What changes this improper fraction, 5/3, to a mixed number?

I am a fifth grade math teacher.  This is the way I explaing it to my 11-year old students... nearly all of them understand it the first time around.  I also use the following terminology when I tutor high-school and college students, simply because it is honestly the easiest way to understand fractions.

First of all, you have to understand that math is a language.  It is one of the few truly universal languages.  As such, you have to understand that everything in math can take multiple forms.  In reality, what you are doing when you "solve" any math problem is simply TRANSLATING the "expression" (act of conveying an idea to someone outside myself) into another format... which retains the same meaning, but just looks different.  It's the same idea as translating from English into Spanish (or any other language).  What makes or breaks a math student is the ability to be able to recognize the information as it is seen in different forms on paper.

In the case of fractions, what most students don't understand is that the fraction itself is actually telling you what to do - it is a form of communication.  In the case of your "improper" fraction, 5/3 tells the reader that an "operation" (something we do) needs to be performed.  Operations, in math, are always things we do to numbers.  The expression 5/3 contains two number symbols and one "operation" (action) symbol.  The fraction bar in a fraction (whether the fraction is proper or improper) always tells the reader that the writer of the expression intends the reader to divide something (convenient, since the bar physically "divides" the two numbers into their own little spaces on the paper, huh?)

When you see an expression with one number over a fraction bar... all of which is over another number (some sort of fraction), you always need to think DIVIDE.  You can re-write the "fraction" into a division problem that looks more familiar to you.  You can use what I call the long-division "house" format.  The top number (numerator) will always go inside the house, and the bottom number (denominator) will always go outside the house.  When you translate the "fraction" into a familiar-looking operation, you can simply work out the long division and further translate the original question into either a decimal (standard format) or a mixed number.

The only difference between the expression 1/3 and 5/3 is that 1/3 means that it is a value less than one... when you work out the long division problem, you will find that the number "translates" to approximately 0.333 (notice the zero in front of the decimal... that means that there are ZERO wholes in the value... it is less than one).  When you work out 5/3, it "translates" to approximately 1.666 - Notice that this is MORE than one.

Either way, the fraction bar is a "command" in math.  It tells the reader to divide.  Hope this helps with fractions in general!