"Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?" — "Nine." — "How come?" — "Because 4 plus 5 is 9." — "But you cannot add boxes to pencils!" — "Why?"
Indeed, why? Why you can multiply boxes and pencils, but cannot add? This is sort of self-evident for most adults, but how do you explain it to an elementary-school student?
I came up with an approach calling a box a "bunch of things" (Common Core likes to use the word "group"), so a box by itself has no meaning, what does have meaning is that it groups, combines, ties together several things that we are actually interested in, say pencils. If each box combines the same amount of things, then we can define and use multiplication as quick addition of the same number of things.
Similar approach can be used to explain why you cannot add two tens of flowers and five flowers as 2 + 5 = 7, because it is not clear seven of what we are getting. First, we "unbunch" two tens into one, two, three, ..., twenty flowers, then add five flowers to them, so we can count them, twenty five flowers. It just so happens that by having ten-based positional system we can simply write 5 to the right of 2 to get the correct number, it won't work if we had two dozen flowers instead of two tens.
Another phrase commonly used is that you can add "like things", things that are similar. All this kinda works, but does not feel perfect, does not seem rigorous, persuasive enough. Does anyone have a better idea, approach, script to explain to kids why adding apples to apples is ok, but adding apples to apple crates is not?