This activity is intended to supplement Geometry, Chapter 2, Lesson 2.
Time required: 40 minutes
In this activity, students will write logical statements related to the given conditional statement. They will explore whether the statements are true or false and find counterexamples for false statements. These explorations will involve the slopes of parallel and perpendicular lines and lengths of collinear and noncollinear segments.
Topic: Inductive & Deductive Reasoning
Write the inverse, converse, and contrapositive statements corresponding to a given conditional statement.
Use a counterexample to prove that a statement is false.
Teacher Preparation and Notes
This activity is designed to be used in a high school or middle school geometry classroom.
Before beginning this activity, students should be familiar with the terms inverse, converse, and contrapositive.
Students will discover the following concepts:
Parallel lines have slopes that are equal; perpendicular lines have slopes that are opposite reciprocals (the product of the slopes is -1).
The Segment Addition Postulate states that AB+BC=AC if B is between A and C and the points are collinear. If the points are not collinear, then AB+BC>AC.
If desired, teachers can explore which of the statements in the activity are also biconditional statements (definitions that are always true).
This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8746 and select COLSEG.8xv and NOCOLSEG.8xv.
Problem 1 – Slopes of lines
To begin, students should open a new Cabri Jr. file.
Step 1: Students will need to construct two parallel lines. First, a line needs to be constructed using the Line tool and a point not on the line using the Point tool.
Step 2: Using the Parallel tool, a line parallel to the existing line should be constructed through the point not on the original line.
Step 3: Students will find the slope of both lines by using the Slope tool (Measure > Slope). Students can now use the Hand tool to drag the original line or the point and observe the results.
What is true of the slopes of parallel lines?
Students will next construct perpendicular lines in a new Cabri Jr. file.
Step 4: Again construct a line and a point not on the line. Have students select the Perp. tool to construct a line perpendicular to the existing line through the point not on the original line.
Step 5: Students will find the slope of both lines. They should drag the original line or the point and observe the results.
Step 6: Have students select Calculate to find the product of the slopes. They should again drag the line to observe the results.
What is true of the slopes of perpendicular lines?
Note: If desired, students can display the equations of both lines using the Coord. & Eq. tool instead of the slopes.
Problem 2 – Collinear and noncollinear segments
Step 2: Students will drag the points with the Hand tool and observe the changes in the lengths.
Students should record their observations on the worksheet and write a conditional statement.
Students should record observations on the worksheet and write a conditional statement.
A. If two lines are parallel, then the slopes of the lines are equal.
Converse: If the slopes of the lines are equal, then the two lines are parallel.
Inverse: If the two lines are not parallel, then the slopes of the lines are not equal.
Contrapositive: If the slopes of the lines are not equal, then the two lines are not parallel.
Each conditional statement is true.
B. If two lines are perpendicular, then the slopes of the lines are equal to -1.
Converse: If the slopes of two lines are equal to -1, then the lines are perpendicular.
Inverse: If two lines are not perpendicular, then the slopes of the lines are not equal to -1.
Contrapositive: If the slopes of two lines are not equal to -1, then the lines are not perpendicular.
Each conditional statement is true.
Each conditional statement is false.