Calculate exponential and logistic population growth, and identify carrying capacity as the ceiling that logistic growth approaches.
Exponential growth produces a J-shaped curve: a population grows by a constant rate, so its size compounds like interest in a bank account, doubling again and again under the assumption of unlimited resources. This is real and observable, especially over a short time -- bacteria in a fresh nutrient dish, or a species introduced to a new habitat with abundant food and no predators -- but it can never continue indefinitely in a finite environment.
Logistic growth produces an S-shaped curve instead: growth starts off fast, like exponential growth, but slows as the population approaches its carrying capacity (K) -- the maximum population size an environment can sustainably support given its food, water, space, and other resources. Limiting factors can be density-dependent (their effect gets stronger as crowding increases -- food shortage, disease, competition) or density-independent (they hit regardless of population size -- a wildfire, an unusually harsh winter).
Mastery looks like: They can calculate several steps of exponential growth, read a leveling-off data table to estimate carrying capacity, and correctly classify limiting factors as density-dependent or density-independent.
Common stumbles: Confusing carrying capacity with the population's current size at some earlier point; thinking exponential growth is purely theoretical rather than a real short-term pattern.