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How Many Triangles Can You Construct?

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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
2......4 (1 times 4)
3......16 (4 times 4)
4......64 (16 times 6)

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. 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. Wrap up the class by having a discussion on any patterns the noticed with the number of segments (or shapes) and the total length (or area).

appicon Fractal Tool 

Assessment Options 

  1. Use the How Many Triangles? Activity Sheet as a form of assessment.
  2. Use the Fractal Tool to display the Koch Snowflake. Go to Stage 2 of the snowflake, and ask students to hypothesize the number of segments and total length in Stage 3.


  1. Have students use a straightedge and compass to construct a fractal of their choice. They can attempt to reproduce one discovered using the Fractal Tool, or come up with one of their own.
  2. Have students use the internet to research about fractals. Some good keywords include: Fractals, Mandelbrot, and Cantor.

    Questions for Students

    Refer to the Instructional Plan for Questions.

    Teacher Reflection 

    • Describe your students' level of enthusiasm. What could you change about this lesson to make it more engaging?
    • How can you incorporate technology to help students find patterns? 
    • How can you modify this lesson to help high and low-level achievers?
      Unit Icon

      Building with Triangles


      Engage students in a study of triangles and their properties.


      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 develop experiments to test them.

      NCTM Standards and Expectations

      • Investigate, 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.

      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.