# MA.6.AR.1.3

Evaluate algebraic expressions using substitution and order of operations.

### Examples

Evaluate the expression , where a=-1 and b=15.

### Clarifications

Clarification 1: Within this benchmark, the expectation is to perform all operations with integers.

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

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

• Base
• Coefficient
• Exponent
• Expression

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 5, students translated written real-world and mathematical descriptions into numerical expressions, evaluated multi-step numerical expressions used order of operations involving combinations of the four arithmetic operations and parentheses with whole numbers, decimals and fractions. They also determined and explained whether an equation involving any of the four operations is true or false. In grade 6, students will use substitution to evaluate algebraic expressions, including exponents and integer coefficients. The values being substituted will also be integers. This benchmark extends to grade 7 where students will evaluate more complex numerical expressions with rational coefficients, apply laws of exponents and generate equivalent linear expressions.
• Substitution is the process in which a symbol or variable is replaced by a given value. In prior grades students have found missing terms in equations and then substituted the value back in to check it they are correct.
• For example, students may have seen something such as 2+? =10 and decided that 8 is the value that should replace the ?.
• This prior experience can be used to connect prior student understanding with new learning. In grade 6, instead of seeing a symbol like a question mark or a box for a missing value, we use letters called variables.
• An algebraic expression is built from integer constants, variables and operations whereas an equation is the statement of two equivalent expressions. This benchmark is specifically addressing substitution.
• Depending on the given expression, students may see opportunities to start generating equivalent expressions before substituting the value of the variable(s). This is a way students can demonstrate flexible thinking and the understanding of patterns and structure in mathematical concepts (MTR.5.1).

### Common Misconceptions or Errors

• Students may incorrectly generate equivalent expressions using the order of operations.
• If more than one variable is present in a given expression, students may incorrectly substitute one value in for all given variables or apply the wrong value to each of the variables. To address this misconception, students can use colors (pens, pencils, markers) to keep track of which variable and location correspond to each given value.
• If students try to generate an equivalent expression before substituting integer values, they may try to combine unlike terms (constants with variables or unlike variable terms).

### Strategies to Support Tiered Instruction

• Instruction includes building a foundation for the meaning of substitution by introducing algebraic expressions in a single variable with an exponent of 1 and representing the expression with algebra tiles. Then the variable tile can be replaced with the appropriate number of unit tiles to represent the provided value and the expression can be evaluated. The algebraic expression should be represented simultaneously to draw connections between the concrete and abstract representations. Use this foundation to build towards using algebraic strategies to evaluate expressions that cannot be represented with algebra tiles.
• For example, when evaluating 3$b$ + 5 when $b$ = −4, students can represent this as shown below.
• Teacher provides opportunities for students to use different colored pencils to represent different variables and use the coordinating color to replace the variable with its assigned value before utilizing the order of operations to evaluate.
• For example, if evaluating the expression −5$a$2 + $c$, where $a$ = −3 and $c$ =−12, the teacher can color coordinate as shown below.
−5$a$2 + $c$ ; $a$ = −3 and $c$ = −12
−5(−3)2 + (−12)
−5(9) + (−12)
−45 + (−12)
−57
• Teacher models how students can use colors (pens, pencils, markers) to keep track of which variable and location correspond to each given value.

Instructional Task 1 (MTR.5.1)
To compute the perimeter of a rectangle you add the length, l, and width, w, and double this sum.
• Part A. Write an expression for the perimeter of a rectangle.
• Part B. Use the expression to find the perimeter of a rectangle with length of 15 feet and width of 8 feet.

Instructional Task 2 (MTR.6.1
To determine the distance traveled by a car you multiply the speed the car traveled by the amount of time the car was traveling at that speed. The scenario can be represented as st, where s is speed, and t is time. What is the distance traveled by a car that travels at an average speed of 75 miles per hour for 20 minutes?

### Instructional Items

Instructional Item 1
If x = 3, find 3x + 8.

Instructional Item 2
If = 8, find 1 p − 3.

Instructional Item 3
Evaluate the expression −5 a2 + c, where a=−3 and c= −12.

*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.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
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.3: Solve an expression using substitution with no more than two operations.

## Related Resources

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

## Formative Assessments

Substitution Resolution:

Students are asked to evaluate formulas for given values of the variables.

Type: Formative Assessment

Exponent Priorities:

Students are asked to evaluate numerical expressions with exponents.

Type: Formative Assessment

## Lesson Plans

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

Going The Distance:

This lesson provides a hands-on activity where students can apply solving one-step multiplication and division equations to a real-world problem. The lesson focuses on the relationship between distance, rate, and time. The students will also represent data on graphs and draw conclusions and make interpretations based on the graphs.

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

The "Power" of Exponents:

The students will use square tiles or cubes to explore the meaning of "squared" and "cubed" numbers. They will model squared and cubed numbers to build a strong foundation for working with exponents. The students will learn the connections between area and volume models and the related positive exponents. This is an introductory lesson.

Type: Lesson Plan

Using Nets to Find the Surface Area of Pyramids:

In this lesson, students will explore and apply the use of nets to find the surface area of pyramids.

Type: Lesson Plan

It's Hip 2b^2 eXponent^s:

Meaning of Exponents... Students will write and simplify numerical and algebraic expressions with natural number exponents. Bases are limited to positive integers.

Type: Lesson Plan

Sticks and Stones May Break My Bones:

In this Model Eliciting Activity, MEA, "Sticks and Stones May Break My Bones", teams of students work as forensic anthropologists and use equations to determine the height and gender of persons to whom a series of newly discovered bones may belong.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

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

## Original Student Tutorials

Order of Operations with Fractions:

Evaluate numerical expressions with fractions using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

Order of Operations with Decimals:

Evaluate numerical expressions with decimals using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

Order of Operations with Whole Numbers: Part 2:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

This is part 2 of a series on evaluating expressions with whole numbers.

Type: Original Student Tutorial

Order of Operations with Integers:

Evaluate numerical expressions with integers using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

MacCoder’s Farm Part 1: Declare Variables:

Explore computer coding on the farm by declaring and initializing variables in this interactive tutorial. You'll also get a chance to practice your long division skills.

Type: Original Student Tutorial

Order of Operations with Whole Numbers:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Teaching Idea

Unlock an effective teaching strategy for using patterns to help students make generalizations when adding integers in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

## Tutorials

How to Evaluate an Expression with Variables:

Learn how to evaluate an expression with variables using a technique called substitution.

Type: Tutorial

How to Evaluate Expressions with Two Variables:

This video demonstrates evaluating expressions with two variables.

Type: Tutorial

Thinking About the Changing Values of Variables and Expressions:

Explore how the value of an algebraic expression changes as the value of its variable changes.

Type: Tutorial

How to Evaluate an Expression Using Substitution:

In this example, we have a formula for converting a Celsius temperature to Fahrenheit.

Type: Tutorial

Introduction to Order of Operations:

Students will evaluate expressions using the order of operations.

Type: Tutorial

Order of Operations: PEMDAS:

Work through a challenging order of operations example with only positive numbers.

Type: Tutorial

Order of Operations :

Work through a challenging order of operations example with only positive numbers.

Type: Tutorial

Order of Operations :

This video will show how to evaluate expressions with exponents using the order of operations.

Type: Tutorial

Examples of evaluating variable expressions:

Examples of evaluating variable expressions

Type: Tutorial

How to evaluate an expression using substitution:

In this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!

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

## Virtual Manipulative

Order of Operations Quiz:

In this activity, students practice solving algebraic expressions using order of operations. The applet records their score so the student can track their progress. This activity allows students to practice applying the order of operations when solving problems. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

## STEM Lessons - Model Eliciting Activity

Sticks and Stones May Break My Bones:

In this Model Eliciting Activity, MEA, "Sticks and Stones May Break My Bones", teams of students work as forensic anthropologists and use equations to determine the height and gender of persons to whom a series of newly discovered bones may belong.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

## MFAS Formative Assessments

Exponent Priorities:

Students are asked to evaluate numerical expressions with exponents.

Substitution Resolution:

Students are asked to evaluate formulas for given values of the variables.

## Original Student Tutorials Mathematics - Grades 6-8

Order of Operations with Decimals:

Evaluate numerical expressions with decimals using the order of operations and properties of operations in this interactive tutorial.

Order of Operations with Fractions:

Evaluate numerical expressions with fractions using the order of operations and properties of operations in this interactive tutorial.

Order of Operations with Integers:

Evaluate numerical expressions with integers using the order of operations and properties of operations in this interactive tutorial.

Order of Operations with Whole Numbers:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

Order of Operations with Whole Numbers: Part 2:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

This is part 2 of a series on evaluating expressions with whole numbers.

## Computer Science Original Student Tutorials

MacCoder’s Farm Part 1: Declare Variables:

Explore computer coding on the farm by declaring and initializing variables in this interactive tutorial. You'll also get a chance to practice your long division skills.

## Student Resources

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

## Original Student Tutorials

Order of Operations with Fractions:

Evaluate numerical expressions with fractions using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

Order of Operations with Decimals:

Evaluate numerical expressions with decimals using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

Order of Operations with Whole Numbers: Part 2:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

This is part 2 of a series on evaluating expressions with whole numbers.

Type: Original Student Tutorial

Order of Operations with Integers:

Evaluate numerical expressions with integers using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

MacCoder’s Farm Part 1: Declare Variables:

Explore computer coding on the farm by declaring and initializing variables in this interactive tutorial. You'll also get a chance to practice your long division skills.

Type: Original Student Tutorial

Order of Operations with Whole Numbers:

Evaluate numerical expressions with whole numbers using the order of operations and properties of operations in this interactive tutorial.

Type: Original Student Tutorial

## Tutorials

How to Evaluate an Expression with Variables:

Learn how to evaluate an expression with variables using a technique called substitution.

Type: Tutorial

How to Evaluate Expressions with Two Variables:

This video demonstrates evaluating expressions with two variables.

Type: Tutorial

Thinking About the Changing Values of Variables and Expressions:

Explore how the value of an algebraic expression changes as the value of its variable changes.

Type: Tutorial

How to Evaluate an Expression Using Substitution:

In this example, we have a formula for converting a Celsius temperature to Fahrenheit.

Type: Tutorial

Introduction to Order of Operations:

Students will evaluate expressions using the order of operations.

Type: Tutorial

Order of Operations: PEMDAS:

Work through a challenging order of operations example with only positive numbers.

Type: Tutorial

Order of Operations :

Work through a challenging order of operations example with only positive numbers.

Type: Tutorial

Order of Operations :

This video will show how to evaluate expressions with exponents using the order of operations.

Type: Tutorial

How to evaluate an expression using substitution:

In this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!

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

## Virtual Manipulative

Order of Operations Quiz:

In this activity, students practice solving algebraic expressions using order of operations. The applet records their score so the student can track their progress. This activity allows students to practice applying the order of operations when solving problems. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

## Parent Resources

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