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Inequalities in Triangles

From NCTM "Illuminations".  Students use pasta to create models of triangles and non-triangles and investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students will measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle.  Includes detailed lesson plan, student activity sheet, assessment options and lesson extensions. This exercise is a hands-on, visual illustration of the Triangle Inequality Theorem, and largest angle opposite longest side theorem.
Weblink address (URL): 
Inequalities in Triangles

Standards & Objectives

Learning objectives: 

Students will:

  • Investigate the relationship between the largest side and the sum of the remaining sides in a triangle.
  • Investigate the relationship between the largest side and the largest angle in the triangle.
  • Use the triangle inequality to solve problems involving triangles.
  • Use the inequality for sides and angles in a triangle to solve problems involving triangles.
Essential and guiding questions: 
  • If the sum of the measures of the small and medium sides of the triangle is greater than the measure of the large side of the triangle, why can it be concluded that the sum of the measures of any other pair of sides of the triangle will be greater than the measure of the remaining side?
  • Is it possible to have a triangle having the sum of the measures of the small and medium sides equal to the measure of the large side?
  • The inequality for sides and angles of a triangle states that the longest side of the triangle must always be opposite the greatest angle of the triangle and that the shortest side of the triangle must always be opposite the smallest angle of the triangle. Why would it be impossible to draw a triangle where the longest side of the triangle was not opposite the greatest angle of the triangle?

Lesson Variations

Blooms taxonomy level: 
Understanding
Extension suggestions: 
  • Discuss the relationship between the triangle inequality and vector addition. Use the following diagram to illustrate.
  • Have students create a poster of their discoveries involving inequalities in triangles. 

Helpful Hints

Materials:

  • Long, thin pasta (such as spaghetti or linguine)
  • Rulers
  • Protractors
  • The Triangle Inequality Activity Sheet 
  • Inequalities for Sides and Angles of a Triangle Activity Sheet 

References

Contributors: 
Lisa Leach
Originator: 
Ayers Institute for Learning & Innovation