Algebra Tiles Explorations of al-Khwārizmī ’s Equation Types: Modeling al-Khwārizmī’s More Complicated Problems

Following his treatment of the solution procedures for each of the six cases of equation, al-Khwārizmī went on in his treatise to present six problems “which will serve to bring the subject nearer to the understanding, to render its comprehension easier, and to make the arguments more perspicuous” [Rosen 1831, 35]. These problems were followed by a large collection of “various questions” that further illustrated how al-Khwārizmī’s procedures for “calculating by completion and reduction must always necessarily lead you to one of these cases” [Rosen 1831, 35]. In the examples below, we illustrate how to model such problems using algebra tiles. We have selected these particular examples in part because they avoid the use of fractions and mixed numbers, yet show the complexity of the equations that al-Khwārizmī treated. Because some problems involve the difference of two quantities, note that it is very natural to use subtraction within the algebra tile models. Following al-Khwārizmī’s advice to his readers that they should “reduce this [equation] according to the rules, which I have above explained to you” [Rosen, 1831, 51], we have omitted the algebra tile representations for some of the later details in our last two examples.

• Example 11: A problem leading to the case “squares = numbers”
• Example 12: A problem leading to the case “squares = numbers"
• Example 13: A problem leading to the case “squares = roots”
• Example 14: A problem leading to the case “squares and numbers = roots”
• Example 15: A problem leading to the case “squares and numbers = roots”