Getting Started: Investigating Exponents

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Objective

SWBAT find the value of expressions involving negative exponents.

Big Idea

This lesson provides a foundation for students to fluently work with geometric sequences and exponential growth.

Launch

10 minutes

NOTE: Successful completion of this lesson requires that the students have worked with the laws of exponents in grade 8.  This refers to being able to multiply and divide bases that are raised to an exponent power (8.EE.1)

Ask students to discuss their answers to the warm-up problems with their partners.  Students should verbally justify the reason for the answer that they wrote down (MP3).  Once students have come to an agreement within their partnership, ask some students to share their ideas with the whole class.

For the first question, many students will have 3^0.  Question students to find out the meaning of this expression.  Ask them how they can use the original expression to make sense of this new expression (MP2).  Some students will think of the original expression as 27/27 others will see it as a number divided by itself.  In either case, expect students to reason that the value of the expression is 1.  

Ask students they think this might be a general rule: any number raised to the zero power will always be equal to one.  Can you think of an example where it is not one?  Can you think of several examples to show that x^0 is equal to one for all values of x?  Have students do a think pair share around these questions.  Once students have had time to think, have them craft a statement with their partners about zero exponents.  

For the second question, have students share their answers and defend their thinking (MP3).  Some students may think that 2^-1 will be a negative number.  Again, you can encourage them to think about the number that they are actually dividing in order to come up with a decimal answer.  Do not tell them that the answer is 1/2 yet as they will come to this understanding during the investigation.

Investigation

10 minutes

Give each student a copy of the exploring_exponents_investigation.  Allow students time to complete the first table only.  Instruct students to complete the entire second column using the law of exponents and complete the values they know from the third column.

Students should be able to complete the bottom 4 rows using their knowledge of positive exponents and zero exponents. Watch for mistakes on the last column.  Remind students they are subtracting -1 from 2. (2 – (-1)).

Allow students to think-pair-share on any pattern that seems to exist in the third column.  Ask, “What do you notice about the last 4 rows in the table? What appears to be happening to the values as you look up/down the last column of the table?”  

Allow students time to make predictions on the possible values that should be placed in the last column.  Remind students that they can have access to graphing calculators  if they need them (MP5) to solve.  Show students or have students determine the missing values.  Discuss the possible reasons for the values and look for patterns (MP7).  Do not explain how negative exponent values are determined.

Have students complete the second chart (without a calculator) and fill in all the values they know.  Let them discuss answers in groups and then out loud.  Focus the discussion on how they are determining their negative exponents (if they have) or what pattern exists in the bottom 5 rows of their table (MP8).

Discussion

15 minutes

Allow students to calculate the values for the numbers raised to negative exponents and do a Think-Pair-Share on the patterns that seem to exist.  Ask students how the pattern changed from the first table to the second table? How is it the same? How do you think you determine the value of a number raised to a negative exponent?

Once students have determined how numbers raised to negative exponents are calculated, ask students to create a statement on how to find the value of any number that is raised to a negative exponent.

After students have had time to create their statements have them read the statement to a small group of 2 or 3 peers.  The group should give the student feedback on their statement (MP3).  If time allows, have one or two students share out their ideas with the whole class and extract the most important points from each of the statements.

As a final check for understanding, have students complete the four problems on the attached practice.  Two of the problems involve variables and will require students to transfer their thinking to more abstract expressions (MP2).  If students complete these questions on whiteboards it will give you instant feedback on who is understanding the concepts and who requires more work.

Closure

5 minutes

Students can be more creative and will have to think more flexibly on this exit ticket.  The questions are open ended and so they allow for many different solutions.  Student responses will certainly give insight into who is grasping the concept and who needs more work.  If students were struggling on the practice problems in the previous section you may want to give them more guided instruction during this ticket out the door.