Algebra Tiles Explorations of al-Khwārizmī ’s Equation Types: Classroom Implementation and Concluding Remarks

It is worth noting that al-Khwārizmī gave geometrical demonstrations for his solution methods for the compound equations (Types 4–6) in order to “point out the reasons for halving” [Rosen 1831, 13]. The figures he gave for equations of Type 5 and Type 6 do not, however, seem to be translatable to an algebra tile representation. Despite this and other limitations of algebra tiles (e.g., the impossibility of yielding an accurate representation of situations involving fractions), the translation of al-Khwārizmī’s verbal descriptions of his procedures to algebra tiles representations offers a multi-representational approach to equation solving that encourages diversity in algebraic reasoning and can promote reflection about its teaching.

The modeling activity proposed in this article can be implemented in an exploratory approach that prompts students to make meaningful and accurate connections between al-Khwārizmī's excerpts and the corresponding algebra tiles representation. After a basic introduction to the use of algebra tiles, the class can work individually or in small groups on Examples 1–10. To guide this work, students can be provided with a two-columned worksheet that gives the excerpts from al-Khwārizmī in the left column and asked to generate algebra tiles representations that model those excerpts in the corresponding “algebra tiles workspace” in the right column; the tables provided in this article for those examples thus serve as an “answer key” for this activity. The initial use of a multiplication mat that physically separates the dimension tiles from the binomial rectangles could be helpful in preventing any potential confusion about the dimension tiles (linear type quantities) and the binomial rectangles (area type quantities). Other obstacles or challenges that the students might encounter for particular equation types (e.g., the necessity of using either the subtraction symbol or “\(-x\)” tiles for Type 5 equations) have also been mentioned in earlier sections of this article. Whole-class discussion and instructor demonstrations may be valuable at certain junctures to overcome those challenges. Instructors who wished to extend the exploration further could then have students engage with the more complicated problems presented in Examples 11–14.

Depending on the student audience and the goals of the instructor, students could additionally be prompted to reflect on and discuss various aspects of the activity. For example, pre-service and practicing secondary teachers could be asked to focus on various components of the algebra tiles representations, such as the role of the dimension tiles, the emerging binomial rectangles, the area conservation principle, and the dynamics of viewing the binomial rectangle area alternately as a sum and as a product. Students in a history of mathematics course might be prompted to reflect on the differences between the mathematical context and assumptions that informed al-Khwārizmī’s work and those that inform today’s algebraic practices that are perhaps better captured by the algebra tile representations. Students in both these groups, as well as high school students and others, might also profitably discuss specific mathematical questions that emerge from the activity, such as whether and why Type 4 and Type 6 equations always have one positive and one negative solution.

As desired, the symbolic representation of al-Khwārizmī’s equation types and procedures could furthermore be brought into the exploration and juxtaposed against both al-Khwārizmī’s verbal descriptions and their algebra tile representations. A discussion of the roles of these different types of representations in supporting student learning of algebra—and especially the pedagogical benefits to be derived from the concrete visualization afforded by algebra tile manipulatives—could be especially fruitful with prospective and practicing secondary mathematics teachers. In these and other ways, the use of algebra tile manipulatives to explore al-Khwārizmī’s procedures opens up new avenues of influence for his Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala.