Illustrative Math Alignment: Grade 8 Unit 3

Topic

Equations

Description

Equivalent Equations result from changing an equation while maintaining equality, often by performing the same operation on both sides. For example, x - 3 = 5 becomes x = 8 after adding 3 to both sides. This term demonstrates how equations can be manipulated without altering their solution, a key concept in solving equations.

Topic

Equations

Description

A False Equation is mathematically inaccurate. For example, 2 + 1 = 4 is false, and x + 3 = x has no solution, as no value of x satisfies the equation. This term contrasts true equations and emphasizes the need to verify solutions, highlighting logical consistency in mathematics.

Topic

Equations

Description

An Identity Equation is always true for all variable values. Examples include x = x and (x + 1)2 = x2 + 2x + 1. Identity equations often use the symbol ≡ to indicate their universal truth. This term introduces a special class of equations, linking algebraic manipulation and properties of equivalence.

Topic

Equations

Description

An Inequation is a mathematical statement that shows two expressions are not equal, as in 2 + 1 ≠ 4 or x + 3 ≥ 0, which includes inequalities. This term broadens the discussion to include inequalities, expanding the types of relationships between expressions.

Topic

Equations

Description

Isolating the Variable refers to the process of solving an equation by focusing on and isolating the variable of interest. For example, x + 3 = 5 becomes x = 2 by subtracting 3 from both sides. This term highlights a key step in solving equations, emphasizing strategies for simplifying equations to find solutions.

Topic

Equations

Description

The Left Side of the Equation is the expression on the left of the equals sign. For example, in 2x + 3 = x - 5, the left side is 2x + 3. This term provides clarity in identifying components of equations, aiding in understanding structure and solving strategies.

Topic

Equations

Description

A Linear Equation is an equation where variables have an exponent of 1. Examples include x + 2 = 7, 2x + 3y = 4, and y = -2x + 10. Graphs of two-variable linear equations are straight lines. This term introduces a fundamental class of equations with specific properties, crucial for algebra and graphing.

Topic

Equations

Description

The Addition Property of Equality allows the addition of the same quantity to each side of an equation while maintaining equality. For example, if a = b, then a + c = b + c. This concept ensures equations can be manipulated while preserving their equality, demonstrated by solving x - 2 = 3 as x = 5 after adding 2 to both sides. This term introduces the basic manipulation of equations, forming the foundation for solving linear equations and maintaining equality.

Topic

Equations

Description

A Literal Equation includes more than one variable and often represents formulas, such as A = πr2, V = l*w*h, and SA = 2πr*h + 2πr2. This term connects to real-world applications and highlights the versatility of equations in representing formulas and relationships.

Topic

Equations

Description

A Multi-Step Equation involves more than two mathematical operations to solve. For example, 4x + 2 = 2x + 8 simplifies to x = 3 after subtraction and division steps. This term emphasizes solving more complex equations, showing sequential steps and reinforcing logical progression in algebra.

Topic

Equations

Description

The Multiplication Property of Equality allows multiplication by the same non-zero quantity on both sides of an equation, maintaining equality. For example, if a = b, then a * c = b * c. Solving x/2 = 3 yields x = 6 by multiplying both sides by 2. This term builds on equality principles and demonstrates another method for solving equations by manipulating terms.

Topic

Equations

Description

A Non-Linear Equation includes non-linear terms such as polynomials of degree 2 or higher, exponential, logarithmic, or trigonometric terms. Examples include y = x2 - 4x + 5 and y = sin(x). This term expands understanding beyond linear relationships, introducing diverse mathematical models and their applications.

Topic

Equations

Description

A Numerical Expression consists of numbers and operations without variables. Examples include 2 + 3 = 5 and 2(4 + 3) = 14. This term highlights fundamental operations, foundational for understanding and solving equations.

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

A One-Step Equation involves a single mathematical operation to solve. For example, x + 2 = 3 simplifies to x = 1 by subtraction. This term introduces the simplest form of equations, essential for beginners to build confidence and understanding in solving.

Topic

Equations

Description

A Polynomial Equation consists of a polynomial expression equal to zero. Examples include x2 + 3x + 2 = 0 and x3 - 2x2 + 4x - 10 = 0. This term defines a key class of equations, critical for algebra and calculus studies.

Topic

Equations

Description

A Quadratic Equation is a polynomial of degree 2. Examples include x2 - 2 = 23 and 2x2 + 3x + 2 = 0. Graphs of such equations are parabolas. This term extends understanding of polynomial equations, connecting algebra to geometry through parabolas.

Topic

Equations

Description

The Reflexive Property of Equality states that any quantity is equal to itself. Examples include 5 = 5 and x + 2 = x + 2. This term reinforces the foundational principle of equality, essential for logical reasoning in mathematics.

Topic

Equations

Description

Right Side of the Equation. The expression to the right side of the equals sign in an equation. 2x + 3 = x - 5 (right side highlighted)

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

An Algebraic Equation is another term for a polynomial equation, represented as a sum of terms equal to zero. Examples include x2 + 3x + 2 = 0 and x3 - 2x2 + 4x - 10 = 0. It highlights the structure of polynomial equations using constants, variables, and powers. This term connects to the topic by defining the types of equations learners encounter when solving or simplifying expressions.

Topic

Equations

Description

Roots of an Equation. The solution or solutions to an equation. y = -x2 + 7x - 10; solutions: x = 2, x = 5

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Solution. A value for the unknown that turns a conditional equation into a true equation. For example, x = 5 is a solution to x + 2 = 7

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Solving an Equation. The process of finding the value(s) for a variable that result in a true equation. For example, x + 2 = 3 has a solution of x = 1

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Subtraction Property of Equality. The property that allows the subtraction of the same quantity from each side of an equation while maintaining the equality. For example, x + 2 = 3 becomes x = 1

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Symmetric Property of Equality. The property that states that an equation can be written two ways, with both sides interchanged. If a = b, then b = a.

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

The Unknown. In a conditional equation, the variable is also referred to as the unknown. 3x = 12, where x is the unknown

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Transitive Property of Equality. The property that states that two quantities equal to a third quantity are equal to each other. If a = b and b = c, then a = c

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

True Equation. A mathematically accurate equation. If variables are involved, the true equation is for the values of the variable that make the equation true. Examples: 2 + 1 = 3; (2 + 3)2 = 25; x + 3 = 9 for x = 6

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Two-Step Equation. A two-step equation is an equation that involves two mathematical operations to solve. The example given is 2x + 2 = 8, where the first operation is subtraction, leading to 2x = 6, followed by division to solve for x = 3. This concept extends the understanding of solving equations by introducing a sequence of operations, crucial for solving more complex equations.

Topic

Equations

Description

Variable Expression. A variable expression is made up of numbers, operations, and variables. Examples include 2x + 3 = 5 (expression on the left), 4 = 3x + 7 (expression on the right), and 4x + 3 = x - 5 (expressions on both sides). This connects variables to equations and highlights their role in forming relationships within mathematical contexts.

Topic

Equations

Description

An Algebraic Expression consists of numbers, operations, and variables. Unlike equations, expressions may not have an equality. Examples include 2x + 3, 4 = 3x + 7, and 4x + 3 = x - 5, demonstrating their versatility in representing mathematical relationships. This term distinguishes between expressions and equations, forming the basis for manipulating algebraic forms before solving equations.

Topic

Equations

Description

Visual Model of an Equation. A visual model of an equation represents the numbers, operations, and equality using visuals. The example shows 1 + 2 = 3 with blocks illustrating the balance of the equation. This visual approach reinforces the foundational concept of equality and provides an intuitive way to understand equations, especially for visual learners.

Topic

Equations

Description

The Approximately Equal To Sign (≈) indicates that two expressions are close in value but not equal. Examples include estimating sums (58 + 49 ≈ 100), square roots (√50 ≈ 7), and function values (f(x) ≈ 25). This term relates to the topic by introducing estimation techniques, valuable for checking solutions or simplifying problems.

Topic

Equations

Description

Proportional To indicates a linear relationship between two variables. For example, distance (d) is proportional to time (t), written as d ∝ t. This term links linear equations to real-world relationships, reinforcing practical applications of mathematical concepts.

Topic

Equations

Description

Similar To. When two geometric shapes are similar, corresponding sides are proportional. ΔABC ~ ΔDEF; AB ∝ DE

The topic of this video is closely tied to Equations, providing an essential understanding of its fundamental concepts. This video explores the mathematics behind this topic by delving into how key principles are applied in practical contexts.

Topic

Equations

Description

Congruent To signifies that two geometric shapes are congruent, meaning their corresponding sides and angles are equal. For example, triangles ABC and DEF are congruent, so side AB equals side DE. This term introduces geometric equality, tying equations to geometric properties and congruence criteria.

Topic

Equations

Description

Assigning Values to Variables involves assigning a numerical value to a variable for evaluating functions. For instance, let x = 3 in f(x) = 2x + 8 yields f(3) = 14. This term supports understanding how to evaluate equations and functions by substituting values, a key step in problem-solving.

Topic

Equations

Description

A Conditional Equation is true only for specific variable values. Examples include x + 1 = 4 (x = 3), x2 - 25 = 0 (x = ±5), and 2x - 5 = x + 9 (x = 14). This term supports solving equations by emphasizing that solutions depend on satisfying the given conditions.

Topic

Equations

Description

A Constant Term in an algebraic equation is a number without a variable. Examples include 2 in x2 + 3x + 2 = 0, -10 in x3 - 2x2 + 4x - 10 = 0, and 5 in x4 - 2x3 + 4x2 - x + 5 = 0. This term explains the role of constants in equations, crucial for identifying and isolating terms when solving equations.

Topic

Equations

Description

The Division Property of Equality allows division by the same non-zero quantity on both sides of an equation, maintaining equality. For example, if a = b, then a / c = b / c, where c ≠ 0. Solving 5x = 15 yields x = 3 by dividing both sides by 5. This term builds on equality principles, showing another method to isolate variables and solve equations.

Topic

Equations

Description

Equality refers to a mathematical statement where two expressions are equal. Examples include 1 + 2 = 3 and x + 3 = 7, where both sides represent the same value. This term defines the core concept of equations, ensuring all manipulations preserve equality to find solutions.

This is the transcript for the video of same title. Video contents: In this episode of Algebra Applications, two real-world explorations of inequalities are developed: Hybrid Cars. An examination of the equations and inequalities that involve miles per gallon (mpg) for city and highway traffic reveals important information about hybrid cars and those with gasoline-powered engines. Floods in Venice. Venice experiences a great deal of flooding, and with the expected rise of sea levels over the next century, this ancient city is in peril. Through a series of inequalities, students analyze the impact of flooding.

Linear Expressions, Equations, and Functions

This is the transcript for the video of same title. Video contents: In this episode of Algebra Applications, students explore various scenarios that can be explained through the use of systems of equations. Such disparate phenomena as profit and loss, secret codes, and ballistic missile shields can be explored through systems of equations.

This is the transcript for the video of same title. Video contents: Profit and loss are the key measures in a business. A system of equations that includes an equation for income and one for expenses can be used to determine profit and loss. Students solve a system graphically.

This is the transcript for the video of same title. Video contents: Secret codes and encryption are ideal examples of a system of equations. In this activity, students encrypt and decrypt a message.

To see the complete collection of video transcriptsy, click on this link.

This is the transcript for the video of same title. Video contents: A ballistic missile shield allows you to shoot incoming missiles out of the sky. Mathematically, this is an example of a linear-quadratic system. Students graph such a system and find the points of intersection between a line and a parabola.

This is the transcript for the video of same title. Video contents: An overview of the key topics to be covered in the video.

To see the complete collection of video transcriptsy, click on this link.

This is the transcript for the video of same title. Video contents: Why do rivers meander instead of traveling in a straight line? In going from point A to point B, why should a river take the circuitous route it does instead of a direct path? Furthermore, what information can the ratio of the river's length to its straight-line distance tell us? In this segment the geological forces that account for a river's motion are explained. In the process, the so-called Meander Ratio is explored. Students construct a mathematical model of a meandering river using the TI-Nspire. Having built the model, students then use it to generate data to find the average of many Meander Ratios. The results show that on average the Meander Ratio is equal to pi.

This is the transcript for the video of same title. Video contents: Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video introduces students to systems of linear equations in two or three unknowns. To solve these systems, the host illustrates a variety of methods: four involve the TI-Nspire (spreadsheet, graphs and geometry, matrices and nSolve) and two are the classic algebraic methods known as substitution and elimination, also called the linear combinations method. The video ends with a summary of the three possible types of solutions. Concepts explored: equations, linear equations, linear systems

This is the transcript for the video of same title. Video contents: In this Investigation we solve a linear system.

To see the complete collection of video transcriptsy, click on this link.