Ways of Adding Fractions

Date: 03/19/99 at 23:53:21
From: demarie
Subject: Fractions

If you need to calculate 1/3 + 1/6, what do you do since the 
denominators are different? Do you add the denominators?

Date: 03/20/99 at 09:24:13
From: Doctor Derrel
Subject: Re: Fractions

Remember that a fraction is composed of three things:

The first is the whole thing, also called the unit or the one. It 
might be the distance from 0 to 1 on the number line. It might be a 
whole square. It might be the kind of one cup measurement you use in 
cooking.

The second is how many equal pieces the whole thing is divided into. 
The number of equal pieces is the denominator. If you have a measuring 
cup that you would use for flour, it holds one cup exactly. (This is 
the type of measuring cup that you fill so the flour is heaped up, and
then you scrape off the stuff that is above the cup.) Some of these 
cups are divided into four equal parts by lines that are drawn on the 
inside or outside. Cups like these are divided into fourths, so the 
denominator is 4. You might get one from your kitchen and see how many 
parts it is divided into.

The third thing is how many of those equal pieces you are interested 
in. That is the numerator. If you are interested in three of the four 
parts in the cup, you would be interested in 3/4 of the cup. Now, it 
is possible for the numerator to be larger than the denominator. This 
happens when you are interested in more pieces than are in the whole 
thing. For example, a pancake recipe might call for 3/2 cups of flour. 
(But the cookbook would probably write it as 1 1/2 cups; 3/2 is what 
is called an "improper fraction," while 1 1/2 cups is what is called a 
"mixed number.")

So, does it make sense to add the denominator? Let me suggest 
something to you. Go into your kitchen and get all of the measuring 
cups you can find that have fractions marked on them, like 1/3 or 1/4 
or 1/2. Next, write a fraction equation, like 1/4 + 1/2. Solve that by 
putting a quarter cup (1/4) of water in a one-cup measure and then 
pouring it in a glass, and then getting a half cup (1/2) of water and 
pouring that in the same glass. Finally, pour the water in the glass 
back into the measuring cup to see what you get.

Now make a table on paper and write down several equations and the 
answers you get. Some of these might be kind of hard, so work on this 
with a friend. Put these equations in a table in this order and figure 
out how much the answer is. Then, look for patterns.

1/4 cup of water plus 1/4 cup of water = ?/? cup of water
1/4 cup of water plus 2/4 cup of water = ?/? cup of water
1/4 cup of water plus 3/4 cup of water = ?/? cup of water
1/4 cup of water plus 4/4 cup of water = ?/? cup of water

1/4 cup of water plus 1/3 cup of water = ?/? cup of water
1/4 cup of water plus 2/3 cup of water = ?/? cup of water
1/4 cup of water plus 3/3 cup of water = ?/? cup of water

1/4 cup of water plus 1/2 cup of water = ?/? cup of water
1/4 cup of water plus 2/2 cup of water = ?/? cup of water

1/3 cup of water plus 1/3 cup of water = ?/? cup of water
1/3 cup of water plus 2/3 cup of water = ?/? cup of water
1/3 cup of water plus 3/3 cup of water = ?/? cup of water

1/3 cup of water plus 1/2 cup of water = ?/? cup of water
1/3 cup of water plus 2/2 cup of water = ?/? cup of water

If you do not want to play with water, take strips of paper all the 
same length and fold separate strips into 2 parts, 3 parts, 4 parts, 
5 parts, 6 parts, 8 parts, 9 parts, 10 parts, or 12 parts. Make the 
parts as equal as you can. You should have 9 strips, each representing 
one type of fraction. Draw lines in the folds so that it is easy to 
see each of the equal pieces that makes up the whole fraction strip. 
Write on each strip what it is: for example, halves, thirds, fourths, 
fifths, sixths, eighths, nineths, tenths, or twelfths. Then, try 
adding various fractions by lining the strips up side by side. For 
example,  1/4 + 1/2 would look like:

+---------+---------+---------+---------+
|||||||||||    1/4  |         |         |
+---------+---------+---------+---------+
          +-------------------+-------------------+
          |||||||||||||||||||||    1/2            |
          +-------------------+-------------------+

You can see from the fraction strips that 1/4 + 1/2 is 3/4. So, rather 
than trying to figure out how to add the fractions just by looking at 
the numbers, use your fraction strips or draw little pictures to help 
you.

- Doctor Derrel, The Math Forum
  http://mathforum.org/dr.math/