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

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MA.912.AR.1.1 : Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.
  • MA.912.AR.1.2 : Rearrange equations or formulas to isolate a quantity of interest.
  • MA.912.AR.1.3 : Add, subtract and multiply polynomial expressions with rational number coefficients.
  • MA.912.AR.1.4 : Divide a polynomial expression by a monomial expression with rational number coefficients.
  • MA.912.AR.1.5 : Divide polynomial expressions using long division, synthetic division or algebraic manipulation.
  • MA.912.AR.1.6 : Solve mathematical and real-world problems involving addition, subtraction, multiplication or division of polynomials.
  • MA.912.AR.1.7 : Rewrite a polynomial expression as a product of polynomials over the real number system.
  • MA.912.AR.1.8 : Rewrite a polynomial expression as a product of polynomials over the real or complex number system.
  • MA.912.AR.1.9 : Apply previous understanding of rational number operations to add, subtract, multiply and divide rational algebraic expressions.
  • MA.912.AR.1.10 : Solve mathematical and real-world problems involving addition, subtraction, multiplication or division of rational algebraic expressions.
  • MA.912.AR.1.11 : Apply the Binomial Theorem to create equivalent polynomial expressions.
Cluster 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.)
Grade:
912
Strand
Algebraic Reasoning
Cluster Access Points

This cluster includes the following 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 : Factor a quadratic expression.
  • 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.
Cluster Resources

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

Original Student Tutorials
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.

  • 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.

  • 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.

  • Solving a Literal Linear Equation: Students are given a literal linear equation and asked to solve for a specific variable.

  • 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.

  • 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.

  • Rewriting Numerical Expressions: Students are asked to rewrite numerical expressions to find efficient ways to calculate.

  • Multiplying Polynomials - 2: Students are asked to multiply polynomials and explain if the product of two polynomials always results in a polynomial.

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

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

  • Finding Missing Values: Students are asked to rewrite quadratic expressions and identify parts of the expressions.

  • Multiplying Polynomials - 1: Students are asked to multiply polynomials and explain if the product of polynomials always results in a polynomial.

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

  • 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.

  • 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.

  • What Happens?: Students are asked to determine how the volume of a cone will change when its dimensions are changed.

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

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.

  • 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.

  • 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.

  • 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.

  • 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.

  • Find your Formula!: 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.

  • 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.

  • 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.

  • 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.

  • Wine Glass Lab: Resonance and the Wave Equation: This activity is designed to help students understand the concept of resonance through the application of the wave equation to sounds produced by a singing wine glass.

  • Sorting Equations and Identities: This lesson is intended to help you assess how well students are able to:

    • Recognize the differences between equations and identities.
    • Substitute numbers into algebraic statements in order to test their validity in special cases.
    • Resist common errors when manipulating expressions such as 2(x – 3) = 2x – 3; (x + 3)2 = x2 + 32.
    • Carry out correct algebraic manipulations.

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

  • Modeling Situations With Linear Equations: This lesson unit is intended to help you assess how well students use algebra in context, and in particular, how well students:

    • Explore relationships between variables in everyday situations.
    • Find unknown values from known values.
    • Find relationships between pairs of unknowns, and express these as tables and graphs.
    • Find general relationships between several variables, and express these in different ways by rearranging formulae.
  • 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.

    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.

  • Matching Trinomials with Area Models_2023: Matching Trinomials with Area Models_2023

  • Motion: Speed and Velocity: In this lesson students should be able to :

    • Identify appropriate SI units for measuring speed.
    • Compare and contrast average speed and instantaneous speed.
    • Interpret position-time graphs.
    • Calculate the speed of an object using slopes.
  • Acceleration: In this lesson students will learn to:

    1. Identify changes in motion that produce acceleration.
    2. Describe examples of objects moving with constant acceleration.
    3. Calculate the acceleration of an object, analytically, and graphically.
    4. Interpret velocity-time graph, and explain the meaning of the slope.
    5. Classify acceleration as positive, negative, and zero.
    6. Describe instantaneous acceleration.
  • Laying Tiles for Polynomial Addition and Subtraction Renovation: In this lesson students will learn how to add and subtract polynomials.

  • Falling for Gravity: Students will investigate the motion of three objects of different masses undergoing free fall. Additionally, students will:

    • Use spark timers to collect displacement and time data.
    • Use this data to calculate the average velocity for the object during each interval.
    • Graph this data on a velocity versus time graph, V-t. They find the slope of this graph to calculate acceleration.
    • Calculate the falling object's acceleration from their data table and graph this data on an acceleration versus time graph, a-t.
    • Use their Spark timer data paper, cut it into intervals, and paste these intervals into their displacement versus time graph.
  • Riding the Roller Coaster of Success: Students compete with one another to design and build a roller coaster from insulation tubing and tape that will allow a marble to travel from start to finish with the lowest average velocity. In so doing, students learn about differences between distance and displacement, speed and velocity, and potential and kinetic energy. They also examine the Law of Conservation of Energy and concepts related to force and motion.

  • Taming the Behavior of Polynomials: This lesson will cover sketching the graphs of polynomials while in factored form without the use of a calculator.

  • 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.

  • 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.

    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.

  • 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.

  • 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?"

  • 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 applying surface area concepts and algebra through modeling.

    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.

  • 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.

     

  • 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).

  • 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.

  • Using algebra tiles and tables to factor trinomials (less guess and check!): This lesson addresses factoring when a = 1 and also when a > 1.  Part 1 (Algebra Tiles) contains examples when a = 1 and a >1. Part 2 (tables) contains only examples when
    a > 1. 

    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.

Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
  • Quadrupling Leads to Halving: 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.

  • Radius of a Cylinder: 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.

  • Solving a literal equation: Students will learn to solve a literal equation. 

  • Division of Polynomials: This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.

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.

Virtual Manipulative
  • Graphing Lines: Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the slope and various types of line equations can be explored.