Triangle Inequality and Side-Angle Relationships
Lesson 3 of 10
Objective: Students will be able to explain the triangle inequality and use their understanding of the relationship between sides and angles in triangles to solve problems.
In this Warm-Up, I ask students to apply their knowledge of special quadrilateral properties and use the given information in each problem to solve. In the first problem, I sometimes see students who hesitate to write out equations relating the side lengths of the parallelogram--I think this is because students end up writing a system of equations that they may or may not remember how to solve.
As I circulate the room, I stress the importance of taking risks to put their thinking out there (MP1); I tell students that I will not let them talk to their group until everyone has at least marked their diagram with symbols that show the properties of parallelograms. Students then discuss in their groups where, inevitably, at least person sees and shares out a solution path. This is the kind of problem that is great for whole-class discussion because it encourages multiple solution paths in addition to having students explain why they chose to discuss the sides of the parallelogram as opposed to the angles and diagonals--important thinking required for proof.
I like to have students apply their understanding of the triangle sum theorem by displaying a simple picture that requires them to reason about the angles that surround all the vertices of a triangle. In this discussion, my goal is to move students beyond considering small cases (for example, equilateral triangles or one or two numerical examples) so they will think generally about any type of triangle.
In the debrief of this problem, I want to be sure to choose a student who is willing to talk through their thinking and how they considered different cases to then conjecture and prove why the sum of these angles will always equal 900 degrees.
I like to have a whole-class discussion about triangle inequality. Students have such a good, intuitive sense about geometric relationships, which is why it is important to build momentum around their "gut" feelings so they will make arguments with the goal of convincing others (MP3).
I ask students whether it is possible for two different sets of three given lengths to form triangles (1 cm, 3 cm, and 100 cm; 6 inches, 8 inches, and 14 inches). I give students about a minute in their groups to discuss with the goal of convincing others that they are correct.
When I facilitate this whole-class discussion, particularly the second set of lengths (6 inches, 8 inches, and 14 inches), I know I need to make it safe to be wrong. To start this discussion, I ask students to state whether they believe this set of side lengths can create a triangle, count to three, and ask students to state their answer loudly (yes or no). Once students hear that there are many people on their side, they feel more safe to engage in the argument. The discussion can be rather robust, with many different students wanting to agree or disagree with other's reasoning. Some of the best arguments I have seen have involved sketches, mental images of a drawbridge, and even constructions to show whether the two "shorter" sides can actually meet to form a triangle.
After a lively whole-class discussion, I know I need to give students some quiet time to really check their own understanding. I project sets of three side lengths and ask students to determine whether they will make triangles:
- 3, 5, 10
- 15, 3, 10
- 4, 0.5, 2
- 56, 92, 60
- 19, 20, 40
- 0.04, 0.5, 0.47
- 100, 200, 300
- 3, 3, 6
After giving the students some time to work, I go over the answers with the whole-class so that students can get a sense of their own understanding.
During this closing time, we take notes as whole class, capturing our ideas about the triangle inequality from our previous whole-class discussion.
We also talk about how the sides and angles in a triangle relate to each other, with the biggest angle forming the largest side, and the shortest side being opposite the smallest angle. I also add some new terms to the students triangle angle vocabulary, like exterior angle and remote interior angles.
As a wrap-up to the day's learning, we prove that the exterior angle of a triangle is equal to the sum of the remote interior angles.