Let $X$ be a variety. In order to define the Chow ring of $X$, there seem to be two different ways of defining rational equivalence of $i$-cycles $Z \sim Z'$.
Definition 1: $Z \sim Z'$ iff there is a subvariety $V \subseteq X$ of dimension $i+1$ and a rational function $\varphi \in K(V)$ such that $Z - Z' = \text{div}(\varphi)$. This is the definition at the Wikipedia article for adequate equivalence relation, for example.
Definition 2: $Z \sim Z'$ iff there is a subvariety $V \subseteq \mathbb{P}^1 \times X$ and points $a,b \in \mathbb{P}^1$ such that $(V \cap \{a\} \times X) - (V \cap \{b\} \times X) = Z - Z'$. I've seen this stated both with and without the additional hypothesis that $V$ is flat over $\mathbb{P}^1$. This is the definition at the Wikipedia page for Chow Ring, for example.
Of course, I'd love to see a proof in as much generality as possible, but I will be satisfied with a proof in the case when $X$ is a smooth projective variety over $\mathbb{C}$ and $Z,Z'$ Cartier divisors on $X$.
EDIT I should add my attempts: it's not hard to see that the first definition implies the second, by taking $V$ equal to the graph of the rational function on the dimension $i + 1$ subvariety and taking the pushforward.
For the second direction, we need to show (in the case of divisors) that $V$ somehow is related to the graph of a non-constant rational function on $X$. But I don't see why this must be true: surely there are examples where $\mathbb{P}^1 \times \{x\}$ for some $x \in X$ is not just a point! (A dumb example is $\mathbb{P}^1 \times W$ for some subvariety $W$ - this only shows that $W$ is rationally equivalent to $W$)