Wash the surface of the concrete to remove any dirt or debris.
Texture the surface of the concrete to make adhesion with the bonding material easier. Rent a concrete grinder from an equipment rental shop or home-improvement store. Put on the face mask and safety goggles and then, starting in the corner of the slab, move the grinder across the entire concrete slab, lightly scratching the surface. Sweep up any debris remaining from the texturing.
Spread a layer of bonding adhesive onto the concrete surface. The adhesive will create a bond between the two layers of concrete and reduce the possibility of cracks developing in the new surface. The adhesive will also keep water from gathering between the layers and causing damage.
Mix the sand-mix concrete, aiming for a consistency that is slightly thicker than a regular concrete mix. Use a spade to spread a layer of the new concrete onto the original slab. Level the concrete using a screed board. This is a flat, straight two-by-four piece of lumber with an attached handle. For the area that slopes, use a thicker layer of concrete. Float the surface of the concrete with a wood float. This is a mason's trowel with a wooden blade rather than a steel one. Use a sweeping motion to smooth the new sand-mix surface.
Cut the control joints into the new concrete to allow the new surface room to expand and contract with temperature changes. Place the joints in the same area as those that were cut in the original surface. Cover the slab with plastic and allow it to cure for week before applying a sealant to protect the new, level concrete surface.