Illustrative Math Alignment: Grade 8 Unit 2

Dilations, Similarity, and Introducing Slope

Lesson 12: Using Equations for Lines

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Definition--Direct Variation	Direct Variation 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 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.	Applications of Linear Functions
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--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--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--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--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 Formula	Slope Formula Topic Slope Definition The Slope Formula calculates the rate of change. Description The Slope Formula is typically represented as This formula is a cornerstone in algebra, enabling students to analyze linear equations effectively.	Slope
Definition--Slope Concepts--Slope-Intercept Form	Slope-Intercept Form Topic Slope Definition The slope-intercept form is an equation of a straight line represented as y = m x + b, where m is the slope and b is the y-intercept. Description The slope-intercept form is a fundamental concept in algebra, providing a straightforward way to graph linear equations. It is widely used in various fields, from economics to physics, to model linear relationships and predict outcomes based on given data. Understanding this form is crucial for students as it lays the foundation for more complex mathematical concepts and real-world applications.	Slope-Intercept Form
Definition--Slope Concepts--Slopes of Parallel Lines	Slopes of Parallel Lines Topic Slope Definition Parallel lines are lines in the same plane that never meet; they have equal slopes. Description Understanding the slopes of parallel lines is essential in geometry and algebra, as it helps in identifying and proving parallelism in shapes and graphs. This concept is applied in various fields, including architecture and engineering, where maintaining parallelism is crucial for structural integrity and design. In mathematics education, this concept aids in developing logical reasoning and problem-solving skills.	Slope
Definition--Slope Concepts--Slopes of Perpendicular Lines	Slopes of Perpendicular Lines Topic Slope Definition Perpendicular lines intersect at a right angle, and their slopes are negative reciprocals of each other. Description The concept of perpendicular slopes is vital in geometry, as it helps in determining the orthogonality of lines. This principle is used in various applications, such as designing perpendicular intersections in roadways and creating right-angle joints in construction. In education, it enhances students' understanding of geometric properties and their ability to solve related problems.	Slope
Definition--Slope Concepts--Steepness	Steepness Topic Slope Definition Steepness is a measure of how steep a line is, typically calculated as the absolute value of the slope. Description Steepness is a key concept in understanding the inclination of lines and surfaces. It is commonly used in fields like geology and civil engineering to assess the gradient of terrains and structures. In math education, learning about steepness helps students grasp the concept of slope and its practical implications, enhancing their analytical skills and understanding of real-world phenomena.	Slope
Definition--Slope Concepts--The Rise Over the Run	The Rise Over the Run Topic Slope Definition The rise over the run describes the change in y over the change in x in a linear relationship. Description The concept of "rise over run" is foundational in understanding how to calculate the slope of a line. It is used in various disciplines, such as physics and economics, to model relationships and predict trends. In education, mastering this concept is crucial for students as it forms the basis for graphing linear equations and understanding the behavior of lines in a coordinate plane.	Slope
Definition--Slope Concepts--Undefined Slope	Undefined Slope Topic Slope Definition Undefined slope occurs when a vertical line is present, meaning there is no change in x. Description An undefined slope is a critical concept in mathematics, particularly in graphing. It indicates a vertical line where the change in x is zero, making the slope calculation impossible. This concept is important in various fields, such as computer graphics and data analysis, where vertical lines can represent boundaries or limits. Understanding undefined slopes helps students in identifying and interpreting vertical lines in graphs.	Slope
Definition--Slope Concepts--Zero Slope	Zero Slope Topic Slope Definition Zero slope describes a horizontal line where there is no change in y as x changes. Description A zero slope is a key concept in algebra and geometry, representing a horizontal line. It is used in various applications, such as designing flat surfaces and analyzing constant relationships in data. In education, understanding zero slopes helps students in graphing and interpreting horizontal lines, enhancing their ability to analyze linear relationships. For a complete collection of terms related to Slope click on this link: Slope Collection.	Slope
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
Desmos Activity: Slope As Rate of Change	Desmos Activity: Slope As Rate of Change Use this activity to explore slope as a rate of change. In this Desmos activity, the slope of the line is the rate (cost per pound) for purchasing fruit. Students manipulate the slider for m to see the impact on the cost.	Slope and Ratios and Rates
Desmos Activity: Slope As Rise Over Run	Desmos Activity: Slope As Rise Over Run This Desmos activity allows students to explore slope as the ratio of the rise over the run. Click Preview to launch the activity. In the activity students click on the points to create a line segment connecting two points. Use the background grid to determine the rise and the run and calculate the slope.	Slope
Desmos Activity: Slope As Rise Over Run 2	Desmos Activity: Slope As Rise Over Run 2 In this activity, students input different values for the coordinates to create a new line. They can then measure the rise and run to calculate the slope.	Slope
Formulas--Slope Formula	Formulas \| Slope Formula The formula for the Slope Formula. This is part of a collection of math formulas. —PRESS PREVIEW TO SEE THE IMAGE— To see the complete collection of formulas, click on this link. The following section includes background information on slope. This background also includes video resources and accompanying transcripts. What Is Slope? Watch this video to learn about the slope formula. (The transcript is also included.)	Slope
Instructional Resource--Strategy Pack--The Slope of a Line	Instructional Resource \| Strategy Pack \| The Slope of a Line Learn different strategies for finding the slope of a line. 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.	Slope
INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Point-Slope Form	INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Point-Slope Form 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. Note: The download is a PPT file. This is part of a collection of Desmos tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Linear Functions, Graphs of Linear Functions and Point-Slope Form
INSTRUCTIONAL RESOURCE: Math Examples 19	INSTRUCTIONAL RESOURCE: Math Examples--Graphing Linear Functions in Slope-Intercept Form The complete set of 13 examples that make up this set of tutorials. This is part of a collection of math examples for 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.	Applications of Linear Functions and Graphs of Linear Functions
INSTRUCTIONAL RESOURCE: Math Examples 25	INSTRUCTIONAL RESOURCE: Math Examples--Line From Two Points The complete set of 29 examples that make up this set of tutorials. This is part of a collection of math examples for 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.	Applications of Linear Functions and Graphs of Linear Functions
INSTRUCTIONAL RESOURCE: Math Examples 26	INSTRUCTIONAL RESOURCE: Math Examples--Linear Equations in Standard Form This PowerPoint includes the 22 Tutorials on the topic of converting Linear Equations in Standard Form to Slope-Intercept Form. This is part of a collection of math examples for 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.	Applications of Linear Functions and Graphs of Linear Functions
INSTRUCTIONAL RESOURCE: Math Examples 30	INSTRUCTIONAL RESOURCE: Math Examples--Midpoint Formula The complete set of 20 examples that make up this set of tutorials. This is part of a collection of math examples for 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.	Midpoint Formula
INSTRUCTIONAL RESOURCE: Math Examples 36	INSTRUCTIONAL RESOURCE: Math Examples--Point-Slope Form The complete set of 8 examples that make up this set of tutorials. This is part of a collection of math examples for 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.	Applications of Linear Functions and Graphs of Linear Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Point-Slope Form	In this Slide Show, create a template for determining the equation of a line given a point and the slope of the line. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Applications of Linear Functions, Graphs of Linear Functions and Point-Slope Form
INSTRUCTIONAL RESOURCE: Tutorial: The Point-Slope Form	<h1>INSTRUCTIONAL RESOURCE: Tutorial: The Point-Slope Form</h1><p>In this Slide Show, learn about thepoint-slope form. </p><h3> This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, <a href="https://www.media4math.com/instructional-resources?field_la_display_tit… on this link.</u></strong></a> </h3> <h3> Note: The download is a PPT file. </h3>< <h2> Library of Instructional Resources </h3>	Applications of Linear Functions, Graphs of Linear Functions and Point-Slope Form
INSTRUCTIONAL RESOURCE: Tutorial: Equations of Parallel and Perpendiclar Lines (SAT Prep)	INSTRUCTIONAL RESOURCE: Tutorial: Equations of Parallel and Perpendiclar Lines (SAT Prep) This slide show shows how to write equations in slope-intercept form for parallel and perpendicular lines. This slide show also includes some sample SAT-style questions. 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.	Slope-Intercept Form
INSTRUCTIONAL RESOURCE: Tutorial: Point-Slope Form (SAT Prep)	INSTRUCTIONAL RESOURCE: Tutorial: Point-Slope Form (SAT Prep) This slide show shows how to use the point-slope form to write a linear equation in slope-intercept form. This slide show also includes some sample SAT-style questions. 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.	Point-Slope Form
Interactive Crossword Puzzle--Slope Intercept Form	Interactive Crossword Puzzle--Slope Intercept Form This interactive crossword puzzle tests knowledge of key terms on the topic of the slope intercept form. 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.	Slope-Intercept Form
Math Example--Geometric Transformation--Transformations: Example 1	Math Example--Geometric Transformation--Transformations: Example 1 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 10	Math Example--Geometric Transformation--Transformations: Example 10 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 11	Math Example--Geometric Transformation--Transformations: Example 11 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 12	Math Example--Geometric Transformation--Transformations: Example 12 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 13	Math Example--Geometric Transformation--Transformations: Example 13 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 14	Math Example--Geometric Transformation--Transformations: Example 14 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 15	Math Example--Geometric Transformation--Transformations: Example 15 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 16	Math Example--Geometric Transformation--Transformations: Example 16 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 17	Math Example--Geometric Transformation--Transformations: Example 17 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 18	Math Example--Geometric Transformation--Transformations: Example 18 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 19	Math Example--Geometric Transformation--Transformations: Example 19 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 2	Math Example--Geometric Transformation--Transformations: Example 2 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 20	Math Example--Geometric Transformation--Transformations: Example 20 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 21	Math Example--Geometric Transformation--Transformations: Example 21 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 22	Math Example--Geometric Transformation--Transformations: Example 22 This is part of a collection of math examples that focus on geometric transformations.	Definition of Transformations