MA.6.AR.1.4

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
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

This benchmark is part of these courses.
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

Alternate version of this benchmark for students with significant cognitive disabilities.
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

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessments

Property Combinations:

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

Type: Formative Assessment

Generating Equivalent Expressions:

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

Type: Formative Assessment

Equivalent Expressions:

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

Type: Formative Assessment

Equivalent Exponents:

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

Type: Formative Assessment

Equal Sides, Equivalent Expressions:

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

Type: Formative Assessment

Identifying Equivalent Expressions:

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

Type: Formative Assessment

Associative and Commutative Expressions:

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

Type: Formative Assessment

Lesson Plans

Order Matters:

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

Type: Lesson Plan

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.

Type: Lesson Plan

Expressions and Art:

Students will design a color-by-expression picture in order to practice evaluating algebraic expressions including substitution and order of operations.

Type: Lesson Plan

Equivalent Expressions with Candies:

In this lesson, students use small round candies and equation mats to explore the equivalency of pairs of expressions. Then they show pairs of expressions are equivalent using properties of operations. For those that are not equivalent, they provide a counterexample.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

Expressions, Phrases and Word Problems, Oh My!:

This lesson focuses on using appropriate mathematical language when reading or writing expressions, with students applying this knowledge to translate written phrases into algebraic expressions and vice versa. Students will analyze word problems for key words and write the representative expressions.

Type: Lesson Plan

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.

Type: Lesson Plan

You Can Never Have Too Many Shoes!:

This lesson teaches Least Common Multiples.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

The Distributive Property:

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

Type: Lesson Plan

Original Student Tutorial

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.

Type: Original Student Tutorial

Problem-Solving Tasks

Rectangle Perimeter 2:

Students are asked to determine if given expressions are equivalent.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Tutorials

Applying Arithmetic Properties with Negative Numbers:

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

Type: Tutorial

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.

Type: Tutorial

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.

Type: Tutorial

How to Use the Distributive Property with Variables:

Learn how to apply the distributive property to algebraic expressions.

Type: Tutorial

How to Simplify an Expression by Combining Like Terms:

Students will simplify an expression by combining like terms.  

Type: Tutorial

How to Combine Like Terms:

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

Type: Tutorial

Combining Like Terms Introduction:

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

 

Type: 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.

Type: Tutorial

Video/Audio/Animation

How to Use the Distributive Property to Factor Out the Greatest Common Factor:

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

Type: Video/Audio/Animation

MFAS Formative Assessments

Associative and Commutative Expressions:

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

Equal Sides, Equivalent Expressions:

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

Equivalent Exponents:

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

Equivalent Expressions:

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

Generating Equivalent Expressions:

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

Identifying Equivalent Expressions:

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

Property Combinations:

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

Computer Science Original Student Tutorials

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.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Original Student Tutorial

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.

Type: Original Student Tutorial

Problem-Solving Tasks

Rectangle Perimeter 2:

Students are asked to determine if given expressions are equivalent.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Tutorials

Applying Arithmetic Properties with Negative Numbers:

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

Type: Tutorial

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.

Type: Tutorial

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.

Type: Tutorial

How to Use the Distributive Property with Variables:

Learn how to apply the distributive property to algebraic expressions.

Type: Tutorial

How to Simplify an Expression by Combining Like Terms:

Students will simplify an expression by combining like terms.  

Type: Tutorial

How to Combine Like Terms:

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

Type: Tutorial

Combining Like Terms Introduction:

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

 

Type: 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.

Type: Tutorial

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Tasks

Rectangle Perimeter 2:

Students are asked to determine if given expressions are equivalent.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task