Teaching Fractions and the Referent Unit

Seeing his wife hard at work in the kitchen, her husband decides to give her a hand and asks if he can help. "It would be wonderful if you could peal half these potatoes for me", she replies handing him a bag of potatoes.  This is what he did.

Whenever we use a fractional term, either in he classroom or in everyday life, we have to be clear about the referent, or the one, to which the fraction refers. Sadly, this is a key concept in teaching fractions which is almost never taught.

Ask a group of fourth graders if they would rather have half the money in your left hand or a quarter of the money in your right hand and they will nearly always say they would prefer half the money in your left hand. When you open you hands to reveal a quarter in your left hand and a $20 bill in your right hand the look on their faces is priceless.

When we add or subtract fractions this is a fairly straight forward activity if we have the one to which the fractions refer at hand. If we are just doing this procedurally without manipulatives then we need to make sure that children know that both fractions refer to the same one.

When we multiply or divide with fractions, however, things get significantly more difficult. In multiplication, for example, 1/2 x 1/4 can be better understood if we make sure the one to which each fraction is known. A half of a quarter of a piece of paper is an eight of the piece of paper. The referent for  the quarter and the eighth is the piece of paper but the referent for the half is the quarter piece of paper.

A similar situation is true in division such as in 1/2 ÷ 1/4.  How many quarters  are there in the first half of a football game, There are two. Here, the referent for the 1/2 and 1/4 is football game and the referent for the 2 is the 1/4 of a football game. This always seems to be such a much more important thing to teach than "change the sign and invert the second fraction"