# MA.7.AR.2.2

Write and solve two-step equations in one variable within a mathematical or real-world context, where all terms are rational numbers.

### Clarifications

Clarification 1: Instruction focuses the application of the properties of equality. Refer to Properties of Operations, Equality and Inequality (Appendix D).

Clarification 2: Instruction includes equations in the forms px±q=r and p(x±q)=r, where p, q and r are specific rational numbers.

Clarification 3: Problems include linear equations where the variable may be on either side of the equal sign.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Equation
• Rational Number

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

Students are building on their ability to write and solve one-step equations in grade 6 with an emphasis on operations with linear expressions being critical as students then write and solve two-step equations in grade 7 and multi-step linear equations in grade 8 (MTR.5.1).
• Instruction includes real-world contexts as well as linear equations where the variable may be on either side of the equal sign (MTR.7.1).
• Instruction includes students verbalizing or writing the Properties of Operations and Properties of Equality (see Appendix D) used at each step to their solution.
• Use models or manipulatives, such as algebra tiles, bar diagrams or balances, to conceptualize equations (MTR.2.1).
• Algebra Tiles

• Bar Diagrams

• Balances

• Avoid a particular order when solving and allow students to proceed in multiple ways that are mathematically accurate.
• For example, in the equation 4($x$ + 7) = 12, students may choose to divide both sides of the equation by 4 or use the Distributive Property with the 4. Compare the various strategies and ask students to determine which will be most efficient given different problem stems (MTR.3.1).

### Common Misconceptions or Errors

• Some students may incorrectly use the addition and subtraction properties of equality on the same side of the equal sign while solving an equation. To address this misconception, use manipulatives such as balances, algebra tiles or bar diagrams to show the balance between the two sides of an equation (MTR.2.1).
• Students may incorrectly identify the constants and the coefficients within a real-world context of the problem.

### Strategies to Support Tiered Instruction

• Teacher provides opportunities for students to practice solving equations using the addition and subtraction properties of equality using an interactive computer equation balance, manipulatives and other visual representations.
• Teacher provides support for students in identifying the coefficients and constants within a real-world context of the problem. Present students with examples of real-world problems that can be solved with equations.
• For example, Cameron’s fish tank can hold 12 gallons of water and he adds 2.5 gallons of water a minute. If there are already 3.4 gallons of water in the tank, for how many minutes can Cameron fill his tank without overflowing?
• Teacher provides opportunities for students to comprehend the context or situation by engaging in questions (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• What do you know from the problem?
• What is the problem asking you to find?
• Can you create a visual model to help you understand or see patterns in your problem?
• Teacher provides opportunities for students to use algebra tiles to co-solve provided equations with the teacher without the need of writing the equation first.
• Teacher provides opportunities for students to co-write an algebraic equation with the teacher without requiring students to solve the equation.
• Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?
• Teacher models the use manipulatives such as balances, algebra tiles or bar diagrams to show the balance between the two sides of an equation.

A plumber has been called in to replace a broken kitchen sink. The material needed costs \$341.25 and the total expected cost of the job is \$424.09. How many hours will the plumber need to work in order to get the job completed?
• Part A. What questions would need to be answered to approach this problem? Is there enough information given to solve the problem? Why or why not?
• Part B. The average rate for a plumber in Florida is \$20.71 per hour. Write and solve an equation to determine how many hours the plumber will be working.

The length of the rectangle is twice its width. The perimeter of the rectangle totals 45 feet. What is the width of the rectangle?

### Instructional Items

Instructional Item 1
What is the exact value of $x$ in the equation $\frac{\text{7}}{\text{9}}$ = $\frac{\text{2}}{\text{3}}$$x$ − 7?

Instructional Item 2
What is the value of $z$ in the equation 5.6(3$z$ − 2) = 11?

*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.
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (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))
7812020: Access M/J Grade 7 Mathematics (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.7.AR.2.AP.2a: Set up two-step equations in one variable based on real-world problems.
MA.7.AR.2.AP.2b: Solve two-step equations in one variable based on real-world problems, where all terms have positive integer coefficients.

## Related Resources

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

## Formative Assessments

Solve Equations:

Students are asked to solve two equations involving rational numbers.

Type: Formative Assessment

Write and Solve an Equation:

Students are asked to write and solve a two-step equation to model the relationship among variables in a given scenario.

Type: Formative Assessment

Squares:

Students are asked to write and solve an equation of the form p(x + q) = r in the context of a problem about the perimeter of a square.

Type: Formative Assessment

Algebra or Arithmetic?:

Students are asked to compare an arithmetic solution to an algebraic solution of a word problem.

Type: Formative Assessment

## Lesson Plans

A Florida Vacation:

Students will calculate sales tax to plan a family vacation budget. Through collaborative learning activities and discussions, students will understand the concept of sales tax as a civic responsibility and recognize the importance of considering sales tax in their financial planning to contribute to their community’s public service and infrastructure in this integrated lesson plan.

Type: Lesson Plan

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

Lost in Translation? Verbal and Algebraic Representations of Expressions and Equations:

This lesson is designed to assist in teacher facilitation of student understanding related to the concept of translating between verbal and algebraic two-step equations.

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

Relating Surface Area and Volume:

Students will recognize that while the surface area may change, the volume can remain the same. This lesson is enhanced through the multimedia CPALMS Perspectives Video, which introduces students to the relationship between surface area and volume.

Type: Lesson Plan

## Original Student Tutorials

Professor E. Qual Part 2: Two-Step Equations & Rational Numbers:

Practice solving and checking two-step equations with rational numbers in this interactive tutorial.

This is part 2 of the two-part series on two-step equations. Click HERE to open Part 1.

Type: Original Student Tutorial

Professor E. Qual Part 1: 2 Step Equations:

Professor E. Qual will teach you how to solve and check two-step equations in this interactive tutorial.

This is part 1 of a two-part series about solving 2-step equations. Click HERE to open Part 2.

Type: Original Student Tutorial

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Water Flow Modeling for Archeology Research:

Submerge yourself in math as a hydrogeologist describes calculations used to investigate water flow questions related to ancient shell rings.

Type: Perspectives Video: Expert

## Perspectives Video: Teaching Ideas

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

Programming Mathematics: Algebra, and Variables to control Open-source Hardware:

If you are having trouble understanding variables, this video might help you see the light.

Type: Perspectives Video: Teaching Idea

## Tutorial

Basic Linear Equation Word Problem:

This video shows how to construct and solve a basic linear equation to solve a word problem.

Type: Tutorial

## Video/Audio/Animations

Solving Motion Problems with Linear Equations:

Based upon the definition of speed, linear equations can be created which allow us to solve problems involving constant speeds, time, and distance.

Note: This video exceeds basic expectations for the mathematical concept(s) at this grade level. The video is intended for students who have demonstrated mastery within the scope of instruction who may be ready for a more rigorous extension of the mathematical concept(s). As with all materials, ensure to gauge the readiness of students or adapt according to student's needs prior to administration.

Type: Video/Audio/Animation

Solving Problems with Linear Equations:

The video explains the process of creating linear equations to solve real-world problems.

Type: Video/Audio/Animation

## MFAS Formative Assessments

Algebra or Arithmetic?:

Students are asked to compare an arithmetic solution to an algebraic solution of a word problem.

Solve Equations:

Students are asked to solve two equations involving rational numbers.

Squares:

Students are asked to write and solve an equation of the form p(x + q) = r in the context of a problem about the perimeter of a square.

Write and Solve an Equation:

Students are asked to write and solve a two-step equation to model the relationship among variables in a given scenario.

## Original Student Tutorials Science - Grades K-8

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

## Original Student Tutorials Mathematics - Grades 6-8

Professor E. Qual Part 1: 2 Step Equations:

Professor E. Qual will teach you how to solve and check two-step equations in this interactive tutorial.

This is part 1 of a two-part series about solving 2-step equations. Click HERE to open Part 2.

Professor E. Qual Part 2: Two-Step Equations & Rational Numbers:

Practice solving and checking two-step equations with rational numbers in this interactive tutorial.

This is part 2 of the two-part series on two-step equations. Click HERE to open Part 1.

## Student Resources

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

## Original Student Tutorials

Professor E. Qual Part 2: Two-Step Equations & Rational Numbers:

Practice solving and checking two-step equations with rational numbers in this interactive tutorial.

This is part 2 of the two-part series on two-step equations. Click HERE to open Part 1.

Type: Original Student Tutorial

Professor E. Qual Part 1: 2 Step Equations:

Professor E. Qual will teach you how to solve and check two-step equations in this interactive tutorial.

This is part 1 of a two-part series about solving 2-step equations. Click HERE to open Part 2.

Type: Original Student Tutorial

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

## Tutorial

Basic Linear Equation Word Problem:

This video shows how to construct and solve a basic linear equation to solve a word problem.

Type: Tutorial

## Video/Audio/Animations

Solving Motion Problems with Linear Equations:

Based upon the definition of speed, linear equations can be created which allow us to solve problems involving constant speeds, time, and distance.

Note: This video exceeds basic expectations for the mathematical concept(s) at this grade level. The video is intended for students who have demonstrated mastery within the scope of instruction who may be ready for a more rigorous extension of the mathematical concept(s). As with all materials, ensure to gauge the readiness of students or adapt according to student's needs prior to administration.

Type: Video/Audio/Animation

Solving Problems with Linear Equations:

The video explains the process of creating linear equations to solve real-world problems.

Type: Video/Audio/Animation

## Parent Resources

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