F.1.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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Formative Assessments
Formative Assessments
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CPALMS: Students are asked to describe defining properties of linear functions.

Explaining Linear Functions

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CPALMS: Students are asked to describe a linear function, its graph, and the meaning of its parameters.

What Am I?

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CPALMS: Students are asked to identify a function as either linear or nonlinear and to justify their decision.

Linear or Nonlinear?

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CPALMS: Students are asked to provide an example of a nonlinear function and explain why it is nonlinear.

Nonlinear Functions

Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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CPALMS: This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Equation Grapher

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CPALMS: This resource allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Graphing Lines

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CPALMS: This tutorial will help students to graph a line using its slope and y-intercept, or to identify the slope and y-intercept from a linear equation written in slope-intercept form.

Graph a Line Using Y-Intercept and Slope

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CPALMS: This resource provides guided practice for writing and graphing linear functions.

Linear Equations

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CPALMS: This manipulative will help students to explore the world of lines. Students can investigate the relationships between linear equations, slope, and graphs of lines.

Graphing Lines

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CPALMS: In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. It allows them to explore linear equations, slopes, and y-intercepts and their visual representation on a graph.

Slope Slider

Assessment Questions
Assessment Questions
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Multiple Choice
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Multi-Select
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Open Response
Videos of Expert Instruction
Videos of Expert Instruction
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Students will build fluency with predicting input/outputs for nonlinear functions from patterns based on nonlinear tables and graphs. The representations used here provide a structure for analyzing rates of change. This work develops students' understanding that nonlinear functions can be predictable. Video and lesson materials are available.
“Using Tables and Graphs to Make Predictions” by LearnZillion
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In this video (approx. 3:30 min.), Sal uses a real-world context to model assigning variables to the unknown quantities within the problem. He explains how to write the equation in slope intercept form to confirm that it is an example of a function.

“Linear and Nonlinear Functions: Word Problems” by Khan Academy

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This video (approx. 5:15 min) shows examples of linear functions who’s graphs do and do not pass through the origin. Vertical translation is modeled. The y-intercept and it’s x-coordinate of zero is demonstrated. Explanation of y=mx + b termed “slope-intercept form” is given.
“Slope-Intercept Form” by WhyU
Hook/Performance Tasks
Hook/Performance Tasks
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Illustrative Mathematics: Students explore the differences between linear and non-linear functions and then contrast the two. This reinforces the properties of linear functions. Encourage discussion about noticeable differences and relating them back to the equation and the graph.

Introduction to Linear Functions

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Real World Application
Real World Application
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In this video (approx. 6:30 min), Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Slope and Deep Sea Sharks

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