Illustrative Math Alignment: Grade 8 Unit 3

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.

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.

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. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

This is part of a collection of definitions related to the topic of systems of equations. The focus of most of the terms is linear systems.

Review key vocabulary on the topic of equations with this interactive and printable word search puzzle.

Related Resources

Learn different strategies for solving one-step addition equations. The Strategy Packs provide alternate ways of solving the same problem, giving your students different approaches to the same problem. The goal of the Strategy Packs is to encourage your students to think strategically when solving math problems.

Note: The download is a PPT file.

Learn different strategies for solving one-step multiplication equations. The Strategy Packs provide alternate ways of solving the same problem, giving your students different approaches to the same problem. The goal of the Strategy Packs is to encourage your students to think strategically when solving math problems.

Note: The download is a PPT file.

The complete set of 13 examples that make up this set of tutorials.

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The complete set of 42 examples that make up this set of tutorials.

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The complete set of 12 examples that make up this set of tutorials.

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The complete set of 21 examples that make up this set of tutorials.

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The complete set of 39 examples that make up this set of tutorials. NOTE: The download is a PPT file.

This slide show defines non-linear systems, showing examples of linear and quadratic systems.

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This slide show shows how to solve a system using the elimination method.

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This slide show shows how to solve a system using the substitution method.

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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.

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In this Slide Show, learn how to solve a linear system by the elimination method.

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This tutorial goes over the steps of solving a two-step equation by using the properties of equality. Note: The download is a PDF version of this tutorial.

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