Chapter 29: Perfect-Square Trinomials: Situation 23 From the MACMTLC–PTM Situations Project

While teaching about factoring perfect-square quadratic trinomials, a teacher realized that the students had the impression that a trinomial is a perfect-square quadratic trinomial if and only if the first and last terms of the trinomial are perfect squares. That is, they believed that the “middle term” is irrelevant. The teacher wanted to construct a counterexample that illustrated the importance of the middle term.

To generate an appropriate counterexample, the teacher must be able to recognize and factor perfect-square quadratic trinomials, which entails understanding the properties of the components of the trinomial that identify it as a perfect-square quadratic trinomial. Then, the teacher must be able to produce a trinomial the components of which lack at least one of those properties.