Help naming a childhood card game

I played a card game all the time as a kid which my Grandma called "One 3, Two 3s, a 3 & a 4"

Basically you dealt cards (7 I think) and the 1st round was one 3... Draw a card and throw one away... try to make 3 of a kind or a run of 3 to win

2nd round was two 3s... Same again but you need two lots of 3 of a kind or run of 3 to win

3rd and final round was 3 and a 4... You needed 3 of a kind or a run of 3 and 4 of a kind or a run of 4 to win

That was it...

I don't know if my Grandma made it up or if it was a variant of something... Google is no help... I plan to play with my daughter when she's a bit older but I'd love to know if others play this or played as kids