The Product Rule and the Power of Product Rule of Exponents

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Objective

SWBAT apply the Product Rule or the Power of Products to Simplify Exponential Expressions.

Big Idea

To Expand! or Take the Shortcut! for an equivalent answer.

Warm Up

10 minutes

I use this Warm Up to assess students' prior knowledge of the rules for simplifying exponential expressions.  It takes about 10 minutes for students to complete the warm up, and for me to review problems with the students to introduce the lesson.

I use today's Warm Up to clarify when to apply the Product Rule or the Power Rule of Products with exponents. In this lesson, I emphasize results that represent equivalent answers when using the shortcut rules (for exponents).  I want my students to consider expanding the exponential expressions as a meaningful alternative when simplifying expressions with exponents.

I am careful to focus my students attention on clearly identifying what part of the expression is being raised to a power. Then, students can consider expanding the expression or using repeated multiplication to simplify.

I demonstrate how I will review today's Warm Up in the video below.

Guided Practice

20 minutes

Today's Guided Practice is an extension of the Warm Up. I structure the lesson in this way to allow my students more opportunities to discern the difference between the Exponents of Products and the Exponents of Powers.  My goal is for students to develop a deep understanding of the meaning of exponents, by firmly grasping the meaning of each property.

In the Guided Practice, students will expand the expressions using repeated multiplication and compare their results to the shortcut rules.  By using this process, students explore for themselves the meaning of The Laws of Exponents.  The Laws of Exponents should not just be memorized, but the meaning and concepts should drive students to remember how to apply the laws.  

I allow students to attempt the problems in Table 1 for three to four minutes. Then, I randomly call on students to share their responses by writing them on the board.  As a whole class we review the responses, and discuss any answers that are challenged by other students. As usual, I expect students challenges and responses to be considerate. Every student answer is valued, whether right or wrong.  Because this is a lesson that often raises student misconceptions, I am careful to enforce this norm during today's lesson.

In my classroom, students learn early in the year that I teach more from wrong answers than right answers. So, even if they struggle with Table 1, then know that I will expect them to persevere (MP1). As we move on to Table 2, I once again allow students three to four minutes to complete their work. Again, I will have students show their work on the front board.  After the responses to Table 2 are discussed, I will ask students to summarize the difference between simplifying exponential expressions in Table 1 and Table 2. 

On the second page of the Guided Practice, I introduce the students to the names for the Laws of the Exponents that were applied in each table.  I give students about three minutes to create their own six examples involving the Product Rule, the Power of Product Rule, and the Power of a Power Rule. Then, I will ask students to trade papers with their elbow partner for another three minutes to simplify each other's problem sets.

After simplifying each other's example problems, the partners will review the problems for another three minutes to try to come to a consensus on the answers to the expressions.  When students are unable to agree on the correct simplified form of an expression, I will encourage them to any of ask another group for help, before I assist them.

 

     

 

Independent Practice

10 minutes

After providing students with the rules to simplify exponential expressions by expanding or using the shortcut rule, I assign an Independent Practice Problem Set on exponential expressions.  Today's Independent Practice is a self-check for students,  as well as a formative assessment for me. Exponential expressions can be simplified using different methods that result in an equivalent correct answer.  My goal is for each student to work the problem in their own way. They will compare their results with those of other students during today's Exit Activity. 

I provide students about 10 minutes to work individually on simplifying the 12 exponential expressions. The expressions are meant for students to practice working with exponents, products, and powers to build confidence in the properties of exponents introduced in the Warm Up and the Guided Practice.

I instruct students to work alone without the assistance of other students or myself.  I find it is helpful to set a visual timer for my students to help them monitor their own time.  

 

Exit Slip

10 minutes

I use the Exit Slip in this lesson as a peer assessment. I begin by asking students to find another student in class that has an equivalent answer, but used a different method.  I want students to recognize that when working with exponents, different approaches can be taken to simplify an expression.  

I allow students to use any method with correct reasoning, and work shown. Some students may use the shortcut rules to simplify exponents, while others may simplify by expanding the expressions. 

Here are two examples of the exit activity:

  • The two students comparing their work in Group 1 have different answers, but when comparing their work, they realized that the first method was incorrect.  This is a common mistake made by students to multiply the base times the exponent instead of expanding it based from the meaning of an exponent.  Two to the third power is eight, not six.  
  • In Group 2, the answers of both students are equivalent, and the students agree that their responses are correct.  When comparing the two different methods that each of them used, one student expanded the expression before simplifying.  The other student applied the power rule of exponents.  Both students in the final step applied the product property of exponents by adding the exponents of like bases that were being multiplied.