Reason about linear equations and inequalities in one variable
• Explain each step in solving a simple equation in one variable (A-REI.A.1).
• Create and solve linear equations in one variable, including equations with coefficients represented by letters (A-REI.B.3).
• Create and solve linear inequalities in one variable (A-REI.B.3).
As preparation for the work with two-variable equations, inequalities and systems in this unit, students must have a strong foundation in working with equations and inequalities in one variable and a clear understanding of what an equation is and what it means for a number to be a solution to the equation (it makes the two sides equal). This section gives students opportunities to practice reasoning, manipulating and solving equations and inequalities.
Tasks
1
How does the solution change?
WHAT: Students are given four equations with the variable x and the constant a. Assuming a is positive, students are asked to explain, for each equation, what would happen to the solution if a were increased.
WHY: The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level. By asking students to reason about solutions without explicitly solving them, the task gets at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them (MP2).
2
Same solutions?
WHAT: Students are given six one-variable equations and are asked to find which equations have the same solution with the instruction to give reasons that are not dependent on solving the equations.
WHY: It is easy for students to lose sight of what it means for two equations to be equivalent they are trained to immediately start following a step by step procedure to solve an equation. The purpose of this task is to provide an opportunity for students to look for structure when comparing equations and reason about their equivalence (MP7). Note that although it is possible to show that two equations are equivalent without solving them, it is more difficult to give reasons why they are not equivalent, even though they do not appear to be. Thus, in the end, confirmation of the solution is achieved by solving the equations, and students get practice with their solving skills.
3
Reasoning with linear inequalities
WHAT: In this task, a sample student solution is given to an inequality. There are, however, two mistakes in the solution and students must find the mistakes and explain why they are mathematically incorrect, and then give a correct solution (MP3).
WHY: The purpose of this task is to focus students on seeing the process of solving an equation or inequality as a special kind of proof. This task provides an opportunity for students to understand why multiplying both sides of an inequality by a negative number changes the direction of the inequality. The errors presented in this activity are logical missteps in the deduction, not just a failure to follow rules.