Equal and Equivalent
In this lesson students are introduced to the idea of equivalent expressions. Two expressions are equivalent if they have the same value no matter what the value of the variable in them. Students use diagrams where the variable is represented by a generic length to decide if expressions are equivalent, and they show that expressions are not equivalent by giving values of the variable that make them unequal. They identify simple equivalent expressions using familiar facts about operations.
- Draw a diagram to represent the value of an expression for a given value of its variable.
- Explain (in writing) that some pairs of expressions are equal for one value of their variable but not for other values.
- Justify (orally, in writing, and through other representations) whether two expressions are “equivalent”, i.e., equal to each other for every value of their variable.
Let's use diagrams to figure out which expressions are equivalent and which are just sometimes equal.
Graph paper in addition to grids printed with the tasks may or may not be necessary. It is recommended to have some on hand just in case.
- I can explain what it means for two expressions to be equivalent.
- I can use a tape diagram to figure out when two expressions are equal.
- I can use what I know about operations to decide whether two expressions are equivalent.
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.
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