PA.A.2.3
PA.A.2.3 Identify graphical properties of linear functions including slope and intercepts. Know that the slope equals the rate of change, and that the yintercept is zero when the function represents a proportional relationship.
In a Nutshell
Extending the understanding of linear data, students will be able to determine the rate of change between two ordered pairs or models, such as graphs and tables and identify it as the slope of a line. Students will be also able to determine the yintercept of a linear function when the x value is zero. Students will understand that the origin in a proportional relationship represents a linear function with a yintercept of zero.
Student Actions

Teacher Actions


Develop a Deep and Conceptual Understanding when identifying properties of the slope and yintercepts of a linear function using models, such as graphs, tables, and equations.

Develop Ability to Make Conjectures, Model and Generalize when examining patterns of change between coordinate points, in tables and/or graphs, to draw conclusions about properties of slope for any linear function, including a negative rate of change.

Develop Accurate and Appropriate Procedural Fluency when using efficient and accurate procedures with strategies to determine the constant of proportionality (unit rate or slope) for various mathematical situations of linear functions, including proportional relationships.
 Develop the Ability to Communicate Mathematically when translating the idea of slope to rate of change in various mathematical situations and demonstrate understanding through verbal, written, and visual explanations.


Use and connect mathematical representations by allowing students to develop different strategies for calculating and modeling slope both graphically and mathematically.

Pose purposeful questions that allow students to view slope in a contextual problem as a rate of change between 2 variables; instead of only a value that can be written as a fraction.
 Implement meaningful tasks that promote reasoning and problemsolving by giving students ample examples of linear equations and their graphs to explore proportional vs nonproportional relationships. Allow students to look for patterns to discover why graphs are classified as either proportional or nonproportional and determine how these patterns are connected to linear functions.

Key Understandings

Misconceptions


Understand that slope is the constant rate of change between two variables and can be represented by similar triangles.

Understand the yintercept of a linear function is the location where the line intersects the yaxis and the xvalue is zero.

Interpret the slope, including both positive and negative rates of change, and yintercept in the context of the given mathematical situations.

Know a proportional relationship is a linear function that contains a yintercept at y=0.

Be able to describe what makes a relationship proportional and in turn, nonproportional, and identify the properties of each representation when given the data in a table, equation, graph, or mathematical situation.


Students confuse which value within a linear equation is the yintercept.

When calculating the slope from a graph, students sometimes will not pay attention to the "direction" of the rise or the run which will sometimes cause the sign of the slope to be wrong.

Knowledge Connections

Prior Knowledge

Leads to


Determine and compare the unit rates (constant of proportionality, slope, or rate of change) (7.A.2.1)

Distinguish between proportional and nonproportional relationships (7.A.1.1)

Recognize that the graph of a proportional relationship is a line through the origin and the coordinate (1,r), where r is the slope and the unit rate (constant of proportionality, k) (7.A.1.2)


Analyze, use and apply mathematical models and other data sets to calculate and interpret slope and the x and yintercepts of a line (A1.A.4.1).

Analyze and interpret mathematical models involving lines that are parallel, perpendicular, horizontal, and vertical (A1.A.4.2).

OKMath Framework Introduction
PreAlgebra Introduction
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