Completely clean the cement sidewalk for the installation of the slate tile. Spray it with water from an outside hose and use a whisk broom to scrub the surface free of debris. Rinse it with clean water from the house and allow the concrete to dry before you do your layout.
Lay out your sidewalk. If you have a straight sidewalk you can simply measure for the center and snap a chalk line across the length of the sidewalk to define the center and give you a reference point, but it is more complicated with a curving sidewalk. In this case you will need to mark the center point of two curves and snap a chalk line between them (or multiple curves if this is the case).
Install the slate tile on top of the cement. Spread a layer of thinset onto the sidewalk with a notched trowel. Spread the thinset in multiple directions to ensure it bonds completely with the concrete. Use the center line as a reference and install one side of the sidewalk first down the length before moving onto the opposite half. Only spread enough thinset for half a dozen tiles at a time.
Push the tile into the thinset mortar and move it around slightly to ensure the bond between adhesive and slate. Make any necessary straight cuts with a tile wet saw. For curved cuts to match the curvature of the sidewalk, mark the backs of the tile with a pencil using the existing sidewalk as a reference point and cut the waste out with an angle grinder.