Fractions

The Key Fraction Rule

We know that  is the answer to a division problem:

represents the amount of pie an individual child receives when  pies are shared equally by  children.

What happens if we double the number of pie and double the number of kids? Nothing! The amount of pie per child is still the same:

For example, as the picture shows,  and  both give two pies for each child.

And tripling the number of pies and the number of children also does not change the final amount of pies per child, nor does quadrupling each number, or one trillion-billion-tupling the numbers!

This leads us to want to believe:

Key Fraction Rule

(at least for positive whole numbers ).

We say that the fractions and are equivalent.

Example: Fractions equivalent to 3/5

For example,

yields the same result as

and as

Think / Pair / Share

Write down a lot of equivalent fractions for , for , and for 1.

Example: Going Backwards

is the same problem as:

Most people say we have cancelled or taken a common factor 4 from the numerator and denominator.

Mathematicians call this process reducing the fraction to lowest terms. (We have made the numerator and denominator smaller, in fact as small as we can make them!)

Teachers tend to say that we are simplifying the fraction. (You have to admit that  does look simpler than .)

Example: How Low Can You Go?

As another example,  can certainly be simplified by noticing that there is a common factor of 10 in both the numerator and the denominator:

We can go further as 28 and 35 are both multiples of 7:

Thus, sharing 280 pies among 350 children gives the same result as sharing 4 pies among 5 children!

Since 4 and 5 share no common factors, this is as far as we can go with this example (while staying with whole numbers).

Mix and Match: On the top are some fractions that have not been simplified. On the bottom are the simplified answers, but in random order. Which simplified answer goes with which fraction? (Notice that there are fewer answers than questions!)

Think / Pair / Share

Use the “Pies Per Child Model” to explain why the key fraction rule holds. That is, explain why each individual child gets the same amount of pie in these two situations:

• if you have  pies and  kids, or
• if you have  pies and  kids.