# Exponentials Lesson 7 (Teachers)

### Finding the Height at Any Time

The students build on what they have learned to discover how to find the height of the beanstalk at any time.

##### Episode 1: Making Sense

Josh and Arobindo critique other students’ reasoning about what the height of the beanstalk should be at Day 2.5. They then use a timeline to create their own argument for what the height should be at that time.

##### Episode 2: Exploring

The students create several different expressions for the height of the beanstalk at Day 2.5 and justify them using the timeline.

##### Episode 3: Repeating Your Reasoning

Arobindo and Josh use their timeline to find the height of the beanstalk on Day 11/3.

##### Episode 4: Repeating Your Reasoning

Josh and Arobindo use their timeline to find the height of the beanstalk on Day 17/28.

##### Episode 5: Reflecting

The students reflect on how to find the height of the beanstalk at any time. They illustrate their process by finding the height of the beanstalk on Day 1.68.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Understand any positive exponent quantitatively.
• Use the timeline to explain the relationship between fractional exponents and roots.

Common Core Math Standards

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.

As in the previous lesson, Josh and Arobindo leverage their method of partitioning a timeline to reason about the growth of a beanstalk at non-integer days. In this lesson, the time periods are more complex than in the previous lesson. For example, Josh and Arobindo must make sense of the growth of the beanstalk over 17/28 of one day, and later figure out the height at Day 1.68. As before, they partition the growth rate to correspond with the relative size of the time period, choosing to partition the timeline into 28 equal sections, for example, and then defining the growth over each period as a corresponding root of 3 (for this example, they used the 28th root of 3).

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…. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.”

Episode 1 from this lesson features Arobindo and Josh critiquing the reasoning of two fictional students, Denzel and Isabella. Denzel has conflated the growth factor for the height. Arobindo and Josh argue that Denzel is incorrect, offer a counterexample, and use Denzel’s incomplete reasoning to argue for the true height of the beanstalk [2:35]. They also analyze Isabella’s claim and verify that it is correct, using methods they developed in previous lessons. In Episode 2, Arobindo and Josh evaluate several expressions and verify that they are all equivalent ways of representing the same quantity, which is the height of the beanstalk on Day 2.5. Episodes 3 and 4 feature Josh and Arobindo using their previously-established results, including their equation for the height of the beanstalk on any given day and their method of partitioning the timeline to reason about roots of the growth factor, to argue about the height of the beanstalk on Day 11/3 and on Day 17/28.