Teenage pregnancy is a social problem in Nigeria, whereby girls between the ages of 10-14 become pregnant by sexual intercourse after ovulation or first menstrual period. This article involves the fractional order mathematical model formulation describing the societal problem of teenage pregnancy in the sense of Caputo. The positivity, existence and uniqueness results of the model were established, and the two equilibria, which are the teenage pregnancy-free and teenage pregnancy-present equilibrium solutions of the model are presented. The graphical illustrations showing the behavior of the model variables when the basic reproduction number $R_{pr}$ is less and greater than unity are displayed, using the numerical technique of Fractional Multi-Stage Differential Transform Method (FMSDTM) in comparison with the Runge-Kutta fourth order method (RK4) via the maple computational software. In addition, simulations involving the effect of rehabilitation is observed not to lessen $R_{pr}$ below unity, which shows that further mitigation measures are needed to halt teenage pregnancy problems in Nigeria.