Mark the trench line with a can of spray paint. Run the trench from the downspout to a low point away from your home.
Dig a trench 6 inches wide along the trench line you marked. Dig the trench at least 10 inches deep, and slope the trench down away from the house at a rate of 1/8-inch for every foot of length.
Dig a hole at the end of the trench about 18 inches deep and 3 feet wide. Place large, flat stones or gravel into the hole.
Lay out all the pipes and pipe fittings next to the trench
Slide a downspout adapter into a 3-by-4 inch reducer coupling. Attach the reducer coupling to the end of the downspout.
Position a 90-degree PVC elbow into the trench below the coupling. Measure the distance between the end of the coupling and the end of the elbow, and cut a length of PVC pipe to fit.
Cover the surfaces where the pipe meets the elbow with purple PVC primer and cement. Twist the pieces together and hold them firmly for a few seconds as the cement cures.
Slide the downspout adapter onto the pipe you cemented to the elbow (don't glue it).
Place an elbow at each bend in the trench, and measure the distance between the elbows. Cut lengths of PVC pipe to fit.
Connect the first section of pipe to the elbow in the trench with the purple primer and cement. Check the slope of the pipe with a level to make sure it's sloping away from the house. Deepen the trench underneath the pipe if necessary.
Continue installing the pipe until you reach the hole at the end of the trench. Connect different lengths of pipe with standard PVC couplings. Place rocks or gravel over the end of the pipe in the hole.
Backfill the trench.