Type of Class:
          Standard High School Geometry

Related VA SOL: G.6

Time Frame:
         Ninety minute block class

Objectives:

• Students will form a conjecture about the triangle inequality.
• Students will be able to know whether a triangle exists using the triangle inequality.
• Students will be able to solve practical problems using the triangle inequality.

Materials:

• Computer that hooks up to a LCD projector
• Geometer’s Sketchpad
• Chalkboard
• Rulers
• Compass

Procedures:

1)      Word Problem to motivate the class for the lesson:

a)      Sarah fractured her elbow playing softball.  When her cast was removed, the angle at which she could extend her arm was 135 degrees.  Two weeks later, her arm extended to an angle of 133 degrees.  Draw a representation of Sarah’s elbow movement for both measurements.  Did her flexibility (the length at which her arm can move) increase or decrease?  Why? 

2)      In a Geometer’s Sketchpad sketch window, construct an isosceles triangle.  Measure the lengths of its congruent sides.  Construct another isosceles triangle with its pair of congruent sides the same length as the first triangle.                     

a)      Questions directed to the class

i)        Are the third sides congruent as well?  Why or why not?

• (1)   Teacher measures the third length after class discussion

ii)       Other than measuring the length of the side, how can we verify which side is longer?

iii)     If the angle is larger, the corresponding length will be longer or shorter?

3)      Repeat this process with different triangles while asking students similar questions

4)      Construct two more isosceles triangles

a)      Measure the interior angles of the triangle before the sides.

i)        Ask students to justify which side is going to be larger based on the angle measurements.

5)      Ask students to get into groups of two or three. Provide each group with paper, ruler, and a compass.  Ask the students to construct a triangle with two congruent sides, and the third sides not congruent.  Ask them to find another triangle where the angle opposite to the non-congruent side of a triangle is larger than the angle of the first triangle’s corresponding third side, but the length is less.

6)      Using the knowledge of everything learned so far, ask the students to come up with a conjecture, for example:

a)      Suppose that two sides of one triangle are congruent to two sides of another triangle, and the included angle of one triangle is larger than the included angle of the other.  Then the third side of the triangle with the larger included angle is longer than the third side of the other triangle.

7)      Examples to be done by students in same groups. Encourage students to write and justify their answers on the board.

a)      Which segment is longer and why?

 q _?_ p

b)      Which angle is longer and why?

 _?_

c)      Which segment is longer and why?

CE _?_ DB

8)      Word Problem

a)      Sarah fractured her elbow playing softball.  When her cast was removed, the angle at which she could extend her arm was 135 degrees.  Two weeks later, her arm extended to an angle of 133 degrees.  Draw a representation of Sarah’s elbow movement for both measurements.  Did her flexibility (the length at which her arm can move) increase or decrease?  Why? 

Assessment: Assigned homework to be started in class if time permits

Suggestions/Comments: If a computer hookup for the Sketchpad portion of the lesson is not available, having the students draw the triangles with rulers in groups will also work.  Passing out compasses and rulers before the lesson begins will cut time down so that students are not wandering around searching for these items during class.