That's my favourite explanation so far. That's how I would put it to the student: Yes, that's a perfectly correct operation. When you take 1/3 of the first tab

That's my favourite explanation so far. That's how I would put it to the student: Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples: * At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table. * Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables. The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3" The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6" What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.