Re: fix concrete patio slope
The ground beneath the concrete slab has likely settled toward the foundation which will cause water infiltration of the foundation ... which is not a good thing. You mentioned your new surface will have a slop away from the foundation which is a good thing to do.
If you were to build up the sloped existing concrete area with sand for the new surface this won't last long since you really should address the soil settling issue first. Sand will also allow water to percolate through and you will still have water infiltrating the foundation.
It's unclear as to condition of the rest of the concrete slab and what area you live .... if you have winter ground frost ... this will also affect the new surface you want. If the existing slab is moving around underneath the new surface you will have problems within a year or two.
The proper way ... though not the cheapest ... would be to remove the existing concrete and build up the settled area with well tamped soil first. Then ... depending on the area you're in ... lay ground cloth covered by a tamped gravel base. The ground cloth will isolate the gravel from the soil which also helps with water drainage and helps to prevent frost heaving if applicable. Then you're ready to lay the new surface.