Arrange the nori, cucumber, crab strips, and avocado on a platter. Put the sesame seeds, sugar, and salt in small bowls and fill a medium bowl with water. Cover both sides of the rolling mat with plastic wrap.
In a small bowl, stir together the rice vinegar, sugar, and salt. Pour the mixture over the rice and stir to coat it.
Lay a sheet of nori on the mat, shiny side down. To keep the rice from sticking to your hands, dip them in the water. Cover the nori almost to the edges with rice. Scatter on the sesame seeds. Flip the nori over (the rice should stay adhered to the nori as you do this).
Arrange avocado, crab strips, and cucumber on the nori about an inch from the edge closest to you.
Roll up the ingredients by slowly folding the mat over the filling. Tuck in the ingredients as you work. When you've formed a log, use the mat to press and shape it. Use the remaining ingredients to make more rolls.
Dip a sharp knife in hot water and use it to slice each roll into six pieces (an adult's job). Tip: First cut it in the center, then cut each half into thirds. Serve the sushi with soy sauce, pickled ginger, and wasabi on the side.