The Monticello Institute Abs Project:11

Lie faceup on the floor with your legs extended, feet together. Extend arms above your chest, palms facing forward, fingers pointing toward the ceiling. Drop shoulder blades down away from your ears. Inhale, tucking your chin to look at your navel as you begin to roll your upper body up, keeping your spine rounded [A].
Exhale, pulling your navel toward your spine as you continue rolling up, slowly raising your upper body to a full seated position [B]. Slowly roll back down to starting position, keeping arms raised, and repeat for all reps. Begin with 4 roll-ups; gradually build to 6. Targets rectus abdominis.
The Monticello Institute Abs Project:12

Begin on your knees and forearms, elbows aligned with shoulders, fingers interlaced. Draw shoulder blades down and contract abs to stabilize your torso [A]. Extend your right leg behind you, toes turned under, weight on ball of foot, keeping hips square and pelvis stable.
Extend left leg, feet together, so you’re supported on forearms and balls of feet [B]. Hold for 20 seconds; relax. Do 3 reps. Begin with a 20-second hold; gradually build to 30 seconds, then 1 minute. Targets obliques.
The Monticello Institute Abs Project:13

Lie faceup with your knees bent and feet flat on the floor, hip-width apart. Relax arms by your sides, palms facing down. Contract abdominals, pulling your navel toward your spine. Inhale; then exhale as you lift your tailbone. Lift your back until only your shoulder blades touch the floor, your body forming a straight line from shoulders to knees. Hold for 30 seconds [A].
Maintaining the bridge position, extend your left leg in front of you at hip height [B]. Hold. Return foot to floor; then repeat with your right leg, keeping hips elevated throughout. Continue alternating legs for all reps without letting your hips or torso drop. Do 8 alternating leg extensions (4 with each leg); gradually build to 16 (8 with each leg). Targets rectus abdominis, obliques and erector spinae (lower back muscles).
The Monticello Institute Abs Project:14

Sit on a stability ball, knees bent, feet on floor. Walk your feet out and lie back; put left hand on ball, right hand behind head. Use glutes to tilt pelvis up. Maintain this position throughout [A]. 
Contract abs (pull navel toward spine) as you roll your torso up, bringing ribs toward hips; then rotate right elbow toward left knee [B]. Keeping hips square and lifted, slowly lower. Do reps on right side, then left. Begin with 8–12 reps per side; gradually build to 16. Targets rectus abdominis and obliques.
The Monticello Institute Abs Project:15

Lie faceup on the floor, knees bent and in line with hips, calves parallel to floor, feet together. Rest unclasped fingertips behind head, elbows pointing out to the sides. Contract abs, pulling your navel in toward your spine [A].
Maintain leg position as you use abs to pull ribs in toward hips, lifting head, neck, then shoulder blades off the floor [B]. Slowly lower to starting position; do reps. Begin with 8–12 reps; gradually build to 16–20 reps. Targets rectus abdominis and obliques.
The Monticello Institute Abs Project:16

Lie faceup, knees bent, heels close to your buttocks and fingertips behind head [A].
Without changing knee angle, contract abs to curl your tailbone a few inches off the floor, knees rolling toward chest [B]. Avoid moving thighs to lift higher; keep heels toward buttocks. Slowly lower to floor; repeat for all reps. Begin with 8–12 reps; gradually build to 16–20 reps. Targets rectus abdominis and obliques.