Cumulative vocabulary models for infants and toddlers were developed from models of learning that predict trajectories associated with low, average, and high vocabulary growth rates (14 to 46 months). It was hypothesized that models derived from rates of learning mirror the type of exchanges provided to infants and toddlers by parents and caregivers. Slow and rapid growth models assume that rates of acquiring new vocabulary are proportional to (a) the child's current vocabulary and (b) the difference between maximum achievable vocabulary and the child's current vocabulary. These models produce (a) exponential growth and (b) bounded exponential growth, respectively. The third model (interactive engagement) results from the interaction of the two previous models and describes vocabulary growth trajectories with a logistic function. It describes the three empirical growth trajectories for average, slow, and rapid growth in the 14 to 46 month study, and for percentile growth rate trajectories of infants and toddlers between 16 and 30 months in a second study. A substantive outcome to the interactive engagement model is that it provided physically meaningful predictions when it was extended forward in time to 60 months and backward in time to 10 months. Although meaningful predictions of achievement can be made based on polynomial growth models (quadratic, and cubic) that fit empirical data, physically meaningful growth trajectories are obtained for logistic-model predictions beyond the empirical data. Excellent agreement between empirical data and model predictions provide evidence that additional birth through age three exchanges by parents and caregivers will narrow socioeconomic status gaps.