Part of what I have been trying to say when it comes to baptism, in order to show that it involves the entire family, is to understand the mindset of the First Century Jew when it came to the sign of circumcision. Both baptism and circumcision are signs of the covenant of grace (often called the New Covenant) and pointing us to the gospel (even in the Old Testament).
Here I would like to show the requirements given to Abraham in Genesis 17 for circumcision. Please note, this is given before the Mosaic Covenant which included the ceremonial rituals of the temple. So the requirements predate that covenant, and therefore were not negated as the ceremonial law was in the New Testament. In other words, the requirements and instructions of the covenant still remain even though the sign has changed.