A fraction by definition represents part of a whole.
The whole can be any number like 1 or a larger number can represent one whole such as 22 or 5 such that a fraction of 5/22 is a portion of the whole which is 22/22.
{jatex} 22 \div 22 {\jatex}
Usually 1 is represented as the whole but 22/22 or 7/7 is also a whole and so 25/22 and 2/7 are part of these wholes respectively.
In this way the denominator represents one whole part and the numerator a part of that whole part. Eg. As above there are 22 parts that make up a whole, when that whole is represented by 22.
A fraction represents a quantity and an amount just like a simple number does. However it represents a ratio of the part to the whole. That it is indicates a relationship between the part to the whole. A fraction can be represented as a decimal number. A decimal represents a notation and an amount that shows the portion of a whole.
A fraction represents an amount.
The amount usually represented by a fraction or a decimal is the amount left over when one number cannot equally go into another number. This occurs when things cannot be all dealt with in whole number outcomes. Eg. When one whole number cannot be divided equally by another whole number, the result is the amount of times it divided in(or goes in), plus the left over which is a portion of a whole unit that is left over that can’t go in fully any further as a whole unit but can as a part of a whole unit ie. a fraction of it.
When that number does not go equally into that other number a certain amount of times, the portion that does not fit is represented as a portion of the base interval or number
A fraction is a ratio, not directly a number in this form but it can be turned into a number or a decimal number. The fraction tells the story of how the numerator relates to the whole(which is the denominator). So 5 is an amount or quantity relative to the whole(13).
This ratio between numbers can be turned into a number which uses the two relations by defining how many times the numerator goes into the whole/denominator.
Thus the above fraction, the numerator is the focal point and seeing that it is not 13/13 which is a whole and rather 5/13 then it must go in 0 times because 13/13 would go in exactly once. So 5/13 represents a number smaller than 0 because it’s a portion of 1 so it must like between 1 and 0. It’s clearly not zero because that would be 0/13. Conveniently this can be worked out by dividing the part by the whole ie 5 divided by 13 which gives a number as 0.38461538 which is a portion of 1 and is between 0 and 1. So this number is the relationship that 5/13 has with 13/13 or 1.
It’s a part of 1 one but only the first 0.38461538 of it.
If 5/13 is 0.38461538 then 1/3 must be that number divided by 5 ie 1/3 is 0.07692308
Which means 13 of those must equal 1 and it does!
All fractions are about how an amount, represented by a fraction(which as the whole amount encoded in its denominator) relates to 1 - ie how smaller or bigger it is that 1.
This is key. Fractions should be considered as mechanisms to reason about 1(the whole) and parts of it.