Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 65225
Associative and Commutative Expressions
Students are asked to write expressions equivalent to a given one by using the Associative and Commutative Properties.
Subject(s): Mathematics
Grade Level(s): 6
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, equivalent, algebraic expressions, Associative Property, Commutative Property
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_AssociativeAndCommutative_Worksheet.doc
MFAS_AssociativeAndCommutative_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Associative and Commutative Expressions worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
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 writes expressions that are not equivalent to the given expression. For example, to create a new expression, the student:
Changes additions to multiplications.
page 1 of 3 Attempts to add unlike terms.
Questions Eliciting Thinking
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?
Instructional Implications
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
Misconception/Error
The student has an intuitive understanding of the Commutative and Associative Properties of Addition but cannot formally distinguish between them.
Examples of Student Work at this Level
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.
Questions Eliciting Thinking
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?
Instructional Implications
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.
page 2 of 3 Provide additional opportunities to use properties to write expressions equivalent to given expressions.
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 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.
Questions Eliciting Thinking
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?
Instructional Implications
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 (6.EE.1.4).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Associative and Commutative Expressions worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.6.EE.1.3:
Description
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to
the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression
24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce
the equivalent expression 3y.
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
By applying properties of operations to generate equivalent expressions, students use properties of operations that
they are familiar with from previous grades’ work with numbers — generalizing arithmetic in the process.
page 3 of 3