# Exponentials Lesson 4 (Teachers)

### Exploring the Equations for Other Magical Beanstalks

The students explore the growth of new magical beanstalks. While all the beanstalks are similar in that they grow by a consistent factor each day, the factors they grow by are different and their heights on Day 0 are not always 1 cm. Josh and Arobindo explore how these changes affect the equations that model their growth.

##### Episode 1: Making Sense

The students use the timeline they made in the last lesson to determine the height of the beanstalk on Day 12.

##### Episode 2: Exploring

Arobindo and Josh create a timeline for a new magical beanstalk. This beanstalk’s height also increases by a factor of 3 each day, but it starts with a height of 2 cm on Day 0 instead of 1 cm. The students use their timeline to create an expression that represents the height of this new beanstalk on Day 12.

##### Episode 3: Repeating Your Reasoning

Josh and Arobindo use their new timeline to create an expression that represents the height of the new beanstalk on Day 100.

##### Episode 4: Repeating Your Reasoning

The students use their timeline to create an equation that gives the height of the new beanstalk on any day.

##### Episode 5: Reflecting

Josh and Arobindo are given an equation that models the growth of a magical beanstalk. They explain how the beanstalk is growing.

##### Episode 6: Reflecting

Arobindo and Josh are given equations that model the growth of magical beanstalks. They use these equations to compare and contrast how the beanstalks are growing.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Create equations that model the height of various magical beanstalks at any time.
• Interpret the symbols in the equation y = a(bx) in terms of the beanstalk context. Specifically, students should understand that x is elapsed time, b is the factor by which the beanstalk grows in a unit of time, bx is the factor by which the beanstalk has grown over x time, a is the initial height, and a(bx) (also known as y) is the height of the beanstalk at time x.

Common Core Math Standards

• CCSS.MATH.CONTENT.HSA.SSE.A.1Interpret expressions that represent a quantity in terms of its context.

This lesson features several opportunities for Josh and Arobindo to interpret expressions arising out of the beanstalk context. They examine two related expressions, 1 × 312 and 2 × 312, and reason that the difference in coefficients reflects the difference in starting heights for two different beanstalks.
• CCSS.MATH.CONTENT.HSA.SSE.A.1.BInterpret complicated expressions by viewing one or more of their parts as a single entity.

Josh and Arobindo also reason more generally with the equation 2 × 3n = p, noting that p is the height of the beanstalk after n days. For 2 × 3n, they connect the 2 to the initial height and the 3 to the growth rate of the plant over a one-day period.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP8Look for and express regularity in repeated reasoning.

According to the CCSSM, “Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts….As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.” In this lesson, Arobindo and Josh reason about many different beanstalk contexts. By comparing different beanstalks, different starting heights, different growing rates, and different growing periods, Josh and Arobindo construct meaning for the different parameters of exponential functions. For example, they compare two expressions for beanstalks, 1 × 312 and 2 × 312 [Episodes 1 and 2], and reason that the difference between the two is that the first starts out at a height of 1 cm while the second starts at 2 cm [Episode 6, 0:36]. They also examine the differences between the expressions 2 × 312 and 2 × 3100 [Episode 3], and later conclude that, in general, the exponent represents the length of time the plant has been growing [Episode 6, 0:36].