Students in the middle school grades have a difficult time understanding exponents. I use an activity to demonstrate exponential growth, to show them how exponentiation grows rapidly. Anytime that a lesson can be related to the real world, and more importantly to experiences that are directly part of the students’ world, are the most effective.

There is a common idea in a math lesson on the powers of two – that of taking a message, and spreading the idea by telling two people who then tell two people who then tell two people and so on.

In the past, I had done this lesson with a chain email example, as that was a common method of communication. Now, text messaging and tweeting are the latest popular forms of communication. This year, I will use tweets as my example.

The handout to the students is something along these lines:

**Tweets**

**Math7B@VestalMiddleSchool Oct 2012**

*Meet 3.14.13 at 3:14 pm for PI day flash mob#ilovemath7B#pi-rocks@friend1@friend2*

*View conversation Reply Retweet Favorite*

*TWEET FOR TWO….AND TWO FOR TWEET*

*If I start a tweet on day 0 to organize a pi day flash mob, I tweet to two people. Then, the next day, those two retweet to 2 new people. Those then go on to retweet to 2, and so on and so forth. Assume it takes one full day to retweet.*

*How many days do you think it will take to reach:*

*1) VMS team 7B?*

*2) VMS 7th grade?*

*3) The whole middle school?*

*4) The Vestal Central School District?*

*5) The town of Vestal?*

*6) The state of New York?*

*7) All the people in the USA?*

*8) All of the people in the world?*

*Work together with your partner to find some strategies to use to answer the questions above.*

For simplicity sake, the students are asked to assume that it takes a day to re-tweet the message after having received the tweet. Students are challenged to predict how long it would take to reach every student in the seventh grade; to reach every student in the school; to reach every student in the district. From there, they estimate how long it would take to reach every person in the town; in the state; in the country; in the world. They are always surprised on how fast a message can spread in this manner.

The students are paired up and the tweeting scenario is presented. They are given some time to work together to come up with some problem solving strategies to use to solve the problem. In this way, the lesson not only provides instruction on the content of the common core, but also on communication, collaboration and problem solving skills.

It becomes apparent that several problem solving skills are used to tackle this problem. The students need to “make the problem simpler”, “draw pictures”,”choose an operation” and “make a table or chart”. The first day of the lesson is working towards a solution. Towards the end of class on day one, we examine the chart created, and find interesting patterns in the chart. There are several and this always provides a rich experience for the students to appreciate how real life activities can have underlying beautiful mathematical patterns. The students are then assigned the task of finishing the chart for homework, in order to answer the questions of how long it would take to reach all the people in New York State. They are also given the challenge of trying to arrive at a general algebraic expression to express the number of people who have received the tweet, based on the day number, *n*.

There was a movie released several years ago called “Pay it Forward”, which was based on the powers of three. The idea in the movie is that if you receive a good deed, you should pay it forward to three other people, who then in turn pay it forward to three more, and so on and so forth. At the end of this two-day lesson, I show a clip from the movie and challenge them to pay it forward with three people. There is a website, payitforwardday.com, and we anticipate that day to start the chain. In this way, the students can now take this idea and use it on another real life scenario, with beneficial results!

Now the lesson has also morphed into character education!

Later in the year, we will solve a problem using the “Towers of Hanoi” puzzle; at that time the students will see this pattern of the powers of two resurface. They will be surprised to see this beautiful pattern now emerges in the form of a puzzle. Who would have thought there was a connection between tweeting and a puzzle! This lesson will be shared in the near future, so stay tuned!

I really like the idea of using the spread of information as a lesson for exponential behavior. Another good one is how popular franchise businesses expand in terms of how many locations they have. Starbucks is a good example because of how their popularity exploded, but then you can also look at the limits of exponential behavior (limited number of people, limited demand/locations for Starbucks, etc.)

Thanks for your comments, Jesse! I really like your idea to expand this to franchises, and ties to limits on exponentials!

I think there are a few and this dependably gives a rich experience to the students to acknowledge how genuine exercises can have underlying wonderful numerical examples. The students are then doled out the assignment of completing the outline for homework, with a specific end goal to answer the inquiries of to what extent it might take to achieve all the individuals in New York State.

Thanks for your comments Alice!