# Explaining fractions in a consistent way

I'm trying to explain fractions to a keen 8 year old. I could think of two approaches:

(1) Total Parts divided among Some People (Numerator = no. of pizza)

$\frac{1}{2}$ is the slice if 1 pizza is divided among 2 people

$\frac{12}{4}$ is 12 pizza divided among 4 people (each gets 3)

But explaining $\frac{3}{4}$ gets messy. Take 3 pizza, but divide it among 4 people and notice that each gets the shape that looks like 3 quarters.

(2) Some Parts out of Total Parts (Consider only 1 pizza overall)

$\frac{3}{4}$ Take 1 pizza. Divide by 4. Pick 3 parts out of 4 total parts

But explaining $\frac{12}{4}$ is messy. Take 1 pizza. Divide by 4. You need 3 such pizza.

Any suggestions on how to use a consistent explanation ? I want to avoid saying: imagine "Approach 1" for some fractions, and imagine "Approach 2" for some other fractions.

I understand Mathematics Stack Exchange usually has difficult questions, but it also states that it is "Q&A for people studying math at any level". Hence I'm hoping this genuine question is not out of place.

• There is a building with 12 floors. If one family lives on 4 floors, then how many families fit in the building? – KKZiomek Sep 5 '17 at 3:32
• I'd use the second approach and explain $12/4$ as repeating $1/4$ $12$ times. – Math Lover Sep 5 '17 at 3:46
• @MathLover That's nice. So, (x/y) irrespective of proper or improper fractions can be explained as (1/y) times x. – Kaya Toast Sep 5 '17 at 3:52
• @KayaToast Yes. – Math Lover Sep 5 '17 at 3:52
• Whatever you do, please explain the terminology correctly. (I'm not suggesting that you don't.) "numerator" = "count" (ie, "how many?"), "denominator" = "type" (ie, "what kind?"). When a student gets these, your Approaches 1 and 2 should merge nicely. – Blue Sep 5 '17 at 3:54