Laws of Exponents - Negative and Zero Powers

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Objective

SWBAT understand negative and zero exponents by applying students' skills with patterns to see the big picture and make better connections.

Big Idea

Memorization is short term knowledge but understanding endures time.  Create the rules for exponents by expanding the bases to understand.

Lesson Rational

2 minutes

If students are merely asked to memorize the seven laws of exponents then they will probably memorize them for the unit exam but forget them soon afterwards.  Memorization is short term memory for most students.  I try to teach strategies for understanding the laws because students understand what an exponent means and how to expand expressions using exponents.   I do not expect them to memorize seven laws and remember them a year from now when using exponents in algebra I class.  However, I do except them to understand exponents and how to expand expressions to see what math they are really dealing with so in a year they will quickly see the property and remember the rule.  

Bellringer - Warm-up

15 minutes

Write the following expressions on the board with the directions, Generate Equivalent Expressions for each of the following expressions. (No calculators allowed)

 

40x638x152x5y8(3yx4z2)2(15x5y)2

  5x14y

 

The goal here is two fold: one, keep practicing what has been learned so far because not all students understand the content on the first day it was taught; secondly, begin to apply the laws of exponents to simplifying radicals.  Students must think about the law for multiplication (add exponents) in order to realize the square root of x to the fourth power is x squared because x squared times x squared will become x to the fourth power.  I like to use challenging bell ringer questions when possible because students are allowed to talk and work together during bell ringers so no one is alone working through tough questions. The last rational expression is a good example for showing students how useful array models are in multiplication of large numbers without a calculator.  If you create a 15 by 15 rectangle and then divide the sides into 10 + 5 by 10 + 5 and multiply the sections to find the total it is much easier to square 15.  Array models are a part of the elementary standards and are very useful in teaching distributive property as well, so creating familiarity with them now is a good idea. 

 

After about 10 minutes or so, collect these bell ringer papers and choose your favorite "no" to share with the class.  After sharing your favorite no, discuss the answers to all questions - even better is to select students to come to the board and explain them (you have their answers so choose students who are correct).  

 

 

Beginning Today's Activity

35 minutes

Clarifying your learning goals before beginning a lesson is very important.  To watch a short video about clarifying learning intentions and criteria for success click on the Clarifying and Sharing Learning Intentions and Criteria for Success.

Clarify for students that today is focused on extending their current understanding of exponents to include how to use negative exponents and to learn what happens when a number is taken to the zero power.  (write an example on the board to clarify).  Tell students the power today is not in expanding expressions but instead in finding a pattern of change using tables, to write a rule for expressing the properties of negative and zero exponents.

Pass out the activity for today and allow students one minute to read the directions and look at the first few questions on their own - no talking yet.  Ask students if they have any questions about how to complete the tables or what is meant by the pattern arrows.  I sometimes even call on students to summarize the directions just to ensure everyone is clear and then let them work in cooperative groups to complete the first page of the activity.  As students are working, move about the room providing feedback that moves learning forward.  For a short video on how to provide feedback, click on the link 

Providing feedback that moves learning forward

Allow students about 8-10 minutes to complete the tables and then another two-three minutes to answer the question that follows about the pattern.  After about 13 minutes, allow students to use the computers to chat in Edmodo.com as instructed in question three.  Tip:  while not using the computer, I ask students to close the lid of their laptop or if you are using desktop computers, ask them to face away from the monitor unless using the computer.  I start everyone in Edmodo at the same time to chat.  Once students begin to use the computers, I set a timer of three minutes max.  I also monitor their conversations either by my teacher computer or on my phone with the edmodo app.  If anyone gets out of line or off topic, they are off the computer for the day with additional consequences.  After time is up, students should begin to work writing down important ideas shared from other groups and move to the questions on the second page.  At this time,  I also go in and "like" good ideas I want students to notice in particular.   Reviewing all the posts in Edmodo is an effective and efficient source of formative assesment feedback about your groups and what they understand so far.  Sometimes, I pull up the Edmodo posts on my projector board and we discuss the comments as a class.  A few reasons why I might use classroom discussion include:  only a few groups are discussing the important information well and I want to highlight their thinking, no one is discussing the important information and I need to ask some guiding questions of the group to move their thinking along, or someone said something so profound and important that I want to share that thought immediately as it will help with the rest of the activity.

Allow students another 10 minutes to work in collaborative groups to complete the second page which is looking at the same four tables but this time, developing a pattern for moving down the table.  The goal of these first two pages, is to develop the patterns - multiply to move up the table and divide to move down - then from the table notice that any number to the zero power is one.  This page is probably as far as your class will make it on the first day.  Look at my student work sample page two, this student correct identified the pattern of change and completed the tables, yet when asked in the last question about the zero power rule she still said anything to the zero power is zero.  This shows how deeply rooted the zero misconception is for our students.  

Homework: If students do not complete page two in class, assign the remaining questions for homework.  On the following day, students will have time to discuss the zero power in Edmodo.com.  If you finish early enough today to allow two to three minutes of Edmodo conversation time, then great!  Dicuss question seven today before the end of class as a wrap up for the day.  I sometimes collect students papers if I want to check formatively what they wrote down versus what they posted in Edmodo.  If the posts in Edmodo were detailed, however, then I usually do not collect the student papers.

 

Strategy Videos 

Activating students as owners of their own learning

Cooperative Grouping Explained

 

Edmodo Preview