In middle school, students learned that two expressions are equivalent if they have the same value for all values of the variables in the expressions. They wrote equivalent expressions by applying properties of operations, combining like terms, or rewriting parts of an expression.
In this lesson, students learn that equivalent equations are equations with the exact same solutions. Students see that the moves that generate equivalent expressions (for example, applying the distributive property or combining like terms) can also create equivalent equations. Additionally, an equivalent equation can be created by adding the same number to both sides or multiplying both sides by the same non-zero number. Students have seen moves like this before, when solving one-variable equations in middle school. What is new here is an awareness that each of the equations created as a part of the solving process is equivalent to the original equation.
Students also regard equivalent equations as synonymous statements about a relationship. They use context to interpret the solution to equivalent equations, and to think about why it makes sense that equivalent equations have the same solution. In doing so, students reason abstractly and quantitatively (MP2).
The emphasis of this lesson is on equations in one variable. Students will have many opportunities to study equivalent equations in two variables in future lessons.
- Comprehend that “equivalent equations” are equations that have exactly the same solutions, and that multiple equivalent equations can represent the same relationship.
- Determine and explain (orally and in writing) whether two equations are equivalent.
- Identify operations that can be performed on an equation to create equivalent equations.
- Let's investigate what makes two equations equivalent.
- I can tell whether two expressions are equivalent and explain why or why not.
- I know and can identify the moves that can be made to transform an equation into an equivalent one.
- I understand what it means for two equations to be equivalent, and how equivalent equations can be used to describe the same situation in different ways.
Equations that have the exact same solutions are equivalent equations.
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