Unit 7, Lesson 6
A Special Point
Geometry
6.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the question, “Why does the salt pile up to make ridges and a peak?” which will be useful when students study triangle incenters in a later activity. While students may notice and wonder many things about these images, the peak and ridges formed by the salt are the important discussion points.
This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
Launch
If desired, demonstrate the process of pouring salt on a triangle. To do so, cut a triangle out of cardboard. Place a cup or bottle on top of a plate, and set the triangle on top. Pour salt on the triangle slowly and keep pouring after the triangle has reached capacity to show how the salt falls.
Alternatively, show students this video.
Salt is poured on a cardboard triangle. The salt grains stack up in a pyramid shape on the triangle.
Arrange students in groups of 2. Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to share their ideas and consider how a salt grain’s position affects the side to which it would fall.
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the images. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the concept that a grain of salt’s distance to the sides of the triangle is a factor in which direction it would fall does not come up during the conversation, ask students to discuss this idea.
Lesson Synthesis
In this lesson, students proved that an angle bisector is the set of points equidistant from the rays that form the angle, and students used that concept to find the incenter of a triangle. Display an image of a segment and its perpendicular bisector alongside an image of an angle with its angle bisector:
Invite students to compare and contrast angle bisectors and perpendicular bisectors. (Each cuts something in half. A perpendicular bisector cuts a segment in half, while an angle bisector cuts an angle in half. Both structures have to do with points being the same distance away from two objects. A perpendicular bisector is the set of points equidistant from the endpoints of a segment, whereas an angle bisector is the set of points equidistant from the rays that form an angle. Both divide a region into sets of points closer to one object than another object.)
Student Lesson Summary
Salt piles up in an interesting way when poured onto a triangle. Why does that happen?
As the salt piles up and reaches a maximum height, new grains of salt will fall off toward whichever side of the triangle is closest. We can show that points on an angle bisector are equidistant from the rays that form the angle. So salt grains that land on an angle bisector will balance and not fall toward either side. This is why we see ridges form in the salt.
As we might conjecture from the salt example, all three angle bisectors in a triangle meet at a single point, called the triangle’s incenter. To see why this is true, consider any two angle bisectors in a triangle. The point where they meet is the same distance from the first and second sides, and is also the same distance from the second and third sides. Therefore, the point is the same distance from all sides, so the third angle bisector must also go through this point.
In the images, segments and are angle bisectors. This means that, for angle , point is the same distance from ray as it is from ray . In triangle , point is the same distance from all three sides of the triangle—it’s the triangle’s incenter.
Angle QRS bisected by line T. Line from point T to line Q creates 90 degree angle. Line from point T to line S creates 90 degree angle. These segments are equivalent.