The airport sign is signed like airplane: take your outstretched hand, and tuck in your middle and ring fingers. Then move your hand away from your body a short distance, as if flying, and bob it back and forth a couple of times. If you move your hand a bit further up and out, slightly above your head, that's already the sign for fly.
The starting sign for both airport and airplane is actually the I love you sign in ASL.
Teach your child the airport sign in the context of preparations for traveling, or when picking up or dropping off someone close at the airport.
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