 3 or 4 of the pieces. That's 1/5, 2/5, 3/5 or 4/5.

Of course, if you gave a customer 5 out of 5 then that's the same as the whole loaf, so 5/5=1.

If a really fussy customer comes in and asks for 7/13 you know exactly what to do. You take a loaf, chop it into exactly 13 equally sized pieces and then sell the customer 7 of the pieces.

These things are fractions. Mathematicians call them rational numbers.

We can think of these as numbers between the integers. It's pretty easy to see that between any two integers there are an infinite amount of rational numbers. For example, the infinite series of fractions {1/2, 1/3, 1/4, 1/5...} all lie between 0 and 1.

So our list of numbers is expanding.

First we had the positive integers {1,2,3..}

Then we added zero {0,1,2,3..}

Then we added the negative integers {..-3,-2,-1,0,1,2,3..}

And now we've added an infinite number of rational numbers between any two integers.

Rational numbers, or fractions, are just pairs of integers that obey certain rules.

We can multiply two fractions. Here's the rule..

(a/b)*(c/d)=(a*c)/(b*d)

Here's an example (3/4)*(2/5)=(3*2)/(4*5)=6/20

We can add two fractions. Here's the rule..

(a/b)+(c/d)=(a*d+b*c)/(b*d)

Here's an example (3/4)+(2/5)=(3*5+4*2)/(4*5)=23/20

Rational numbers do a very important job, they extend the number system. We have an infinite number of integers, and between any two integers we have an infinite number of rational numbers. That's a lot of numbers, but it's still not enough! That's because rational numbers do not totally fill the gaps between the integers. Amazingly there's still room for other numbers.

That's right. Between every two integers there's an infinite amount of other numbers in addition to the rational numbers. These are yet another type of number. But the name that mathematicians picked for these number is not so creative, they called them irrational numbers!

Content written and posted by Ken Abbott abbottsystems@gmail.com