JP: as you know, altitude problems can strike anyone at anytime. They say children are more susceptible then adults.
To lessen the possibility of altitude sickness and increase our chances of making it we..... camped one night in the Alabama Hills (5000 feet), one night at LBSL (10300 feet) and one night at Iceberg Lake (12600 feet). This plan was intentional, hoping to avoid altitude sickness concerns. Two of my children had some difficulty with the altitude, although you never know how much is combination of altitude sickness, dedydration, fatigue and the rigors of the hike.
Certainly, you must go slower and have patience with them. Everything takes longer.
A nice hike is out of Tuttle creek campground to the church up the tuttle creek canyon. An interesting place and not too far. See http://www.whitneyportalstore.com/cgi-bin/ubb/ultimatebb.cgi?ubb=get_topic;f=1;t=000195