Sarah demanded that Avraham banish (geresh) Hagar and Ishmael, while Avraham sent (shalach) them away because Sarah saw Ishmael making sport of Isaac (laughing at the idea that Isaac would inherit Avraham) for he claimed a double portion as Avraham's first born son.
The Midrash relates that they would go out into the field, argue over the inheritance and Ishmael would take his bow and arrows and shoot them at Isaac claiming to be pretending to shoot at him but actually intending on slaying Isaac!
(Here, it seems as if Ishmael is amusing himself at the expense of Isaac!)
Another Midrash relates that Ishmael slandered Sarah [as having committed adultry with Abimelech] saying Isaac was conceived through Abimelech and not Avraham. This slander he heard from the scoffers of the generation.
Rabbi Hirsch comments on 21.6 Va-tomer Sara, tsehok asa li Elokim…” (21:6).
This verse is translated by Rabbi Samson Raphael Hirsch as follows:
“And Sarah said: God hath made me into a laughing-stock, all that hear will laugh at me.”
The Talmud Sanhedrin 89b relates:
R. Levi said [in explanation of 'after these words' B'rashith - Genesis 22.1]; After Ishmael's words to Isaac. Ishmael said to Isaac: 'I am more virtuous than thee in good deeds, for thou wast circumcised at eight days, [and so couldst not prevent it], but I at thirteen years'. 'On account of one limb wouldst thou incense me!' he replied: 'Were the Holy One, blessed be He, to say unto me, Sacrifice thyself before Me, I would obey', Straightway, G-d did tempt Abraham.
It seems to me that a strong argument can be made for Ishmael mocking the halacha of circumcision on the eighth day from birth; that is, he mocked the covenant of circumcision (Rashi's illicitness) from which he was ultimately destined to be excluded from, by the prophetic words of Sarah!
This mockery of the covenant of circumcision seems to me a likely scenario given that HaShem's sense of justice is middah keneged middah (measure for measure), as Sarah spoke prophetically, "cast out this bondwoman, for the son of this bondwoman shall not inherit with my son, with Isaac [with laughter]!"