Lesson 14

In this lesson, students see more examples of inequalities in a context. This time, many inequalities involve negative coefficients. This illustrates the idea that solving an inequality is not as simple as solving the corresponding equation. After students find the boundary point, they must do some extra work to figure out the direction of inequality. This might involve reasoning about the context, substituting in values on either side of the boundary point, or reasoning about number lines. All of these techniques exemplify MP1: making the problem more concrete and visual and asking “Does this make sense?” In this lesson, the emphasis is on reasoning about the context.

It is important to understand that the goal is not to have students learn and practice an algorithm for solving inequalities like “whenever you multiply or divide by a negative, flip the inequality.” Rather, we want students to understand that solving a related equation tells you the lower or upper bound of an inequality. To know whether values greater than or less than the boundary number make the inequality true, it's best to either think about the context or test some values that are above and below the boundary number. This way of reasoning about inequalities will serve students well long into their future studies, whereas students are very likely to forget a procedure memorized for a special case.

Several activities suggest providing students with blank number lines to use for scratch work. One way to accomplish this is to print a line with unlabeled, evenly-spaced tick marks, and place these into sheet protectors. Students can write on these with dry erase markers and wipe them off.