Exponentials Lesson 5 Episode 1 (Teachers)

Making Sense

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Josh and Arobindo review the mathematical relationships on a timeline that represents the growth of a beanstalk that represents the growth of a beanstalk that triples in height each day and has a height of 1 cm on Day 0.

Episode Supports

Students’ Conceptual Challenges

Early in the video [1:14], there is more evidence that Josh and Arobindo can compose new multiplicative and additive relationships. For example, Arobindo discusses 243 as being times 3, five times, while Josh says that over a two-day period, the growth factor is 9. Josh also says that the growth rate being 9 means “it increases by 9,” [2:35]. Taken literally, this would indicate an additive relationship between the heights of the beanstalk on two successive days (e.g., growing from 2 cm to 2 + 9 = 11 cm) instead of a multiplicative relationship (e.g., growing from 2 cm to 2 × 9 = 18 cm). This could indicate confusion between multiplicative and additive relationships, though he does immediately clarify afterwards. Other students who have not yet cleared this conceptual hurdle might not be so quick to clarify their meaning.

Focus Questions

For use in a classroom, pause the video and ask this question:

[Pause the video at 4:36] Consider the explanation given by Josh for why the beanstalk grows by a factor of 9 over any two-day period. Do you agree or disagree with his argument? How do multiplicative growth factors over individual days contribute to the overall growth?

Supporting Dialogue

After watching the video, ask your students to talk with each other to summarize the discussion about groups of groups of…. In the video, Josh and Arobindo use this language to argue about how they know the growth factor over a given time period. For example, you might ask students to argue about the growth factor for the height of the plant from Day 3 to Day 6 and ask them to try to use the groups of framing in their argument.