ALEX Classroom Resource

Students:

• can analyze linear equations in the form y=mx + b as representing a line where m represents the rate of change, called the slope of the line when graphed. and b is the initial value, called the y-intercept when graphed.
• Can create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
• Can explain why any two points on a given line will have the same slope.
• Can graph linear relationships on a coordinate plane when given in multiple contexts.

Students know:

• how to graph points on a coordinate plane.
• Where to graph the initial value/y-intercept.
• Understand how/why triangles are similar.
• how to interpret y=mx equations.

Students are able to:

• create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
• point out similar triangles formed between pairs of points and know that they have same slope between any pairs of those points.
• Show that lines may share the same slope but can have different y-intercepts.
• Interpret a rate of change as the slope and the initial value as the y-intercept.

Students understand that:

• slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
• When given an equation in the form y = mx + b it generally symbolizes that you will have lines with varying y-intercepts. even when the slope is the same.
• you can use the visual of right triangles created between points on a line to explain why the slope is a constant rate of change.

Students:

• Analyze linear equations in the form y=mx + b as representing a line where m represents the rate of change, called the slope of the line when graphed. and b is the initial value, called the y-intercept when graphed.
• Create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
• Explain why any two points on a given line will have the same slope.
• Graph linear relationships on a coordinate plane when given in multiple contexts.

Students know:

• how to graph points on a coordinate plane.
• Where to graph the initial value/y-intercept.
• Understand how/why triangles are similar.
• how to interpret y=mx equations.

Students are able to:

• create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
• point out similar triangles formed between pairs of points and know that they have the same slope between any pairs of those points.
• Show that lines may share the same slope but can have different y-intercepts.
• Interpret a rate of change as the slope and the initial value as the y-intercept.

Students understand that:

• Slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
• When given an equation in the form y = mx + b it generally symbolizes that there will be lines with varying y-intercepts. even when the slope is the same.
• Use of the visual of right triangles created between points on a line to explain why the slope is a constant rate of change.

Learning Objectives:
M.8.9.1: Define linear functions, nonlinear functions, slope, and y-intercept.
M.8.9.2: Recall how to solve problems using the distributive property.
M.8.9.3: Recognize linear equations.
M.8.9.4: Identify ordered pairs.
M.8.9.5: Recognize ordered pairs.
M.8.9.6: Define similar triangles, intercept, slope, vertical, horizontal, and origin.
M.8.9.7: Recognize similar triangles.
M.8.9.8: Generate the slope of a line using given ordered pairs.
M.8.9.9: Analyze the graph to determine the rate of change.
M.8.9.10: Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
M.8.9.11: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
M.8.9.12: Graph a function given the slope-intercept form of an equation.
M.8.9.13: Recognize that two sets of points with the same slope may have different y-intercepts.
M.8.9.14: Graph a linear equation given the slope-intercept form of an equation.

Students:

• Describe the comparison of linear functions qualitatively and quantitatively by discussing and analyzing the rates of change (slopes), initial values (y-intercepts), and any points of intersection.
• Tell the difference between functions that are linear and those that are non-linear by analyzing information in a variety of contexts.

Students know:

• how to find rates of change and initial values for function represented multiple ways.
• how to graph functions when given an equation, table, or verbal description.

Students are able to:

• identify the differences between functions represented in multiple contexts.
• Tell the differences between linear and nonlinear functions.

Students understand that:

• Converting to different representations of functions can assist in their comparisons of linear functions qualitatively and quantitatively.

Learning Objectives:
M.8.15.1: Define rate of change.
M.8.15.2: Recognize linear and nonlinear functions.
M.8.15.3: Recall how to read/interpret information from a table.
M.8.15.4: Identify algebraic expressions.
M.8.15.5: Recall how to name points on a Cartesian plane using ordered pairs.
M.8.15.6: Compare and contrast the differences between linear and nonlinear functions.

Students:
Given functions represented in various ways(algebraically, graphically, numerically in tables, or by verbal descriptions).

• Use key features to compare the functions.
• Explain and justify the similarities and differences of the functions.

• Given a variety of functions in equation form, use logical reasoning to justify their classification as linear or non-linear by interpreting the relationships in the expressions.

Students know:

• Techniques to find key features of functions when presented in different ways.
• Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).
• Characteristics of linear and nonlinear functions.

Students are able to:

• Accurately determine which key features are most appropriate for comparing functions.
• Manipulate functions algebraically to reveal key functions.
• Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.
• Compare functions based on their properties.

Students understand that:

• Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
• Different representations of functions may aid in comparing key features of the functions.
• Functions are relationships between two variables that have a unique characteristic, that being, for each input there exists exactly one output.
• Functions can be represented in a variety of ways (graphs, tables, and equations), each of which provides unique perspectives of the relationship between the variables.
• Linear functions have a defining characteristic of a unit rate or slope that other nonlinear functions do not have.