Exponentials Lesson 5 Episode 2 (Teachers)

Exploring

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The students use their timeline to find the height of the beanstalk on Day ½.

Episode Supports

Students’ Conceptual Challenges

1. Early in the video, Arobindo introduces the idea of square roots, and Josh says “the square root of 3 times the square root of 3 equals 3” [0:55].The previous lessons have been building toward this moment—linking fractional exponents with square roots. While Josh and Arobindo seemingly have made this connection, your students might not have done so yet. Consider using the focus questions and supporting dialogue prompts below to assess your students’ understanding of square roots so they will also be ready to form this important connection.
2. A hypothetical student, Jack, is convinced that the height of the beanstalk on Day ½ is 2 cm. His reasoning seems to be that since ½ is halfway between Day 0 and Day 1, the height should be halfway between the height of the beanstalk on Day 0 and on Day 1 [2:20]. This reflects additive reasoning for the heights, and is a common challenge for students. You may want to ask students if they agree with Jack or not before continuing the video.

Focus Questions

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

1. Pause the video at 1:46] Can you rephrase Josh’s description of square roots? Remind a partner what a square root means. Have you worked with or seen square roots before?
2. [Pause the video at 1:46] Consider the expression Arobindo wrote: 1 • 3½. Why did Arobindo substitute ½ for x in his expression for the height of the plant on Day ½? Why does this make sense in the context of the growing beanstalk?
3. [Pause the video at 2:33] Jack suggests that the height on Day ½ is 2 cm. Does this makes sense given what Arobindo and Josh have discovered about this growing beanstalk?
4. [Pause the video at 7:08] Arobindo claims that the height of the beanstalk on Day ½ will be 3½, and the height of the beanstalk on Day 1½ will be 3. Do you agree or disagree with Arobindo? Why do you think = 3½, and = 3?

Supporting Dialogue

1. Before watching the video, give your students time to reason about finding the height of the beanstalk on Day ½. Encourage them to discuss with a partner their ideas about how they might represent this height. You can prompt them to make estimates, to draw a timeline, or to use any other representations they’ve been working with, such as expressions or equations.
2. Consider pausing the video at [4:47]. Ask students to analyze Arobindo’s argument for why the height of the beanstalk on Day ½ shouldn’t be 2 cm. If need be, have them rewatch from 3:00. You can prompt them to consider his argument that the times 3 pattern seems to work from Day 0 to Day 1, Day 0.5 to Day 1.5, and Day 1 to Day 2, but that a doubling or times 2 pattern doesn’t seem to be consistent. Ask them to share with a partner why that consistency matters in this context.
3. At the end of the video [8:45], Arobindo argues that the height is being multiplied by  every half day, and that this “is constant.” Ask your students what they think he means by that—what is constant?