Math

Sierpinski’s Triangle

Reporting Category:Geometry, Fractals and Engineering

Primary SOL:3.14Shapes

4.10Geometric Terms

4.12 Polygons

4.15 Patterns

5. 17Patterns

Henrico 21Research and Information Fluency

Communication and Collaboration

Critical Thinking and Problem Solving

Creativity and Innovation

Children’s EngineeringSTEM / STREAM / STEAM

This lesson adapted and/or developed by Dwayne Cabaniss and Judy Fitzpatrick with the support of a generous 2011-2012 grant from the MathScience Innovation Center in Richmond, VA ( For more information on fractals go to Fractal Keys

Materials (used with all students in the “Student/Teacher Actions” section below): Per student

20 cm equilateral triangle worksheet

inch/cm rulers

colored pencils/markers

scissors

Data recording sheet

Protractors

Calculators

Math journals

Vocabulary:recurring (repeating), self-similar (each part is similar to the whole), zooming (grows bigger or smaller at the same scale), stage, centimeter, inch, midpoint, equilateral triangle, Sierpinski’s Triangle, fractal, fraction, greatest common factor, lowest terms

Student/Teacher Actions – What should students be doing? What should teachers be doing to facilitate learning?

  1. Distribute blank triangle worksheets and data recording sheets. Tell students they will record data as they create what is called a Sierpinski Triangle pattern, and that this is an example of a fractal (discuss or introduce fractals as needed; refer to for complete definition; fractals are self-similar, recursive, scale invariant geometric figures). Have students measure the length of each side of the triangle (20 cm), but don’t specify inches or cm. Have students discuss “ballpark” measurements. Approximately how many cm are in one inch? (Optional: also have students measure and compare angles)
  2. Have students mark the midpoint of each side, using a straightedge. Students should then create a new equilateral triangle inside the first by connecting the midpoints. Next, students should use a red marker/pencil to color in the middle triangle. How many new triangles were created (4)? What fraction of the original triangle was shaded (1/4)? What fraction remained unshaded (3/4)? Have students use the GCF to determine the fraction in lowest terms for each stage. Tell students that the “red” stage represents stage one. Have students measure the sides of the new triangles to prove that these, too, are equilateral. Compare metric and customary measurements. Review concepts of geometric transformations and similarity: how are the new triangles oriented to the original? How can we describe them mathematically?
  3. Repeat the process of finding midpoints and creating new triangles for each of the 3 unshaded triangles. For this stage (stage 2), students should color each new triangle purple. Complete the data tracking sheet for this stage. Have students discuss what patterns they notice, both visually and numerically. How can we describe this pattern or give directions for making a Sierpinski triangle using mathematical vocabulary (option: have students record responses in math journals)?
  4. Repeat the process for stage 3, shading the new “middle” triangles green and completing the data chart. Have students discuss how many stages might be possible, and what limitations would prevent further stages (the weight of the lines, the size of the paper, markers “bleeding”, etc.).
  5. OPTIONAL: Continue the process for stages 4 and 5, discussing numerical patterns. An evaluation and comparison of perimeter and area are also possible at this point.
  6. Have students carefully cut out the original triangle (not the smaller ones they’ve created). One by one, have students recreate a larger version of the Sierpinski pattern by laying their colored triangles on the floor or a large table. Encourage students to discuss what patterns they notice. If introducing fractals, discuss the concepts of self-similarity, recursion, and scale invariance at this point.
  7. To close the activity, have students share or write summaries using the vocabulary introduced/reviewed throughout the lesson.

Assessment

  • Circulate and observe students as they complete their Sierpinski patterns. Coach or assist as needed.
  • Have students turn in their completed data charts and check for correct identification of number patterns.
  • Have students complete written explanations of their observations in paragraph form or as journal entries.

Extensions and Connections (for all students)

  • Students research Sierpinski and other patterns he discovered.
  • Students compare the area and perimeter of the Sierpinski triangle, assuming that the colored sections represent portions that are “removed” or cut out of the original triangle, and that each new triangle’s sides are added to the overall perimeter of each stage.
  • Have students create fractal tiles (similar to tessellations) using a repeated process of dividing or multiplying similar shapes.

We want your input! Upon completion of this lesson, please contact Dwayne Cabaniss and Judy Fitzpatrick with any comments, questions or other feedback regarding the use of fractals as a teaching tool. Or, you may complete the survey at:

Stage / # of traingles / # of new uncolored triangles / # new uncolored traingles
All new trianges / Side Length / Perimeter of each New Triangle
0 / 1 / 1 / 1/1 / 8 / 24
1 / 4

Think Question:

  1. What is happening to the side length?
  1. Can you write a formula to determine the side length of any triangle in the Sierpinski triangle at any length?
  1. What is happening to the perimeter?
  1. Can you write a formula to determine the perimeter of a Sierpinski triangle of any given dimension?