Mathematics: Calculating the Distance Between a Point and a Line

TItle: Calculating the Distance Between a Point and a Line – by Hand and Using TI-89/TI-Voyage 200 Calculator Technology
Discipline(s) or Field(s): Mathematics
Authors: Theresa Adsit, James Meyer, Gary Wardall, University of Wiscnsin-Green Bay
Submission Date: December 12, 2007

Executive Summary

The 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.

Mathematics: Exploring Difficulties with Combining Rational Expressions

Title: Lesson Study in Exploring Difficulties with Combining Rational Expressions
Discipline(s) or Field(s): Mathematics
Authors: Laura Schmidt and Diane Christie, University of Wisconsin-Stout
Submission Date: February 28, 2007

Executive Summary:

Learning Goals. The overall learning goal is to have students be able to add and subtract rational expressions. Students will first combine expressions with common denominators, then find a common denominator to combine expressions with unlike denominators. Long-term goals not directly assessed by the lesson are to ease anxiety when dealing with fractions and to have students realize the connection between adding/subtracting rational numbers and adding/subtracting rational expressions.

Lesson Design. The lesson reviewed addition and subtraction of fractions, demonstrated addition and subtraction of simple rational expressions, and worked up to difficult examples. The lesson began with three examples of rational numbers, one with common denominators and two with un-like denominators, followed by rational expressions with common denominators. Examples of rational expressions with un-like denominators started out simple and increased in difficulty level. The number of expressions to be added increased along with the difficulty in the factorization of the denominators. The examples were chosen so that the answers could be rewritten in reduced forms at the end to remind students to check that final step in their answers. Due to the anxiety that this lesson has caused in the past, hard examples were presented by the end so that students could be exposed to more difficult problems.

Major findings about student learning. The findings showed that even though students were successful at the beginning problems in the homework, they were intimidated by the “difficult look” of the later homework problems and simply did not attempt them. This was evident in the analysis of the homework where the amount of incomplete problems drastically increased at a certain problem when the difficulty level was higher. In the revised lesson, more difficult examples were used, and it was stressed that the steps remain the same even though it looked much harder than previous examples. Several days later when the students had to use the lesson to solve equations involving rational expressions their confidence level was greater and the majority of students got the correct answers.

Below are links to the lesson plan and the materials used to teach it.