Equivalent Expressions

Getting Started

The student does not understand what it means for expressions to be equivalent.

The student does not demonstrate an understanding of the meaning of equivalent. The student:

• Identifies expressions that are not equivalent. For example, the student identifies as equivalent:
  
  • 5n and 6n – 1 because “6n – 1 = 5n.”
  • 2d – d and 2 because “2d – d = 2.”
  • 5x + 3y and 8xy because “they each involve an x and a y and 5 + 3 = 8.”
• Does not recognize 10t – 7t and 3t as equivalent.
• Tries to “solve” expressions as if they are equations.
• States that it is not possible to determine whether or not the expressions are equivalent because the value of the variable is not known.

What does it mean for two expressions to be equivalent?

What does 5n mean? What does 6n – 1 mean?

What do you know about the properties of operations (e.g., the Commutative Property, the Associative Property, and the Distributive Property)?

What is the difference between an expression and an equation? Can an expression be solved?

Do you have to know what the variable represents in order to determine if the expressions are equivalent?

Explain what it means for two expressions to be equivalent (e.g., two expressions are equivalent if they always result in the same number when evaluated for the same values of the variables). Demonstrate the equivalence of two expressions (e.g., 10t – 7t and 3t) by evaluating each for the same value of t. Explain to the student that it is not possible to evaluate each expression for every possible value of t to check for equivalence, but properties of operations can be used to verify equivalence. For example, explain that 10t – 7t = 3t by the Distributive Property [e.g., 10t – 7t = (10 – 7)t = 3t].

Review the properties of operations and the associated vocabulary as needed (e.g., equivalent, expression, factor, term, coefficient, variable, constant, distribute). Emphasize the Distributive Property and how it can be used to both expand and factor expressions to generate equivalent expressions. Model using the Distributive Property to generate equivalent expressions. Emphasize that the properties of operations ensure that expressions are equivalent for every value of the variable(s). Provide opportunities for the student to use properties of operations to generate equivalent expressions.

Make explicit the difference between an expression and an equation. Explain that expressions can be evaluated for particular values of the variable but cannot be “solved.” Demonstrate evaluating an expression and solving an equation and make clear the distinction.

Consider administering other MFAS tasks.

Making Progress

The student is unable to identify equivalent expressions using properties of operations.

The student “tests” the expressions by evaluating them for a particular value of the variable and concludes that only 10t – 7t and 3t are equivalent.

Can you determine if the expressions are equivalent without replacing the variable with a number and evaluating each expression? Explain.

Can you think of any properties you could use to justify your choices? Explain.

Confirm that the student’s approach is consistent with the definition of equivalent expressions; however, he or she only tested the expressions for one possible value of the variable. Clarify that expressions can be shown to not be equivalent by evaluating each for particular values of the variable and showing that they result in different numbers. Yet, showing that expressions are equivalent requires using definitions and properties. Show the student a pair of expressions such as 2x and 3x or 2x and . Evaluate the pairs of expressions for a particular value for which they are equal (e.g., x = 0 and x = 2, respectively). Then evaluate the pairs of expressions for another value to show that they are not always equal and are not, therefore, equivalent.

Review the properties of operations and the associated vocabulary as needed (e.g., term, coefficient, variable, constant, distribute). Emphasize the Distributive Property and how it can be used to both expand and factor expressions to generate equivalent expressions. Model using the Distributive Property to generate equivalent expressions. Emphasize that the properties of operations ensure that expressions are equivalent for every value of the variable(s). Provide opportunities for the student to use properties of operations to generate equivalent expressions.

Consider administering other MFAS tasks.

Got It

The student provides complete and correct responses to all components of the task.

The student explains that only 10t – 7t and 3t are equivalent. The student explains the equivalence by stating that 10t and 7t are like terms and, when subtracted, equal 3t. The student also uses the concept of “like terms” to explain why the other pairs of expressions are not equivalent. The student may also evaluate each pair of expressions for a particular value of the variable to show that they are not equivalent.

Why can’t you subtract one from 6n to get 5n? What would you have to subtract from 6n to make it equivalent to 5n?

Given 2d – d, why don’t you just remove the d from 2d when you subtract?

Review the properties of operations and challenge the student to explain why 10t – 7t = 3t using the Distributive Property.

Consider administering MFAS task Property Combinations.