# Logarithms Lesson 2 (Teachers)

### Justifying the Power Rule for Exponential Expressions

Josh and Arobindo write two equivalent exponential expressions that represent the growth of a magical beanstalk over several consecutive time periods of an undetermined length. In doing so, they justify a rule for rewriting exponential expressions of a certain form.

##### Episode 1: Making Sense

The students explore the growth of the beanstalk between Day 2 and Day 12. They mark in several mathematical relationships on a number line that represents this growth and write several exponential expressions.

##### Episode 2: Exploring

Josh and Arobindo explore the growth of the beanstalk between Day 5 and Day 19. They write two equivalent expressions to represent this growth and use a number line to justify their expressions.

##### Episode 3: Repeating Your Reasoning

The students represent the growth of the beanstalk over 101 2-day periods on a number line. They use their number line to justify two equivalent expressions that describe the growth over that period.

##### Episode 4: Repeating Your Reasoning

The students use a number line to explore the growth of the beanstalk over an unknown number of 1.5-day periods of time. They develop two equivalent exponential expressions to represent this growth.

##### Episode 5: Reflecting

Josh and Arobindo write a general equation that describes the equivalence of two exponential expressions.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Justify the power rule for exponential expressions (3a)b = 3ab, using a quantitative understanding of exponential expressions in the beanstalk context. Specifically, students can interpret 3a as the factor by which the beanstalk increases over an a-day period. They can interpret both (3a)b and 3ab as the factor by which the beanstalk increases in b a-day periods.

Common Core Math Standards

• CCSS.Math.Content.8.EE.A.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = (1/3)3 = 1/27.

This lesson utilizes a context that is familiar to Josh and Arobindo, the magic growing beanstalk. Instead of reasoning about the growth over several 1-day periods, the students are tasked with reasoning about how the beanstalk grows over several multi-day periods, for example two seven-day periods. In doing so, Arobindo and Josh create expressions to model the growth of the beanstalk and reason why pairs of expressions must be equal. In the case of the seven-day periods, they demonstrate why 314 must be the same as (37)2.
• CCSS.Math.Content.HSA.SSE.B.3.C. Use the properties of exponents to transform expressions for exponential functions.

Throughout this lesson, Josh and Arobindo utilize exponential expressions to represent the growth of a magic beanstalk. They draw meaning from the context to create equivalent expressions. For example, when reasoning about the growth of the plant over 101 2-day periods, they argue that one way of understanding the context is to show the growth factor over 2 days and then multiply that by itself 101 times, which yields the expression (32)101. They then argue that 101 2-day periods is the same as 202 days, so the growth factor is also 3202. Later, they argue against a fictional student who claims that (30.5)= 34.5 by noting the left side can be rewritten as 3.05 × 3.05 × 3.05 × 3.05, which is the same as 32.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP2Reason 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 provides several opportunities for students to engage in this math practice. In Episode 1 [5:30], Arobindo is asked to represent 59,049 in a different way. He responds by counting the number of factors of 9 that are visible in the diagram representing the growth of a magic beanstalk. Each 9 represents the growth rate over two days, so Arobindo reasons that since it takes five groups of 2 days to make 10 days, it must take five factors of 9 to make 59,049. Episode 4 features the students reasoning about the growth rate over n periods of 1.5 days. The pair reason that 1.5n represents both 1.5n many days (so 1 period of 1.5n days) and also n many 1.5-day periods. Their quantitative reasoning about how these two quantities are equal enables to them to reason abstractly about why 31.5n = (31.5)n.