The involvement of students in the development of such models is directly aimed at both building their self-confidence in approaching such situations while illustrating the central role the mathematics plays in both personal and professional decision making. The mathematically rich contexts students have an opportunity to explore range from designs in arts and crafts to managing lottery winnings to piloting ships in a fog to bungee jumping. The exploration, reasoning, and solving of problems in these areas, plus others dealing with sports, sunlight hours, and manufacturing settings, are structured around central themes uniting models and functions. The six units are directly focused on developing students' abilities to use functions in the development of and application of models. The models, which are used to understand, manipulate, predict, and control outcomes in quantitative settings, begin with the study of the role that proportionality plays in a wide variety of settings involving mathematics.
Direct proportion is the essence out of which linear functions grow.
Students are carefully introduced to this concept, extending the work that they have done in their previous study of secondary algebra and geometry. This in-depth look at direct proportion provides a solid base for examining the role of change in linear situations, situations which are characterized by arithmetic growth. This ties sequences from discrete mathematics to ideas of relative change that provide a basis for a general understanding of change, in the terms of slope and unitary growth, for those who would continue to calculus. It also provides a development of regression and change from a trend-line standpoint for those continuing to statistics. This arithmetic growth is contrasted with multiplicative growth illustrated by geometric sequence settings that grow to exponential functions in the second unit.