# 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

## 2 Answers

You can explain 3/4 as being three 1/4 pizzas. The denominator of a feaction generally represents howany parts we will break 1 into, the numerator counts howany of these parts we have. With this in mind it is not too much of a leap for a kid to conceptualize 5/4 as "five pieces of pizza each of which is one quarter of a whole pie".

I would reccomend first practicing examplesthe where the numerator is less than. The denominator. You can draw pictures of arrangements of quarter circles and ask the kid to tell you in each case how many quarters there are. Finally draw a whole pizza and an extra quarter pizza slice next to it. If the kid is used to counting quarters they should be able to extrapolate that there are 5/4 of a pizza presented to them because that is how many quarter slices they see in front of them.

Approach two's 12/4 looks inconsistent for a 8-year-old indeed;

the 3/4 scenario for approach one may be explained using easily-divisible objects like sticks or cups of water (sticking 3 sticks together to get '3' and divide between '4' people equally).

Hope it helps, wish the kid will be a great mathematician.