In order to define the chow group of a variety $X$ one considers cycles modulo rational equivalence where two cycles $A_0, A_1\in Z(X)$ are called rationally equivalent if there exists a cycle on $\mathbb{P}^1\times X$ whose restriction to $\{t_0\}\times X$ and $\{t_1\}\times X$ are $A_0, A_1$ respectively.
I want to use this definition to give some intuition for the behaviour of the cohomology ring in Schubert calculus. This means that the variety $X$ is simply the Grassmannian. Is it possible in this restricted context to describe $\mathbb{P}^1\times X$ and rational equivalence without having to introduce the machinery of schemes? I suspect that there may be some relation to Segre varieties but am not sure I truly understand what $\mathbb{P}^1\times X$ means.