Games with Paper

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SWBAT to model exponential change as they cut and stack paper

Big Idea

We can model the immense numbers created in the exponential process and find hands on activities that help all learners.

Start Up

15 minutes

I came across Jeff Root's page and fell in love with this activity. The launch here is simple. Jeff writes:

Get a big sheet of paper and a good paper cutter.  Cut the sheet in half, and stack the pieces together.  Turn the pile 90 degrees, cut it in half again, and stack the cut pieces again.  Keep doing this until you've cut and stacked the sheet 100 times.

That's it. That's the entire introduction. I set up different size sheets and scissors and set them to work. As they get to work, they notice how impossible this task really is. I let each group discover this at their own pace, but react immediately when they are ready for the next step. 

I ask questions like, "I noticed you have stopped cutting the paper. What's up? Do you need a larger sheet? Maybe that one was too small?" 

Sometimes students will try a larger sheet and other times they believe the size of the sheet won't matter. If they decide that the paper size won't make a difference, I ask them to start proving it by estimating the answer to this question and explaining their reasoning:

"How high would the stack go if we were able to complete 100 cuts?"

I usually treat this estimation reflection as an individual activity.


25 minutes

We start the group investigation by sharing and discussing the students' estimations and observations about the process undertaken in the Start Up. I begin with some light questions, like "how many cuts did you get?" I record these amounts on the board so that we can compare them as a class. We talk about the results and what they mean. 

Then I start to ask about their estimation, "so how high would the stack get? How did you describe this height? Did you compare it to something and say 'I think it will be as tall as me,' or did you give it a number?"

After we talk about their ideas, I ask the question that is the focal point of this investigation:

What would you need to know in order to answer this question precisely?

The obvious answer is the thickness of the paper, but my students will think of many clever and interesting ideas. I sometimes list them all out and then ask the class to "pick one" from the list and tell me why that one is the most important. This opens up the possibility for a brief debate. 

When they are ready, I hand out Question and Info and get the class started on the problem posed for them. I love to see how they approach the problem and what they reach as an answer. They can try and use a calculator but will have to estimate their work. This will lead to some variation in answers and be a source of rich conversation in the summary. 


20 minutes

During the Investigation, some students recognize immediately that this is a problem of exponential growth, but most students are tripped up by the decimals:

.01, .02, .04, .08, ....

To help some students, I ask them "when will it reach a meter? Will it ever be exactly a meter tall?" If possible, I also push other students by saying, "is there a function for this?"

Teacher's Note: The function f(x) = .01(2^(x-1)) is very similar to their chess investigation. It is also possible to change the difficulty of the Investigation by asking about inches, by asking for a graph, etc. 

I use the summary as a time to address all the ideas that came up during the investigation. I also use this summary to add meaning to their answer of .01 x 2^99 mm. I like the comparisons made by Root on his site (see However, I would build up to the answer and have the students involved as I go. I plan to ask my students to find when the height of the stack would pass ____________ , using different key points from Jeff Root's chart:

I plan to take my students through this journey by giving them the approximate distance across the USA and asking, how many cuts will it take to exceed this number? We would repeat this process until students understand that the size of this stack of paper seems to reach past the observable universe!