Standard #: MA.6.AR.1.4


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Apply the properties of operations to generate equivalent algebraic expressions with integer coefficients.


Examples


Example: The expression 5(3x+1) can be rewritten equivalently as 15x+5.

Example: If the expression 2x+3x represents the profit the cheerleading team can make when selling the same number of cupcakes, sold for $2 each, and brownies, sold for $3 each. The expression 5x can express the total profit.



Clarifications


Clarification 1: Properties include associative, commutative and distributive.

Clarification 2: Refer to Properties of Operations, Equality and Inequality (Appendix D).



General Information

Subject Area: Mathematics (B.E.S.T.)
Grade: 6
Strand: Algebraic Reasoning
Standard: Apply previous understanding of arithmetic expressions to algebraic expressions.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Coefficient 
  • Distributive Property
  • Expression
  • Integer

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 5, students evaluated multi-step numerical expressions using order of operations including parentheses, whole numbers, decimals and fractions. In grade 6, variable terms are introduced to create algebraic expressions with whole number coefficients. In grade 7, rational number coefficients will be used in variable expressions.
  • Instruction focuses on students using the properties of operations to generate equivalent algebraic expressions.
  • Within this benchmark, all variables will have an exponent value of 1. Laws of exponents are introduced in grade 7.
  • Students should understand if values are equivalent, they are worth the same amount.
    • For example, a 1 dollar bill, 4 quarters, 10 dimes, 20 nickels, 2 half-dollars or 100 pennies have equal values and can be represented as 1b = 4q, 1b = 10d, 1b = 20n, 1b = 2h or 1b = 100p, respectively.
  • Instruction includes generating equivalent expressions and comparing with other students’ solutions. Discussion focuses around looking for similarities and relationships between students’ approaches (MTR.4.1).
  • Instruction includes the use algebra tiles or other concrete representations to provide visual representation of the properties of operations.
    Properties of operations
  • Students should be encouraged to generate equivalent expressions in ways that are mathematically accurate and make sense to the individual. Students should be able to justify their process using properties of operations.
  • Instruction includes having expressions with more than one variable as well as nested grouping symbols.
    • Expression with more than one variable
      −4x + 3x − 2y
    • Nested grouping symbols
      −2[3(x − 3)]

 

Common Misconceptions or Errors

  • Students may incorrectly think there is only one right answer or equivalent value to be found. Although the most simplified version of an expression is an equivalent expression, it may not be the only equivalent expression.
  • Students may incorrectly perform operations involving nested grouping symbols. Students can be taught or reminded that each grouping symbol is treated similarly to parentheses.

 

Strategies to Support Tiered Instruction

  • Instruction includes providing students with algebra tiles and two different expressions. Students should represent the provided expressions with the algebra tiles and determine if the two expressions are equivalent or not by comparing the number of each tile.
  • Teacher provides opportunities for students to use algebra tiles to represent various given expressions, manipulate the tiles to show the expression a different way and then write the corresponding expression.
  • Instruction includes building a foundation for the properties of operations by modeling the associative, commutative, and distributives properties with algebra tiles for numeric and algebraic expressions and allowing the students to manipulate the algebra tiles as well.
  • Instruction includes providing students with two different expressions already represented with algebra tiles and then allowing the students to manipulate the algebra tiles to determine if they are equivalent. Students should be allowed to justify their reasoning for why the expressions are equivalent, or not.
  • Teacher reminds students that each grouping symbol is treated similarly to parentheses.

 

Instructional Tasks

Instructional Task 1 (MTR.2.1, MTR.5.1)

Look at the following expressions.

  • Part A. Simplify the expressions shown above.
  • Part B. Can any grouping symbols be removed from the expressions without changing the value of the expression?

 

Instructional Task 2 (MTR.2.1)

How can you show the two expressions, 2(x + 5) and 2(x) + 2(5), are equivalent? Explain.

 

Instructional Items

Instructional Item 1 

Tamika is selling chocolate candy bars for $1 and bags of popcorn for $3 at the school fair. She sells y candy bars and (y − 7) bags of popcorn. The expression 1 y + 3(y−7) represents Tamika’s total sales. Rewrite the expression in a different but equivalent form.

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.



Related Courses

Course Number1111 Course Title222
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))


Related Access Points

Access Point Number Access Point Title
MA.6.AR.1.AP.4 Use tools or models to combine like terms in an expression with no more than four operations.


Related Resources

Formative Assessments

Name Description
Property Combinations

Students are asked to identify expressions equivalent to a given expression and justify their responses.

Generating Equivalent Expressions

Students are asked to write equivalent expressions using the Distributive Property.

Equivalent Expressions

Students are asked to determine if pairs of expressions are equivalent and to justify their responses.

Equivalent Exponents

Students are asked to identify expressions equivalent to a given exponential expression and justify their responses.

Equal Sides, Equivalent Expressions

Students are asked to generate and justify an expression equivalent to a given one using the properties of operations.

Identifying Equivalent Expressions

Students are asked to identify expressions equivalent to a given expression and justify their responses.

Associative and Commutative Expressions

Students are asked to write expressions equivalent to a given one by using the Associative and Commutative Properties.

Lesson Plans

Name Description
Power of a Right Triangle: Day 1 Proving Pythagoras

In this first of three lessons on the Pythagorean Theorem students work to prove the Pythagorean theorem and verify that the theorem works.

Extending the Distributive Property

In this lesson, students will build upon their arithmetic experiences with the distributive property to equate algebraic expressions through a series of questions related to real world situations and the use of manipulatives. Activities include the use of Algebra Tiles for moving the concrete learner to the abstract level and the use of matching cards.
This is an introductory lesson that only includes producing equivalent expressions such as 3(2 + x) = 6 + 3x.

Collectively Collecting

In this lesson, students will examine and experience collecting like terms through an analogy to real world situations and the use of manipulatives. Activities include the use of Algebra Tiles for transitioning a concrete experience to the abstract level and a card-matching activity.

Can You Find the Relationship?

In this lesson students will first define in their own words what the greatest common factor (GCF) and least common multiple (LCM) mean. They will take this understanding and apply it to solving GCF and LCM word problems. Students will then illustrate their understanding by creating posters based on their word problems. There are examples of different types of methods, online games, a rubric, and a power point to summarize this two-day lesson.

Have You Met Your Match?-Understanding Equivalent Expressions

In this lesson, students will use the properties of operations to generate and identify equivalent algebraic expressions. Students will apply their prior knowledge of the distributive property and combining like terms to create equivalent algebraic expressions. The hands-on memory "match" game will engage students and support student understanding of algebraic thinking.

Digesting the Distributive Property

This lesson will show the student how to use the distributive property to express a sum of two whole numbers 1-100.

You Can Never Have Too Many Shoes!

This lesson teaches Least Common Multiples.

Finding the Greatest Crush Factor

This lesson uses a real-life approach to exploring the use of Greatest Common Factors (GCF). The students will utilize math practice standards as they analyze math solutions and explain their own solutions.

Factoring out the Greatest

This lesson teaches students how to find the GCF and LCM by factoring. This is a different method than is normally seen in textbooks. This method easily leads to solving GCF word problems and using the distributive property to express a sum of two whole numbers.

Power of a Right Triangle: Day 1 Proving Pythagoras

In this first of three lessons on the Pythagorean Theorem students work to prove the Pythagorean theorem and verify that the theorem works.

Order Matters

Students will analyze a Scratch program and compare its computerized algorithm to the mathematical order of operations, in this lesson plan.

The Distributive Property

Introductory lesson on the distributive property using word problems as context for area models.

Original Student Tutorial

Name Description
MacCoder's Farm Part 4: Repeat Loops

Explore computer coding on the farm by using IF statements and repeat loops to evaluate mathematical expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Problem-Solving Tasks

Name Description
Rectangle Perimeter 2

Students are asked to determine if given expressions are equivalent.

Rectangle Perimeter 3

The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.

Distance to School

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

Equivalent Expressions

Students are asked to use properties of operations to match expressions that are equivalent and to write equivalent expressions for any expressions that do not have a match.

Tutorials

Name Description
Applying Arithmetic Properties with Negative Numbers

In this video, you will practice using arithmetic properties with integers to determine if expressions are equivalent.

The Distributive Law of Multiplication over Addition

Learn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property.

The Distributive Law of Multiplication over Subtraction

Learn how to apply the distributive property of multiplication over subtraction. This is sometimes just called the distributive property or distributive law.

How to Use the Distributive Property with Variables

Learn how to apply the distributive property to algebraic expressions.

How to Simplify an Expression by Combining Like Terms

Students will simplify an expression by combining like terms.  

How to Combine Like Terms

This tutorial is an explanation on how to combine like terms in algebra. 

Combining Like Terms Introduction

This is an introduction to combining like terms in this tutorial.

 

Why aren't we using the multiplication sign?

Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here.

Video/Audio/Animation

Name Description
How to Use the Distributive Property to Factor Out the Greatest Common Factor

Learn how to apply the distributive property to factor numerical expressions.

Student Resources

Original Student Tutorial

Name Description
MacCoder's Farm Part 4: Repeat Loops:

Explore computer coding on the farm by using IF statements and repeat loops to evaluate mathematical expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Problem-Solving Tasks

Name Description
Rectangle Perimeter 2:

Students are asked to determine if given expressions are equivalent.

Rectangle Perimeter 3:

The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.

Distance to School:

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

Equivalent Expressions:

Students are asked to use properties of operations to match expressions that are equivalent and to write equivalent expressions for any expressions that do not have a match.

Tutorials

Name Description
Applying Arithmetic Properties with Negative Numbers:

In this video, you will practice using arithmetic properties with integers to determine if expressions are equivalent.

The Distributive Law of Multiplication over Addition:

Learn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property.

The Distributive Law of Multiplication over Subtraction:

Learn how to apply the distributive property of multiplication over subtraction. This is sometimes just called the distributive property or distributive law.

How to Use the Distributive Property with Variables:

Learn how to apply the distributive property to algebraic expressions.

How to Simplify an Expression by Combining Like Terms:

Students will simplify an expression by combining like terms.  

How to Combine Like Terms:

This tutorial is an explanation on how to combine like terms in algebra. 

Combining Like Terms Introduction:

This is an introduction to combining like terms in this tutorial.

 

Why aren't we using the multiplication sign?:

Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here.



Parent Resources

Problem-Solving Tasks

Name Description
Rectangle Perimeter 2:

Students are asked to determine if given expressions are equivalent.

Rectangle Perimeter 3:

The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.

Distance to School:

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

Equivalent Expressions:

Students are asked to use properties of operations to match expressions that are equivalent and to write equivalent expressions for any expressions that do not have a match.



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