MA.8.AR.2.1

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Solve multi-step linear equations in one variable, with rational number coefficients. Include equations with variables on both sides.

Clarifications

Clarification 1: Problem types include examples of one-variable linear equations that generate one solution, infinitely many solutions or no solution.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Algebraic Reasoning
Standard: Solve multi-step one-variable equations and inequalities.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Coefficient
  • Linear Equation
  • Rational Number

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 7, students wrote and solved two-step equations in one variable within a mathematical or real-world context, where all terms are rational numbers. In grade 8, students solve multi-step linear equations in one variable, with rational number coefficients, including equations with variables on both sides. In Algebra 1, students will write and solve linear equations in one variable in a real-world context, with rational number coefficients. 
  • In this benchmark, students work with linear equations, which is foundational for the work with both linear equations and nonlinear equations throughout all future mathematics courses.
  • Instruction includes the use of manipulatives, drawings, models, properties of operations and properties of equality.
    • Algebra Tiles
      3x-12=-2x+13 on Algebra Tiles.
    • Balance
      Balance
  • Problem types involve multi-step problems that require the use of the distributive property, combining like terms, and variables on both sides of the equation.
  • Since there are variables on both sides of the equation, instruction includes discovering that one-variable equations can result in three possible solution sets. The possible solutions are one solution, no solution or infinitely many solutions. This benchmark provides a foundation for  when students are working with systems of equations and two-variable equations.

 

Common Misconceptions or Errors

  • Students may incorrectly apply the distributive property by multiplying the monomial to only one of the terms in the parentheses. To address this misconception, emphasize that it is the distributive property of multiplication over addition to help support student understanding.
  • Students may incorrectly apply the rules of integer arithmetic as they distribute when working with the operations of negative numbers and applying the distributive property of multiplication over addition.
  • Students may incorrectly think that you will always need a variable that equals a constant as a solution. To address this misconception, provide examples that show a constant equal to a variable as a solution, a constant equal to a constant or a non-valid equality statement.

 

Strategies to Support Tiered Instruction

  • Teacher provides opportunities to use manipulatives to demonstrate using the distributive property as repeated addition of the given expression.
    • For example the expression 3(x − 4) can be represented as adding (x − 4) three times together.
      Algebra Tiles
  • Instruction includes support with relating that if the solution is in the form x = a, there is only one solution. If the solution is in the form a = a , there are infinitely many solutions. If the solution is in the form a = b, where a and b are different numbers, there are no solutions. Teacher co-creates a graphic organizer with examples of one, no solutions, and infinitely many solutions. Demonstrate using substitution to help students make sense of the solutions.
  • Teacher co-creates an anchor chart for multiplying negative integers for students that incorrectly apply the rules of negative integers as they distribute.
  • Teacher provides examples for students that need additional support for distributive property by using the area model (like the one shown below).
    2(x + 4)
    2(x + 4)
    2(x + 4) = (x + 4) + (x + 4) = 2x + 8
  • Instruction includes emphasizing that it is the distributive property of multiplication over addition to help support student understanding.

 

Instructional Tasks

Instructional Task 1 (MTR.1.1, MTR.4.1)
Part A. How many solutions does the equation, 2y + 7 = 7 + 2y have? Explain your reasoning to another student and justify your answer.
Part B. How many solutions does the equation, 2(y + 3) + 1 = 2(3.5 + y) have? Explain your reasoning to another student and justify your answer.
Part C. What do you notice about the equations in Part A and Part B?

Instructional Task 2 (MTR.1.1, MTR.4.1)
For each equation, state whether there is no solution, one solution, or infinitely many solutions. Explain your reasoning.
Table

 

Instructional Items

Instructional Item 1
Solve for x:
a. −3.5(10x − 2) = −176.75
b. 15(2x − 10) + 4x = − 3(15x + 4)

 

*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.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 - 2024, 2024 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.AR.2.AP.1a: Identify the steps to solve a given multi-step equation in one variable, with integers coefficients. Include equations with variables on both sides.
MA.8.AR.2.AP.1b: Solve multi-step equations in one variable, with integers coefficients. Include equations with variables on both sides.

Related Resources

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

Formative Assessments

Linear Equations - 2:

Students are asked to solve a linear equation in one variable with rational coefficients.

Type: Formative Assessment

Linear Equations - 1:

Students are asked to solve a linear equation in one variable with rational coefficients.

Type: Formative Assessment

Equation Prototypes:

Students are asked to write examples of equations with one solution, no solutions, and infinitely many solutions.

Type: Formative Assessment

Counting Solutions:

Students are asked to identify and explain whether given equations have one solution, no solutions, or infinitely many solutions.

Type: Formative Assessment

Linear Equations - 3:

Students are asked to solve a linear equation in one variable with rational coefficients and variables on both sides of the equation.

Type: Formative Assessment

Lesson Plans

How Will the Ninja Capture the Valuable Princess?:

This lesson allows students to solve two-step equations involving a mythological story line in which the princess (variable) is protected by a bodyguard (number added or subtracted in an equation) and by a protector (number multiplied or divided by the variable). The three characters live in a castle, surrounded by the moat (equal sign) and an innocent bystander that lives outside the castle (number on the opposite side of the variable). However, Ninjas are infiltrating the castle to steal the “valuable” princess. Using this story line, students must then decide who the Ninja must eliminate first to get to the princess. This lesson can also be used to solve equations with like terms on the same side and equations with the same variable on each side.

Type: Lesson Plan

Vertical Angles: Proof and Problem-Solving:

Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements.

Type: Lesson Plan

Parallel Lines:

Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel.

Type: Lesson Plan

What's My Function?:

Students will determine function rules that have been written on cards taped to their backs. They will suggest input values and peers will provide output values to help them determine their function. They will then graph their functions for additional practice.

Type: Lesson Plan

Angles, angles, everywhere!:

This is an introductory lesson that allows students to explore complementary and supplementary angles and their relationships. Students will measure the degrees for sets of angles. They will then use their knowledge of straight, right, obtuse, and acute angles to make connections to complementary, supplementary, adjacent, and vertical angles. This lesson provides opportunities for kinesthetic, auditory, and visual learning.

Type: Lesson Plan

In Whose Best Interest is Interest?:

Students will use simple interest to explore real world examples of interest rates. Students will explore loan rates, CD rates and compare benefits of different rates versus different terms of loans.

Type: Lesson Plan

Functions With Vertical Line Test!:

This is an introductory lesson on functions. Students will review how to graph ordered pairs and learn how to use a function table. In addition, students learn the difference between functions and nonfunctions and how to distinguish between the two.

Type: Lesson Plan

Company Charges:

In this lesson, students will learn how to write and solve linear equations that have one solution, infinitely many solutions, and no solutions. As the students decipher word problems, they will recognize which elements of equations affect the number of possible solutions. This lesson is guided by a PowerPoint presentation.

Type: Lesson Plan

The Ins and Outs of Polygons:

In this lesson, students will explore how to find the sum of the measures of the angles of a triangle, then use this knowledge to find the sum of the measures of angles of other polygons. They will also be able to find the sum of the exterior angles of triangles and other polygons. Using both concepts, students will be able to find missing measurements.

Type: Lesson Plan

The Speeding Ticket (Part 1: Solving Linear Equations with One Variable):

"The Speeding Ticket" lesson uses real world application to create and solve linear equations and tables with one variable numerically, verbally, and algebraically. The student will also learn the relationship between the independent and dependent variables.

Type: Lesson Plan

Method to My Mathness:

In this lesson, students will complete proof tables to justify the steps taken to solve multi-step equations. Justifications include mathematical properties and definitions..

Type: Lesson Plan

The Variable Stands Alone:

Students will practice and create problems solving linear equations that involve one solution, no solution, infinitely many solutions. There will be class discussion so students can verbalize their thoughts. In addition, students will create their own real-world problems that can be used for the next day’s extension exercise.

Type: Lesson Plan

What Are My Test Scores?:

Students will define what a function is at the beginning of class. They will activate prior knowledge by playing "Four Corners." After that, there will be an investigation of what a function is and how it relates to the total questions on an assessment and the point value for each question. Then, students will analyze different assessments. and present their findings. For homework, students will have practice problems.

Type: Lesson Plan

How Many Degrees?:

This lesson facilitates the discovery of a formula for the sum of the interior angles of a regular polygon. Students will draw all the diagonals from one vertex of various polygons to find how many triangles are formed. They will use this and their prior knowledge of triangles to figure out the sum of the interior angles. This will lead to the development of a formula for finding the sum of interior angles and the measure of one interior angle.

Type: Lesson Plan

Justly Justifying:

Students will review the properties used in solving simple equations through a quiz-quiz-trade activity. As a class, they will then associate these properties with individual steps in solving equations. The students will then participate in a Simultaneous Round Table to practice their justifications. Finish the lesson with a discussion on the different methods that students could use to acquire the correct answer. The following day, students will take a short quiz to identify misconceptions.

Type: Lesson Plan

Kissing “V”s:

This lesson uses a paper-cutting activity to teach vertical angles. The lesson provides examples in which students must solve equations to find missing measures.

Type: Lesson Plan

Solving Linear Equations in One Variable:

This lesson unit is intended to help you assess how well students are able to:

  • Solve linear equations in one variable with rational number coefficients.
  • Collect like terms.
  • Expand expressions using the distributive property.
  • Categorize linear equations in one variable as having one, none, or infinitely many solutions.

It also aims to encourage discussion on some common misconceptions about algebra.

Type: Lesson Plan

Original Student Tutorials

Multi-Step Equations: Part 5 How Many Solutions?:

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • [CURRENT TUTORIAL] Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 4 Putting it All Together:

Learn alternative methods of solving multi-step equations in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • [CURRENT TUTORIAL] Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-step Equations: Part 3 Variables on Both Sides:

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • [CURRENT TUTORIAL] Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 2 Distributive Property:

Explore how to solve multi-step equations using the distributive property in this interactive tutorial. 

This is part two of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • [CURRENT TUTORIAL] Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 1 Combining Like Terms:

Learn how to solve multi-step equations that contain like terms in this interactive tutorial. 

This is part one of five in a series on solving multi-step equations.

  • [CURRENT TUTORIAL] Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Justifiable Steps:

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Teaching Idea

Solving Equations using Zero Pairs:

Unlock an effective teaching strategy for teaching solving equations using zero pairs in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Problem-Solving Tasks

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

Tutorial

Solve a Consecutive Integer Problem Algebraically:

This video will show how to solve a consecutive integer problem.

 

Type: Tutorial

MFAS Formative Assessments

Counting Solutions:

Students are asked to identify and explain whether given equations have one solution, no solutions, or infinitely many solutions.

Equation Prototypes:

Students are asked to write examples of equations with one solution, no solutions, and infinitely many solutions.

Linear Equations - 1:

Students are asked to solve a linear equation in one variable with rational coefficients.

Linear Equations - 2:

Students are asked to solve a linear equation in one variable with rational coefficients.

Linear Equations - 3:

Students are asked to solve a linear equation in one variable with rational coefficients and variables on both sides of the equation.

Original Student Tutorials Mathematics - Grades 6-8

Multi-Step Equations: Part 1 Combining Like Terms:

Learn how to solve multi-step equations that contain like terms in this interactive tutorial. 

This is part one of five in a series on solving multi-step equations.

  • [CURRENT TUTORIAL] Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Multi-Step Equations: Part 2 Distributive Property:

Explore how to solve multi-step equations using the distributive property in this interactive tutorial. 

This is part two of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • [CURRENT TUTORIAL] Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Multi-step Equations: Part 3 Variables on Both Sides:

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • [CURRENT TUTORIAL] Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Multi-Step Equations: Part 4 Putting it All Together:

Learn alternative methods of solving multi-step equations in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • [CURRENT TUTORIAL] Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Multi-Step Equations: Part 5 How Many Solutions?:

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • [CURRENT TUTORIAL] Part 5: How Many Solutions?

 

Original Student Tutorials Mathematics - Grades 9-12

Justifiable Steps:

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

Student Resources

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

Original Student Tutorials

Multi-Step Equations: Part 5 How Many Solutions?:

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • [CURRENT TUTORIAL] Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 4 Putting it All Together:

Learn alternative methods of solving multi-step equations in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • [CURRENT TUTORIAL] Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-step Equations: Part 3 Variables on Both Sides:

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial. 

This is part five of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • [CURRENT TUTORIAL] Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 2 Distributive Property:

Explore how to solve multi-step equations using the distributive property in this interactive tutorial. 

This is part two of five in a series on solving multi-step equations.

  • Click HERE to open Part 1: Combining Like Terms
  • [CURRENT TUTORIAL] Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Multi-Step Equations: Part 1 Combining Like Terms:

Learn how to solve multi-step equations that contain like terms in this interactive tutorial. 

This is part one of five in a series on solving multi-step equations.

  • [CURRENT TUTORIAL] Part 1: Combining Like Terms
  • Click HERE to open Part 2: The Distributive Property
  • Click HERE to open Part 3: Variables on Both Sides
  • Click HERE to open Part 4: Putting It All Together
  • Click HERE to open Part 5: How Many Solutions?

 

Type: Original Student Tutorial

Justifiable Steps:

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

Problem-Solving Tasks

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

Tutorial

Solve a Consecutive Integer Problem Algebraically:

This video will show how to solve a consecutive integer problem.

 

Type: Tutorial

Parent Resources

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

Problem-Solving Tasks

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task