A Random Walk, Part 2

Here’s a short summary of the Random Walk activity:

A student takes a random walk on a number line determined by flips of a fair coin: one step forward with a flip of a heads, and one step back with a flip of tails. After students propose questions about an individual’s random walk, the whole class, joined by other students if possible, conducts a simultaneous random walk. Students arrive in various locations on the number line and then walk into columns to create a human bar graph that records their final location. Students then investigate the theoretical probabilities involved in the random walk.

After the initial random walk of one student,  I have asked students to record what they noticed and what they wondered.  Here are a few of their thoughts:

• How often will I get heads or tails?
• Where will I arrive after a certain number of flips?
• How long will it take me to get to the other end of the room?
• What are my chances of getting to the other end of the room?
• It would take forever just to walk a couple of feet!

I tell students that in order to investigate these and other questions we are all going to do a random walk at the same time. Before we do this, I again ask,”what would you wonder?” Students are interested in where everyone would end up – and the walk itself would look like.  One prediction is that everyone will end up back in the middle because we all have an equal chance of getting heads and tails.

After having students record their thoughts about where they could end up, and how many people might end up at each location, we go out to a hallway or other suitable location.  Everyone lines up, and we do individual walks all at once (on cue, one flip at a time.)  Students have a calculator to generate their motion forward or back, or we have already flipped coins and written down our individual directions.  Here’s an example of what it looks like “from above,” after each step, for 5 flips…..and after the 6th flip; then we march together to form a human histogram. 6 flips is a good number to make an interesting histogram.The result is a surprise for many students – and something they consider worth explaining. Usually without my even asking, students return to the class with lots of noticing and wonderings. Why did the largest (but often not the majority) of people end up back in the center, at 0? Since there are 7 possible locations one can arrive at, why is the probability of reaching each final location not 1/7? I encourage students to investigate and explain the results using whatever approach they wish.

Many students turn to making a systematic list of possible sequences of coin flips.  (Below is a list for 4 flips.)

I encourage students to create a list of possible flip sequence for 2, 3 and 4 flips. In creating such lists, there is often discussion about whether a sequence like TTHH is “different” than that of HTTH, since they are both “2 heads and 2 tails”.  I have found the context of walking (“I had a different walk but ended up in the same place”) helps students think about why each of these possibilities should be listed separately when thinking about the probability of arriving at any of the possible locations.

After the students create lists organized like the one shown above,  they notice that the list looks similar to the human histogram they had created – but there is more that students want to know.   They have a lot of fun doing the walk and watching it happen, but most importantly it leaves students asking more questions.

To be continued…