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 should record their observations on the worksheet part A. They will write the converse, inverse, and contrapositive of the statement and determine the truth of each.
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?
Students should record their observations, write conditional statements, and determine the truth of the statements on the worksheet in part B.
On a new Cabri Jr. file, students will construct two lines that have the same y−intercept. If the axes are not currently showing, they should select Hide/Show > Axes.
Step 7: Students should start by placing a point on the y−axis using the Point > Point On tool.
Step 8: Make sure that students construct two different lines with the same y−intercept (the point previously created on the y−axis) and find the slopes of both lines.
Note: If desired, students can display the equations of both lines using the Coord. & Eq. tool instead of the slopes.
Step 9: Direct students to drag the lines and the y−intercept point and observe the changes in the coordinates and slopes.
Students will write conditional statements on the worksheet part C. They should sketch counterexamples for any false statements.
The screenshot at right shows a counterexample for the converse statement “If two lines have different slopes, then they have the same y−intercept.”
Problem 2 – Collinear and noncollinear segments
Step 1: Distribute the file COLSEG to students. Points A, B, and C are constructed on a line and the lengths AB, BC, and AC are displayed.
Step 2: Students will drag the points with the Hand tool and observe the changes in the lengths.
Be sure that students drag all three points, and change the order of the points (they should investigate situations where B is not between the other two points).
Students should record their observations on the worksheet and write a conditional statement.
Step 3: Distribute the file NOCOLSEG to students. Points A, B, and C are not collinear. The lengths AB, BC, and AC are displayed.
Step 4: Students should now drag the points and observe the changes in the distances. If desired, students can use the Calculate tool to find the sum of AB+BC.
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.
C. Converse: If two lines have different slopes, then the lines have the same y−intercept.
Inverse: If two different lines do not have the same y−intercept, then the lines do not have different slopes.
Contrapositive: If two lines do not have different slopes, then the different lines do not have the same y−intercept.
Each conditional statement is false.
A. Answers will vary for distances of AB, BC, and AC.
The lengths AB and BC add up to equal AC when B is between A and C.
If A, B, and C are collinear and B is between A and C, then AB to +BC=AC.
B. Answers will vary for distances of AB, BC, and AC.
If A, B, and C are not collinear, then the sum of the lengths of AB and BC is greater than AC.