The Pathways approach to solving word problems helps students gain confidence in their ability to visualize and model (with formulas and graphs) how covarying quantities change together. To illustrate what we mean, consider a problem to determine the maximum volume of a box formed by cutting squares from the four corners of an 8.5” by 11” sheet of paper and folding up the sides. It is common for curriculum to only ask students to find one unknown value of x for which the volume f(x) is a maximum. The Pathwaysapproach asks students to first visualize the quantities (attributes of the box that have a numerical value) and to imagine how varying quantities, such as the volume of the box f(x) and length of the side of a square cutout x, vary together.
This critical step of first identifying the varying quantities and visualizing how they are related and change together allows students to construct the funtion formua and graph and use them meaningfully. As students imagine the square’s cutout length varying from 0 to 4.25 inches they are able to visualize the box’s volume increasing to a maximum value and then descreasing back to 0 inches. Our data has revealed that only 25% of non-Pathways precalculus students while 80% of Pathways students are able to define this formula.
The Pathways materials introduce all function types by examining the patterns of change involving two varying quantities and modeling this variation with function definitions (see above) and graphs. Students also learn and practice methods for simplifying expressions and solving equations in the context of exploring attributes of the function model (e.g., roots, the y-intercept, the maximum value). This approach helps students appreciate the usefulness of methods for solving and simplifying, making it much more likely that they access and effectively apply these methods when solving novel problems.