This is one of my favorite fractals to investigate with students. There are so many neat patterns to discuss, and it’s connected to so many other mathematical ideas: probability, algebra, exponents, volume, area, and more.

Materials:

• Sierpinski triangle grid – or triangle grid paper
• Pencil
• Markers, colored pencils, or crayons

The Launch:

Do a notice and wonder with an image of the Sierpinski triangle pattern:

Some possible notices and wonders can be found in the table below:

The focus question here is: How much of the whole triangle is shaded/not shaded?

The Investigation:

Provide students with Sierpinski triangle grid or triangle grid paper. Review with students how to create each stage by finding the midpoint of each side, connecting the midpoints, then shading the new (upside down) triangle. this is different from the sample above where the whole triangle starts off shaded and the new triangles have no color.

As students create each stage, they should record the area of the triangle that is shaded and the area of the triangle that is not shaded.

Students can record this in a table like the one below:

Stage	Area Shaded	Area Not Shaded
0
1
2
3

Students may have noticed some patterns in the not shaded, but may be having trouble identifying the pattern for shaded. Remind students that the area of the whole triangle for stage 0 was 1. Ask students how they could figure out the area of the shaded portion for each stage.

The Closing

As students share their work, ask questions to keep them thinking and looking for more patterns, like:

• What patterns did you notice?
• How can you generalize the patterns?
• How might we use these general patterns to predict the area not shaded/shaded for any stage?
• What “What if…?” questions do you have to share?

Highlight student thinking that pushes students toward generalizing the pattern. Ask students to restated what a student just said, to clarify what another group stated, and to question themselves and the mathematics they’re investigating.