Page 151 - IM_FL_Geometry_Print Sample

Lesson 8: A Special Point
• Locate and identify properties of the incenter of a triangle Lesson Narrative
In the previous lesson, students learned what it means to measure the distance between a point and a line. In this lesson, students build on that concept to discover that an angle bisector is the set of points that are equidistant to the two rays that form the angle. To build physical intuition for this concept, students rst observe that pouring salt on a triangle forms ridges that meet at a peak, and the ridges appear to make angle bisectors. Students go on to prove that the angle bisector is the set of points equidistant to the two rays that form the angle, and that the three angle bisectors in a triangle coincide at a point equidistant to all three of the sides. This special point is introduced as the incenter of the triangle.
The concept that an angle bisector separates points closer to one ray from points closer to the other ray is analogous to the concept that the perpendicular bisector of a segment separates points closer to one endpoint from points closer to the other endpoint. Students will discuss this connection towards the end of the lesson, hinting towards an upcoming lesson about circumcircles.
Students create viable arguments and critique the reasoning of others when they use what they know about triangle congruence to prove facts about angle bisectors. (MP3).
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to explore and solve problems. We recommend making technology available.
Required Materials Geometry toolkits
Required Preparation
If dynamic software is available having it as an option when students explore the distance from the angle bisector to the rays that form angle might be helpful for some students.
Student Learning Goals
• Let’s see how salt piles up on a triangle. 8.1 Salt Pile
Warm Up: 5 minutes
The purpose of this warm-up is to elicit the question, “Why does the salt pile up to make ridges and a peak like that?” which will be useful when students study distance to sides of a triangle in a later activity. While students may notice and wonder many things about these images, the peak and ridges formed are the important discussion points.
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Teacher Guide