Algebra Tiles Explorations of al-Khwārizmī’s Equation Types

This article offers an interactive method of visualizing the algebraic solution methods presented by Muhammad ibn Musa al-Khwārizmī (780–850 CE) in his highly influential text Al-Kitab almukhtasarfi hisab al-jabr wa’l-muqabala (Compendium on Calculating by Completion and Reduction). Using algebra tile manipulatives in as much accordance as possible with al-Khwārizmī's completely verbal descriptions of those methods, the multi-representational module explores all six of his equation types: squares = roots; squares = numbers; roots = numbers; roots and squares = numbers; squares and numbers = roots; roots and numbers = squares.  Originally developed for use in a history of mathematics course, the audience may include students from a diversity of backgrounds and levels, including preservice and practicing secondary mathematics teachers, as well as high school students. Certain parts of the activity could also be used with preservice and practicing elementary mathematics teachers. Unlike other activities that prompt these various audiences to explore al-Khwārizmī's algebraic procedures via modern symbolic representations (e.g., [Otero 2019]), the activity proposed in this article offers a new perspective by providing students with a concrete way to visualize those procedures geometrically.

This article begins with an overview of al-Khwārizmī’s Compendium on Calculating by Completion and Reduction, before introducing the basic use of algebra tiles to represent the quantities he used in that treatise. The next two sections of the article then present and discuss examples of how to model al-Khwārizmī’s two types of equations (simple and compound). For instructors who wish to go into more depth with their students, a section that illustrates how algebra tiles could be used to model al-Khwārizmī’s procedures for calculating by completion and reduction in more complicated problems is also included. The article closes with some suggestions and advice for classroom implementation.

Figure 1. A page from al-Khwārizmī's algebra text that corresponds to page 15 of the classic
19th-century translation into English by Friedrich August Rosen (1805–1837) [Rosen 1831].
Image retrieved from Convergence’s Mathematical Treasures collection.