Would you rather have half the money I have in my right hand or a quarter of the money I have in my left hand?
We use the language of fractions constantly in our lives from the time we are old enough to ask for half a cup of orange juice or asked to share a candy bar with a brother or sister so that we have half each. Ask any six year old how old she is and you will most likely get the reply "six and half". Everything is just fine until we come across fractions in school and then all hell breaks loose and we leave common sense at the classroom door.
Remember the rigmarole we were taught to find 1/2 divided by 1/4? I seem to remember "Change the sign and flip the second fraction" as being the most common piece of nonsense we were taught in 4th grade. We never questioned the wisdom of this piece of sage advice because we never had a clue why it worked. It was part of the magic of math; it just worked when we had to use it on an exam question. Never mind the fact that we had no idea when to use it; "was it half of a quarter or how many times does a half go into a quarter or ........?"
How much easier life is now that we use mathematical tools such as the fraction strips in the picture. 1/2 divided by 1/4 actually means how many 1/4s in a 1/2 as in "how many 1/4s in the first 1/2 of a football game?". How many yellow pieces fit in one pink piece above?
As for the 'which handful of money would you rather have' question; it depends on how much money I have in each hand. The size of any fraction depends on the size of the whole to which it refers.
This is the topic of my math class this afternoon.