Sort the slate tiles by shape, size and color to create your pattern. Lay out the slate pieces beside the existing concrete patio in a pattern of your choice. If you're working with snapped slate, fit smaller pieces in the gaps between large pieces for a puzzle-like design. Planning out where each stone should go will enable you to work quickly once the mortar is mixed.
Scrub the existing concrete with a solution of 1 part detergent and 4 parts water. Rinse thoroughly with a hose and scrub again with concrete stain remover if the surface has any grease stains.
Mix 3 parts fine-grained sand, 1 part cement and water in a wheelbarrow until it is stiff -- not runny.
Place two buckets of water near the concrete. One bucket is used to keep the mortar moist, while the other contains clean water to wipe off any spatters with a moist cloth.
Apply 1 inch of mortar on the concrete evenly over the space for two tiles. Moisten the tiles and place them into the corner in your prearranged pattern. Set a level over the stones to check that they're level and press them deeper into the mortar to adjust them as needed.
Continue to set the slate pieces with spacers in between for even joints. Wipe off any spatters as you go.
Mix sanded grout with 1 part cement and 1 part sand and water, using a mixer rod until the batch is stiff.
Moisten the slate with a sponge and remove the spacers. Tamp grout into the joints until the spaces are filled. Finish the joints with a concave jointing tool to remove rough edges and give the surface a neat appearance.