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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 8 Unit 3

Linear Relationships

Lesson 6: More Linear Relationships

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topic
Definition--Linear Function Concepts--Increasing Linear Function 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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