Sweetie, 3-1/2 months is very little new growth ( about 1-1/2 inches). Your mom may not go for that. You may get more support in transitioning. Have you researched that option? If you can prove that you can take care of your hair while doing that, your mom may let you cut it after you gain some length.
Good luck and congrats on your decision to embrace your natural hair. Natural hair is soooooo versatile. Maybe If you can show your mom what some of the ladies on YouTube who have similar hair to yours are doing with their hair, it will help some.
My hair: thick, medium/coarse, high porosity in the crown naturally, normal porosity everywhere else.
Poo: Loreal Ever Curl Sulfate Free poo
Rinse out: Aussie Moist/Tresemme Flawless Curls
Leave-in: JC Hydrate Smoothing leave-in/Deva One C
DC: Crece Pelo Treatment/Camille Rose Algae Deep cond/Curl Junkie Rehab
Stylers: Super Wet original gel/Extreme Wetline gel
Oils: EVCO, Castor Oil
Moisturizer: Oyin Hair Dew
Sealant: Oyin pomades or KBB Butter Love