# Q&A: How to Explain Improper Fractions to a Kid

*Cross-posted from here.*

## Question

When explaining fractions to my kids, I’ve used the analogy that $\frac{a}{b}$ means “you want $a$ out of every group of $b$ (of the thing you’re finding a fraction of).”

E.g. $\frac{3}{4}$ of a pizza means “you want $3$ out of every group of $4$ (pizza slices) of a pizza (that’s been sliced up to let you make groups of 4)”. So if you sliced up the pizza into $8$ slices, you’d have $2$ groups of $4$, so if you took $3$ from each group of $4$, you’d have $6$ slices, which is your $\frac{3}{4}$ of the original pizza that was cut up into 8 slices.

This analogy works pretty well for fractions that are $\leq 1$, like $\frac{3}{4}$, $\frac{5}{7}$, $\frac{11}{11}$, etc. But it seems to break down (or at least I can’t find a way to extend it) when considering fractions that are $>1$, like $\frac{8}{4}$. The extended analogy for $\frac{8}{4}$ would be “you want $8$ out of every group of $4$ pizza slices (of a pizza that’s been sliced up to let you make groups of $4$).”

Is there a way to extend the analogy that $\frac{a}{b}$ means “you want $a$ out of every group of $b$” to account for when $a > b$ (i.e. that it effectively means ${a}\div{b}$)?

## Answer

I think the heuristic

still makes sense if you emphasize that you **want** $8$ slices of pizza out of every group of $4,$ even if it’s not possible to take that many slices.

If you *want* more slices than there are available in the group, then what you’re really saying is you want more than a full pizza.

This naturally leads to the follow-up question *“Then how many full pizzas do you want?”* and the answer to that question is to divide $8 \div 4.$

Here’s how I imagine this actually being explained to a kid. (I’ll refer to “pancakes” instead of “pizzas” since it’s easier to imagine oneself hungry for multiple pancakes than for multiple pizzas.)

**Kid:**How can you take $8$ slices out of every $4$ slices?? That doesn't make sense. There are only $4$ slices. You can't take more than $4$ of those slices.**You:**You're right. You can't actually take $8$ of those $4$ slices, because there are only $4.$ But you can*want*$8$ slices that are the same size as those $4$ slices.- Imagine that we're having pancakes for breakfast. Normally, you don't even want a full pancake. So I cut up a single pancake into $4$ pieces and ask you how many of those pieces you want.
- But today you're really hungry and you want to eat not only the full pancake (all $4$ pieces), but also ANOTHER full pancake! You want to eat two full pancakes instead of just one.
- But I don't know this, so I bring out a pancake that's cut into $4$ pieces and ask you how many of those pieces you want. You have to answer my question with a number. What number can you tell me, that indicates you want more than I'm offering you?
- Well, you just tell me how many pieces
*of those size*you want. You want the entire pancake that I'm offering you ($4$ pieces), and another full pancake (another $4$ pieces of the same size). So you should tell me that you want $4+4=8$ of those pieces.