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J. Gregory, B. Kronholm, and J. White, Iterated rascal triangles, Aequationes Math. (2023), https://doi.org/10.1007/s00010-023-00987-6. (Accepted 2023 and published August 2024.
the Rascal Triangle
the Rascal Triangle
The Rascal Triangle is a result of inquiry based learning when middle school students were asked to complete rows of an unfinished Pascal's Triangle, but unexpectedly constructed the Rascal Triangle instead. This student inquiry based learning model was repeated in 2015 and these students discovered additional patterns and identities in the Rascal Triangle. We show the Rascal Triangle is one member of an infinite family of number triangles we call Iterated Rascal Triangles. These Iterated Rascal Triangles are unified by a single recursive formula and we will show how they are connected directly to Pascal's triangle. Further student based learning produced additional identities in the Rascal Triangle and the discovery of a larger family of "Rascal-like" number triangles or Generalized Rascal Triangles. The Rascal Triangle is a basis for continued undergraduate projects or research with the prospect of added identities and connections to Pascal's Triangle.
The Rascal Triangle and its related structures offer an inviting yet challenging foundation for undergraduate research. Because it grows out of the familiar setting of Pascal’s Triangle, students can connect with the idea quickly. From there, inquiry-based research becomes a natural fit—students start with open-ended questions, experiment with patterns, and make discoveries through guided exploration. Some projects might focus on proving new identities, while others could look for links between Iterated Rascal Triangles and known combinatorial objects. The work stays open-ended, so there is always something new to uncover. Along the way, students gain experience with the true process of mathematics of noticing something unexpected, asking why it happens, and building theory one step at a time.
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