In an earlier lesson, students learned that equivalent equations are equations with the same solution. They practiced identifying and writing equivalent equations by performing some acceptable moves. Students also recognized that, when solving one-variable equations, they are essentially writing a series of equivalent equations that lead to the solution.
This lesson serves two main goals. The first goal is to further develop the idea of equivalent equations. Students think about and articulate how they know that the equations produced using acceptable moves are indeed equivalent. The process is an opportunity to practice constructing logical arguments and critiquing the reasoning of others (MP3).
The second goal is to refamiliarize students with equations with no solutions (which they encountered in grade 8), and with a move that might appear acceptable (dividing each side of an equation by the same variable expression) but would in fact lead us to the wrong conclusion.
- Explain (orally and in writing) why performing certain operations on an equation may create equivalent equations but performing other operations may not.
- Understand that dividing by a variable is not used in solving equations because it can lead to equations that have fewer solutions than the original equation.
- Understand that equations that are not true for any value of the variable(s) do not have solutions.
- Let’s think about why some steps for rewriting equations are valid but other steps are not.
- I can explain why some algebraic moves create equivalent equations but some do not.
- I know how equivalent equations are related to the steps of solving equations.
- I know what it means for an equation to have no solutions and can recognize such an equation.
Equations that have the exact same solutions are equivalent equations.
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