Reviewing Exponent Rules with "The Zombie Game" and the Alice Portfolio

I was looking for some ideas over the weekend and ran across this blog post by Sarah Hagan (Math=Love). The post includes alot of great ideas but the one that I borrowed came from yet another blog post by Nathan Kraft. I have really gotten so many great ideas from the teachers I follow on Twitter and through my Feedly blog reader! I am so thankful for teachers who are so willing to share their resources and ideas with others.

So, today was the first day back after Spring Break and I wanted to review problems using exponent rules. I borrowed some individual white boards from Ms. Whitt and gave one to each student. I had my students go write their names on my board and put 4 x's under each name. The last student with an x under his/her name wins. I gave the class a problem to work and had them put their markers down after a certain period of time. Any student who gets an answer correct gets to go erase an 'x' from any person's name. The "zombie" part comes when students have all 4 of their x's erased. Even though they can no longer win the game, they can still erase x's from others.  I will give the winners 5 bonus points on the next quiz grade. Ms. Whitt and I did go over how to work each problem afterwards.

I altered what my students will be doing for the Alice portfolio. Due to time constraints we chose to skip some of the activities that were mentioned in the portfolio list in the book. I created my own Alice Portfolio assignment. I borrowed some items from the Meaningful Math version of the portfolio. I do not necessarily think that mine is better; it just was a better fit for me this year since we had to cut some of the activities.

IMP Alice - Stranger Pieces of Cake and Confusion Reigns

Today I first had my students do Stranger Pieces of Cake. I started off by reading over the into and giving them 5 minutes to think about number 1. When they were asked to put their findings on the ipads and project them (using airplay) I realized that they were in need of help. I did have one girl say that 2 to the 3/5 power would be like Alice's height increasing by 60 percent. I was excited that she used that description but I needed to lead them to a point where we could explain the problem using ideas we had already learned in Alice...so that we could develop the rule for fractional exponents.

So...I then asked them if they could figure out a problem where they either raised a power to a power or multiplied powers with the same base and the answer was 2^(3/5). Luckily, at least one group in each class eventually came up with 2^(1/5) * 2^(1/5) * 2^(1/5). We related this back to the activity where we developed the rule for exponents that are unit fractions which eventually got us to the idea that the 5th root of 2^3. Anyway...I had to do ALOT of leading through this activity but I really believe that the students have a better chance of remembering the rule for fractional exponents after they have had their "hands on" trying to figure it out themselves.

Confusion Reigns is a good activity that makes students just rethink through a few of the rules they have learned. I LOVE the first problem because so many students didn't pay any attention to the fact that they were adding 2 powers with the same base instead of multiplying. I hope this helps them to pay closer attention to the operation signs in the future.

Going from the "Context" to the "Abstract"

I have been mulling over the advice of a veteran IMP teacher (Michael Reitemeyer). He mentioned in an email how he tries to use the context of the lessons to introduce topics but then moves as quickly as possible to the abstract. Sonya and I were discussing today how we would like to develop around 8-10 "homework/classwork" problems that reiterate the concept of the day/lesson. This is not appropriate for every lesson but there are several that it would work well with. For instance, each time we develop/investigate a new exponent law in Alice we could assign 8-10 practice problems. There is a point in Overland Trail where I wish I had given them more problems in which they practiced graphing lines without a context.  The problems they will see on standardized tests won't always have a story to go with them.

Now...having said all of the above...I have been practicing solving equations with my 5th period.  We have been working through a packet together of different types of equations. Today I once again heard some grumblings. I told them that they didn't realize how much they liked the textbook we were using (IMP Meaningful Math Algebra). They were talking about LOVING the textbook. They enjoy the math so much more when there is a story/context to help make sense of it all. I have especially found this to be true with my students who struggle with the math.

IMP Alice Day 8 - Having Your Cake and Drinking Too

When I saw that the teacher's guide allowed 60 minutes for this activity I thought there was no way. I thought we would get through it quicker...haha! We didn't even finish and we had OVER an hour. This activity helps the students to explore and HOPEFULLY discover the rule for dividing powers with the same base. I love it! It is really revealing the lack of number sense that my students have. The questions are not really that difficult...that is why I thought we would breeze right through.

I have allowed myself to "sit" on this activity and not rush it because I really believe it might help to cement the concept. I am going to be absent tomorrow from my classes and I have some good exploration worksheets on exponents to leave for my students. However, it has reminded me how blessed we are to have found a book that combines the exploration with the context. Someone had a wonderful imagination!

**Update 3/7/15** Original post was made 3/16/15
This year when I taught this lesson I took the time to help the students to explore #2. I did not let anyone blurt out an answer when we first started looking at it. I have been using the random integer generator on the TI-83 (which is awesome but I just now figured out how to do it!!) to call on students. In my classes I just happened to call on students who were unsure of what to do so I just asked them to start by making a guess. I told them to tell me a number of ounces of cake and a number of ounces of beverage and then I showed them how to "test" their answer to see if this would give an answer where Alice's height was multiplied by 8. Then I randomly called on more students. The first 3 students guessed more beverage than cake (ugh!) so I asked the next student to make a "conjecture" on whether or not more beverage than cake would EVER allow Alice's height to be multiplied by 8. My 2nd class arrived at the correct number of ounces much faster than my 1st one did but I feel that modeling to the students how to "guess and check" was valuable. I am always telling them not to ever erase their guesses. I want them to learn to look back over the ideas that didn't work in order to help them to identify new theories that might work!! Anyway...the day I taught this lesson this year I felt really good about what we had accomplished:)

IMP Alice Days 6-7 - Piece After Piece, Many Meals for Alice, and In Search of the Law

For Many Meals for Alice and In Search of the Law I really put my Ipads to use. I am learning better ways to implement using the Ipads. I gave each group a few minutes to brainstorm on each problem and then told them that at the end of the allotted time they had to put SOMETHING on the Ipad (using the Educreations app). The Airserver app allows you to project up to 10 Ipads at a time so I have an Ipad in each group (sometimes more). I had told my students that if you have an Ipad you HAVE to be mirroring the screen. I can see all the screens on my computer. This has been a tremendous help for keeping my students from using the Ipads for purposes other than the math! When I turn on my projector where the students can see everyone else's work it is interesting. I am still amazed how my students will approach problems so differently when I don't stand at the board and show them an example of what to do (which causes them to all try to do it my way). We have an easy way to compare and critique the work without it being totally obvious whose group it came from. Then they are able to edit their work when it is needed. Formative assessment on the fly!!

I had a student volunteer to "teach" - she was great!

The context given in All About Alice provides such a neat alternative to simply just giving the students the rules of exponents. Piece after Piece establishes the rule for multiplying powers with the same base. Some of my students wanted to ADD 2^3 and 2^5 so to help them I told them to remember that we are always looking at what we are MULTIPLYING Alice's height by. I am interested to see how other IMP teachers handle this...

Many Meals for Alice establishes the rule for raising a power to a power. The original power is for the number of ounces of cake she eats at each meal - #1 is 3 ounces of base 2 cake so it is actually 2^3. Then the students figure out what is happening to her height after certain numbers of meals. In the end they have hopefully developed the idea that (2^3)^4 means that Alice ate 4 meals where she had 3 ounces of base 2 cake. It takes a few times of looking at it to start making the connection. I LOVE #3.

In Search of the Law actually has the students to explore exponent rules that I have not usually put much thought into. It took me a minute or two of exploring myself - I actually had to tell my students to start working on #1 while I went to read the teacher's guide! When I glanced over the activity the day before I taught it I didn't realize it was going to throw me a curve. Anyway...this activity is actually not difficult AT ALL. It gives the students several scenarios to just work with exponents.