What does the Pasuk mean when it writes that Sarah saw Yishmael "Metzachek"?
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 and bowing before HaSh-m, 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 (to 21:10): 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 Yishmael 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, Bereishis 39:17, and Shmuel II 2:14, respectively.
Rashi learns this from Sarah's comment (to 21:10).
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: Yishmael's mocking 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 heard from his mother. 1
As Chazal say in Sukah (56b) 'What a child says over in the street he (probably) heard from his father or mother.
QUESTIONS ON RASHI
Rashi writes: "'Scoffing' (Metzachek) - An expression of idolatry... of immorality... of murder...." The Midrash Rabah (53:11) presents these as three separate opinions; why does Rashi combine them?
Mizrachi: If the Torah does not specify the type of Tzechok that Yishmael did, he must have committed all of them; thus, the opinions in the Midrash do not argue.
Gur Aryeh: All three of these sins are deviations from the normal world order, just as Tzechok is something that is not real, or does not fit with the norm. 1 Immorality (Giluy Arayos) is the epitome of Tzechok; 2 the other two opinions come to add idolatry and murder to the category as well. 3
To explain - a person laughs (Tzocheik) when something deviates from its norm, in a circumstance when it should have followed the norm. (EK)
'Laughter and light-headedness lead a person to sin' (Avos 3:13); also see Maharal (Derech Chayim loc. cit).
Idolatry constitutes depending for one's existence upon worthlessness. Murder is removing one's fellow from reality. (EK)
Rashi writes: "An expression of idolatry... of immorality...." If Yishmael was guilty of such grievous sins, why did only Sarah protest, and not Avraham?
Gur Aryeh: Avraham did not wish to banish Yishmael; he thought that upon receiving rebuke, he would do Teshuvah. But Sarah knew through Ru'ach ha'Kodesh that this was not to be; rather, Yishmael would leave the correct path altogether.
Rashi writes: " An expression of idolatry...." Would Avraham have allowed idolatry into his house?!
Gur Aryeh: It was no less of a disgrace for Avraham that Yishmael himself was part of his household! Yet, this was HaSh-m's plan; Yishmael, who would end up as an idolater, was Avraham's son.
