Algebra Tiles Explorations of al-Khwārizmī’s Equation Types: Representing al-Khwārizmī’s Quantities with Algebra Tiles

Since a quantitative approach is prevalent in al-Khwārizmī’s text, in order to set up the historically based portion of the activity, I first introduced algebra tiles to my students in a manner which shows how the basic quantities involved in today’s quadratic equations can be represented as products. For that purpose, each quantity is derived using a multiplication mat by placing the two dimension tiles (one on top and the other on left) respectively:²

Note that negative quantities are included in the algebra tile representations. In the Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala, however, al-Khwārizmī used only positive integers and fractions as the coefficients and solutions of equations. The system of numbers used by al-Khwārizmī was thus different from the one that is used in present-day classrooms. Interestingly, however, al-Khwārizmī did provide rules for multiplying negatives [Rosen 1831, 25–26]:

If the instance be, “ten and thing to be multiplied by thing less ten, then you say, thing multiplied by ten is ten things positive; and thing by thing is a square positive; and minus ten by ten is a hundred dirhems negative; and minus ten by thing is ten things negative. You say, therefore, a square minus a hundred dirhems; for, having made the reduction, that is to say, having removed the ten things positive by the ten things negative, there remains a square minus a hundred dirhems.

The following table illustrates how the computation in this excerpt might be modeled using algebra tiles to represent the negative quantities:

An interesting observation to discuss with students is how the so-called “zero principle” manifests itself as the appearance of the zero pairs “the ten things positive” and “the ten things negative.” Another interesting point to observe is the relevance to the well-known “difference of squares” identity \((A+B)(A-B) = A^2 - B^2\) to this computation and how algebra tiles involving negative quantities can be used to represent and reinforce it.

Notes

2. These figures were generated using the online algebra tiles applet mathies.ca/learningTools.php.