Exponentials Lesson 3 (Teachers)

This lesson can help students:

• Create an equation that models the height of the beanstalk at any time.
• Justify the equation they have written by appealing to quantitative relationships in the context.

• CCSS.MATH.CONTENT.HSF.BF.A.1. Write a function that describes a relationship between two quantities.
  
  In this lesson, Arobindo and Josh work together to create a symbolic representation for the growing beanstalk. They start by creating expressions and equations that relate the height of the beanstalk to a specific period of elapsed time. For example, in Episode 2 they state the height of the plant on Day 25 would be 3²⁵, and in Episode 3 they write out the equation Height of the plant on Day 100 = 3¹⁰⁰. By the end of this lesson, they create an equation that relates the height of the plant for any given day, y = 3^x.
  
• CCSS.MATH.CONTENT.HSF.LE.A.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  
  Throughout this lesson, Josh and Arobindo work from the description of a beanstalk that grows exponentially. They know the beanstalk triples in height every day that it grows, and they work to abstract that context into a symbolic representation. They use expressions to represent specific instances of the relationship, and eventually write the function y = 3^x, which relates the height of the plant, y, to the number of days it has been growing, x.
  
• CCSS.MATH.CONTENT.HSF.LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context.
  
  In Episode 3, Josh and Arobindo examine and explain the various components of the equation Height of the plant on Day 100 = 1 • 3¹⁰⁰. They point out that the 1 represents the initial height, the 3 represents the growth rate of the beanstalk over one day, and the 100 represents the number of days the beanstalk has been growing. In Episode 4, they interpret the parameters of an exponential function in the beanstalk context.

CCSS.MATH.PRACTICE.MP2. Reason abstractly and quantitatively.

According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” This lesson requires Josh and Arobindo to reason both abstractly and quantitatively. In Episode 1, the students create a timeline and annotate their work to describe mathematical relationships they’ve noticed. This work requires them to reason quantitatively about the beanstalk and how it grows. For example, the students note that the beanstalk is nine times taller on Day 2 than it is on the initial day, and they support their argument by using circles to represent groups of heights of the beanstalk on Day 0 and on Day 1 [2:40]. They also reason abstractly, using symbols and mathematical operations to describe the exponential growth. This can be seen in Episode 2, when Josh and Arobindo inscribe “× 3⁵” to indicate the growth of the beanstalk over 5 days, and then compose multiple instances of “× 3⁵” to discuss a growth rate of 3²⁵ over a 25-day period.