Getting Started |
| Misconception/Error The student does not understand what it means for expressions to be equivalent. |
| Examples of Student Work at this Level The student is unable to identify both the first and third expressions as equivalent to (2x + 7) + (5y – 3) and the second expression as not equivalent. The student may also incorrectly apply or cite properties of operations to justify choices.
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| Questions Eliciting Thinking What does it mean for two expressions to be equivalent?
What do you know about the Commutative Property? Do you see any example of its use?
What do you know about the Associative Property? Do you see any example of its use? |
| Instructional Implications 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 such as 2x + y and y + 2x by evaluating each for the same value of x and y. Explain to the student that it is not possible to evaluate each expression for every possible value of x and y to check for equivalence; however, properties of operations can be used to verify equivalence (e.g., 2x + y = y + 2x by the Commutative Property of Addition).
Review the properties of operations and the associated vocabulary as needed (e.g., equivalent, expression, factor, term, coefficient, variable, constant, distribute). Emphasize the Commutative and Associative Properties and demonstrate their uses in a variety of examples either singly or in combination. Model using these properties to both generate equivalent expressions and justify that two expressions are equivalent. 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.
Emphasize 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. However, to show 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. Guide the student to apply properties of operations to determine equivalence.
Consider implementing the MFAS task Associative and Commutative Expressions. Then consider implementing Property Combinations again. |
Moving Forward |
| Misconception/Error The student correctly identifies all expressions equivalent to the given one but does not correctly apply properties of operations. |
| Examples of Student Work at this Level The student identifies (5y – 3) + (7 + 2x) and 7 + (5y + 2x) – 3 as equivalent and (3 – 5y) + (7 + 2x) as not equivalent to the given expression. However, the student:
- Tests the expressions by evaluating them at a particular value of the variable rather than apply properties of operations.
- Cites the wrong property or omits a property in justifying the equivalence.
 
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| Questions Eliciting Thinking 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. |
| Instructional Implications If needed, make clear 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. However, to show 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. Guide the student to apply properties of operations to determine equivalence.
Review the properties of operations and the associated vocabulary as needed (e.g., equivalent, expression, factor, term, coefficient, variable, constant, distribute). Emphasize the Commutative and Associative Properties and demonstrate their uses in a variety of examples either singly or in combination. Model using these properties to both generate equivalent expressions and justify that two expressions are equivalent. 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 implementing the MFAS task Associative and Commutative Expressions. Then consider implementing Property Combinations again. |
Almost There |
| Misconception/Error The student does not provide complete and clear justifications. |
| Examples of Student Work at this Level The student identifies (5y – 3) + (7 + 2x) and 7 + (5y + 2x) – 3 as equivalent to the given expression and (3 – 5y) + (7 + 2x) as not equivalent to the given expression. However, the student does not provide complete justifications. For example, the student says:
- (5y – 3) + (7 + 2x) is equivalent by the Commutative Property of Addition. The student does not recognize that the Commutative Property was applied twice.
- 7 + (5y + 2x) – 3 is equivalent by the Commutative and Associative Properties but cannot explain how each was used.
 
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| Questions Eliciting Thinking How was the Commutative Property used to rewrite the given expression as (5y – 3) + (7 + 2x)? Can you explain step-by-step?
How were the Commutative and Associative Properties used to rewrite the given expression as 7 + (5y + 2x) – 3? |
| Instructional Implications Model showing how to transform the given expression, (2x + 7) + (5y – 3), into (5y – 3) + (7 + 2x) step-by-step, justifying each step with a property. For example, say that (2x + 7) + (5y – 3) = (5y – 3) + (2x + 7) by the Commutative Property since the order of the addition of (2x + 7) and (5y – 3) was changed. Then (5y – 3) + (2x + 7) = (5y – 3) + (7 + 2x) by the Commutative Property since the order of the addition of 2x and 7 was changed. Ask the student to provide a similar justification to show that (2x + 7) + (5y – 3) is equivalent to 7 + (5y + 2x) – 3.
Provide additional opportunities to determine if two expressions are equivalent by applying more than one property of operations. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student says the expressions (5y – 3) + (7 + 2x) and 7 + (5y + 2x) – 3 are each equivalent to the given expression and provides clear and complete justifications such as:
- (2x + 7) + (5y – 3) = (5y – 3) + (2x + 7) by the Commutative Property of Addition and (5y – 3) + (2x + 7) = (5y – 3) + (7 + 2x) by the Commutative Property of Addition.
- (2x + 7) + (5y – 3) = (7 + 2x) + (5y – 3) by the Commutative Property of Addition and (7 + 2x) + (5y – 3) = 7 + (2x + 5y) – 3 by the Associative Property of Addition.
The student explains that (3 – 5y) + (7 + 2x) is not equivalent to (2x + 7) + (5y – 3) since subtraction is not commutative or by evaluating each expression for particular values of x and y and showing that they result in different numbers. |
| Questions Eliciting Thinking Is subtraction commutative? Why or why not?
What if the subtraction in the original expression had been a division? Is (2x + 7) + (5y ÷ 3) equivalent to 7 + (2x + 5y) ÷ 3? |
| Instructional Implications Challenge the student to determine if (2x + 7) + (5y ÷ 3) is equivalent to 7 + (2x + 5y) ÷ 3 and to justify his or her decision. |
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