# Exponentials Lesson 2 (Teachers)

### Creating a Timeline

Arobindo and Josh create a timeline that shows the growth of the beanstalk over several days. They find several mathematical relationships among the heights and among the days on their timeline.

##### Episode 1: Making Sense

The students explore a timeline that shows the release of movies in the Marvel Cinematic Universe. They then create a timeline that represents the growth of the beanstalk.

##### Episode 2: Exploring

Josh and Arobindo mark in several important mathematical relationships on the timeline they created to represent the growth of the beanstalk.

##### Episode 3: Repeating Your Reasoning

The students explore what segments of various lengths on the timeline represent in terms of both the passage of time and height.

##### Episode 4: Reflecting

Josh and Arobindo compare the Marvel Cinematic Universe timeline to the timeline they have created to represent the growth of a magical beanstalk.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Model the growth of the beanstalk and an exponentially-scaled timeline.
• Understand that same-sized segments on an exponentially-scaled timeline represent growth by the same factor.
• Find the factor by which the beanstalk grows over several time periods and illustrate these factors on the timeline.
• Describe the difference between a linearly-scaled timeline and an exponentially-scaled timeline.

Common Core Math Standards

CCSS.MATH.CONTENT.HSF.LE.A.1Distinguish between situations that can be modeled with linear functions and with exponential functions.

This lesson features two different timelines. The first timeline showcases release dates of movies in the Marvel Cinematic Universe. This timeline essentially acts a model of a linear function, relating distance on the timeline with periods of time. The second timeline showcases the height of a growing beanstalk to the period of time it has been growing. Arobindo and Josh reason about what one-inch segments mean in each timeline, and notice that for the beanstalk, such a segment represents both an additive increase in time and a multiplicative increase in the beanstalk’s height.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP4Model with mathematics.

According to the CCSSM, mathematically proficient students “are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.” Throughout this lesson, Josh and Arobindo model with mathematics. In Episode 1, they superimpose a partitioning scheme on a given timeline to accurately place the date of a movie in between two already-placed movies [1:32]. In Episode 3, Arobindo and Josh work to coordinate three different quantities to make sense of the growing beanstalk. As they do so, they use relationships in their model to reason about how many times taller the beanstalk would be after five days [1:27]. Finally, in Episode 4, the students explicitly state relationships between the three quantities in their timeline model for the beanstalk: the length of a segment on the timeline itself, a corresponding additive change in the number of days the beanstalk has been growing, and a corresponding multiplicative change in the height of the beanstalk.