A tooth-positioning appliance comprising: attachments (12A, 12B) adapted to be bonded to and protrude from selected teeth; a first appliance segment (80, 82) having: (a) a thin shell with recesses for removably engaging a first set of teeth; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the first appliance segment for removably engaging attachments (12A, 12B) once bonded to and protruding from selected teeth of the first set of teeth; and (c) a vertical slot extending horizontally on the thin shell of the first appliance segment; a second appliance segment (80, 82) having: (a) a thin shell with recesses for removably engaging a second set of teeth, wherein the first set of teeth and the second set of teeth are separated by a space; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the second appliance segment for removably engaging attachments once bonded to and protruding from selected teeth of the second set of teeth; and (c) a thin elongated tab extending laterally from the second appliance segment spanning the space and in sliding engagement with the vertical slot of the first appliance segment; and a tractive element or pushing element connectable between the first appliance segment and the second appliance segment exerting a force to change the size of the space between teeth as the tab slides with respect to the vertical slot. attachments (12A, 12B) adapted to be bonded to and protrude from selected teeth; a first appliance segment (80, 82) having: (a) a thin shell with recesses for removably engaging a first set of teeth; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the first appliance segment for removably engaging attachments (12A, 12B) once bonded to and protruding from selected teeth of the first set of teeth; and (c) a vertical slot extending horizontally on the thin shell of the first appliance segment; (a) a thin shell with recesses for removably engaging a first set of teeth; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the first appliance segment for removably engaging attachments (12A, 12B) once bonded to and protruding from selected teeth of the first set of teeth; and (c) a vertical slot extending horizontally on the thin shell of the first appliance segment; a second appliance segment (80, 82) having: (a) a thin shell with recesses for removably engaging a second set of teeth, wherein the first set of teeth and the second set of teeth are separated by a space; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the second appliance segment for removably engaging attachments once bonded to and protruding from selected teeth of the second set of teeth; and (c) a thin elongated tab extending laterally from the second appliance segment spanning the space and in sliding engagement with the vertical slot of the first appliance segment; and (a) a thin shell with recesses for removably engaging a second set of teeth, wherein the first set of teeth and the second set of teeth are separated by a space; (b) tooth-clasping elements comprising holes or recesses in the thin shell of the second appliance segment for removably engaging attachments once bonded to and protruding from selected teeth of the second set of teeth; and (c) a thin elongated tab extending laterally from the second appliance segment spanning the space and in sliding engagement with the vertical slot of the first appliance segment; and a tractive element or pushing element connectable between the first appliance segment and the second appliance segment exerting a force to change the size of the space between teeth as the tab slides with respect to the vertical slot.