# Exponentials Lesson 1 (Teachers)

### Investigating the Growth of Magical Beanstalks

Arobindo and Josh explore the growth of magical beanstalks. These beanstalks’ heights increase by the same factor each day. In this lesson, the students draw pictures that show these factors. They also explore how the beanstalks are growing over several days.

##### Episode 1: Making Sense

Josh and Arobindo explore the growth of a beanstalk whose height quadruples each day. They create drawings that show the growth of the beanstalk over several days.

##### Episode 2: Making Sense

The students watch a video of the beanstalk growing. They describe what they notice about how it is growing.

##### Episode 3: Exploring

Arobindo and Joshua use an applet to determine how a new beanstalk is growing.

##### Episode 4: Exploring

The students determine the factor by which the beanstalk is increasing over several time periods.

##### Episode 5: Repeating Your Reasoning

Josh and Arobindo draw pictures that show the relationships they found in the previous episode.

##### Episode 6: Reflecting

Arobindo and Josh mark in mathematical relationships on a picture another student has drawn to show the factors by which a magical beanstalk is growing.

##### Episode 7: Reflecting

The students describe how a beanstalk is growing between Days 100 and 102.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Illustrate the mathematical relationship between the height of a magical beanstalk that grows exponentially at different times in a picture.
• Find the factor by which the beanstalk grows over different periods of times (e.g., find the growth factor over 2 days) and justify their answers.

Common Core Math Standards

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

In this lesson, Josh and Arobindo investigate how a magic beanstalk grows over several days. Each day that the beanstalk grows, its height triples. Josh and Arobindo make predictions about how tall the beanstalk will be after growing for some number of days, and use math drawing and applets to check their predictions. They eventually link their predictions about the beanstalk to the exponential function y = 3x.
• CCSS.MATH.CONTENT.HSF.LE.A.1.AProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Arobindo and Josh use the idea of multiplication as groups of to make sense of the beanstalk’s exponential growth rate over multiple days. They initially reason day by day, recognizing that if the plant triples each day and starts out at 1 cm, then after one day it will be 3 × 1 cm = 3 cm tall, and after another day it will be 3 × 3 × 1 cm = 9 cm tall. Later, they are able to compose these units of time to reason that over the first two days, the beanstalk will increase in height by a factor of 9. By the end of the lesson, Arobindo and Josh reason that this increase by a factor of 9 will hold for the growth of the beanstalk over any two-day period.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP5Use appropriate tools strategically.

According to the CCSSM, “Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet…or dynamic geometry software.” In this lesson, Josh and Arobindo utilize dynamic applets to investigate the growing rate of the beanstalk. For example, in Episode 2 they watch as a beanstalk grows higher and make the observation that the “growth rate is growing” [1:22]. Later, in Episode 3, they use applets to see how the height of the beanstalk on a later day can be composed of groups of heights of the beanstalk on an earlier day. In Episode 4, Arobindo and Josh create mathematical drawings to reason about the beanstalk and annotate their timeline with circles to show groups of heights of the beanstalk [1:18].