The basis of this lesson is a geometry problem that allows students to think about many concepts related to triangles and to apply a diverse set of theorems. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles.
The Triangle Classification Problem can be stated as follows:
Line segment AB is drawn in a plane. Find all points C in the plane such that triangle ABC is:
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- isosceles
- scalene
- equilateral
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Prior to teaching this lesson, you should experience the problem for yourself. Print one copy of the Triangles activity sheet and divide the plane into regions according to the classifications by sides and by angles. You can see the solution under Assessment Options below, or you can self‑assess your work using the Triangle Classification online activity, but try to solve the problem on your own before looking at either solution. You will be able to pose better questions if you have thought about the problem yourself before giving it to students.
In addition, copy the activity sheet to a transparency to display on the overhead projector, for use during class discussions.
To begin the class, ask students the following questions:
- How can you classify triangles according to their angles? [right, acute, obtuse]
- How can you classify triangles according to their sides? [equilateral, isosceles, scalene]
After the introduction, place an overhead copy of the activity sheet on the projector. Read the directions aloud to students, and then ask students to suggest a point C that would create a right triangle. Call on a volunteer to come to the overhead projector and, without any explanation, place a dot on the transparency. Then, allow the class to discuss. Would the point form a right triangle? How do you know? During this discussion, ask questions to prompt and further student thinking, but be careful not to insert comments of your own.
After a brief discussion, distribute the Triangles activity sheet to all students. Answer any questions that students have regarding the activity. Once all questions have been answered and students are ready, allow them to work for 1‑2 minutes individually to identify the various types of triangles. (You will not need much time for this. During the discussion about right triangles, most students will have begun to think about the points that form the other types of triangles, too.)
For the next 3‑5 minutes, allow students to share their thoughts with a partner. During these discussions, students will often realize any errors that they made. In addition, two students working together will find most, if not all, of the points that form each type of triangle.
Spend the remaining time in class discussing the student discoveries. Allow a different student to indicate which points form each of the six different types of triangles. During this discussion, be sure to review theorems that are needed to solve this problem. For instance, the circle with the midpoint of AB as its center represents all right triangles with AB as the hypotenuse; this is true because of the following theorem: "An angle inscribed in a semicircle is a right angle."
To accompany this discussion, you may want to use the Triangle Classification online activity for demonstration purposes. The Show buttons will indicate the paths that create right and isosceles triangles. (The Hide buttons remove the triangles but leave the paths.) After all four paths are drawn, you can use the Show Random Triangle button to explore the regions where acute and obtuse triangles occur.
