Illustrative Math Alignment: Grade 8 Unit 3

Linear Equations and Linear Systems

Lesson 6: Strategic Solving

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Definition--Equation Concepts--Division Property of Equality	Division Property of Equality Topic Equations Definition The Division Property of Equality states that if you divide both sides of an equation by the same nonzero value, the equality remains true. Description The Division Property of Equality is a key principle in algebra. It states that for any real numbers a, b, and c (where 𝑐 ≠ 0), if a = b, then a ÷ c = b ÷ c This property is used to solve equations by isolating variables. For example, to solve 3x = 12 divide both sides by 3 to get x = 4.	Applications of Equations and Inequalities
Definition--Equation Concepts--Equality	Equality Topic Equations Definition Equality is a mathematical statement that asserts that two expressions are equal. Description Equality is a foundational concept in mathematics. It indicates that two expressions have the same value, represented by the symbol "=". For example, in the equation 2 + 3 = 5, both sides are equal. Equality is used to form equations and solve problems. In real-world applications, equality is used in accounting, engineering, and data analysis to ensure balance and accuracy. Understanding equality helps students develop logical reasoning and problem-solving skills.	Applications of Equations and Inequalities
Definition--Equation Concepts--Equation	Equation Topic Equations Definition An equation is a mathematical statement that asserts the equality of two expressions. Description Equations are central to mathematics, representing relationships between quantities. They consist of two expressions separated by an equals sign. For example, 2x + 3 = 7 is an equation that can be solved to find x. Equations can be linear, quadratic, or polynomial, among others.	Applications of Equations and Inequalities
Definition--Equation Concepts--Equations in One Variable	Equations in One Variable Topic Equations Definition Equations in one variable involve a single variable and can be solved to find its value. Description Equations in one variable are fundamental in algebra. They typically take the form of ax + b = 0, where x is the variable. Solving these equations involves isolating the variable to determine its value. For example, solving 2x + 3 = 7 yields x = 2	Applications of Equations and Inequalities
Definition--Equation Concepts--Equations in Two Variables	Equations in Two Variables Topic Equations Definition Equations in two variables involve two variables and describe a relationship between them. Description Equations in two variables are essential in algebra and coordinate geometry. They typically take the form of ax + by = c and represent lines in a coordinate plane. For example, the equation 2x + 3y = 6 can be graphed as a line.	Applications of Equations and Inequalities
Definition--Equation Concepts--Equivalent Equations	Equivalent Equations Topic Equations Definition Equivalent equations are equations that have the same solutions. Description Equivalent equations are a key concept in algebra. They may look different but yield the same solutions. For instance, 2x + 3 = 7 and 4x + 6 = 14 are equivalent because both have the solution x = 2. Transformations such as addition, subtraction, multiplication, or division can produce equivalent equations.	Applications of Equations and Inequalities
Definition--Equation Concepts--False Equation	False Equation Topic Equations Definition A false equation is an equation that is not true for any value of the variable(s). Description False equations are equations that do not hold true for any value of the variable(s). For example, the equation x + 2 = x + 3 is false because there is no value of x that makes both sides equal. Identifying false equations is important in verifying the validity of mathematical statements.	Applications of Equations and Inequalities
Definition--Equation Concepts--Identity Equation	Identity Equation Topic Equations Definition An identity equation is true for all values of the variable(s). Description Identity equations are equations that hold true for all values of the variable(s). For example, the equation 2(x + 1) = 2x + 2 is an identity because it is true for any value of x. These equations are used to express mathematical identities and properties. In the equation above, the identity results from the use of the distributive property.	Applications of Equations and Inequalities
Definition--Equation Concepts--Inequation	Inequation Topic Equations Definition An inequation is a mathematical statement that shows the inequality between two expressions. Description Inequations, or inequalities, are statements that compare two expressions using inequality symbols such as >, <, ≥, and ≤. For example, x + 3 > 5 indicates that 𝑥 x must be greater than 2. Inequations are used to represent constraints and conditions in mathematical models. Inequations sometimes involve the inequality symbol: ≠.	Inequalities
Definition--Equation Concepts--Isolating the Variable	Isolating the Variable Topic Equations Definition Isolating the variable involves manipulating an equation to get the variable alone on one side. Description Isolating the variable is a fundamental technique in algebra used to solve equations. It involves performing operations to both sides of an equation to get the variable by itself. For example, solving 2x + 3 = 7 involves subtracting 3 and then dividing by 2 to isolate x, resulting in x = 2.	Variables and Unknowns
Definition--Equation Concepts--Left Side of the Equation	Left Side of the Equation Topic Equations Definition The left side of the equation refers to the expression on the left side of the equals sign. Description The left side of an equation is the part of the equation that appears before the equals sign. For example, in the equation 2x + 3 = 7 the left side is 2x + 3. Understanding the left side of the equation is crucial for solving and balancing equations. In real-world applications, recognizing the left side of an equation helps in setting up and solving problems accurately. It is essential for students to understand this concept to manipulate and solve equations effectively.	Applications of Equations and Inequalities
Definition--Equation Concepts--Linear Equation	Linear Equation Topic Equations Definition A linear equation is an equation that does not have an variables raised to a power higher than one. A linear equation can have one or more variables. Description Linear equations are fundamental in algebra and describe relationships are summarized below.	Applications of Linear Functions
Definition--Equation Concepts--Literal Equation	Literal Equation Topic Equations Definition A literal equation is an equation that involves two or more variables. Description Literal equations involve multiple variables and are used to express relationships between them. For example, the formula for the area of a rectangle, A = l•w is a literal equation involving the variables l and w. Solving literal equations often involves isolating one variable in terms of the others.	Applications of Equations and Inequalities
Definition--Equation Concepts--Multi-Step Equation	Multi-Step Equation Topic Equations Definition A multi-step equation requires more than one step to solve. Description Multi-step equations involve multiple operations to isolate the variable. For example, solving 3x + 2 = 11 requires subtracting 2 and then dividing by 3 to find x=3. These equations are common in algebra and require a systematic approach to solve. In real-world applications, multi-step equations are used in complex problem-solving scenarios such as engineering and finance. Understanding how to solve multi-step equations helps students develop critical thinking and problem-solving skills.	Solving Multistep Equations
Definition--Equation Concepts--Multiplication Property of Equality	Multiplication Property of Equality Topic Equations Definition The Multiplication Property of Equality states that if you multiply both sides of an equation by the same nonzero value, the equality remains true. Description The Multiplication Property of Equality is a fundamental principle in algebra. It states that for any real numbers a, b, and c (where 𝑐 ≠ 0), if a = b, then ac = bc This property is used to solve equations by isolating variables. For example, to solve x/3 = 4 you multiply both sides by 3 to get x = 12.	Applications of Equations and Inequalities
Definition--Equation Concepts--Nonlinear Equation	Nonlinear Equation Topic Equations Definition A nonlinear equation is an equation that graphs as a curve and does not form a straight line. Description Nonlinear equations are equations that involve variables raised to powers other than one or involve products of variables. For example, the equation y = x2 is nonlinear because it graphs as a parabola. These equations are used to model more complex relationships than linear equations.	Applications of Equations and Inequalities
Definition--Equation Concepts--Numerical Expression	Numerical Expression Topic Equations Definition A numerical expression is a mathematical phrase involving numbers and operation symbols, but no variables. Description Numerical expressions consist of numbers and operations such as addition, subtraction, multiplication, and division. For example, 3 + 4 × 2 is a numerical expression. These expressions are evaluated to find their value. In real-world applications, numerical expressions are used in everyday calculations such as budgeting, measuring, and data analysis. Understanding numerical expressions helps students perform arithmetic operations and develop computational skills.	Numerical Expressions
Definition--Equation Concepts--One-Step Equation	One-Step Equation Topic Equations Definition A one-step equation requires only one operation to solve. Description One-step equations are the simplest type of equations in algebra. They involve a single operation to isolate the variable. For example, solving x + 3 = 7 requires subtracting 3 from both sides to find x = 4. These equations are used in basic problem-solving scenarios and form the foundation for understanding more complex equations. Understanding one-step equations helps students develop confidence in solving algebraic problems and prepares them for advanced algebraic concepts.	Solving One-Step Equations
Definition--Equation Concepts--Polynomial Equation	Polynomial Equation Topic Equations Definition A polynomial equation is an equation that involves a polynomial expression. Description Polynomial equations involve expressions that include terms with variables raised to whole-number exponents. For example, the equation x2 − 4x + 4 = 0 is a polynomial equation. These equations can be linear, quadratic, cubic, etc., depending on the highest power of the variable.	Polynomial Functions and Equations
Definition--Equation Concepts--Quadratic Equation	Quadratic Equation Topic Equations Definition A quadratic equation is a polynomial equation of degree 2, typically in the form ax2 + bx + c = 0. Description Quadratic equations are fundamental in algebra and involve variables raised to the second power. For example, the equation x2 − 4x + 4 = 0 is quadratic. These equations can be solved using methods such as factoring, completing the square, and the quadratic formula.	Quadratic Equations and Functions
Definition--Equation Concepts--Reflexive Property of Equality	Reflexive Property of Equality Topic Equations Definition The Reflexive Property of Equality states that any value is equal to itself. Description The Reflexive Property of Equality is a basic principle in mathematics. It states that for any value a, a = a This property is used to justify steps in solving equations and proving mathematical statements. In real-world applications, the reflexive property underlies the concept of identity and is fundamental in logical reasoning and proofs. Understanding this property helps students build a strong foundation in algebra and develop rigorous mathematical arguments.	Applications of Equations and Inequalities
Definition--Equation Concepts--Right Side of the Equation	Right Side of the Equation Topic Equations Definition The right side of the equation refers to the expression on the right side of the equals sign. Description The right side of an equation is the part of the equation that appears after the equals sign. For example, in the equation 2x + 3 = 7 the right side is 7. Understanding the right side of the equation is crucial for solving and balancing equations.	Applications of Equations and Inequalities
Definition--Equation Concepts--Roots of an Equation	Roots of an Equation Topic Equations Definition The roots of an equation are the values of the variable that satisfy the equation. Description The roots of an equation are the solutions that make the equation true. For example, the roots of the quadratic equation x2 − 4x + 4 = 0 are x = 2 because substituting 2 into the equation satisfies it. Finding roots is a fundamental task in algebra.	Applications of Equations and Inequalities
Definition--Equation Concepts--Solution	Solution Topic Equations Definition A solution is the value(s) of the variable(s) that satisfy an equation. Description The concept of solution is fundamental in equations, referring to the values that make the equation true. For example, in the equation x + 2 = 5 the solution is x = 3 because substituting 3 in place of x results in a true statement. Solutions can exist for various types of equations, whether single-variable, multi-variable, linear, or nonlinear.	Applications of Equations and Inequalities
Definition--Equation Concepts--Solving an Equation	Solving an Equation Topic Equations Definition Solving an equation involves finding the value(s) of the variable(s) that make the equation true. Description Solving an equation is a key skill in algebra, where one determines the values of variables that satisfy the equation. For example, in the equation 2x + 3 = 7 one can find that x = 2 by isolating the variable through algebraic manipulations. Different techniques such as substitution, factoring, or using the quadratic formula may apply depending on the complexity of the equation.	Applications of Equations and Inequalities
Definition--Equation Concepts--Subtraction Property of Equality	Subtraction Property of Equality Topic Equations Definition The Subtraction Property of Equality states that if you subtract the same value from both sides of an equation, the equality remains true. Description The Subtraction Property of Equality is a fundamental principle in algebra. It states that for any real numbers a, b, and c, if a = b, then a − c = b − c This property is used to solve equations by isolating variables. For example, to solve x + 3 = 7 you subtract 3 from both sides to get x = 4.	Applications of Equations and Inequalities
Definition--Equation Concepts--Symmetric Property of Equality	Symmetric Property of Equality Topic Equations Definition The Symmetric Property of Equality states that if a = b, then b = a. Description The Symmetric Property of Equality is a basic principle in mathematics. It asserts that the equality relation is symmetric, meaning that if one quantity equals another, then the second quantity equals the first. For example, if x = y then y = x This property is used to justify steps in solving equations and proving mathematical statements.	Applications of Equations and Inequalities
Definition--Equation Concepts--The Unknown	The Unknown Topic Equations Definition The unknown is the variable in an equation that needs to be solved for. Description The unknown in an equation represents the value that needs to be determined. For example, in the equation x + 3 = 7 x is the unknown. Identifying and solving for the unknown is a core aspect of algebra. In real-world applications, finding the unknown is crucial for solving problems in various fields such as science, engineering, and finance. Understanding how to identify and solve for the unknown helps students develop problem-solving skills and apply mathematical concepts to real-life situations.	Variables and Unknowns
Definition--Equation Concepts--Transitive Property of Equality	Transitive Property of Equality Topic Equations Definition The Transitive Property of Equality states that if a = b and b = c, then a = c. Description The Transitive Property of Equality is a fundamental principle in mathematics. It states that if one quantity equals a second quantity, and the second quantity equals a third, then the first and third quantities are equal. For example, if x = y and y = z then x = z This property is used to justify steps in solving equations and proving mathematical statements.	Applications of Equations and Inequalities
Definition--Equation Concepts--True Equation	True Equation Topic Equations Definition A true equation is an equation that holds true for the given values of the variable(s). Description A true equation is an equation that is valid for specific values of the variable(s). For example, the equation 2 + 3 = 5 is true because both sides are equal. Identifying true equations is important in verifying the correctness of mathematical statements. In real-world applications, recognizing true equations helps in ensuring the accuracy of mathematical models and solutions. Understanding true equations helps students develop critical thinking and analytical skills.	Applications of Equations and Inequalities
Definition--Equation Concepts--Two-Step Equation	Two-Step Equation Topic Equations Definition A two-step equation requires two operations to solve. Description Two-step equations involve performing two operations to isolate the variable. For example, solving 2x + 3 = 7 requires subtracting 3 from both sides and then dividing by 2 to find x = 2. These equations are common in algebra and require a systematic approach to solve. In real-world applications, two-step equations are used in problem-solving scenarios such as calculating costs or determining measurements. Understanding how to solve two-step equations helps students develop critical thinking and problem-solving skills.	Solving Two-Step Equations
Definition--Equation Concepts--Variable Expression	Variable Expression Topic Equations Definition A variable expression is a mathematical phrase involving variables, numbers, and operation symbols. Description Variable expressions consist of variables, numbers, and operations such as addition, subtraction, multiplication, and division. For example, 3x + 4 is a variable expression. These expressions are used to represent quantities and relationships in algebra.	Variable Expressions
Definition--Equation Concepts--Visual Models for Equations	Visual Models for Equations Topic Equations Definition Visual models for equations use graphical representations to illustrate the relationships between variables. Description Visual models for equations include graphs, charts, and diagrams that represent the relationships between variables. For example, a graph of the equation y = 2x + 3 shows a straight line with a slope of 2 and a y-intercept of 3. These models help in understanding and interpreting equations.	Applications of Equations and Inequalities
INSTRUCTIONAL RESOURCE: Tutorial: The Equals Sign Is Not an Operator!	INSTRUCTIONAL RESOURCE: Tutorial: The Equals Sign Is Not an Operator! In this presentation, learn the basics of equations and dispel the misconception that the equals sign is an operator. The tutorial will ground the student in the properties of equality, the structure of an equation, and the important role of the equals sign. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. < Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Equations and Inequalities
INSTRUCTIONAL RESOURCE: Tutorial: Addition and Subtraction Equations	INSTRUCTIONAL RESOURCE: Tutorial: Addition and Subtraction Equations Review the basics of addition and subtraction equations with this interactive. This is part of a collection of math tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Addition Expressions and Equations and Subtraction Expressions and Equations
INSTRUCTIONAL RESOURCE: Tutorial: Interactive: Basic Addition and Subtraction Vocabulary, Part 2	INSTRUCTIONAL RESOURCE: Tutorial: Interactive: Basic Addition and Subtraction Vocabulary, Part 2 This interactive reviews key first and second grade vocabulary on addition and subtraction. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Addition Expressions and Equations and Subtraction Expressions and Equations
INSTRUCTIONAL RESOURCE: Tutorial: Introduction to Addition and Subtraction Equations	INSTRUCTIONAL RESOURCE: Tutorial: Introduction to Addition and Subtraction Equations Explore the basics of addition and subtraction equations with this interactive. This is part of a collection of math tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Addition Expressions and Equations and Subtraction Expressions and Equations
INSTRUCTIONAL RESOURCE: Tutorial: Visual Models for Equations	INSTRUCTIONAL RESOURCE: Tutorial: Visual Models for Equations An equation shows a relationship between two quantities. The download is a PDF. This is part of a collection of math tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Algebra Tiles--Expressions and Equations and Applications of Equations and Inequalities
INSTRUCTIONAL RESOURCE: Tutorial: What Is an Equation?	INSTRUCTIONAL RESOURCE: Tutorial: What Is an Equation? An equation shows a relationship between two quantities. Note: The download is a PDF. This is part of a collection of math tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Equations and Inequalities and Variables and Unknowns
INSTRUCTIONAL RESOURCE: Tutorial: What Is an Identity?	INSTRUCTIONAL RESOURCE: Tutorial: What Is an Identity? An identity is a special type of equation. It includes at least one variable and an identity is true for all values of the variable. Note: The download is a PDF. This is part of a collection of math tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Equations and Inequalities
INSTRUCTIONAL RESOURCE: Tutorial: Variables and Variable Expressions	INSTRUCTIONAL RESOURCE: Tutorial: Variables and Variable Expressions In this Slide Show, learn about variables and variable expressions. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. < Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Variable Expressions, Applications of Equations and Inequalities and Variables and Unknowns
Interactive Crossword Puzzle--Equations 2	Interactive Crossword Puzzle--Equations 2 Review key vocabulary on the topic of equations with this interactive and printable crossword puzzle. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Equations and Inequalities, Solving One-Step Equations and Solving Two-Step Equations
Interactive Crossword Puzzle: Equations	Interactive Crossword Puzzle: Equations Review key vocabulary on the topic of equations with this interactive and printable crossword puzzle. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Equations and Inequalities
Interactive Math Game--DragNDrop--One-Step Equations	Interactive Math Game--DragNDrop Math--One-Step Equations In this drag-and-drop game, match a one-step equation to its solution. Includes Addition, subtraction, multiplication, and division one-step equations. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Solving One-Step Equations
Interactive Math Game--DragNDrop--Two-Step Equations	Interactive Math Game--DragNDrop Math--Two-Step Equations In this drag-and-drop game, match a two-step equation to its solution. Includes all four operations. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Solving Two-Step Equations
Interactive Math Game: Equation Sort 2	Interactive Math Game: Equation Sort 2 Use this math game to review one-step equations. Students are shown four equations and are asked to correctly label the equation with drag-and-drop. The game provides feedback. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Equations and Inequalities and Solving One-Step Equations
Interactive Math Game: Equation Sort 3	Interactive Math Game: Equation Sort 3 Use this math game to review two-step equations. Students are shown four equations and are asked to correctly label the equation with drag-and-drop. The game provides feedback. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Solving Two-Step Equations
Interactive Math Game: Math Riddles--Expressions and Equations	Interactive Math Game: Math Riddles--Expressions and Equations In this Math Riddles Game, have your students review vocabulary around the topic of expressions and equations. The Math Riddles games are useful for practicing: Math Vocabulary, Key Concepts, Critical Thinking. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Numerical and Algebraic Expressions
Interactive Word Search Puzzle: Equations	Interactive Word Search Puzzle: Equations Review key vocabulary on the topic of equations with this interactive and printable word search puzzle. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 1	Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 1 This is part of a collection of math clip art images about equations and inequalities.	Applications of Equations and Inequalities