Illuminations: Building with Triangles

Building with Triangles

How Many Triangles Can You Construct?

 Students identify patterns in a geometrical figure (based on triangles) and build a foundation for the understanding of fractals.

Learning Objectives

 Students will: identify patterns in a geometrical figure build a foundation for the understanding of fractals make hypotheses and then develop experiments to test them

Materials

 How Many Triangles? Activity Sheet Ruler, pencils, or fine-line markers Writing paper

Instructional Plan

 Distribute and follow the directions on the How Many Triangles? activity sheet. Initially, students should attempt the activity sheet individually. You may wish for students to work together after they have had a chance to work independently. Ask the following questions to stimulate a whole class discussion: How did your triangle change? How did you find out the number of triangles that were possible? What did you notice about the number patterns? Solutions to the Activity Sheet: Students should see the following pattern emerge for Triangle 1: Stage...Number of Triangles 1......1 2......4 3......16 4......64 Students should see the following pattern emerge for Triangle 2: Stage...Number of Shaded Triangles (and Reason) 1......3 (3 to the power of 1) 2......9 (3 to the power of 2) 3......27 (3 to the power of 3) 4......81 (3 to the power of 4) Ask students if they have heard the term fractal previously. Students who are familiar with the term will know that a fractal is a geometric shape that can be split into parts, where the parts are smaller versions of the original geometric shape. Introduce the Fractal Tool, which allows students to explore and create their own fractals.

NCTM Standards and Expectations

 Geometry 3-5Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes. Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

1 period

NCTM Resources

Principles and Standards for School Mathematics

Web Sites

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