Genesis 21:9

And Sarah observed the son of Hagar the Mizreitha, whom she bare to Abraham, mocking with a strange worship, and bowing to the Lord.

‘METZACHEIK’ (MAKING SPORT). This refers to worshipping idols, murder and sexual immorality. He [Ishmael] quarrelled with Isaac about inheritance, saying, “I am the first-born and will take a double portion.” They then went into the field, and Ishmael took his bow and shot arrows at Isaac, just as you say, As a madman who casteth firebrands, arrows and death, so is the man who deceiveth his neighbor, and saith, Am I not in sport? It is from Sarah’s complaint to Abraham — for the son of this bondwoman shall not be heir, etc. — that you learn [that they were quarrelling about the inheritance]. All this is Rashi’s language.
Here too, the Rabbi writes all the different opinions, [mentioned in the following]: “We have been taught: Rabbi Shimon the son of Eleazar said, ‘There are four interpretations of Rabbi Akiba which I interpret differently, and my interpretation seems more acceptable than his. Rabbi Akiba interpreted: “And Sarah saw the son of Hagar the Egyptian, whom she had borne unto Abraham, making sport. Making sport is but a designation for idolatry, etc.” But I say Heaven forbid that such be in the house of the righteous one! Is it possible that he, of whom it was written, For I have known him, to the end that he may command his children and his household, etc., will have in his household idolatry, sexual immorality, and murder? Making sport mentioned here is but a designation for the inheritance. When Isaac was born and everyone rejoiced, Ishmael said to them, “Fools, I am the first-born, and I take a double portion.” From the complaint of our mother Sarah to Abraham you learn [that making sport refers to the inheritance]. And my interpretation seems more acceptable than that of Rabbi Akiba.’”
The expression of the Rabbi, “that Ishmael quarrelled with Isaac about the inheritance,” also does not appear correct for if so, this must have happened much later when Isaac was grown up, and Ishmael would then have been too big for his mother to carry him on her shoulder. Our Rabbis have also said that Ishmael was seventeen years old [at the time he left his father’s house]. If so, this happened at the time when Isaac was weaned, [and Isaac was thus too young for Ishmael to quarrel with him about the inheritance].
Rabbi Abraham ibn Ezra said in line with the literal interpretation of Scripture that metzachek means “playing,” as is normal for every boy, and she was jealous of him because he was bigger than her son.
The correct interpretation appears to me to be that this event took place on the day that Isaac was weaned, and Sarah saw Ishmael mocking Isaac or the great feast. It is for this reason that the verse says, And Sarah saw the son of Hagar the Egyptian — rather than Ishmael — making sport. Similarly, she said, Cast out this bondwoman and her son, for she said: “The slave who mocks his master is deserving of death or stripes, but I want only that you cast him out from before me, and that he should in no way inherit your belongings together with my son, who is the son of the mistress.” She also told Abraham to cast out his mother, as the boy was unable to leave her for he would die if he were to leave his mother.

מצחק, in the meantime Ishmael was already 16 years old. Sarah did not want him to remain in the same household any longer so that he would not get ideas about sharing in the inheritance with Yitzchok.

את בן הגר המצרית, she assumed that the reason Ishmael had made disparaging remarks about Yitzchok was due to his mother putting him up to it, or to the Egyptian genes of his mother coming into play here. We have a saying in Sukkah 56 that the prattle of children in public reflects either what they picked up from their father or what they picked up from their mother. In the case of Ishmael, he certainly did not overhear his father make such disparaging remarks.

מצחק, making fun of the big party Avraham had given to mark the weaning of Yitzchok, He claimed that surely Sarah must have become pregnant from Avimelech. The reason he had not made such remarks already at the time when Yitzchok had been born, was because he had only overheard wicked gossip about this at a later stage, and now he repeated what he had heard.

ותרא...מצחק. Ishmael was belittling Yitzchok for having a father who was so much older than he.