Model with inequalities in two variables
• Identify constraints from a context, choose relevant variables and model the context with an inequality or system of inequalities (A-CED.A.3$^\star$).
• Identify coordinates pairs or points in the plane as solutions or non-solutions and interpret them in terms of the context (A-CED.A.3$^\star$).
• Graph solution sets to a linear inequality or system of inequalities (A-REI.D.12).
In Grades 6–8 students learned about inequalities in one variable, and about equations and systems of equations in two variables. Here they tie these together and study inequalities, or systems of inequalities, in two variables. The emphasis is on modeling and interpreting solutions or non-solutions. Students also represent inequalities by shaded regions and interpret points in the plane; however, graphing should not be overemphasized or reduced to a procedure that does not engage the meaning of the inequality.
WHAT: Students are given a graph with two lines, four shaded regions formed by the lines, and two marked points. They write a system of equations or inequalities to represent each of the four regions and two points. They then pick their own points to verify that they satisfy the inequalities, and finally given an argument why and entire region satisfies the inequality.
WHY: This task presents the typical system of inequality problem in a less conventional manner by presenting the solution set graphically and asking students to identify the corresponding system. The task gives students an opportunity make connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities The last part of the problem requires the student to make a general argument without using specific numbers and instead to recognize the structure of the inequalities (MP7). The task could be used in many instructional settings, but having students share their thinking and respond to each others' arguments would provide a rich learning experience (MP3).
WHAT: Students are given constraints on the maximum number of people and the maximum allowable weight on a tourist fishing boat. They write inequalities, graph the solution sets, and decide whether three different groups can safely rent the boat given these two inequalities.
WHY: This problem ties together most of the big ideas from this unit; students must create inequalities to represent the given situation, solve a system of inequalities to arrive at a solution set, and make sense of non-realistic solutions (cannot have negative or non-whole number people on the boat) (MP4).
WHAT: In a diagnostic assessment preceding this activity, students are given clues for the location of buried treasure on a coordinate grid in the form of inequalities (A-REI.D.12). During the main students play the game Give Us The Clue! where they set the location of a treasure and their partner has to find its location based on student generated clues.
WHY: By focusing on an individual point and asking students to design inequalities that include that point, this activity helps students understand the inequality has having solution set made up of points in the plane. Furthermore, since the inequality is the only thing the student can give their partner, each student must share a common understanding of what it means for a point to be a solution to an inequality and communicate that meaning precisely (MP6).