He will usually do homework, but only the barest minimum ... So here’s my question (finally): I was thinking of offering him a large reward if he turns his grades around — think airplane ...

or "I'm proud of you". Set small rewards, like extra playtime, a walk, or their favorite dessert. Help them with homework when they have questions, but avoid doing it for them. Ask about what they ...

You could even create your own homework reward chart and come up with treats you know that they'd love - a trip to their favourite pizza place, maybe, or a night where they get to control the TV?