Illustrative Math Alignment: Grade 8 Unit 3

Video Tutorial: Linear Functions: Negative Slope, Zero y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a negative slope and a zero y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Negative y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a negative y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Positive y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a positive y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Zero y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a zero y-intercept.

Topic

Linear Functions

Definition

A constant function is a linear function of the form f(x) = b, where b is a constant. The graph of a constant function is a horizontal line.

Description

Constant functions are a fundamental concept in linear functions. They represent scenarios where the output value remains unchanged, regardless of the input value. This is depicted graphically as a horizontal line, indicating that the function's rate of change is zero.

In real-world applications, constant functions can model situations where a quantity remains steady over time. For example, a flat fee service charge that does not vary with usage can be represented as a constant function.

Topic

Linear Functions

Definition

Converting from standard form to slope-intercept form involves rewriting a linear equation from the form Ax + By = C to the form y = mx + b, where m is the slope and b is the y-intercept.

Description

Converting linear equations from standard form to slope-intercept form is a key skill in algebra. This conversion allows for easier graphing and interpretation of the equation's slope and y-intercept.

Topic

Linear Functions

Definition

A decreasing linear function is a linear function where the slope is negative, indicating that as the input value increases, the output value decreases.

Description

Decreasing linear functions are important in understanding how variables inversely relate to each other. The negative slope signifies a reduction in the dependent variable as the independent variable increases.

Real-world examples include depreciation of assets over time or the decrease in temperature as altitude increases. These functions help model scenarios where an increase in one quantity results in a decrease in another.

Topic

Linear Functions

Definition

Direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other, expressed as y = kx, where k is the constant of variation.

Description

Direct variation is a fundamental concept in linear functions, illustrating how one variable changes proportionally with another. The constant of variation, k, represents the rate of change.

In real-world scenarios, direct variation can model relationships such as speed and distance, where distance traveled varies directly with time at a constant speed. Understanding this concept is crucial in fields like physics and engineering.

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.

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.

Topic

Linear Functions

Definition

An identity function is a linear function of the form f(x) = x, where the output is equal to the input for all values of x.

Description

The identity function is a basic yet crucial concept in linear functions. It represents a scenario where the input value is always equal to the output value, graphically depicted as a 45-degree line passing through the origin.

In real-world applications, the identity function can model situations where input and output are directly proportional and identical, such as converting units of the same measure. This also introduces the concept of identity, which is fundamental to mathematics.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

In this Algebra Application, students develop a linear mathematical model based on the maximum heart rate during exercise based on age. Using this model, students investigate heart rate for moderate and vigorous workouts. The math topics covered include: Mathematical modeling, Linear functions, Families of functions, Computational thinking. The culminating activity is for students to create a simple program (using either a spreadsheet or a programming language like Python) to develop an exercise chart. This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.

In this Algebra Application, students study the direction between diameter and circumference of a circle. Through measurement and data gathering students analyze the line of best fit and explore ways of calculating pi. The math topics covered include: Mathematical modeling, Linear functions, Data gathering and analysis, Ratios, Direct variation. This is a great back-to-school activity for middle school or high school students. This is also a great crossover activity that ties algebra and geometry.

In this Slide Show, apply concepts of linear functions to the context of circumference vs. diameter.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of cost vs. time data and graphs.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of cricket chirps.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of distance vs. time data and graphs.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of Hooke's law.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of momentum and impulse.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of rectangular perimeter.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of saving money.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of speed and acceleration.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of converting Celsius and Fahrenheit temperatures.

Note: The download is a PPT file.

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In this Slide Show, use the Desmos graphing calculator to explore the point-slope form. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore linear equations in slope intercept form. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore linear equations in standard form. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore absolute value functions. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore linear functions. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, learn about graphs of linear functions.

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In this Slide Show, learn how to convert from linear equations in standard form to slope intercept form.

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In this Slide Show, learn about linear function models.

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The complete set of 32 examples that make up this set of tutorials.

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

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The complete set of 29 examples that make up this set of tutorials.

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This PowerPoint includes the 22 Tutorials on the topic of converting Linear Equations in Standard Form to Slope-Intercept Form.

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This set of 21 tutorials provides different examples of linear functions in function table and graph form.

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This set of tutorials provides 32examples that involve finding the equation of a line parallel or perpendicular to a given line and through a given point.

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The complete set of 8 examples that make up this set of tutorials.

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In this Slide Show, learn how graph data in a scatterplot using the graph window. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.