You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).
You can only add two things if they have the same units, as per "dimensional analysis" [1].
So this is an entirely meaningless statement:
[students]/[seats at table 1] + [students]/[seats at table 2]
But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):
([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])
Which simplifies to:
[students]/[total seats] + [students]/[total seats]
Now that's a statement with meaning!
Since I know that
[seats at table 1]/[total seats] = 1/2
[seats at table 2]/[total seats] = 1/2
I've just derived the calculation that I really wanted to do:
(1/3)(1/2) + (1/3)(1/2) = (2/6)
[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
> What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.
Explaining this in a classroom setting may be quite challenging indeed.