# Exponentials Lesson 6 Episode 5 (Teachers)

### Reflecting

No captions Captions
Stop the video above first if it is playing.

The students reflect on the values they can plug into the equation they created that gives the height of the beanstalk on Day x, y=1(3x).

### Episode Supports

Students’ Conceptual Challenges

In this episode, Josh and Arobindo are asked to consider if it makes sense to let x be a negative number in the equation y = 1 • 3x. Initially, the pair seem to believe that it would not make sense [0:16]. Over the course of the episode, they shift their thinking to making sense of a negative exponent as relating to the number of days before Jack was given the beanstalk. Notice how the contextual representation of a growing beanstalk provides opportunities for Josh and Arobindo to make sense of the mathematics, and ultimately helps them make sense of negative exponents.

Focus Questions

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

1. [Pause the video at 0:16] The supporting dialogue prompt at the end of Lesson 4 asked students to consider the different kinds of values that x could be in an exponential equation. Consider using the following focus question to bring students’ attention back to this important idea:

What numbers can we “plug in” to x in the equation y = 1 • 3x?
2. [Pause the video at 0:45] What does it mean to use a negative number in place of x in the equation y = 1 • 3x? Remember that Josh and Arobindo initially said you couldn’t use negative numbers, but then they changed their minds to say it would be weird and perhaps a fraction. Discuss with a partner what Arobindo and Josh might mean by those claims.

Supporting Dialogue

The students initially thought there were restrictions on using negative numbers for x. After watching the video, ask your students to restate Josh and Arobindo’s explanation for making sense of negative exponents. Ask them to consider why it makes sense to think of a negative exponent representing some number of days before Jack got the beanstalk. Ask them to discuss why it makes sense for the expression 3-1 to equal 1/3. Encourage them to discuss these ideas through the lens of the context, rather than using or making up any rules about negative exponents.