Illustrative Math Alignment: Grade 8 Unit 3

Linear Relationships

Lesson 5: Introduction to Linear Relationships

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Definition--Linear Function Concepts--Increasing Linear Function	Increasing Linear Function Topic Linear Functions Definition An increasing linear function is a linear function where the slope is positive, indicating that as the input value increases, the output value also increases. Description Increasing linear functions are essential in understanding how variables positively relate to each other. The positive slope signifies an increase in the dependent variable as the independent variable increases. Real-world examples include income increasing with hours worked or the rise in temperature with the increase in daylight hours. These functions help model scenarios where an increase in one quantity results in an increase in another.	Slope-Intercept Form
Definition--Linear Function Concepts--Increasing Linear Function	Increasing Linear Function Topic Linear Functions Definition An increasing linear function is a linear function where the slope is positive, indicating that as the input value increases, the output value also increases. Description Increasing linear functions are essential in understanding how variables positively relate to each other. The positive slope signifies an increase in the dependent variable as the independent variable increases. Real-world examples include income increasing with hours worked or the rise in temperature with the increase in daylight hours. These functions help model scenarios where an increase in one quantity results in an increase in another.	Slope-Intercept Form
Definition--Linear Function Concepts--Line of Best Fit	Line of Best Fit Topic Linear Functions Definition A line of best fit is a straight line that best represents the data on a scatter plot, showing the trend of the data points. Description The line of best fit is a crucial concept in statistics and data analysis. It helps in identifying the trend and making predictions based on the data. In real-world applications, the line of best fit is used in various fields such as economics, biology, and engineering to analyze trends and make forecasts. For example, it can be used to predict future sales based on past data.	Graphs of Linear Functions
Definition--Linear Function Concepts--Line of Best Fit	Line of Best Fit Topic Linear Functions Definition A line of best fit is a straight line that best represents the data on a scatter plot, showing the trend of the data points. Description The line of best fit is a crucial concept in statistics and data analysis. It helps in identifying the trend and making predictions based on the data. In real-world applications, the line of best fit is used in various fields such as economics, biology, and engineering to analyze trends and make forecasts. For example, it can be used to predict future sales based on past data.	Graphs of Linear Functions
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--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--Linear Function	Linear Function Topic Linear Functions Definition A linear function is a function that can be graphed as a straight line, typically written in the form y = mx + b, where m is the slope and b is the y-intercept. Description Linear functions are one of the most fundamental concepts in mathematics. They describe relationships where the rate of change between variables is constant, represented graphically as a straight line. This simplicity makes them a central topic in algebra and calculus.	Slope-Intercept Form
Definition--Linear Function Concepts--Linear Function	Linear Function Topic Linear Functions Definition A linear function is a function that can be graphed as a straight line, typically written in the form y = mx + b, where m is the slope and b is the y-intercept. Description Linear functions are one of the most fundamental concepts in mathematics. They describe relationships where the rate of change between variables is constant, represented graphically as a straight line. This simplicity makes them a central topic in algebra and calculus.	Slope-Intercept Form
Definition--Linear Function Concepts--Linear Function Tables	Linear Function Tables Topic Linear Functions Definition Linear function tables display the input-output pairs of a linear function, showing how the dependent variable changes with the independent variable. Description Linear function tables are useful tools for understanding and analyzing linear relationships. They provide a clear way to see how changes in the input (independent variable) affect the output (dependent variable).	Applications of Linear Functions and Graphs of Linear Functions
Definition--Linear Function Concepts--Linear Function Tables	Linear Function Tables Topic Linear Functions Definition Linear function tables display the input-output pairs of a linear function, showing how the dependent variable changes with the independent variable. Description Linear function tables are useful tools for understanding and analyzing linear relationships. They provide a clear way to see how changes in the input (independent variable) affect the output (dependent variable).	Applications of Linear Functions and Graphs of Linear Functions
Definition--Linear Function Concepts--Linear Function Tables	Linear Function Tables Topic Linear Functions Definition Linear function tables display the input-output pairs of a linear function, showing how the dependent variable changes with the independent variable. Description Linear function tables are useful tools for understanding and analyzing linear relationships. They provide a clear way to see how changes in the input (independent variable) affect the output (dependent variable).	Applications of Linear Functions and Graphs of Linear Functions
Definition--Linear Function Concepts--Point-Slope Form	Point-Slope Form Topic Linear Functions Definition Point-slope form of a linear equation is written as y − y1 = m(x−x1 ), where m is the slope and (x1 ,y1) is a point on the line. Description The point-slope form is a versatile way to express linear equations, especially useful when you know a point on the line and the slope. It allows for quick construction of the equation of a line.	Point-Slope Form
Definition--Linear Function Concepts--Point-Slope Form	Point-Slope Form Topic Linear Functions Definition Point-slope form of a linear equation is written as y − y1 = m(x−x1 ), where m is the slope and (x1 ,y1) is a point on the line. Description The point-slope form is a versatile way to express linear equations, especially useful when you know a point on the line and the slope. It allows for quick construction of the equation of a line.	Point-Slope Form
Definition--Linear Function Concepts--Rate of Change	Rate of Change Topic Linear Functions Definition Rate of change in a linear function is the ratio of the change in the dependent variable to the change in the independent variable, often represented as the slope m in the equation y = mx + b. Description Rate of change is a fundamental concept in understanding linear functions. It describes how one variable changes in relation to another, and is graphically represented by the slope of a line.	Slope
Definition--Linear Function Concepts--Rate of Change	Rate of Change Topic Linear Functions Definition Rate of change in a linear function is the ratio of the change in the dependent variable to the change in the independent variable, often represented as the slope m in the equation y = mx + b. Description Rate of change is a fundamental concept in understanding linear functions. It describes how one variable changes in relation to another, and is graphically represented by the slope of a line.	Slope
Definition--Linear Function Concepts--Slope-Intercept Form	Slope-Intercept Form Topic Linear Functions Definition Slope-intercept form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept. Description Slope-intercept form is one of the most commonly used forms of linear equations. It provides a clear way to understand the slope and y-intercept of a line, making it easier to graph and interpret. In real-world applications, slope-intercept form is used in various fields such as economics, physics, and engineering to model linear relationships. For example, it can represent the relationship between cost and production levels in business.	Slope-Intercept Form
Definition--Linear Function Concepts--Slope-Intercept Form	Slope-Intercept Form Topic Linear Functions Definition Slope-intercept form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept. Description Slope-intercept form is one of the most commonly used forms of linear equations. It provides a clear way to understand the slope and y-intercept of a line, making it easier to graph and interpret. In real-world applications, slope-intercept form is used in various fields such as economics, physics, and engineering to model linear relationships. For example, it can represent the relationship between cost and production levels in business.	Slope-Intercept 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--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--Linear Function Concepts--x-Intercept	x-Intercept Topic Linear Functions Definition The x-intercept is the point where a graph crosses the x-axis, indicating the value of x when 𝑦 y is zero. Description The x-intercept is a key concept in understanding the behavior of linear functions. It represents the point where the function's output is zero, providing insight into the function's roots and behavior. In real-world applications, the x-intercept can be used to determine break-even points in business, where revenue equals costs, or to find the time at which a process starts or stops.	Slope-Intercept Form
Definition--Linear Function Concepts--x-Intercept	x-Intercept Topic Linear Functions Definition The x-intercept is the point where a graph crosses the x-axis, indicating the value of x when 𝑦 y is zero. Description The x-intercept is a key concept in understanding the behavior of linear functions. It represents the point where the function's output is zero, providing insight into the function's roots and behavior. In real-world applications, the x-intercept can be used to determine break-even points in business, where revenue equals costs, or to find the time at which a process starts or stops.	Slope-Intercept Form
Definition--Linear Function Concepts--y-Intercept	y-Intercept Topic Linear Functions Definition The y-intercept is the point where a graph crosses the y-axis, indicating the value of y when x is zero. Description The y-intercept is a fundamental concept in understanding the behavior of linear functions. It represents the initial value of the function when the input is zero, providing insight into the function's starting point. In real-world applications, the y-intercept can be used to determine initial conditions in various scenarios, such as the starting balance in a bank account or the initial position of an object in motion.	Slope-Intercept Form
Definition--Linear Function Concepts--y-Intercept	y-Intercept Topic Linear Functions Definition The y-intercept is the point where a graph crosses the y-axis, indicating the value of y when x is zero. Description The y-intercept is a fundamental concept in understanding the behavior of linear functions. It represents the initial value of the function when the input is zero, providing insight into the function's starting point. In real-world applications, the y-intercept can be used to determine initial conditions in various scenarios, such as the starting balance in a bank account or the initial position of an object in motion.	Slope-Intercept Form
Definition--Slope Concepts--Delta x	Delta x Topic Slope Definition Delta x is the change in the x-coordinate in a linear relationship. Description The term Delta x is fundamental to understanding slope as it represents the horizontal change in a line’s position. In real-world applications, Delta x can represent the distance traveled over time in physics and engineering contexts.	Slope
Definition--Slope Concepts--Delta x	Delta x Topic Slope Definition Delta x is the change in the x-coordinate in a linear relationship. Description The term Delta x is fundamental to understanding slope as it represents the horizontal change in a line’s position. In real-world applications, Delta x can represent the distance traveled over time in physics and engineering contexts.	Slope
Definition--Slope Concepts--Delta x	Delta x Topic Slope Definition Delta x is the change in the x-coordinate in a linear relationship. Description The term Delta x is fundamental to understanding slope as it represents the horizontal change in a line’s position. In real-world applications, Delta x can represent the distance traveled over time in physics and engineering contexts.	Slope
Definition--Slope Concepts--Delta y	Delta y Topic Slope Definition Delta y is the change in the y-coordinate in a linear relationship. Description The term Delta y signifies the vertical change, essential for calculating slope used in graphing and data interpretation. In practical scenarios, Delta y can illustrate changes in temperature over time or the rise in elevation in geography.	Slope
Definition--Slope Concepts--Delta y	Delta y Topic Slope Definition Delta y is the change in the y-coordinate in a linear relationship. Description The term Delta y signifies the vertical change, essential for calculating slope used in graphing and data interpretation. In practical scenarios, Delta y can illustrate changes in temperature over time or the rise in elevation in geography.	Slope
Definition--Slope Concepts--Delta y	Delta y Topic Slope Definition Delta y is the change in the y-coordinate in a linear relationship. Description The term Delta y signifies the vertical change, essential for calculating slope used in graphing and data interpretation. In practical scenarios, Delta y can illustrate changes in temperature over time or the rise in elevation in geography.	Slope
Definition--Slope Concepts--Grade	Grade Topic Slope Definition Grade is a ratio indicating the steepness of a slope. Description The term Grade commonly refers to the angle of elevation or slope expressed as a percentage. In engineering, grades are crucial for designing roads and railways, ensuring safety and efficiency.	Slope
Definition--Slope Concepts--Grade	Grade Topic Slope Definition Grade is a ratio indicating the steepness of a slope. Description The term Grade commonly refers to the angle of elevation or slope expressed as a percentage. In engineering, grades are crucial for designing roads and railways, ensuring safety and efficiency.	Slope
Definition--Slope Concepts--Grade	Grade Topic Slope Definition Grade is a ratio indicating the steepness of a slope. Description The term Grade commonly refers to the angle of elevation or slope expressed as a percentage. In engineering, grades are crucial for designing roads and railways, ensuring safety and efficiency.	Slope
Definition--Slope Concepts--Gradient	Gradient Topic Slope Definition Gradient represents the rate of change of a quantity. Description The Gradient measures how steep a line is, calculated by the ratio of the rise to run. This concept is significant in fields like physics and economics where gradients can represent relationships between variables.	Slope
Definition--Slope Concepts--Gradient	Gradient Topic Slope Definition Gradient represents the rate of change of a quantity. Description The Gradient measures how steep a line is, calculated by the ratio of the rise to run. This concept is significant in fields like physics and economics where gradients can represent relationships between variables.	Slope
Definition--Slope Concepts--Gradient	Gradient Topic Slope Definition Gradient represents the rate of change of a quantity. Description The Gradient measures how steep a line is, calculated by the ratio of the rise to run. This concept is significant in fields like physics and economics where gradients can represent relationships between variables.	Slope
Definition--Slope Concepts--Negative Slope	Negative Slope Topic Slope Definition Negative Slope indicates a decrease in value as x increases. Description The Negative Slope signifies that as one variable increases, the other decreases, often used in economics to depict inverse relationships. This term finds relevance in real-world scenarios like demand curves which slope downwards.	Slope
Definition--Slope Concepts--Negative Slope	Negative Slope Topic Slope Definition Negative Slope indicates a decrease in value as x increases. Description The Negative Slope signifies that as one variable increases, the other decreases, often used in economics to depict inverse relationships. This term finds relevance in real-world scenarios like demand curves which slope downwards.	Slope
Definition--Slope Concepts--Negative Slope	Negative Slope Topic Slope Definition Negative Slope indicates a decrease in value as x increases. Description The Negative Slope signifies that as one variable increases, the other decreases, often used in economics to depict inverse relationships. This term finds relevance in real-world scenarios like demand curves which slope downwards.	Slope
Definition--Slope Concepts--Pitch	Pitch Topic Slope Definition Pitch refers to the steepness of a slope in a real-world context. Description Pitch is commonly applied in construction and design fields to denote the angle of roofs and ramps. This is essential in architecture, especially when dealing with drainage and material use.	Slope
Definition--Slope Concepts--Pitch	Pitch Topic Slope Definition Pitch refers to the steepness of a slope in a real-world context. Description Pitch is commonly applied in construction and design fields to denote the angle of roofs and ramps. This is essential in architecture, especially when dealing with drainage and material use.	Slope
Definition--Slope Concepts--Pitch	Pitch Topic Slope Definition Pitch refers to the steepness of a slope in a real-world context. Description Pitch is commonly applied in construction and design fields to denote the angle of roofs and ramps. This is essential in architecture, especially when dealing with drainage and material use.	Slope
Definition--Slope Concepts--Point-Slope Form	Point-Slope Form Topic Slope Definition Point-Slope Form is a way to express linear equations given the slope and a point on the line. Description The Point-Slope Form is crucial in algebra for identifying linear relationships given the slope and a point on the line. Understanding this concept aids in graphing lines efficiently and is foundational in higher mathematics. For a complete collection of terms related to Slope click on this link: Slope Collection.	Point-Slope Form
Definition--Slope Concepts--Point-Slope Form	Point-Slope Form Topic Slope Definition Point-Slope Form is a way to express linear equations given the slope and a point on the line. Description The Point-Slope Form is crucial in algebra for identifying linear relationships given the slope and a point on the line. Understanding this concept aids in graphing lines efficiently and is foundational in higher mathematics. For a complete collection of terms related to Slope click on this link: Slope Collection.	Point-Slope Form
Definition--Slope Concepts--Point-Slope Form	Point-Slope Form Topic Slope Definition Point-Slope Form is a way to express linear equations given the slope and a point on the line. Description The Point-Slope Form is crucial in algebra for identifying linear relationships given the slope and a point on the line. Understanding this concept aids in graphing lines efficiently and is foundational in higher mathematics. For a complete collection of terms related to Slope click on this link: Slope Collection.	Point-Slope Form
Definition--Slope Concepts--Positive Slope	Positive Slope Topic Slope Definition Positive Slope indicates an increase in value as x increases. Description The Positive Slope indicates that as x grows, y also rises, signifying direct relationships in data analysis. This principle enables predictions in various analytics and trends. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
Definition--Slope Concepts--Positive Slope	Positive Slope Topic Slope Definition Positive Slope indicates an increase in value as x increases. Description The Positive Slope indicates that as x grows, y also rises, signifying direct relationships in data analysis. This principle enables predictions in various analytics and trends. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
Definition--Slope Concepts--Positive Slope	Positive Slope Topic Slope Definition Positive Slope indicates an increase in value as x increases. Description The Positive Slope indicates that as x grows, y also rises, signifying direct relationships in data analysis. This principle enables predictions in various analytics and trends. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
Definition--Slope Concepts--Slope	Slope Topic Slope Definition Slope measures the steepness of a line. Description Slope is a fundamental concept in mathematics, expressing the ratio of vertical rise to horizontal run. This is vital for understanding linear functions, essential in fields such as physics, economics, and data science. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
Definition--Slope Concepts--Slope	Slope Topic Slope Definition Slope measures the steepness of a line. Description Slope is a fundamental concept in mathematics, expressing the ratio of vertical rise to horizontal run. This is vital for understanding linear functions, essential in fields such as physics, economics, and data science. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
Definition--Slope Concepts--Slope	Slope Topic Slope Definition Slope measures the steepness of a line. Description Slope is a fundamental concept in mathematics, expressing the ratio of vertical rise to horizontal run. This is vital for understanding linear functions, essential in fields such as physics, economics, and data science. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope