SWBAT to model exponential change

We can help students understand the abstract world of tiny measurements with a hands on activity.

15 minutes

**Carl Sagan** is my science hero. So I use his work whenever I can. I start this lesson by showing the first few minutes from his **Cosmos Series: Episode 9 the Lives of Stars**. Here it is on youtube:

Source: http://youtu.be/X7L6SZPxgNg (accessed August 12, 2014)

The only edit I make is to mute or "bleep" out the part where he says it takes 90 cuts to reach the size of the atom. I only bleep out the word "90" and ask the class to estimate, "how many cuts would it take before a slice of pie is as large as an atom?"

Talking about their estimation and why they think they are correct is a wonderfully fun conversation. It is a great opportunity to hear them conjecture in mathematics.

25 minutes

Following our discussion of the video, I place different sized sheets on each desk and ask students a question similar to the start up, "How many times will you have to cut the your paper in half before it is as wide as an atom?"

Before I set them to task, we discuss what is needed in order to solve the problem. We list these out and I ask them to only choose the most essential bit of information from the list:

**What is a "cut"?**

Here we agree to cut in a consistent direction so that we can compare our results as a class. We also agree that a cut should cut the paper in half.

I show them an image like this to clarify: vertical cut line.pdf

We also agree to disregard the cut paper in one of the many bins I bring to class for this activity.

**How big is an atom?**

They can find the size of the paper by measuring it, since they only need to know the length and width of the paper to be successful.

I like to make each group responsible for their own atom, so I display this chart and ask groups to pick one atom at a time: Atomic Radii

This chart is from wikipedia: http://en.wikipedia.org/wiki/Atomic_radius#Calculated_atomic_radii

Each number is in picometers. I explain that a picometer is 1 x 10^-12 meters. I briefly talk about what this means and help students understand that this is .000000000001 meters. Since they are not yet familiar with scientific notation, I use this is a *natural* way to start getting them familiar with the number form.

I circulate as they solve the problem for their paper size and their atom.

To record their work, I suggest that students use a table, with one column listing the number of cuts and the other listing the width of the paper (in cm). After a few cuts, they should notice the pattern of exponential decay and see that the paper is just getting 1/2 as large as before. The real challenge is to deal with this as a comparison to a meter.

20 minutes

We have students write out their final guess on the board for the class to review.

At this point in the lesson, students have found their respective answers and recorded their answers on the board. I do this by having students write their atom on a sticky note with its atomic radii and then ask them to place it in a table on the board. The table is simple. It is set up with many different columns with numbers of cuts: 89 cuts, 90 cuts, and so forth.

The first discussion is about the distribution of the cuts. **Did each sheet of paper require a different amount of cuts? Did you expect all the cuts to be the same or different? Why?**

Next we check out calculations again in Carl Sagan's video. I simply replay the video and include the answer of 90 cuts. Then I give students a minute to think about how they would prove to the class that this is (or isn't correct).

Students will generally reason by listing the halving process taking place with each cut

If each student's sheet was 80 mm x 110 mm, the first cut might make it 40 mm x 110 mm and then 20 mm x 110mm and so forth. During the investigation I ask many of them to make record the changes to predict when it would hit their atomic radii.

An advanced technique would be to write out the function.

f(x) = measure of the width of the paper based on the number of cuts x.

f(x) = .5^(x-1) * a, where a is the dimension of the paper and x is the number of cuts.

The discussion from here is meant to add more context to this process. I show this video and talk to the class about their observations: