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Can triangles have more than three segments? An investigation into perpendicular bisectors, angle bisectors, and altitudes of triangles with the use of The Geometer’s Sketchpad By: Carla Castro Grades: 9-10 Class: Geometry Unit: Triangles Time Frame: 2-45 minute class periods Introduction: This lesson is designed for a 9^th-10^th grade Geometry class. This lesson fits into the unit titled “Triangles” in which students learn about the different properties of triangles, for example: classifying triangles, isosceles triangles, Triangle Angle-Sum Theorem, and special segments in triangles. Teachers will need to reserve a computer lab with access to the program The Geometer’s Sketchpad in order to complete this lesson. This lesson is designed for a class that has had previous exposure to The Geometer’s Sketchpad. The Geometer’s Sketchpad is an interactive tool that allows students to create accurate geometric figures with the use of a toolbar and various menus and then manipulate their figures to arrive at relevant conclusions. For this lesson students will construct a variety of triangles and their respective perpendicular bisectors, angle bisectors, and altitudes. Using the manipulations available with The Geometer’s Sketchpad students will arrive at conclusions about the point of concurrency (intersection) of each type of segment. This will lead students to their own discovery of the various theorems about points of concurrency in triangles. Lesson Objectives: q Students will be able to use The Geometer’s Sketchpad to construct the perpendicular bisectors, angle bisectors, and altitudes of a triangle. q Students will be able to discover the location of the point of concurrency of the perpendicular bisectors and altitudes based on the type of triangle they created. q Students will be able to make a conjecture about the distance from the point of concurrency of the angle bisectors to the three sides of the triangle Instructional Materials q Computer lab with enough computers for each student, or for each pair of students q Activity Worksheets with assignments for students (see attached) Lesson Sequence: Students should be told to meet in the computer lab for the 2 days of this lesson. Day 1: q Warm-Up: (5 minutes) o Have the following questions on the board when students walk in. They can write their answers on scrap paper for discussion: § What is a perpendicular bisector? § What is an angle bisector? § What is an altitude of a triangle? § How many of each segment exist in a single triangle? q Whole Group Instruction: (10 minutes) o Discuss the answers to the questions above (most have been discussed in previous class sessions, although they may have been a few weeks in the past) o Write definitions on the board for students to write down in their notes. q Cooperative Learning: (40-55 minutes, will carry into Day 2) o Divide students into pairs o Tell students they are going to be given three activities to complete using The Geometer’s Sketchpad. All students will begin with the perpendicular bisector activity. When they complete it and have it checked by the teacher they will receive the angle bisector activity. When that one is complete and checked they will receive the altitudes of a triangle activity. o Student directions for each activity are on the activity worksheets, along with the analysis questions. o Students should be allowed to work at their own pace through each activity and through the analysis questions, being made aware that all three activities must be completed in the 2 days allotted for this lesson. o Teacher should circulate and assist as needed. Day 2: q Cooperative Learning: (40-55 minutes, continued from Day 1) o Students should come to class prepared to continue working where they left off the previous day. o Teacher should continue to circulate and assist as needed. o If groups finish early they can attempt to complete the graphic organizer with all of their findings and then begin working on a series of sample problems from the textbook that will be assigned for homework. q Closure: (15-20 minutes) o Have students come back together as a class and complete the graphic organizer with all of their findings. o Work through three sample problems for students to see how to apply their findings to actual problems. (Note: The teacher may want to have these problem, and maybe a diagram depending the level of the class, projected on a screen, or on a handout) § Examples: · Where is the point of concurrency of the perpendicular bisectors of a triangle located if the triangle is acute? · In a triangle ABC, segment BD bisects angle ABC, and segment AE bisects angle BAC. If the measure of angle CBD is 55 degrees, find the measure of angle CBA. · In triangle DEF, segment EC is an angle bisector. If the measure of angle CEF is 2x+10, and the measure of angle DEC is x+25, find the measure of angle DEC. Assessment: Assessment of the student’s ability to complete the necessary constructions will take place through observation as the teacher circulates the room during the cooperative learning portion of the lesson. Assessment of the student’s findings and conjectures will be through evaluation of their responses on the activity worksheets, as well as on the completion of their nightly homework assignment. Supplemental Material: q Perpendicular Bisector Activity q Angle Bisector Activity q Altitudes Activity q Concluding Graphic Organizer: Student Copy q Concluding Graphic Organizer: Teacher Copy
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©C. Castro | Last revised 4/22/09 | ccastro@sjc.edu