Students love to be able to solve hard problems (contrary to popular beliefs) and as a visual/hands-on learner, I loved this problem and the hands-on way of solving it. Many of my students have taken this problem home to challenge their parents. (I love when that happens!) In this problem, geometric series and limits are accessible by all students while reviewing fractions and exponents. So, let’s get started making a visual representation of the problem.
Are you ready to solve the problem yet? If not, tear the remaining paper in your hand in the same way and continue until you understand what is going on.
Not every problem is a pencil and paper problem.
Exploring this problem also started me thinking about solving problems differently. I use this problem (with 1/3 + 1/9 + 1/27 + ….) when introducing LIMITS in calculus along with this activity: Have students stand up and walk half way to the wall and then half way again. Ask them when/if they will reach the wall if they can only go half way each time they move towards the wall. The two activities do a great job of introducing the fact that even though you can’t theoretically get rid of all the paper in your hand when tearing it and putting it in piles or can’t really get to the wall when walking only 1/2 way, that you get so close you might as well say you’ve reached the “limit”.
I hope it will help you and your students to start exploring problems differently. This activity can be used to help students understand limits, explore geometric series, learn or use summation notation, and introduce the concept of iteration and fractals.
For this complete lesson (with more problems) in a SMART Notebook file or in a .pdf format, visit http://smartboardsmarty.wikispaces.com/Download+a+Notebook+File#Calculus and download the lesson, “Tear it Up!”
I’d love to give credit to a presenter at a National NCTM conference in the early 1980’s who showed me this idea but I don’t know who it was. If it was you (I think this person was part of a group presentation), thank you.