# Standard 1: Interpret and rewrite algebraic expressions and equations in equivalent forms.

General Information
Number: MA.912.AR.1
Title: Interpret and rewrite algebraic expressions and equations in equivalent forms.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.912.AR.1.AP.1
Identify a part(s) of an equation or expression and explain the meaning within the context of a problem.
MA.912.AR.1.AP.2
Rearrange an equation or a formula for a specific variable.
MA.912.AR.1.AP.3
Add, subtract and multiply polynomial expressions with integer coefficients.
MA.912.AR.1.AP.4
Divide a polynomial expression by a monomial expression with integer coefficients.
MA.912.AR.1.AP.5
Divide polynomial expressions using long division, synthetic division and algebraic manipulation where the denominator is a linear expression.
MA.912.AR.1.AP.6
Solve mathematical and/or real-world problems involving addition, subtraction, multiplication or division of polynomials with integer coefficients.
MA.912.AR.1.AP.7
MA.912.AR.1.AP.8
Select a polynomial expression as a product of polynomials with integer coefficients over the real or complex number system.
MA.912.AR.1.AP.9
Apply previous understanding of rational number operations with common denominators to add and subtract rational expressions.

## Related Resources

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

## 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

Interpreting Basic Tax:

Students are asked to interpret the parts of an equation used to calculate the total purchase price including tax of a set of items.

Type: Formative Assessment

Subtracting Polynomials:

Students are asked to find the difference of two polynomials and explain if the difference of polynomials will always result in a polynomial.

Type: Formative Assessment

Rewriting Numerical Expressions:

Students are asked to rewrite numerical expressions to find efficient ways to calculate.

Type: Formative Assessment

Multiplying Polynomials - 2:

Students are asked to multiply polynomials and explain if the product of two polynomials always results in a polynomial.

Type: Formative Assessment

Determine the Width:

Students are asked to find the width of a rectangle whose area and length are given as polynomials.

Type: Formative Assessment

Students are asked to identify equivalent quadratic expressions and to name the form in which each expression is written.

Type: Formative Assessment

Finding Missing Values:

Students are asked to rewrite quadratic expressions and identify parts of the expressions.

Type: Formative Assessment

Multiplying Polynomials - 1:

Students are asked to multiply polynomials and explain if the product of polynomials always results in a polynomial.

Type: Formative Assessment

Students are asked to find the sum of two polynomials and explain if the sum of polynomials always results in a polynomial.

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

Dot Expressions:

Students are asked to explain how parts of an algebraic expression relate to the number and type of symbols in a sequence of diagrams.

Type: Formative Assessment

What Happens?:

Students are asked to determine how the volume of a cone will change when its dimensions are changed.

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

Solving Quadratic Equations by Completing the square:

Students will model the process of completing the square (leading coefficient of 1) with algebra tiles, and then practice solving equations using the completing the square method. This lesson provides a discovery opportunity to conceptually see why the process of squaring half of the b value is considered completing the square.

Type: Lesson Plan

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

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

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

Following the Law of Sine:

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

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

CollegeReview.com:

This is a model-eliciting activity where students have been asked by a new website, CollegeReview.com, to come up with a system to rank various colleges based on five categories; tuition cost, social life, athletics, education, city population and starting salary upon graduation.

Type: Lesson Plan

Matching Trinomials with Area Models_2023:

Matching Trinomials with Area Models_2023

Type: Lesson Plan

Laying Tiles for Polynomial Addition and Subtraction Renovation:

In this lesson students will learn how to add and subtract polynomials.

Type: Lesson Plan

Taming the Behavior of Polynomials:

This lesson will cover sketching the graphs of polynomials while in factored form without the use of a calculator.

Type: Lesson Plan

How much is your time worth?:

This lesson is designed to help students solve real-world problems involving compound and continuously compounded interest. Students will also be required to translate word problems into function models, evaluate functions for inputs in their domains, and interpret outputs in context.

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.Â  Students will compare the two methods and think about ways to decide which method to use for a particular problem. This lessonÂ connects prior instruction on solving systems of equations graphically with usingÂ 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

Turning Tires Model Eliciting Activity:

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applyingÂ surface area concepts and algebra through modeling.

Type: Lesson Plan

A Rational Representation:

Students will tackle a real-world situation regarding starting a business that requires a rational equation to evaluate the plan. Students will determine a method and set of steps for solving rational equations and then revisit the original scenario and solve using the new method they have synthesized. Students will also explore, through collaborative learning structures, the concept of extraneous solutions.

Â

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

Using algebra tiles and tables to factor trinomials (less guess and check!):

This lesson addresses factoring when a = 1 and also when a > 1.Â  Part 1 (Algebra Tiles) contains examples when a = 1 and a >1. Part 2 (tables) contains only examples when
a > 1.Â

In part 1, students will use algebra tiles to visually see how to factor trinomials (a = 1 and a > 1). In part 2, they will use a 3 x 3 table (a > 1). This process makes students more confident when factoring because there is less guess and check involved in solving each problem.

Type: Lesson Plan

## Original Student Tutorials

Learn to use algebra tiles to model adding polynomial expressions with this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials with Greatest Common Factor:

Learn how to factor polynomials by finding their greatest common factor in this interactive tutorial.

Type: Original Student Tutorial

Identifying and Interpreting Parts of Quadratic Equations in Vertex Form:

Identify parts of quadratic equations in vertex form and interpret them in terms of the context they represent in this interactive tutorial.

Type: Original Student Tutorial

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Polynomials when "a" Does Not Equal 1, Snowflake Method:

Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method:

Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Diamond Game: Factoring Quadratics when a = 1:

Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Identifying Parts of Linear Expressions:

Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.

Type: Original Student Tutorial

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Introduction to Polynomials: Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click here to open Part 2.

Type: Original Student Tutorial

Long Division With Polynomials:

Use long division to rewrite a rational expression of the form a(x) divided by b(x) in the form q(x) plus the quantity r(x) divided by b(x), where a(x), b(x), q(x), and r(x) are polynomials with this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials Using Special Cases:

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiasts

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Type: 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

## Perspectives Video: Teaching Ideas

Perfect Square Trinomials:

Unlock an effective teaching strategy for teaching perfect square trinomials using algebra tiles in this Teacher Perspectives video for educators,

Type: Perspectives Video: Teaching Idea

Making Connections with the Area Model:

Unlock an effective teaching strategy for making connections in area models in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

The Van de Walle Dot Matrix: A tool to support concepts from counting to multiplying polynomials:

Unlock an effective teaching tool that can help students all the way from basic counting principles to multiplying polynomials.

Dot Matrix sheet is available for dowload here.

Type: Perspectives Video: Teaching Idea

Multiplying Polynomials:

Unlock an effective teaching strategy for teaching multiplying polynomials 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

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

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.

The Physics Professor:

Students write explanations of the structure and function of a mathematical expression.

Throwing Horseshoes:

Students evaluate equivalent constructions of the same expression to determine which is the most useful for determining a maximum value.

The Bank Account:

Students explore an expression that calculates the balance of a bank account with compounding interest.

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Increasing or Decreasing? Variation 1:

Students examine variable expression that is a complex fraction with two distinct unit fractions in the denominator. Students are asked to consider how increasing one variable will affect the value of the entire expression. The variable expression is used in physics and describes the combined resistance of two resistors in parallel.

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

## Tutorials

The golden ratio:

In this tutorial, students learn an algebraic approach to understanding Phi, one of the most amazing numbers in mathematics.

Type: Tutorial

Solving a literal equation:

Students will learn to solve a literal equation.

Type: Tutorial

Division of Polynomials:

This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.

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

## Student Resources

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

## Original Student Tutorials

Learn to use algebra tiles to model adding polynomial expressions with this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials with Greatest Common Factor:

Learn how to factor polynomials by finding their greatest common factor in this interactive tutorial.

Type: Original Student Tutorial

Identifying and Interpreting Parts of Quadratic Equations in Vertex Form:

Identify parts of quadratic equations in vertex form and interpret them in terms of the context they represent in this interactive tutorial.

Type: Original Student Tutorial

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Polynomials when "a" Does Not Equal 1, Snowflake Method:

Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method:

Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Diamond Game: Factoring Quadratics when a = 1:

Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Identifying Parts of Linear Expressions:

Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.

Type: Original Student Tutorial

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Introduction to Polynomials: Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click here to open Part 2.

Type: Original Student Tutorial

Long Division With Polynomials:

Use long division to rewrite a rational expression of the form a(x) divided by b(x) in the form q(x) plus the quantity r(x) divided by b(x), where a(x), b(x), q(x), and r(x) are polynomials with this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials Using Special Cases:

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Type: Perspectives Video: Professional/Enthusiast

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

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.

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

## Tutorials

The golden ratio:

In this tutorial, students learn an algebraic approach to understanding Phi, one of the most amazing numbers in mathematics.

Type: Tutorial

Solving a literal equation:

Students will learn to solve a literal equation.

Type: Tutorial

Division of Polynomials:

This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.

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 topic.

## Perspectives Video: Professional/Enthusiast

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Type: Perspectives Video: Professional/Enthusiast

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

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.

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.