How many toothpicks does it take to make an n x n square composed of 1 x 1 squares? The lesson begins with a review of transformations of quadratic functions--vertical and horizontal shifts, and stretches and shrinks. First, students match the symbolic form of the function to the appropriate graph, then given the graphs, students analyze the various transformations and determine the equation for the functions.

How many toothpicks does it take to make an n x n square composed of 1 x 1 squares? The lesson begins with a review of transformations of quadratic functions--vertical and horizontal shifts, and stretches and shrinks. First, students match the symbolic form of the function to the appropriate graph, then given the graphs, students analyze the various transformations and determine the equation for the functions. This review is followed by an activity in which students explore a mathematical pattern that emerges as they build larger and larger squares composed of 1 x 1 squares. The pattern is quadratic, and students determine the mathematical model in several different forms. Students examine the recursive nature of the relationship, and using the model, extend the domain to negative integers. An explicit model for the relation is developed by examining the scatterplot and determining the equation from the transformations. Finally, the class uses graphing calculators to develop another model and verify that all of the models are equivalent. 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 lesson plan is accompanied by video clips illustrating lesson procedures. The user should first locate the Toothpicks and Transformations lesson and then access the appropriate video clips at the PBS TeacherSource website. The video player necessary to view the video clips can be downloaded for free from the site. (author/pk)