I expect you'll do fine and your children should love it. Your obviously informed and will be prepared, based on your posts.
I suggest bleachers or stairs with a loaded pack as a way to get in shape if you have no hills. Be sure and pound yourself on the down the bleachers cycle. The inclines and stair step machines don't give you that shock when going down. Flat running or flat walking don't give to shock effect, either.
Typically the down hill pounding takes a bigger toll on me, and most others I have been with and talked, to than the up. Probably cause that's when the pack is full and you have a tendency to want to go faster.
One can always compensate for a lack of training by going slower on the up. If I don't force myself to go slow the initial down hill day, the calves seize up and ache for the rest of the trip. Poles help to reduce the shock, too.
Warning: Teenagers tend to want to run the down hill and naturally accelerate you along with them.
For future reference to consider: We did four years of spring breaks in a row, March, with my kids and their friends when they were high-school aged (which I think is a fun time of year - you can count on variety of weather from snow to heat and the fresh snow really makes the views spectacular). The rim to rim (rim squared) is possible then - You can go to the North rim pretty easily some years in March (We did in Mid-March 2005 but somebody else broke the drifts. Mid-March 2006 there were a couple blizzards up there). You have to turn around and go back cause the roads not open so you get the rim to rim to rim (rim cubed). We have done a late-May, early-June trip with teenagers in the western canyon when we were hiding out from the heat all day. June is great for getting in the streams (forces you in). Some places get nasty gnats in May-June and the biting-crawly critters are out by then, too.
I have a solo trip planned in April but just to Cottonwood camp and a day trip to Roaring Springs. Then back to the south because the north is closed and I don't feel like huffing up there just to come down again.
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