Lesson 6

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 an upcoming activity. While students may notice and wonder many things about these images, the peak and ridges formed 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).

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.

Then, display the images in the task statement 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 the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the concept of the distance to the sides of the triangles as a factor in the direction a given grain of salt will fall does not come up, ask students to discuss this idea.

In this activity, students show that a point is on an angle bisector if and only if it is equidistant from the rays that form the angle. This concept is essential for the next activity, where students reason that the 3 angle bisectors of a triangle meet at a single point that is equidistant from each side of the triangle.

Students may need to be reminded that an angle bisector divides an angle into 2 congruent halves.

The key point for discussion is that all points equidistant to the two rays are on the angle bisector, and all points on the angle bisector are equidistant to the two rays. Here are some questions for discussion:

• “How does this relate to the salt pile activity?” (If the angle were one of the angles in the triangle in the salt pile, the angle bisector would represent the ridge. The salt that forms the ridge is the same distance from either side, so it doesn’t fall in one direction or the other.)
• “What is the difference between what you showed in the first question and what you showed in the second question?” (In the first question, we showed that if a point is equidistant from the rays that form an angle, then it’s on the angle bisector. In the second, we proved the converse: If a point is on the angle bisector, it’s equidistant from the rays that form the angle.)
• “Have we proven these conjectures for all angles or just this one?” (This works for all angles, because we didn’t rely on any specific measurements or placements. If we drew a new angle, the same arguments would all apply.)

Tell students that what they’re learning will be useful when they construct another special circle in an upcoming lesson.

In a previous lesson, students reasoned that the perpendicular bisectors of a triangle’s sides meet at a single point. Here, students use a similar line of argument, combined with the results from the previous activity about points on angle bisectors, to determine that the 3 angle bisectors of a triangle also meet at a single point. This point is called the triangle’s incenter. The fact that this point is the same distance from all 3 sides of a triangle will lead to the construction of an inscribed circle in the next lesson.

Remind students that they have shown that all the perpendicular bisectors of a triangle met at a single point, and that point was the same distance from all the vertices of the triangle. Tell them they’re going to try and see if something similar is true for the angle bisectors of a triangle.

Give students 2–3 minutes to work. Then pull them back together to make sure all students have correctly sketched segments showing the distance between point \(G\) and the sides of the triangle.

If students are having trouble sketching segments that show the distance from point \(G\) to the sides of the triangle, suggest that they use an index card to estimate a right angle.

Tell students that this point where the angle bisectors meet is called the triangle’s incenter. We will add a theorem about a triangle’s incenter to the reference chart in the next lesson, after we look at a special circle related to incenters.

The goal is to further explore how to visualize the distance between a point and the side of a triangle, in order to strengthen students’ understanding that the incenter of a triangle is the same distance from all 3 of the triangle’s sides. First, ask students, “How does the incenter relate to salt pile?” (This point is the same distance from all the sides of a triangle, so grains of salt that land on this point balance there and don’t fall towards any of the sides.)

Then, display this image for all to see, explaining that the dashed lines are the angle bisectors for triangle \(ABC\).

For each point \(D,E,F,\) and \(G,\) ask students these questions: Which side or sides are closest to the point? Which sides are equidistant from the point, if any? Sample responses:

• Point \(D\) is closest to side \(AC\). None of the distances to the other sides are equal.
• Point \(E\) is closest to side \(BC\). It is equidistant from sides \(AB\) and \(AC\).
• Point \(F\) is equidistant from sides \(AC\) and \(BC\). It is closer to these 2 sides than it is to side \(AB\).
• Point \(G\) is equidistant from all 3 sides.