Today I want to put in a good word for improper fractions. You know, those fractions where the numerator is larger than the denominator, like 5/3 or 289/18.

We don’t usually tell kids about improper fractions until kind of late in their fraction learning trajectory, after they’re comfortable with “normal” fractions like 1/2 and 3/4 and 2/3 and 5/8. So children naturally get used to thinking that a fraction is like a partially filled pie. 3/4, for example, means we have a pie with 4 pieces but there are only 3 left. And then once they’re really good at this idea, we spring 5/4 on them, and the kids think, “Huh? If there are only 4 pieces in the pie to begin with, how is it possible to have 5?”

Even adults can have a hard time with improper fractions.

The conceptual root of this problem (if you’ll allow me to go into math educator mode for a moment) is that when kids see only fractions less than one, they start to think that 3/4 means “3 out of 4 things” which is not *quite* right, because 5/4 is a totally legitimate fraction but “5 out of 4 things” doesn’t make any sense. A better way to think about 3/4 is as three 1/4’s, where it takes four 1/4’s to make a whole. 5/4 then means five 1/4’s, where it takes four 1/4’s to make a whole. When we think of a 1/4 as a unit, we can have as many of them as we want.

So moving back to parent mode, when we give our child chances to see improper fractions, like 5/4, in real life, and when we do this early in their fraction learning trajectory, we’re not only making improper fractions themselves easier, we’re helping them develop a strong and solid understanding of what a fraction is in the first place.

Here are a few ideas for seeing improper fractions in everyday situations:

**Graham Crackers**

Graham crackers are great for introducing fractions to young learners because they break naturally into halves and into fourths, and because those halves and fourths are an identifiable unit. A graham cracker square can be called a half. A small graham cracker rectangle can be called a fourth. Four small rectangles make a whole cracker – that’s why they’re fourths. And it’s not at all inconceivable that you could have 5 or 6 or 7 or more small rectangles: 5/4, 6/4, 7/4, and so on.

**Measuring Cups**

Measuring cups are also great for thinking of fractions as units. It takes three 1/3-cup measuring cups to fill up a 1-cup measuring cup; that’s why it’s called 1/3. 1/2-cup measures and 1/4-cup measures are similar. You could experiment and see how many 1/3-cup measures would fill a glass measuring cup up to the 2 cup line. That’s six thirds (6/3)! How many thirds would it take to fill it all the way to the top, above the line? 7 thirds? 8?*

**Pizza**

Or quesadillas, pies, mini bread loaves… The important thing is that a) you can cut the food into equal-sized servings, and b) you have more than one whole (whole pizza, whole quesadilla, whole pie, whole loaf). If a child can identify a piece of pizza as 1/8 of a pizza, and can count pieces as eights (one pieces is 1/8, three pieces are 3/8, etc.), they can also tell how many eights 10 pieces would be, or 15, or how many eights there are in 2 pizzas.

**Ruler Measurements**

It’s common to use fractions in measurements – a quarter inch, a half centimeter. We have to have a way of naming measurements that are in between whole number measurements. If you’re working on measurement with your child, you’re probably using mixed numbers (e.g., 2 1/2 inches) rather than improper fractions. But go ahead and try to make the leap. If something measures 2 1/2 inches, ask: “How many half inches is that?” or “How many quarter inches is it?” This is more for older children – this is more challenging than pizza where you can see and hold and count an eighth. But it never hurts to ask something that’s beyond the child, and come back to it later if you find it’s beyond the child’s current capacity.

* As an aside, a colorful set of plastic measuring cups and spoons (maybe even one with a 1/8-cup measure) is a great Christmas or birthday gift for a child. It’s not terribly expensive, and opens up opens up all sorts of opportunities for experiences, creativity, and one-on-one time with parents or older siblings *and* (bonus!) they have their very own tool for thinking about and reasoning with fractions in a completely natural context.

It is a great idea to use foods that naturally have fractional parts. Graham crackers are perfect, partly because it is ok for the kids to eat several. We usually cut apples into eighths and have several at one time, so that suits us. Kitkat bars come in halves and are small enough that I can tolerate seeing them eat some improper fraction number of bars. Many other chocolate bars are good for talking about fractions, but I don’t even want them to eat a whole Lindt 70% on their own, let alone 9/8ths of one (or more).

Two other good places to flag fractions, improper and otherwise:

(1) money games: this works especially well for US quarters and half-dollars, since the name of the coin is the same as the fraction of a dollar.

(2) Reading sheet music is good for powers of 2, with half, quarter, eighth, sixteeenth, etc notes. This is a case where American English is strictly superior to British English as the latter doesn’t use fraction names for the notes.

This page (not mine) has some other fraction resources: http://blog.richmond.edu/openwidelookinside/archives/3252

Haha – I wouldn’t want my kid eating an improper fraction of a Lindt bar, either. I guess Halloween with all its mini candy bars could be a good time to use improper fractions with candy, though :).

Thanks for the other suggestions, too – every kid deals with money, and quite a lot have music experience. My husband is a professionally trained musician, and keeping time is the perfect overlap of our two areas of expertise :).