Lesson 15

This lesson serves two main purposes: to reiterate that some solutions to quadratic equations are irrational, and to give students the tools to express those solutions exactly and succinctly.

Students recall that the radical symbol (\(\sqrt{\phantom{3}}\)) can be used to denote the positive square root of a number. Many quadratic equations have a positive and a negative solution, and up until this point, students have been writing them separately. For example, the solutions of \(x^2=49\) are \(x=7\) and \(x=\text-7\). Here, students are introduced to the plus-minus symbol (\(\pm\)) as a way to express both solutions (for example, \(x=\pm7\)).

Students also briefly recall the meanings of rational and irrational numbers. (They will have a more thorough review later in the unit.) They see that sometimes the solutions are expressions that involve a rational number and an irrational number—for example, \(x=\pm \sqrt8+3\). While this is a compact, exact, and efficient way to express irrational solutions, it is not always easy to intuit the size of the solutions just by looking at the expressions. Students make sense of these solutions by finding their decimal approximations and by solving the equations by graphing. The work here gives students opportunities to reason quantitatively and abstractly (MP2).

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)