In Hatchet, why did Brian's mother give him a hatchet? How did Brian feel about the gift?

In Hatchet, Brian’s family is falling apart. His mother is having an affair, something he calls “The Secret,” and she is seeking a divorce from his father, who lives in Canada. Brian is legally forced to spend the school year with his mother, something he detests, and he is only allowed to visit his father in the summer.
Brian’s mother doesn’t know that he is aware of her affair and its part in the divorce. However, the affair is responsible for Brian being distant from her. She attempts to win him over some by going to a hardware store and buying him a hatchet which she says, “The man at the store said you could use it. You know. In the woods with your father." (Chapter 1) The gift is not meant to be a bonding tool for Brian and his dad, but instead can be understood as an attempt to bridge the divide that has grown between Brian and his mom since the divorce.
Brian is not super thrilled about the hatchet. He says it makes him look “hokey,” but he tries it on because he is still trying to figure out how to talk to his mom. Lucky for him, he puts the hatchet on and forgets about it. He wears it on his belt all the way to the airport, on the plane, and through the crash. Without it, it is unlikely that he would have survived—which is probably why the story is called Hatchet.