Card Game Name: 5 points and you’re out
Your game's type is Draw and discard.  Knowing the nationality of your teacher could narrow your search.  Your description sounds a lot like a variant of Yaniv / Jhyap.
Yaniv description in Pagat:

Deal: Five cards are dealt to each player


The objective of the game is to have a lower total value of cards in your hand than the other players. When you think you have achieved this, you call &quot;Jhyap&quot; in Nepal or &quot;Yaniv&quot; in Israel to stop the play and compare hands.


Each game is composed of rounds, and each player's total score is updated after each round. The goal of the game is to keep your score to a minimum. Players with more than an agreed maximum, for example 200 points, drop out of the game, which is then continued until there is only one player left.