# MA.912.AR.1.2

Rearrange equations or formulas to isolate a quantity of interest.

### Examples

Algebra 1 Example: The Ideal Gas Law PV = nRT can be rearranged as  to isolate temperature as the quantity of interest.

Example: Given the Compound Interest formula , solve for P

Mathematics for Data and Financial Literacy Honors Example: Given the Compound Interest formula , solve for t.

### Clarifications

Clarification 1: Instruction includes using formulas for temperature, perimeter, area and volume; using equations for linear (standard, slope-intercept and point-slope forms) and quadratic (standard, factored and vertex forms) functions.

Clarification 2: Within the Mathematics for Data and Financial Literacy course, problem types focus on money and business.

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

## Benchmark Instructional Guide

• Equation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students isolated variables in one-variable linear equations and one-variable quadratic equations in the form $x$= $p$ and  $x$= $q$. In Algebra I, students isolate a variable or quantity of interest in equations and formulas. Equations and variables will focus on linear, absolute value and quadratic in Algebra I. In later courses, students will highlight a variable or quantity of interest for other types of equations and formulas, including exponential, logarithmic and trigonometric.
• Instruction includes making connections to inverse arithmetic operations (refer to Appendix D) and solving one-variable equations.
• Instruction includes justifying each step while rearranging an equation or formula.
• For example, when rearranging $A$ = $P$(1 + $\frac{\text{r}}{\text{n}}$)$n$$t$   for $P$, it may be helpful for students to highlight the quantity of interest with a highlighter, so students remain focused on that quantity for isolation purposes. It may also be helpful for students to identify factors, or other parts of the equations.

### Common Misconceptions or Errors

• Students may not have mastered the inverse arithmetic operations.
• Students may be frustrated because they are not arriving at a numerical value as their solution. Remind students that they are rearranging variables that can be later evaluated to a numerical value.
• Having multiple variables and no values may confuse students and make it difficult for them to see the connections between rearranging a formula and solving a one-variable equation.

### Strategies to Support Tiered Instruction

• Instruction includes doing a side-by-side comparison of solving a multistep equation with rearranging equations and formulas. The teacher should allow students time to understand that the steps in solving both equations are the same.
• For example, solve both equations and note the similarities in solving both types of equations.

• Teacher provides a chart for students to use as a study guide or to copy in their interactive notebook.
• For example, inverse operations chart below.

• Part A. Given the equation $a$$x$2 + $b$$x$ + $c$ = 0, solve for $x$
• Part B. Share your strategy with a partner. What do you notice about the new equation(s)?

• Part A. Given the equation $A$$x$ + $B$$y$ = $C$, solve for $B$
• Part B. Given the equation 7$x$ − 6$y$ = 24, determine the $x$- and $y$-intercepts.
• Part C. What do you notice between Part A and Part B?

### Instructional Items

Instructional Item 1
• Solve for $x$ in the equation 3$x$ + $y$ = 5$x$$x$$y$

Instructional Item 2
• The formula $d$ relating to the translational of motion, where d represents distance, $v$0 represents initial velocity, $v$t represents final velocity, and $t$ represents time. Rearrange the formula to isolate final velocity.

Instructional Item 3
• The area $A$ of a sector of a circle with radius $r$ and angle-measure $S$ (in degrees) is given by  solve for the radius $r$.

*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.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.AR.1.AP.2: Rearrange an equation or a formula for a specific variable.

## Related Resources

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

## Formative Assessments

Solving Formulas for a Variable:

Students are given the slope formula and the slope-intercept equation and are asked to solve for specific variables.

Type: Formative Assessment

Solving Literal Equations:

Students are given three literal equations, each involving three variables and either addition or subtraction, and are asked to solve each equation for a specific variable.

Type: Formative Assessment

Literal Equations:

Students are given three literal equations, each involving three variables and either multiplication or division, and are asked to solve each equation for a specific variable.

Type: Formative Assessment

Solving a Literal Linear Equation:

Students are given a literal linear equation and asked to solve for a specific variable.

Type: Formative Assessment

Surface Area of a Cube:

Students are asked to solve the formula for the surface area of a cube for e, the length of an edge of the cube.

Type: Formative Assessment

Rewriting Equations:

Students are given a literal equation involving four variables and are asked to solve for the variable in the quadratic term.

Type: Formative Assessment

## Lesson Plans

Filled to Capacity!:

This is a lesson where students investigate, compare, dissect, and use the relationship between volume of a cone and cylinder with equal corresponding dimensions.

Type: Lesson Plan

My Geometry Classroom:

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

Free Fall Clock and Reaction Time!:

This will be a lesson designed to introduce students to the concept of 9.81 m/s2 as a sort of clock that can be used for solving all kinematics equations where a = g.

Type: Lesson Plan

Students will investigate the formula for the volume of a pyramid and/or cone and use those formulas to calculate the volume of other solids. The students will have hands-on discovery working with hollow Geometric Solids that they fill with dry rice, popcorn, or another material.

Type: Lesson Plan

Ranking Sports Players (Quadratic Equations Practice):

In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.

Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.

Type: Lesson Plan

My Favorite Slice:

The lesson introduces students to sectors of circles and illustrates ways to calculate their areas. The lesson uses pizzas to incorporate a real-world application for the of area of a sector. Students should already know the parts of a circle, how to find the circumference and area of a circle, how to find an arc length, and be familiar with ratios and percentages.

Type: Lesson Plan

Efficient Storage:

The topic of this MEA is work and power. Students will be assigned the task of hiring employees to complete a given task. In order to make a decision as to which candidates to hire, the students initially must calculate the required work. The power each potential employee is capable of, the days they are available to work, the percentage of work-shifts they have missed over the past 12 months, and the hourly pay rate each worker commands will be provided to assist in the decision process. Full- and/or part-time positions are available. Through data analysis, the students will need to evaluate which factors are most significant in the hiring process. For instance, some groups may prioritize speed of work, while others prioritize cost or availability/dependability.

Type: Lesson Plan

Graphing vs. Substitution. Which would you choose?:

Students will solve multiple systems of equations using two methods: graphing and substitution. This will help students to make a connection between the two methods and realize that they will indeed get the same solution graphically and algebraically.  Students will compare the two methods and think about ways to decide which method to use for a particular problem. This lesson connects prior instruction on solving systems of equations graphically with using algebraic methods to solve systems of equations.

Type: Lesson Plan

Math in Mishaps:

Students will explore how percentages, proportions, and solving for unknowns are used in important jobs. This interactive activity will open their minds and address the question, "When is this ever used in real life?"

Type: Lesson Plan

Don't Take it so Literal:

The purpose of this lesson is to have students practice manipulation of literal equations to solve for the variable of interest. A literal equation is an equation that has more than variable (letter).

Type: Lesson Plan

Survey Says... We're Using TRIG!:

This lesson is meant as a review after being taught basic trigonometric functions. It will allow students to see and solve problems from a real-world setting. The Perspectives video presents math being used in the real-world as a multimedia enhancement to this lesson. Students will find this review lesson interesting and fun.

Type: Lesson Plan

## Perspectives Video: Professional/Enthusiast

<p>Have a need for speed? Get out your spreadsheet! Race car drivers use algebraic formulas and spreadsheets to optimize car performance.</p>

Type: Perspectives Video: Professional/Enthusiast

Equations and Formulas:

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

## Tutorial

Solving a literal equation:

Students will learn to solve a literal equation.

Type: Tutorial

## Video/Audio/Animation

Solving Literal Equations:

Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.

Type: Video/Audio/Animation

## STEM Lessons - Model Eliciting Activity

Efficient Storage:

The topic of this MEA is work and power. Students will be assigned the task of hiring employees to complete a given task. In order to make a decision as to which candidates to hire, the students initially must calculate the required work. The power each potential employee is capable of, the days they are available to work, the percentage of work-shifts they have missed over the past 12 months, and the hourly pay rate each worker commands will be provided to assist in the decision process. Full- and/or part-time positions are available. Through data analysis, the students will need to evaluate which factors are most significant in the hiring process. For instance, some groups may prioritize speed of work, while others prioritize cost or availability/dependability.

Ranking Sports Players (Quadratic Equations Practice):

In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.

Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.

## MFAS Formative Assessments

Literal Equations:

Students are given three literal equations, each involving three variables and either multiplication or division, and are asked to solve each equation for a specific variable.

Rewriting Equations:

Students are given a literal equation involving four variables and are asked to solve for the variable in the quadratic term.

Solving a Literal Linear Equation:

Students are given a literal linear equation and asked to solve for a specific variable.

Solving Formulas for a Variable:

Students are given the slope formula and the slope-intercept equation and are asked to solve for specific variables.

Solving Literal Equations:

Students are given three literal equations, each involving three variables and either addition or subtraction, and are asked to solve each equation for a specific variable.

Surface Area of a Cube:

Students are asked to solve the formula for the surface area of a cube for e, the length of an edge of the cube.

## Student Resources

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

Equations and Formulas:

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

## Tutorial

Solving a literal equation:

Students will learn to solve a literal equation.

Type: Tutorial

## Video/Audio/Animation

Solving Literal Equations:

Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.

Type: Video/Audio/Animation

## Parent Resources

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