I wish I had more time in the teaching year to have students work with partners to do what I did below which is to write and illustrate a story that matches each fraction operation. Google table and Google draw are excellent vehicles for math model illustration. By working with all operations at once, students would be able to better see how the fraction operations are related and different. This might be something I include in upcoming RTI sessions or other math blocks. We'll see if time allows.
This is an example of the kind of work I'm thinking about.
I had a big birthday party for my dad. I bought a giant rectangular cake. My family ate 1/2, how much cake was left?
I wanted to divide the remaining 1/2 of a cake into 1/8 pieces to give to my family members. How many 1/8 pieces can I made with 1/2.
1/2 divided by 1/8
1/2 X 8/1 = 8/2 = 4
Also at the birthday party, I had 6 pieces of red licorice which wasn't enough, so I divided each piece into 1/4 pieces. How many pieces did I end up with.
6 divided by 1/4 = 6/1 divided by 1/4 = 6/1 X 4/1 = 24
To illustrate that looks like this:
Six pieces of licorice cut into 1/4 pieces makes 24 pieces.
It's very easy to think about multiply whole numbers by a fraction. For example, if I wanted to give every child 1/2 an apple for snack and there were 5 children, how many apples would I need altogether.
Mathematically you can calculate 5 X 1/2 = 5/1 X 1/2 = 5/2 = 2 1/2 or 2.5 apples.
It's a bit trickier to think about multiplying two fractions. For example, getting back to my dad's cake. If one of my brother's took his 1/8 piece of cake home and then he gave 1/4 of it to his neighbor, how much of the original cake did his neighbor get. What is 1/4 of 1/8 or 1/4 X 1/8. This is where the area model works well:
My brother had 1/8 of the cake (colored in gray), and he gave his neighbor 1/4 X 1/8 which is colored in yellow. 1/4 of 1/8 is 1/32 of the entire cake.
It's simple to add fractions such as 1/2 + 1/2 + 1/2 + 1/2 + 1/2 (as we see in the apple picture above), in this case you simply add the numerators to get 5/2 and then you can turn it into the mixed fraction of 2 1/2.
It's more difficult to add (or subtract) fractions with different denominators and that's when you have to come up with a same denominator.
For example if my brother ate 1/2 of a chocolate bar, and my sister ate 1/4 of the candy bar, and I ate 1/8. How could we add these fractions to find out how much candy we ate altogether.