Triangles in the Limelight

Certain triangles have become famous! Most accessible to middle school students are the Sierpinski triangle, a fractal students can create themselves, and Pascal’s triangle, a source of patterns and real-world applications. These resources explore the iteration underlying fractals and the significant patterns in both triangles. The resources also offer problems that may engage your students in critical thinking.

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The Sierpinski Triangle Fractal http://arcytech.org/java/fractals/sierpinski.shtml

You may wish to have your students do the first stages of the Sierpinski triangle by hand before seeing this applet, which can move quickly through several iterations of the fractal. What the applet adds is the ability to see at a glance the number of triangles created at each stage—the applet keeps count—and to consider such problems as finding a formula that would give that number at any stage. This is one of the Educational Java Programs from Jacobo Bulaevsky. MSP full record

Fractals: What Comes Next? http://www.figurethis.org/challenges/c23/challenge.htm

Here is a problem based on the Sierpinski triangle: What is the relationship between the number of triangles and the sum of the triangle perimeters in each of the first three iterations of the Sierpinski triangle fractal? In other problems, students are challenged to find the amount of paint needed to cover the triangles created in the first few iterations of the fractal and then formulate a general rule. The activity also includes information about the fractal, such as the fact that it was named after the Polish mathematician Waclaw Sierpinski, who developed it around 1915. MSP full record

Pascal's Triangle (Grades 9-12) http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_1.html?open=activities

Although this resource is marked for grades 9-12, the applet offers activities for middle school students as well. The applet shows the first 27 rows of Pascal’s triangle, more than sufficient for students to determine the rule behind the order of numbers in the triangle. They can also color in, one by one or all at once, the multiples of 2, or 3, or 4, and so forth. Older students can discover the relationship between the number of the row in the triangle and the sum of the numbers in that row. Included are instructions for using the manipulative, background on the triangle, and teaching suggestions. MSP full record

Coloring Multiples in Pascal's Triangle http://www.shodor.org/interactivate/activities/pascal1/index.html

An excellent exercise in finding multiples! The applet shows up to 15 rows of Pascal’s triangle, sufficient to contain numbers in the thousands. Students color numbers in the triangle by rolling a number and then clicking on all entries that are multiples of the number rolled, thereby practicing multiplication tables and division. Each entry is immediately noted as correct or incorrect, and the number of multiples remaining to find is given. Surprising are the number patterns created by the multiples! MSP full record

The Smithville Families http://www.pbs.org/teachersource/mathline/lessonplans/msmp/smithville/smithville_procedure.shtm

In this lesson from PBS Teacher Source, students consider the total number of possible girl/boy combinations in a five-child family. The lesson begins with a review of Pascal’s triangle and the creation of its first eight rows. Next, the Smithville families are generated by each group of students tossing a coin five times--heads for a girl, tails for a boy. Patterns of gender are examined and found, surprisingly, to have a relationship to the number sequences of Pascal’s triangle. The site provides a detailed procedure for the lesson, questions for class discussion, and worksheets. You can watch an online video of the lesson at
http://vvi.onstreammedia.com/cgi-bin/ visearch?user=pbs_mathline&template=template_6-8.html&query
=+probability&grade=6&MathCategory=probability&submit.x=13&submit.y=9&page=3. MSP full record

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Copyright January 2007 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons License.