Shape

Triangle Recipes--Challenge: Spaghetti Triangles

In this lesson, you will investigate the use of alternative technologies and strategies for constructing triangles. The directed activities Construct an equilateral triangle using the Geometers Sketchpad and Construct an equilateral using paper folding revisit the task of constructing an equilateral triangle. The challenge problem and directed activities in this section begin investigation of the question, “How much information (i.e., sufficient information) is needed to construct a triangle?” Each of these constructions is based on a different set of features (See Figure 1.6). The directed activities Side-Angle-Side asks you to apply what you have learned to devise a method for indirectly measuring the length of a segment for which no direct measurement is possible. In each case, your goal is to understand why each construction determines a figure with the proper features. Dry spaghetti cut to length and held in a manner that fixes the relative positions of the various pieces may be used to support the development of meaningful insights.

Figure 1.6: SSS vs. SAS

Materials needed: Dry spaghetti

Fraction Circle pieces, not halves

Part I

v Break several pieces of spaghetti. Sort the broken pieces into piles of congruent length. Discard any that do not have at least two the same length. Mark congruent pieces with a colored marker. For example, all the pieces that are one length are marked blue, all the pieces that are another length are marked red, and all the pieces that are yet another length are marked green mark, etc.

v Select any three pieces that create a triangle. Arrange the pieces to make a triangle. Use a metric ruler to measure the pieces of spaghetti used. Sketch this triangle including measurements of sides. Take three more pieces that are congruent to the first three you chose. Can you create a different triangle with this set of the three pieces? Note: Two triangles are the same if one can be transformed to the other by rotation or reflection. Sketch your triangles.

v Repeat at least 6 times.

	Sketch of first triangle created. Include measurements	Sketch of second triangle created. Include measurements.	Are the triangles different?
First trial
Second trial
Third trial
Fourth trial
Fifth trial
Sixth trial

v Were you able to create different triangles with the same set of three pieces? Why or why not? Explain.

v What conjectures can you and your group make about three specific sides of a triangle? Explain.

Part II

v Select any two pieces of spaghetti and one of the fraction circle pieces. Use the two pieces of spaghetti as two sides of a triangle with the fraction circle piece as the included angle between them. Break a piece of spaghetti piece to complete the triangle. Use a metric ruler to measure the pieces of spaghetti used. Sketch this triangle including measurements of sides and the measure of the included angle.

v Select a second set of pieces (two spaghetti pieces and one fraction circle piece) that are congruent to those already chosen. Can you create a different triangle than the first one you created? Placement of angle may not change.

v Repeat at least 6 times.

	Sketch of first triangle created. Include side and angle measurements	Sketch of second triangle created. Include side and angle measurements.	Are the triangles different?
First trial
Second trial
Third trial
Fourth trial
Fifth trial
Sixth trial

v Were you able to create different triangles with the same set of three pieces? Why or why not? Explain.

v What conjectures can you make about two specific sides and an included angle of a triangle?