Title: Calculating the Distance Between a Point and a Line - by Hand and Using TI-89/TI-Voyage 200 CalculatorTechnologyAuthors: Theresa Adsit, James Meyer, Gary Wardall, University of Wisconsin Green BayContact: Theresa Adsit, adsitt@uwgb.eduDiscipline or Field: MathematicsCourse Name: Elementary Functions: Algebra and TrigonometryDate: 12/20/2007

 Course DescriptionElementary Functions: Algebra and Trigonometry is an introductory Mathematics course preparing students for Calculus. Topics include: the real number system; inequalities; functions and their inverses; exponential and logarithmic functions; trigonometric and inverse trigonometric functions; complex numbers; polynomial and rational functions and systems of equations. The class meets four times per week for 50 minutes each class period. Classes are composed of 35 students meeting in a lecture hall. Classes are generally lecture style interspersed with interaction among students working together in small groups. Calculator use features prominently throughout the course.The topic of the lesson, The Distance Between a Point and a Line, examines an algebraic approach and a calculator based approach to problem solving. The lesson took place in the 10th week of a 15 week term.The class met in the early afternoon. Attendance was good and students came to class on time. There were no significant academic problems. The class atmosphere was noticeably congenial – students were friendly and comfortable with one another and the instructor.The lesson involved significant calculator work. Students were accustomed to in-class calculator exercises and working together on their calculators.

 Executive SummaryThe lesson topic is the distance between a point and a line using an algebraic approach and a calculator based approach to problem solving.Learning Goals: The immediate academic learning goals of this lesson were to develop students’ understanding of the derivation of the point to line distance formula and to develop the ability to apply the point to line distance formula to solve problems. The ongoing academic learning goals of this lesson were to develop the ability to use the calculator to build structures to solve problems involving systems of equations, to develop a greater understanding of the similarities between calculators and other forms of technology, and to further develop strategies for solving multi-step problems.Instructional Design: The lesson was divided into five steps. The first step was instructor led and involved the determination of the shortest distance between a specific point and a specific line using the techniques of algebra and paper and pencil. The second step mimicked the first but rather than using paper and pencil the instructor and students used either a TI 89 or TI-Voyage 200 calculator. During the third step of the lesson the instructor and students then developed the point to line distance formula for any point and any line using the TI 89 or TI-Voyage 200 calculator. The fourth step of the lesson involved the students verifying the formula by using the developed formula along with the point and the line from parts one and two to determine if the developed formula did indeed yield the same results as their previous calculations. Finally, in step five the students worked collaboratively and then independently on an assignment related to the lesson.Major Findings about Student Learning: The students with the assistance of the instructor were able to build the appropriate structures using either a TI 89 or TI-Voyage 200 calculator to solve a problem involving systems of equations and to derive a formula involving systems of equations. The students were collaboratively and individually able to apply the developed formula to other problems in the assignment. Students questioned each other and the instructor more often during the collaborative work period than during the instructor led portion of the lesson. Some students did have an underlying misunderstanding of the benefits of a formula.

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