# Logarithms Lesson 1 (Teachers)

### Justifying the Product Rule for Exponential Expressions

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

##### Episode 1: Making Sense

The students create two equivalent expressions that both represent the height of the beanstalk 5 days after Day 2, and create a number line to represent these expressions

##### Episode 2: Making Sense

Arobindo and Josh create two equivalent expressions that both represent the height of the beanstalk 18 days after Day 2, and create a number line to represent these expressions.

##### Episode 3: Repeating Your Reasoning

The students create an expression that represents the height of the beanstalk any number of days after Day 2, and create a number line to represent this expression.

##### Episode 4: Repeating Your Reasoning

The students create an expression that represents the height of the beanstalk 5 days after an unknown start date, and create a number line to represent this expression.

##### Episode 5: Reflecting

Josh and Arobindo determine whether or not two exponential expressions are equivalent.

##### Episode 6: Reflecting

Arobindo and Josh interpret several exponential expressions using a number line.

##### Episode 7: Reflecting

The students create two equivalent exponential expressions that both represent the height of the beanstalk any number of days after an unknown start date. They justify these expressions using a number line.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Develop a quantitative interpretation of the product of two exponential expressions in the beanstalk context (i.e., the expression 3• 3e can be interpreted as the height of the beanstalk e days after Day p).
• Justify the product rule for exponential expressions 3• 3e = 3(p+e), using a quantitative understanding of the two exponential expressions in the equation 3• 3e and 3(p+e).

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.

In this lesson, Josh and Arobindo continue investigating a magic beanstalk that triples in height every day (the Project MathTalk unit on exponentials features their initial explorations of this context). Their investigation in this lesson centers on describing the growth and height of the beanstalk some number of days after a given day. For example, they initially describe the height of the beanstalk five days after Day 2 as 9 × 35 and as 3(2+5). Throughout the lesson they reason about why expressions such as 3× 318 = 320. In the last episode, the students generalize their work to express the height of the beanstalk some number of days (e) after some other number of days (p) as both 3p+e and 3× 3e
• CCSS.Math.Content.HSN.RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Arobindo and Josh generalize a method for describing the height of a growing beanstalk some number of days after some other number of days. As part of that work, they investigate a specific instance in which the beanstalk has been growing for 0.4 days after Day 7.2. They create the expression (37.2)(30.4) and argue for why it is equivalent to 37.6.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP4Model with mathematics.

According to the CCSSM, “mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace…. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense….” As Josh and Arobindo investigate the growth of a magic beanstalk, they create several number lines and annotate them with expressions and inscriptions to highlight important relationships. For example, in Episode 2, Josh draws several bumps to highlight how many times the plant’s height is being tripled [3:59]. In Episode 3, the students draw arcs to show different spans of time and link those to exponential expressions such as 1 × 3(2+5) [1:35]. Later they adjust this model when working with unknown lengths of time, and are able to effectively model the growth of the beanstalk some number of days after some other unknown number of days by creating the expression 1 × 3(p+e) [Episode 7].