Chapter 8: Product Of Two Negative Numbers: Situation 2 From The MACMTL–CPTM Situations Project

A question commonly asked by students in middle school and secondary mathematics classes is “Why is it that when you multiply two negative numbers together, you get a positive number answer?”

Students are able to visualize the addition and subtraction of integers, but multiplication of integers, particularly signed numbers, seems to be more abstract. Representing multiplication of quantities less than 0 is difficult. The Foci make this abstract concept more concrete by providing multiple ways to think about the multiplication of negative numbers, some of which only suggest that the product of two negative numbers should be positive and some of which establish definitively that the product of two negative numbers is positive via a general proof. Focus 1 applies a repeated addition model to multiplication of negative numbers, but that model is shown to have limitations. Real-world applications often are used to suggest that the product when multiplying two negative numbers should be positive; Focus 2 offers one such application involving an employee's pay. Focus 3 employs a visual approach using scalar properties of vectors to suggest that the product should be positive. In Focus 4, the distributive property and other properties of the real numbers are used to show, for specific cases and in general, that the product of two negative numbers should be a positive number. Focus 5 develops an intuitive, pattern-finding approach to suggest that the product should be positive. A geometric argument based on similar triangles appears in Focus 6. Focus 7 offers an analysis based on some concepts from abstract algebra.