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

by Justin Skycak on

Cross-posted from here.


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 ${a}\div{b}$)?


I think the heuristic

$$\begin{align*} \frac{8}{4} \quad \longleftrightarrow \quad 8 \textrm{ out of every group of } 4 \end{align*}$$

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.