Associative and Commutative Expressions

Getting Started

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

The student writes expressions that are not equivalent to the given expression. For example, to create a new expression, the student:

• Changes additions to multiplications.
  
  
  
  
• Attempts to add unlike terms.
  
  

What does it mean for expressions to be equivalent?

Is x + 2 equivalent to 2x? If you evaluated each expression for x = 5, would you get the same result?

Do you know what the Commutative Property states?

Do you know what the Associative Property states?

Explain what it means for expressions to be equivalent (i.e., the value of each expression is the same when evaluated for the same values of the variables). Ask the student to demonstrate that two expressions [e.g., 2(a + 3) and 2a + 6] are equivalent in specific instances by evaluating each expression for various values of a. Also, provide an example of two expressions that are not equivalent [e.g., 2(a + 3) and 2a + 3] and ask the student to show they are not equivalent by evaluating each expression for a particular value of a. Be sure the student understands that the demonstration that two expressions are equivalent for a variety of values does not constitute a proof that they are equivalent. To prove two expressions are equivalent, properties and theorems must be used.

Provide instruction on the Associative and Commutative Properties and be very clear in describing what each property says both in words and in symbols. For example, explain that the Commutative Property of Addition states that it does not matter the order in which two numbers are added – the sum will be the same. Illustrate this property by writing “a + b = b + a for all values of a and b.” Show the student specific examples of the use of the properties (e.g., 2a + b = b + 2a by the Commutative Property). Demonstrate how the properties can be used to simplify computation [e.g., the expression (44 + 28) + 56 can be rewritten as (28 +44 )+ 56 by the Commutative Property which can then be rewritten as 28 + (44 + 56) by the Associative Property so that it can be evaluated as 28 + 100 = 128].

Provide additional opportunities to use properties to write expressions equivalent to given expressions.

Making Progress

The student has an intuitive understanding of the Commutative and Associative Properties of Addition but cannot formally distinguish between them.

The student writes three expressions that are equivalent to the given expression as a consequence of applications of the Associative and Commutative Properties but cannot describe a specific instance of the use of one or both properties. For example, the student uses each property one or more times to rewrite the given expression but identifies the new expression as resulting from only the use of the:

• The Commutative Property.
  
  
  
  
• The Associative Property.
  
  

Can you show me specifically where you used the Commutative Property?

Can you show me specifically where you used the Associative Property?

Can you show me, step-by-step, how you rewrote the expression and how you used each property?

Review the Associative and Commutative Properties and be very clear in describing what each property says both in words and in symbols. For example, explain that the Commutative Property of Addition states that it does not matter in what order two numbers are added - the sum will be the same. Illustrate this property by writing “a + b = b + a for all values of a and b.” Show the student specific examples of the use of the properties (e.g., 2a + b = b + 2a by the Commutative Property). Demonstrate how the properties can be used to simplify computation [e.g., the expression (44 + 28) + 56 can be rewritten as (28 +44) + 56 by the Commutative Property which can then be rewritten as 28 + (44 + 56) by the Associative Property and then evaluated as 28 + 100 = 128].

Ask the student to write counterexamples that demonstrate why there are no Commutative or Associative Properties of subtraction and division. Refer to the properties by their full names (e.g., the Commutative Property of Addition) to make clear to what operation the properties apply.

Provide additional opportunities to use properties to write expressions equivalent to given expressions.

Got It

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

The student generates several expressions equivalent to the given one using the Associative and Commutative Properties. The student is able to clearly identify specific instances of the use of each property. For example, the student writes:

• (3x + 2y) + 4z = (2y +3x) + 4z by the Commutative Property. 
• (3x + 2y) + 4z = 3x + (2y + 4z) by the Associative Property.
  
     

Can you apply the Commutative property (or Associative property) to subtraction? Why or why not?

Can you apply these properties to addition regardless of the value of the variables? Why or why not?

Model rewriting the expression (3x + 2y) + 4z in several steps clearly justifying each step with the appropriate property. For example,

Challenge the student to identify a sequence of steps and justifications in a similar manner that shows that two given expression are equivalent [e.g., show that (3x + 2y) + 4z is equivalent to (4z + 3x) + 2y] using the least number of steps.

Consider using MFAS task Identifying Equivalent Expressions.