Exponentials Lesson 1 Episode 5 (Teachers)

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1. In the previous episode, Josh and Arobindo briefly displayed evidence of the conceptual challenge for determining the overall growth rate for two successive time periods. In that episode, they added the growth rates rather than multiplied them. In this episode, this challenge seems to still be present, but manifests itself differently. When asked how many times taller is the beanstalk on Day 2 is than Day 0, Arobindo quickly responds with 27 instead of 9 [2:05].
2. This episode also illustrates the challenge students face in accurately linking two mathematical representations of exponential growth. Throughout the episode, Josh and Arobindo are challenged to accurately represent the height of one beanstalk in terms of groups of heights of the beanstalk from an earlier day. This requires quantifying one day’s height in terms of another and abstractly representing that using circles or some other distance-indicating inscription (e.g., the mathematical drawing seen at [3:00] and [5:15]).

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

1. [Pause the video at 0:43 and/or 3:56] Draw a picture that shows the height of the beanstalk on Days 0, 1, and 2 (and Day 3, if pausing at [3:56]). As you work, discuss with a partner any mathematical relationships you notice in your drawing.
2. [Pause the video at 3:00] Arobindo and Josh initially claim the beanstalk is 27 times taller on Day 2 than on Day 0, but then revise that claim to be only 9 times taller. Decide which one you think is correct. Use the visual representation in the video to support your argument.

1. As students draw their beanstalks, ask them to connect the visual representation of exponential growth to the contextual representation. You can introduce the language of groups of as the instructor does in the video. For example, ask students where they see three groups of the height of the beanstalk on Day 1 in Day 2, and where they see 27 groups of the height of the beanstalk on Day 0 in Day 3. Encourage them to use circles or some other inscription that indicates those groups.
2. Ask students to discuss the three different growth rates that seem to be present: A growth rate of 3 over one day, a growth rate of 9 over two days, and a growth rate of 27 over three days. What patterns do they notice, and why might those patterns exist?