# Exponential vs. Linear Models

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Students will compare linear and exponential models and solve problems, observe using tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, and write a function that describes a relationship between two quantities.

Included in this Unit:
1. Task #1: students have to decide if given the opportunity to receive \$1000 a day for 30 days or a penny on the first day, two pennies on the second day, and so on for 30 days, what would they choose? Students can do the calculations themselves individually or with a partner. I like to ask the class to raise their hands and see how many would pick the penny and how many the \$1000. Then I ask the students after the tenth day how much those who chose the \$1000 a day have in total so far (\$10,000) and how much those who chose the penny have in total (\$10.23). At this point those who chose the \$1000 are super happy about their choice. Then I do the same thing after 20 days. By now those who chose the \$1000 option have a total of \$20,000 while those who chose the penny have a total of \$10,485.75. Still, those who chose the \$1000 are happy. But then once the students finish all three days, they can’t believe how much money those who chose the penny have. This is a great introduction to discussing or reviewing exponential growth versus linear growth.
2. Task #2: Given a set of data, students are to determine whether it can be modeled by a linear function or an exponential function. I like to do this task whole class as a guided practice. I let them try to figure it out on their own and talk it out as a class, but I guide their reasoning and answers.
3. Task #3: Students are given two sets of data to analyze and determine whether it can be modeled as a linear function or exponential function. Then the third problem brings them back to task #1 where they have to establish a connection on how the first choice of \$1000 a day represents a linear model while the penny a day that doubles represents an exponential model.
4. Task #4: this is a short review on linear equations, but it covers a lot! Students practice finding slope, writing equations given a point and the slope in point-slope form, slope-intercept form, and standard form. They also practice writing equations of lines that are parallel and perpendicular to another line.
5. Task #5: This is a fun puzzle where students have to unscramble the words and then use selected letters to come up with a phrase.

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16 pages
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