God remembered His promise to Sarah that she will have children, and a son was born to Sarah and Avraham who they named Yitzhak. After eight days, Avraham circumcised his son. As per Sarah, request, Avraham sent away Hagar and Ishmael. When they ran out of water, God provided Hagar and Ishmael with a well and promised that Ishmael would too turn into a great nation. Avraham then made a treaty with Avimelekh.
Why would Sarah want to send Hagar and Ishmael away? While many commentaries criticize Sarah’s action, they still suggest what might have driven Sarah to do it. The Ramban turns our attention to the events preceding the act. The day Yitzhak was weaned from being nursed, they made a great celebration for him. At that point, Sarah noticed Hagar’s son laughing at Yitzhak. Ishmael was mocking and ridiculing his brother Yitzhak, and that’s why Sarah wanted to send them away. Rabbi David Tzvi Hoffman elaborates on this point: as Avraham’s older son, Ishmael felt that he could patronize his younger brother and mock him so that Yitzhak would know that Ishmael is his superior. This was not the way to begin a brotherly relationship.
Rabbi Roy Feldman