What does Genesis 21:10 mean?

On the day of a great feast to celebrate Isaac's being weaned, Sarah suddenly became very angry. The previous verse tells us that she saw Ishmael, now maybe 16, laughing or perhaps mocking. Whatever he was doing apparently triggered her anger over something she was likely already deeply concerned about. This is not surprising, since Sarah has already demonstrated anger and hurt feelings towards Hagar and Ishmael in the past (Genesis 16:4–6). Of course, Sarah is partly responsible for their place in her life, since she was the one who encouraged Abraham to conceive a child with Hagar (Genesis 16:1–2).

Ishmael was Abraham's firstborn son by Sarah's Egyptian slave girl Hagar, and Abraham clearly loved the boy. Sarah's concerns are not entirely unreasonable. Would Ishmael somehow become Abraham's heir? Would his presence in their family keep Sarah's son Isaac from receiving all that was due to him? Sarah decided she could not allow this, even if she was partly responsible for it in the first place.

Apparently in a fury, she demands that Abraham cast out both the slave woman and her son. She refuses even to say their names. She simply declares her objection in direct, impersonal terms: "The son of this slave woman shall not be heir with my son Isaac."

Abraham will not take this well.