As the students walk in the room, I hand them a slip of paper with three triangles on it. Each of the triangles has its sides labeled with congruency marks. These marks help students to identify the triangles based on their sides.
Students are instructed to identify the triangles and write a description of the triangles. I expect some students to try to identify the triangles based on their angles. Since there is no information given about the angles, I don't accept these answers. Instead, we have a discussion about information that is given and information that is assumed. Students often assume an angle is acute because it looks acute. I reiterate that diagrams are only models or sketches, and, we must learn to reason from the given information, only.
In this mini-lesson, students will discover the relationship between the sides of a triangle. This concept is not explicitly included in the high school geometry standards, but is necessary knowledge for students to apply to prove theorems about triangles (G.CO.10).
I hand out pipe cleaners of varying lengths: 6 in, 4 in, 3 in, 2 in, and 1 in, which I cut before the lesson (see Pipe_Cleaner_Modeling_Introduction). Depending on the level of the class, I sometimes have the students practice measurement and precision by having them cut the pipe cleaners themselves (MP5). I give the students a key to ensure they know which colors are which lengths. Since I use the pipe cleaners for multiple classes, Make I tell the student to keep the pipe cleaners straight and remind them I will be collecting the same amount I have handed out.
Each group of students needs at least 3 pipe cleaners of each size. I call on one student to give an example of a potential set of sides. All students in the class will place the same three pipe cleaners together and decide if those three could make a triangle. Note, if two of the pipe cleaners together line up exactly with the third pipe cleaner, the set does not make a triangle (MP4, MP7). I then ask a student to show a second example with opposite results from the first trial.
After we go over the examples, students begin the activity.
In the beginning of the activity, I hand out the Triangle Inequality Theorem Investigation. Students are placed in pairs or small groups. Each group member does his/her own investigation and records the results in the table. Students write down the sizes of the three pipe cleaners and if they form a triangle or not. After each student does 8 trials, group members compare their results. Each student writes down three sets of pipe cleaners that form triangles and three sets that do not form triangles based on other group members’ results. They then work with their groups to come up with a statement about the relationship between the three sides of a triangle.
As students are working, I circulate and help students with the less obvious sets like: 1 in, 2 in and 3 in. If the pipe cleaners are not cut precisely, these sides may appear to form a triangle. I encourage students to come up with their own responses for the question, “Write a rule for the relationship between the sides of a triangle,” even if it is not exactly what you may have been looking for.
Some of my students who are English Language Learners had difficulty writing their response. I have them show me what they mean using their pipe cleaners. It is easier for them to explain their thinking than to write it. However, after explaining and showing me, they are usually able to write a response.
As a whole class, we discuss groups’ results. I ask, "What general statements can be made about the relationship between the sides of a triangle?" We continue the discussion until the students come up with the statement, “The sum of the length of two sides of a triangle must be greater than the length of the third side.” We use the students’ responses to show a way of deciding if sets of sides form triangles or not.
For example: 3, 4, 5 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3
All these statements are true, so 3, 4, 5 works.
7, 4, 3 7 + 4 > 3, 7 + 3 > 4, but 4 + 3 is not greater than 7.
Time permitting, I ask the students to come up with a side that will work with 7, 4, or 3.