Illustrative Math Alignment: Grade 7 Unit 7

Expressions, Equations, and Inequalities

Lesson 9: Dealing with Negative Numbers

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
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
Definition--Exponential Concepts--Exponential Equation	Definition--Exponential Concepts--Exponential Equation This is a collection of definitions related to exponential concepts. This includes exponential functions, equations, and properties of exponents.	Applications of Exponential and Logarithmic Functions
Definition--Functions and Relations Concepts--Parametric Equations	Parametric Equations Topic Functions and Relations Definition Parametric equations are a set of equations that express the coordinates of the points of a curve as functions of a variable called a parameter.	Relations and Functions
Definition--Inequality Concepts--Inequation	Definition--Inequality Concepts--Inequation This is a collection of definitions related to inequalities. This includes inequalities that include numbers only and those with one or two variables.	Inequalities
Definition--Linear Function Concepts--Equations of Parallel Lines	Equations of Parallel Lines Topic Linear Functions Definition Equations of parallel lines are linear equations that have the same slope but different y-intercepts, indicating that the lines never intersect. Description Understanding equations of parallel lines is crucial in geometry and algebra. Parallel lines have identical slopes, which means they run in the same direction and never meet. In real-world applications, parallel lines can model scenarios such as railway tracks or lanes on a highway, where maintaining a consistent distance is essential.	Slope-Intercept Form
Definition--Linear Function Concepts--Equations of Perpendicular Lines	Equations of Perpendicular Lines Topic Linear Functions Definition Equations of perpendicular lines are linear equations where the slopes are negative reciprocals of each other, indicating that the lines intersect at a right angle. Description Equations of perpendicular lines are significant in both geometry and algebra. The negative reciprocal relationship between their slopes ensures that the lines intersect at a 90-degree angle. In real-world applications, perpendicular lines are found in various structures, such as the intersection of streets or the corners of a building, where right angles are essential.	Slope-Intercept Form
Definition--Linear Function Concepts--Linear Equations in Standard Form	Linear Equations in Standard Form Topic Linear Functions Definition Linear equations in standard form are written as Ax + By = C, where A, B, and C are constants, and A and B are not both zero. Description Linear equations in standard form are a fundamental representation of linear functions. They provide a way to express linear relationships in a general form.	Standard Form
Definition--Linear Function Concepts--The Equation of a Line From Two Coordinates	The Equation of a Line From Two Coordinates Topic Linear Functions Definition The equation of a line from two coordinates can be determined by finding the slope between the two points and using it in the point-slope form of a linear equation. Description Finding the equation of a line from two coordinates is a fundamental skill in algebra. It involves calculating the slope between the two points and then using one of the points to form the equation in point-slope or slope-intercept form.	Point-Slope Form
Definition--Quadratics Concepts--Quadratic Equation	Quadratic Equation Topic Quadratics Concepts Definition A quadratic equation is a polynomial equation of degree two, typically in the form ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0.	Quadratic Equations and Functions
Definition--Rationals and Radicals--Radical Equations	Radical Equations Topic Rationals and Radicals Definition Radical equations are equations in which the variable is inside a radical, such as a square root or cube root. Description Radical Equations are a fundamental aspect of Radical Numbers, Expressions, Equations, and Functions. These equations involve variables within radical signs, such as square roots or cube roots. Solving radical equations typically requires isolating the radical on one side of the equation and then squaring both sides to eliminate the radical. For example, to solve $$\sqrt{x+3} = 5$$ one would square both sides to obtain $$x + 3 = 25$$	Radical Functions and Equations
Definition--Rationals and Radicals--Rational Equations	Rational Equations Topic Rationals and Radicals Definition Rational equations are equations that involve rational expressions, which are fractions containing polynomials in the numerator and denominator. Description Rational Equations are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These equations involve rational expressions, which are fractions containing polynomials in the numerator and denominator. Solving rational equations typically requires finding a common denominator, clearing the fractions, and then solving the resulting polynomial equation. For example, to solve $$\frac{1}{x} + \frac{1}{x+1} = \frac{1}{2}$$	Rational Functions and Equations
Definition--Systems Concepts--Simultaneous Equations	Definition--Systems Concepts--Simultaneous 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.	Solving Systems of Equations
Desmos Activity: Equations of Parabolas 1	Derive the equation of a parabola with a vertex at the origin and focus at coordinates (0, f) using a Desmos graphing calculator. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources To see additional resources on this topic click on the Related Resources tab above. Desmos Collection To see the complete collection of Desmos Resources click on this link.	Graphs of Quadratic Functions
Desmos Activity: Equations of Parabolas 2	Derive the equation of a parabola with Focus at coordinates (a, c) and Directrix at y = c using a Desmos graphing calculator. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources To see additional resources on this topic click on the Related Resources tab above. Desmos Collection To see the complete collection of Desmos Resources click on this link.	Graphs of Quadratic Functions
Desmos Activity: Linear Equations in Point-Slope Form	In this graphing calculator activity, have your students explore how to convert linear equations in point-slope to a linear function in slope-intercept form. This Desmos template allows students to explore the effect of changes in the values of coordinates of the point and the slope of the line. A companion downloadable worksheet uses the graphing calculator template to explore the properties of these linear equations and functions. Related Resources To see additional resources on this topic click on the Related Resources tab above. Desmos Collection To see the complete collection of Desmos Resources click on this link.	Point-Slope Form
Desmos Activity: Linear Equations in Standard Form	Desmos Activity: Linear Equations in Standard Form In this graphing calculator activity, have your students explore how to convert linear equations in standard form to a linear function in slope-intercept form. This Desmos template allows students to explore the effect of changes in the values of A, B, and C in the standard form and m and b in the slope-intercept form. A companion downloadable worksheet uses the graphing calculator template to explore the properties of these linear equations and functions. Note: The download is a PDF worksheet. —PRESS PREVIEW TO SEE THE ACTIVITY— To see the complete collection of Desmos activities, click on this link. The following section includes background information on slope. This background also includes video resources and accompanying transcripts.	Standard Form
Elementary Definition--Multiplication and Division Concepts--False Equation	False Equation Topic Multiplication and Division Definition A false equation is a mathematical statement that shows equality between two expressions that are not equal.	Multiplication Expressions and Equations
Elementary Definition--Multiplication and Division Concepts--True Equation	True Equation Topic Multiplication and Division Definition A true equation is a mathematical statement that asserts equality between two expressions that are equal. Description True equations are essential in mathematics as they represent accurate relationships between numbers and expressions. For example, the equation 2 + 3 = 5	Multiplication Expressions and Equations
Elementary Math Definitions--Addition Subtraction Concepts--False Equation	False Equation Topic Addition and Subtraction Definition A false equation is a mathematical statement that shows two expressions that are not equal. Description A false equation is a mathematical statement where two expressions are not equal. For example, the equation 3 + 4 = 8 is false because the sum of 3 and 4 is 7, not 8. Understanding false equations helps students recognize and correct errors in their calculations.	Addition Expressions and Equations
Elementary Math Definitions--Addition Subtraction Concepts--True Equation	True Equation Topic Addition and Subtraction Definition A true equation is a mathematical statement that shows two expressions that are equal. Description A true equation is a mathematical statement where two expressions are equal. For example, the equation 3 + 4 = 7 is true because the sum of 3 and 4 is indeed 7. Understanding true equations helps students recognize and verify the accuracy of their calculations.	Addition Expressions and Equations
Equations Word Search Puzzle 2	Equations Word Search Puzzle 2 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, Solving One-Step Equations and Solving Two-Step Equations
Instructional Resource--Strategy Pack--One-Step Addition Equations	Instructional Resource \| Strategy Pack \| One-Step Addition Equations 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. —PRESS PREVIEW TO LAUNCH THE PRESENTATION— To see the complete collection of Instructional Resources, click on this link. Note: The download is a PPT file.	Solving One-Step Equations
Instructional Resource--Strategy Pack--One-Step Multiplication Equations	Instructional Resource \| Strategy Pack \| One-Step Multiplication Equations 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. —PRESS PREVIEW TO LAUNCH THE PRESENTATION— To see the complete collection of Instructional Resources, click on this link. Note: The download is a PPT file.	Solving One-Step Equations
INSTRUCTIONAL RESOURCE: Tutorial: Adding and Subtracting Rational Numbers	INSTRUCTIONAL RESOURCE: Tutorial: Adding and Subtracting Rational Numbers In this Slide Show, learn how to add and subtract rational numbers. Includes links to several Media4Math videos and a math game. 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.	Rational Expressions and Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + -b = -cx - d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + -b = -cx - d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax + -b = -cx - d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -c	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -c In this interactive, look at the solution to a two-step equation by clicking on various hot spots. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -cx + d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -cx + d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax + b = -cx + d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -cx - d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = -cx - d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax + b = -cx - d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = c	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = c In this interactive, look at the solution to a two-step equation by clicking on various hot spots. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = cx + d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = cx + d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax + b = cx + d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = cx - d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax + b = cx - d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax + b = cx - d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX + By = -C	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX + By = -C In this PowerPoint Presentation, analyze the steps in converting a linear equation in Standard Form to a linear function in Slope-Intercept Form. In this Interactive we work with this version of the Standard Form: -AX + By = -C. This is part of a collection of tutorials for solving different types of equations. 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.	Standard Form
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX + By = C	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX + By = C In this PowerPoint Presentation, analyze the steps in converting a linear equation in Standard Form to a linear function in Slope-Intercept Form. In this Interactive we work with this version of the Standard Form: -AX + By = C. This is part of a collection of tutorials for solving different types of equations. 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.	Standard Form
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = -c	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = -c In this interactive, look at the solution to a two-step equation by clicking on various hot spots. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = -cx + d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = -cx + d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax - b = -cx + d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = c	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = c In this interactive, look at the solution to a two-step equation by clicking on various hot spots. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = cx + d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = cx + d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax - b = cx + d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = cx - d	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax - b = cx - d In this PowerPoint presentation, analyze the solution to a multi-step equation of the form: -ax - b = cx - d. This is part of a collection of tutorials for solving different types of equations. 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.	Solving Two-Step Equations
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX - By = -C	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX - By = -C In this PowerPoint Presentation, analyze the steps in converting a linear equation in Standard Form to a linear function in Slope-Intercept Form. In this Interactive we work with this version of the Standard Form: -AX - By = -C. This is part of a collection of tutorials for solving different types of equations. 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.	Standard Form
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX - By = C	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -AX - By = C In this PowerPoint Presentation, analyze the steps in converting a linear equation in Standard Form to a linear function in Slope-Intercept Form. In this Interactive we work with this version of the Standard Form: -AX - By = C. This is part of a collection of tutorials for solving different types of equations. 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.	Standard Form
INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax^2 + bx + c = 0	INSTRUCTIONAL RESOURCE: Anatomy of an Equation: -ax^2 + bx + c = 0 In this PowerPoint presentation, analyze the steps in solving a quadratic equation with two roots. In this Interactive we work with this version of the quadratic equation: -ax^2 + bx + c = 0. This is part of a collection of tutorials for solving different types of equations. 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.	Polynomial Functions and Equations