Students explore relationships between x-intercepts, factors, and roots of polynomial functions using the graphing calculator. Students also investigate rational functions, identifying the roots and the asymptotes as well as holes (discontinuities) in the graphs. Students then construct boxes of various dimensions using graph paper, collect height and volume data, and create a scatterplot in order to determine the height of the box with the maximum volume. Students can solve this problem using a graphing calculator or by using their own scatterplots drawn by hand. The use of questioning by the teacher and the group work of the students are important features of this lesson. In addition to the lesson plan, the site includes ideas for assessment, teacher discussion, extensions of the lesson, additional resources, and a discussion of the mathematical content. The user should first locate the lesson and then access the appropriate video clips illustrating lesson procedures. (author/pk).

The functions presented in this lesson are built around a real-world exercise to determine the optimal volume of a box to be constructed. Construction and Manufacturing students should immediately see the application of this mathematical principle as they determine the best and most cost effective use of materials in the design process. In marketing, the application might be in determining break-even points for sales, customer use, or other variables. Practice in the use of graphing calculators should also be considered a general career skill. Because of the higher level mathematics in this lesson, career-technical teachers and mathematics teachers might consider teaming up to accomplish the project and present the mathematical concepts effectively.