Lesson Plan
Cassie Banka
Advanced Math Methods
8th Grade Mathematics
Geometry
Lesson Title: Discovering the Triangle Inequality Theorem
Prerequisite Skills: Students should be able to identify the characteristics of a triangle
and understand naming a triangle by its sides (scalene, isosceles and equilateral) and by
its angles (obtuse, acute, right and equiangular).
Kansas Mathematics Curriculum Standards:
8th Grade
Standard 3: The student uses geometric concepts and procedures in a variety of
situations.
Benchmark 1: Geometric Figures and Their Properties—The student recognizes
geometric figures and compares their properties in a variety of situations.
Knowledge Indicator 5: The student knows and describes Triangle Inequality Theorem
to determine if a triangle exists.
Prior Knowledge:
7th Grade
Standard 3: The student uses geometric concepts and procedures in a variety of
situations.
Benchmark 1: Geometric Figures and Their Properties—The student recognizes
geometric figures and compares their properties in a variety of situations.
Knowledge Indicator 3: The student identifies angle and side properties of triangles and
quadrilaterals:
a. sum of the interior angles of any triangle is 180º.
Knowledge Indicator 5: The student identifies corresponding parts of similar and
congruent triangles and quadrilaterals.
Knowledge Indicator 7: The student classifies triangles as:
a. scalene, isosceles, or equilateral;
b. right, acute, obtuse or equiangular.
Knowledge Indicator 8: The student determines if a triangle can be constructed given
sides of three different lengths.
I chose to focus on the Triangle Inequality Theorem because it is a concept usually
presented in a very dry manner. Students are expected to memorize a meaningless
theorem and apply it to countless problems. In my lesson, students discover the theorem
through building various triangles, observing patterns, discussing conjectures and then
applying them. The Triangle Inequality Theorem is a crucial concept to comprehend
prior to continuing on to trigonometry. By creating a conceptual learning environment
where students are physically involved with the mathematics, the teacher creates an
anchor to which students can refer when they forget the technical definitely of the
theorem.
Engage:
The class will review how to classify a triangle by its angles (right, obtuse, acute, and
equiangular triangles) and its sides (scalene, isosceles, and equilateral triangles). The
teacher will then connect students’ prior knowledge to the new concept by stating,
“We’re going to study the sides of a triangle and we need to know what needs to be true
in order to form a triangle.”
Explore:
Students will break dry spaghetti noodles into various lengths, measure them and then
observe whether or not they can form a triangle. They will record the lengths of the three
sides in the table on their worksheet. If students can form a triangle, they must see if they
can build a different triangle with the same legs. During this time, the teacher will
circulate around the room, picking out different triangles to discuss as a class (focusing
on the attempts that did not form triangles).
Explain:
The class will reconvene after most students have attempted 3-5 different triangles to
answer the following questions leading students to discover the Triangle Inequality
Theorem:
1) Was it always possible to build a triangle?
2) If a triangle could be built, could you build a different shaped triangle using
the same three segments?
3) If a triangle could NOT be built, what appeared to be the problem?
4) If a triangle could be built, how do the segments appear to be related?
As a class, students will discuss the answers to the above questions and develop a
conjecture regarding the necessary relationship between the sides of a triangle. Using
student examples, the class will discuss why or why not certain side combinations were
able to form a triangle. Once students compose an informal idea of the Triangle
Inequality Theorem, students will answer questions in a more traditional (ACT-type)
question format.
5) Use a ruler and spaghetti to make lengths 3 cm, 4 cm and 5 cm. Can a triangle
be built?
6) Use a ruler and spaghetti to make lengths 5 cm, 6 cm, and 12 cm. Can a
triangle be built?
7) Suppose a, b, and c are side lengths of any triangle (generalize here). Write an
inequality (use > or <) to relate a, b, and c.
8) Write in words the relationship that must be satisfied by the side lengths of
any triangle. This relationship is called Triangle Inequality.
During these questions, students will formalize their ideas regarding the formation of
triangles to a + b > c. The teacher must ensure that students successfully generalize their
observations verbally and then translate their notions to symbols.
Elaborate/Extension:
Students will answer the following extension question:
9) Recall the terms scalene, isosceles, and equilateral triangles. What type of
triangle were most of the triangles you built in your experiment? Explain why
this was most often the case.
The teacher will then ask students to observe the formal written rule, a + b > c, and state
whether or not the inequality holds for the other sides of the triangle (ie. b + c > a or a + c
> b). Students will then explain why the order of the sides does not matter unlike the
Pythagorean Theorem.
Evaluate:
1. Students will be assessed through direct teacher observation and class discussion.
Prompts and guidance will be given during the explore phase to struggling
students.
2. Students will be assessed based on accurate and thorough completion of the
worksheet.