Student Actions

Teacher Actions

• Develop a deep and flexible conceptual understanding  by examining various models of linear inequalities and discovering relationships between these models.

• Develop accurate and appropriate procedural fluency by connecting models of inequalities (number lines and coordinate graphs)  to algebraic representations and using inverse properties to determine solutions of inequalities.

• Develop strategies for problem-solving by modeling inequalities using numbers lines and coordinate planes. Applying those models to find solutions to the inequalities algebraically will allow students to understand that more than one value will be the solution to the situation.

• Develop mathematical reasoning by drawing connections between various models of a situation involving an inequality as well as interpreting and assessing the reasonableness of solutions in context.

• Develop the ability to communicate mathematically by interpreting and translating, both verbally and graphically, the solutions of inequalities using correct mathematical notation and by explaining their reasoning and solutions to teachers and peers through visual models and/or written descriptives. 

• Use and connect mathematical representations by providing students with opportunities to analyze models of one and two-variable inequalities (verbal, sketches, number lines, coordinate plane) to determine relationships of these different models. 

• Build procedural fluency from conceptual understanding by helping students discover the connections between finding the solution to equations and finding the solutions to inequalities. 

• Pose purposeful questions to guide students’ exploration of a variety of models of inequalities, finding patterns in the models, and connecting them in various contexts. 

• Implement tasks that promote reasoning and problem-solving by introducing increasingly complex problems that involve inequalities, encouraging the use of multiple methods for solving (including relying on models), and expecting students to interpret their solutions in the correct context.

• Elicit and use evidence of student thinking by allowing students to demonstrate their understanding in a variety of ways (verbal, written, and models). 

Key Understandings

Misconceptions 

• Solutions of one-variable inequalities are represented on a number line.

• For < and >, closed circles represent a value included in a solution set. 

• For < or >, open circles represent a value not included in a solution set.

• Greater than and greater than or equal to have solutions shaded to right of a value.

• Less than and less than or equal to have solutions shaded to the left of a value.

• Solutions of two-variable inequalities are represented in the coordinate plane.

• For (include symbol here), a solid graphed line represents solutions included in the shaded region.

• For < or >, a dashed graphed line represents solutions not included in the shaded region.

• Greater than and greater than or equal to include solutions shaded above the graphed line.

• Less than and less than or equal to include solutions shaded below the graphed line.

Example:

For the graph of the inequality y≥−x+1,  any ordered pair that lies in the shaded region and graphed line would be an acceptable solution.

Solution:

• Students may forget to reverse the direction of the inequality symbol when multiplying or dividing by a negative number in the process of solving a one-variable or two-variable linear inequality.

• Students may not correctly shade the solution set on either side of a graphed boundary line in the coordinate plane.

• Students may not consider the shaded region and its boundary line together as acceptable solutions to a graphed linear inequality in the coordinate plane.

• Students may confuse the differences between solid and dashed lines and their meanings in the solutions of a graphed  linear inequality.

• Students may incorrectly represent a solution set of a one-variable inequality on a number line. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form px+q>r and px+q<r, where p, q, and r are rational numbers. (PA.A.4.2)  

• Represent real-world situations using equations and inequalities involving one variable (PA.A.4.3) 

• Solve systems of linear inequalities in two variables, with a maximum of three inequalities; graph and interpret the solutions on a coordinate plane. Graphing calculators or other appropriate technology may be used. (A2.A.1.9)