How Many Will There Be? Triangles Within Triangles

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Students will make mathematical predictions about an infinitely repeating sequence of triangles.

Inspired by the website Visual Patterns.


  1. First, students simply count up triangles in the first three steps (1, 3, and 9). Then they predict what the 4th step will be like.
  2. We reveal that step 4 has 27 triangles. I ask students to predict step 6.
  3. After revealing that step 6 has 243 triangles, we will try to predict all the way up to step 20!
  4. I discuss the pattern of repeating 3s and show how exponents are the key to quickly finding any step. And students confirm that, yes, step 20 has over a billion triangles!


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