Exponentials Lesson 6 (Teachers)

Finding the Height at Any Time between Day 0 and Day 1

The students continue to explore the height of their beanstalk at fractional times of day. In this lesson, they discover how to find the height of the beanstalk at any time between Day 0 and Day 1.

Episode 1: Making Sense

The students create a timeline that shows how the beanstalk is growing over four one-quarter day time periods.

Episode 2: Exploring

Josh and Arobindo use their timeline to find the height of the beanstalk on Day ¾.

Episode 3: Repeating Your Reasoning

The students use their timeline to find the height of the beanstalk on Day ⅚.

Episode 4: Repeating Your Reasoning

Arobindo and Josh find the time when the beanstalk reaches a height of about 1.24573 cm.

Episode 5: Reflecting

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

Episode 6: Reflecting

Josh and Arobindo reflect the possible values for y in the equation they created that gives the height of the beanstalk on Day xy=1(3x).

Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

  • Understand fractional exponents quantitatively, where the exponent is any fraction between 0 and 1.
  • Use the timeline to explain the relationship between these fractional exponents and roots.
  • Use the beanstalk context to reason about the domain and range of the function y = 3x.

Common Core Math Standards

  • CCSS.MATH.CONTENT.8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. 

    Josh and Arobindo work with exponential expressions throughout this unit. In this lesson, they use square, cube, fourth, and sixth roots to reason about the height of a growing magic beanstalk. For a beanstalk that triples in height in one day and is 1 cm high on Day 0, Arobindo and Josh express the height of the beanstalk on Day 5/6 as . They also reason that the plant must have been growing for 3/5 of a day if the height of the beanstalk is 1.24573, since 1.2457 is approximately .
  • CCSS.MATH.CONTENT.HSN.RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. 

    In this lesson, Josh and Arobindo continually make use of a partitioned timeline to reason about the growth of a beanstalk at times between 0 and 1 days. Since the beanstalk triples in growth in one day, the students must reason about how to partition the growth over time periods less than a day. To do so, they partition the growth rate to correspond with the relative size of the time period, and then use roots to describe the growth. For example, for a period of one-fourth of a day, Josh and Arobindo recognize that the growth rate must be the number that when multiplied by itself four times is equal to 3. This is the fourth root of 3, and they use both radical and fractional exponents to represent this number (as  and as 31/4). 

Common Core Math Practices

CCSS.MATH.PRACTICE.MP3Construct viable arguments and critique the reasoning of others. 

According to the CCSM, “Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples…. Later, students learn to determine domains to which an argument applies.”

In this lesson, Josh and Arobindo work together to determine the height of a growing beanstalk at various points throughout the first day it grows. This task requires them to partition a one-day period into fractional days and find the corresponding growth rates for each of those partitions. In the first two episodes, they use previously established results (including recognizing equal growth rates over equal time periods and the use of exponential notation to describe the height of the beanstalk) to argue that after 6 hours (1/4 of a day) the beanstalk should be cm tall [Episode 1, 1:31], and after 18 hours (3/4 of a day) the height would be  cm tall [Episode 2, 2:09]. Later in the lesson, they explore what the types of numbers can be “plugged in” for x and what types of numbers they can get for y in the equation y = 3x, where y is the height of the beanstalk on Day x. In Episode 5, they initially refute the idea that negative numbers can be used for x, arguing that you “can’t have a negative day” [0:18]. Later, they revise this interpretation to think of negative x values as the number of days before Jack got the growing beanstalk [2:00]. In other words, the students make sense of the domain of this exponential function. In Episode 6, they consider the idea of y being negative and zero, and eventually determine that y must be positive. They first decide that it can never have a negative height, and later realize that once the plant was no longer a seed, it never had a height of zero either.