Rich Problems

1. A Clever Method for Factoring Trinomials
   

   Here is an extension of some basic factoring methods. Although factoring has been lessened in importance in recent years, it is extremely helpful in maintaining algebraic skills and extending logical thinking and problem-solving skills. We want to factor a trinomial like 12x² + 7x - 12 as the product of two binomials with as little guess-and-check as possible.

   Learning Objectives:

   • Reinforce knowledge of the distributive property
   • Become proficient at finding all the factors of a natural number
   • Become proficient at finding a common monomial factor
   • Use the grouping method of factoring with proficiency
   • Become more proficient at factoring basic trinomials
   • Learn a clever technique for factoring more complex trinomials
   
   
   
2. Maximizing Profits on Donut Sales
   Students develop a plan to maximize profits on a donut sale using data provided. The parabolic graphs of selling price vs. income and selling price vs. profit are used to explore intercepts, symmetry, and the vertex. The graphs and table allow for a side-by-side comparison of linear and quadratic functions. The goal is for students to link representations of data through tables and graphs and to see the connection to algebraic solutions. A quadratic rule is generated using the idea that income is selling price multiplied by number sold. This problem is suited to an introduction of quadratic functions after a study of quadratic equations. An extension could be for students to gather their own data through a survey for a fund-raising effort in their school and devise a selling plan.