Exponential Functions Introduction Activity

It has been a while since my last post, but I thought I'd share something new that I'm doing in my classroom. I've had a lot going on. I started my masters, my wife had our second child, and we moved into a new (to us) house.

I've been trying my hardest to implement new and engaging activities into my classroom. With so much going on in my own life, it can be difficult to get those things planned. I am pretty proud of this Introduction to Exponential Functions lesson that I did with my Algebra 1 minions. The conversations, noticings, wonderings, and reflections that they had during the short lesson were amazing! The day started with a quiz over exponent properties, and we finished the last 30 minutes with this activity.

I started by putting students in groups of three or four. On each table I placed several pieces of plain 8.5"x 11" pieces of paper and several pieces of plain 11" by 17" pieces of paper. I got the 11" x 17" paper pretty cheap from Staples because it was an open container. I gave my kids a simple instruction, "As a group and without touching the paper, make a prediction about how many times you think you can fold that piece of paper in half." For the sake of better discussion and prediction, I made them come up with a number that they knew was too low and a number that they knew was too high in addition to their prediction. Here is a picture of the results from two classes.

I had to ask them if they'd done this before, their predictions were so good!

Several groups assumed that the number of folds would double! 

I was amazed at the enthusiasm my kids showed for what was a relatively simple task. I think the key was not letting them touch the paper. They had all sorts of theories about why they thought their predictions were accurate. I let each kid fold their own piece of paper to test their theories. The blue numbers above each prediction are the highest number of folds they were able to achieve at their tables.

After they did the 8.5 x 11 size, we discussed as a class some of the different strategies for folding the paper in half. Then I had them repeat this same process for 11x17 piece of paper. Many students noticed that it was twice as big, so they doubled their numbers. There was much more debate about how many times the 11x17 could be folded. The results shocked some of my students. The number of times they could fold the paper didn't change! I asked them why that was so, and several students in each class were able to deduce that it had more to do with the thickness of the paper than the size. I was impressed that some of my kids who normally don't participate were the ones who came up with this idea. 

Once we discovered that the thickness is what matters for the number of folds, we turned our attention to how we could measure the thickness. I started by asking how thick the paper was when we hadn't folded it. We didn't have a precise number, so we decided to say "one sheet" thick. This reinforced the Zero Property of exponents... zero folds means it's 1 sheet thick! I walked them through each fold again and we discussed how the thickness changed each time. Most students were able to deduce that the thickness doubled with each fold, awesome! We plotted the results in a table in Desmos where x is the number of folds and y was the # of sheets thick and observed the graph.

The kids noticed almost immediately that it wasn't linear. So I showed them how when we stack paper, the result is linear. If I have one piece of paper, it is one piece of paper thick and so on and so forth. This seemed to click with them when they saw it compared visually.

I'm planning to follow up on this tomorrow with the first activity in the Desmos bundle for exponents. My goal is to really drive home the point that exponential functions are multiplicative and not additive. Of course, kids weren't using that vocabulary today, but we will get there.

Overall, for a 25 minute mini lesson, this served as a great introduction to exponential functions and I will definitely use it again in the future!