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How Many Triangles Can You Construct?

  • Lesson
Location: Unknown

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

Distribute and follow the directions on the How Many Triangles? activity sheet.

pdficon  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

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.

appicon  Fractal Tool  

What Does it Take to Construct a Triangle?

Students explore the importance of the side lengths of a triangle and when triangles can or cannot be constructed on the basis of these lengths.

What's So Special About Triangles, Anyway?

Students explore ways of building different basic shapes from triangles. They also investigate the basic properties of triangles, as well as relationships among other basic geometric shapes.

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

Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.C.5
    Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Grade 5, Geometry

  • CCSS.Math.Content.5.G.B.4
    Classify two-dimensional figures in a hierarchy based on properties.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.