This is the first of three lessons about linear functions. Students are already familiar with linear equations and their graphs from previous units.
In the first activity, students see that a proportional relationship between two quantities can be viewed as a function. They see that either quantity can be chosen as the independent variable and that the only difference in the equation and the graph is the constant of proportionality, which is visible on the graph as the slope of the line through the origin.
In the next activities, students investigate and make connections between linear functions as represented by graphs, descriptions, and by the equation \(y=mx+b\). They interpret the slope of the line as the rate of change \(m\) of the dependent variable with respect to the independent variable and the vertical intercept of the line as the initial value \(b\) of the function. Students also compare properties of linear functions represented in different ways to determine, for example, which function has the greater rate of change. Consider using the optional activity if students need more practice comparing linear functions represented in different ways.
Lesson overview
 8.1 Warmup: Bigger and Smaller (5 minutes)
 8.2 Activity: Proportional Relationships Define Linear Functions (15 minutes)

8.3 Activity: Is it Filling Up or Draining Out? (10 minutes)
 Includes "Are you Ready for More?" extension problem
 8.4 Optional Activity: Which is Growing Faster? (10 minutes)
 Lesson Synthesis
 8.5 Cooldown: Beginning to See Daylight (5 minutes)
Learning goals:
 Comprehend that any linear function can be represented by an equation in the form \(y=mx+b\), where \(m\) and \(b\) are rate of change and initial value of the function, respectively.
 Coordinate (orally and in writing) the graph of a linear function and its rate of change and initial value.
Learning goals (student facing):

Let’s investigate linear functions.
Learning targets (student facing):
 I can explain in my own words how the graph of a linear function relates to its rate of change and initial value.
 I can determine whether a function is increasing or decreasing based on whether its rate of change is positive or negative.
Glossary:
 Access the complete Grade 8 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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