What does the Pasuk mean when it writes that Sarah saw Yishmael Metzachek (laughing)?
Rashi: "Metzachek" in this context means that he was indulging either in Avodah-Zarah, in adultery or in murder. 1
Targum Yonasan: Yishmael was worshipping Avodah-Zarah but bowing before Hashem, making believe that he served Him.
Seforno and Ramban: He was poking fun at the party that Avraham made, claiming that Yitzchak was the son of Avimelech and not of Avraham.
Rashi #2 (on Pasuk 10): After quarreling with Yitzchak over his father's inheritance (claiming that he was the Bechor and that he was therefore destined to receive a double portion) 2 , he took him out to the field and began shooting arrows at him, claiming that he was only playing. 3
Perhaps even all three. The word "Metzachek" is used by these three sins as we see in Shemos (32:6), in Bereishis (39:17) and in Shmuel 2, 2:14, respectively.
Rashi learns this from Sarah's comment (in Pasuk 10).
Ramban asks that Rashi brought opposing opinions from the Midrash. Ohr Yechezkel Midos p.170 - the primary opinion is that he only quarreled about inheritance. Surely it was a minor matter, for Avraham did not notice it!
Why does the Pasuk refer to him as the son of Hagar and not by his real name?
Ramban: This raised Sarah's hackles, because a slave who mocks his master deserves the death-sentence - though she only demanded that Avraham expel him together with his mother.
Seforno: Because Sarah thought that he said over what he (obviously) heard from his mother. 1
As Chazal say in Succah (56b) 'What a child says over in the street he (probably) heard from his father or mother.
