The pizza analogy that is so useful for teaching fractions doesn’t seem to describe 1/x when x itself is much smaller than one, e.g., how do you divide a pizza into 1/10 pieces, i.e., so the total number of pieces is 1/10? I think the language breaks down.
Say that x = .000001 and we want to consider 1/x. I’d ask something like this: How many millionths do you need to altogether make one? The answer is one million of them. The smaller the size of something, the more of them you need to make one standard unit. The product of x and 1/x is one (x * 1/x = 1) . As one number goes up the other goes down & vice versa, i.e., they are reciprocals, in order to maintain a constant product. If the student understands graphs, reveal the 1/x hyperbola – at least the positive part.
I’d make a formal analogy between the operations of addition and multiplication. This is the way I learned it back in the day…
Additive identity = 0 because a+0=a for all a.
Multiplicative identity = 1 because a*1=a for all a.
Additive inverse of a is -a because a + (-a) = 0 = additive identity.
Multiplicative inverse of a (a<>0) is 1/a because a * (1/a) = 1 = multiplicative identity.
Then you go on to define subtraction and division in terms of the above.
With either operation (addition or multiplication) when one of the pair of inverses goes up, the other goes down & vice versa. In the case of addition their sum is a constant zero. In the case of multiplication their product is a constant one. Hmm, if the student doesn’t comprehend negative numbers either, @LostInParadise, maybe this won’t be so easy!