Uncountable set of uncountable equivalence classes

I am trying to find $A/\sim $, a set A with an equivalence relation $\sim $ such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.

I already tried with $A=\mathbb{R}$ and $x\sim y:=x-y\in \mathbb{Z}$. The equivalence classes are the sets of reals with the same fractional part. You can show that $A/\sim =[0,1)$. Which is easy to show is uncountable with an injection $f:\{0,1{\}}^{\mathrm{\infty}}\to [0,1)$ (this is the standard method I use to show a set is uncountable).

However, I think all elements in my equivalence classes are countable, as they are all natural numbers plus a specific fractional part. Am I right to think like this, does it make my example false ? In case it is false, is there a way to "fix" it?

I am trying to find $A/\sim $, a set A with an equivalence relation $\sim $ such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.

I already tried with $A=\mathbb{R}$ and $x\sim y:=x-y\in \mathbb{Z}$. The equivalence classes are the sets of reals with the same fractional part. You can show that $A/\sim =[0,1)$. Which is easy to show is uncountable with an injection $f:\{0,1{\}}^{\mathrm{\infty}}\to [0,1)$ (this is the standard method I use to show a set is uncountable).

However, I think all elements in my equivalence classes are countable, as they are all natural numbers plus a specific fractional part. Am I right to think like this, does it make my example false ? In case it is false, is there a way to "fix" it?