Introduction | Addition of Fractions | Common Denominators & Equivalent Fractions | Subtraction of Fractions 

Introduction
 CA GR6 NS 1.3, CA GR7 NS 1.2

WHAT IS A FRACTION?

A fraction is a part of a whole thing.  I might be three of ten potatoes - or one of a dozen bananas - or a quarter of a dollar.  We use three things to show a fraction:

We use a number,

and a line under that number,

and a number under that line (the bar of the fraction).

The bottom number (denominator) tells us how many parts are in the whole thing.  The top number (numerator) tells is how many parts we are using.

Let's take a 1-foot piece of wood as an example,  and divide it into fractions.
 

1 ft.
 
Here, we divide it into 2 equal parts.  The shaded part is 1 out of 2 parts, and the
   
fraction is  

Numerator---> Denominator--->
 
Here, there are 4 equal parts and the
       
fraction is .

Notice that when the denominator gets larger, we get more equal parts, but each gets smaller.

 

Here again we have made 4 parts, but  now we are using 2 of them.
       

 

and here we are using 3 of them.  
       

 

For additional instructional material and practice exercises on this subject, go to: www.visualfractions.com/EnterFraction.html.
 

Did you notice that the size of the fractions  and  are the same? Well they are for a good reason.  You know that any number divided by itself equals 1. For  example, 2 ÷ 2 = 1, 5 ÷ 5 = 1, 10 ÷ 10 = 1.  You also know that 1 times any number equals that number.  For example, 2 x 1 = 2, 5 x 1 = 5, and 10 x 1 = 10.  This means that if we multiply the numerator and denominator of any fraction by the same number, we are really multiplying that fraction by 1, so we are not changing its size.  We are only changing the way it looks.  

As you saw from the diagrams, if we start with the fraction of  and multiply the denominator by 2, we get 4 equal and smaller parts. But if we now multiply the denominator by 2, we get twice as many of the smaller parts, and so .

Because the fractions are equal, we call them equivalent fractions.

Since we can make equivalent fractions by multiplying the top and bottom of the fraction by the same number, we can also make equivalent fractions by dividing each by the same number.  For example:

This means that by multiplying or dividing we can change any fraction to all kinds of

equivalent fractions.  Look at the equivalent fractions we can get by starting with  and multiplying the numerator by 2, 3, 4, etc.

Or by dividing the numerator and the denominator of the two fractions by 5

When both the numerator and denominator of a fraction cannot be divided by any number other than 1, the fraction is said to be in lowest terms.

As you will soon see, being able to change fractions to equivalent fractions will make it possible to add and subtract fractions.

Addition of Fractions  (top)
CA GR5 NS 2.3, CA GR6 NS 2.1

Let's start off and try to add  and .

That's easy.  We have 1 whole thing divided into 5 equal parts.  We just take 1 of those equal parts, add it to the other 2 of the equal parts, and we have 3 of the equal parts.  Here is what it looks like.....

So, if the denominators are the same, we can add the numerators.

But what if we want to add ?  We just can't do it off the bat because the first whole thing was divided in 5 equal parts and the second whole thing was divided into 10 equal parts.  It's like, it's okay to add 1 penny and 2 pennies to get 3 pennies.  But to add 1 penny to 1 nickel, you have to change the nickel to pennies, and then get 1 + 5 = 6 pennies.  It's the same thing with fractions.  You first have to change the denominator to be the same so that the size of each is the same.


This brings us back to equivalent fractions  

and common denominators.. 

Now we can add the numerators:

Now let's do Practice Exercise 4-1.

 

Next Page: Common Denominators