Exponentials Lesson 3 Episode 2 (Teachers)

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One conceptual challenge that many students might encounter in this lesson is how to succinctly represent exponential growth using appropriate mathematical notation. There are two important issues students need to grapple with. One of these is the convention of using exponential notation to represent how many times a factor appears in an expression. Josh and Arobindo seem to have already figured this out. Early in the episode they claim that the height on Day 25 would be 3²⁵[0:40]. You can introduce this convention if students aren’t already familiar with it. The second issue is what this notation means. Again, Arobindo and Josh seem to understand that the 3 represents a factor, in this case a growth factor over a period of time equal to one day, and the 25 represents how many times that factor is being multiplied.

For use in a classroom, pause the video and ask these questions:

1. [Pause the video at 0:21] Create a timeline showing how the height of a beanstalk changes from Day 0 to Day 25. Use your timeline to decide how tall the beanstalk will be on Day 25. 
2. [Pause the video at 3:25] Arobindo and Josh have drawn several arcs on their timeline. Some are labeled “+ 5” and others “× 3^5”. Why do they use addition for relationships between days? Why do they use multiplication for relationships between heights? Can you describe how the growth factor of 3⁵ relates to the overall growth of the beanstalk over 25 days?

Josh and Arobindo introduce the notation 3²⁵ [0:47] to describe the height of the beanstalk on Day 25. Ask your students what they think about this notation. Instruct them to discuss with a partner what they think the 3 means and what they think the 25 means. Ask them if they can write an equation (or describe one) to support their thinking.