The usage of denominator

"How will you explain to your students that when you add or subtract fractions you must have a common denominator but you do not need one with multiplication or division? In your explanation, make clear what a common denominator is and how to find it."

Step 1: I would give the definition of a common denominator which is defined as
Common denominator just means that the denominators in two (or more) fractions are common, or the same. example 2/5 +1/5

Step 2: I would explain to the student(s) how to find the common denominator. I will explain there are two ways to find the common denominator you can add you two bottom numbers(denominator) or you can list the multiples.  Example 1/2+2/5  2*5=10   or list multiples 2,4,6,8,10... 5, 10...

Step 3:  I would use a method of measurement to show them why the denominators have to be the same.  When it comes to adding or subracting(take-away) the bottom measurements have to be the same. For example I would draw a pizza graph.  First a pizza with three sides and only 1 slice left (1/3) + another with 6 sides and only 1 slice (1/6) you can't add the denominators because the are not the same.  Use one of the methods in step 2 to get your common denominator.   

When it comes to multiplication you just multiply across.  Changing the denominator will force the fraction to beome improper; which will change the entire problem.