One of the reasons
probability is a favorite attraction is that it can
lead to many surprises along with some useful math
done in interesting ways. The challenge in this
activity is to see if students can make a prediction
about the likelihood of toothpicks landing on lines
when dropped randomly using experimental data.
Target grades: 8-10.
- Make a set of shoe box
tops toothpick tossing ŇarenasÓ.
Note: If you don't have
the boxes, you can demonstrate the experiment by
dropping the toothpicks on a tiled floor. Here I dropped
a bunch on a hardwood floor. (See photo below.) I
highlighted the toothpicks and lines on the floor. An
adjustment will need to be made since the toothpicks
will not be the same as the perpendicular distance
- Draw equidistant,
parallel lines inside the shoebox top. The distance
between the lines should be the same as the length
of the toothpicks. Make one of these for each of
Setting the Stage
Doing the Activity
- Have your students sit
in groups or teams. You are facing them standing
next to your one computer station.
- Tell the students that
they will be answering a question that was
originally posed by a man named Georges-Louis
Leclerc, Comte de Buffon about 250 years ago. But
- Show this video
Stop it after one minute. Ask the students what they
notice and have them take 2 minutes to share with
their neighbor(s). Ask volunteers to share what they
noticed with the class. Repeat with what they
wonder about. (If you are new to this
"notice/wonder" strategy watch this video.)
- Suppose we have a floor
made of parallel strips of wood, each the same
width, and we drop a needle onto the floor. What is
that the needle will lie across a line between two
- Take guesses from your
class. Then ask them to explain their guess.
- Ask them: How might we
proceed to find out which student or group made the
best guess? LetŐs do the experiment to help us find
- Show them an arena for the
- Hand out the activity sheet. Make
sure the students understand the instructions.
- Each group will drop a
total of 100 toothpicks 10 at a time on their
parallel lined arena.
- Students toss the
toothpicks and record them on the student sheet.
When finished each group will share their results
via a spreadsheet projected
onto the white board by the teacher.
Debriefing the activity and what about the
- Students help the teacher
to complete the summary spreadsheet.
- After finishing, ask the
students how they might come up with even a better
or best "guess"? (Theoretical probability is 2/Pi or
approximately .6366.) Who had the best estimate in
the class? Compare the group results with the
individual group guesses to determine who had the
closest estimate. (That is, the group that came
closest to .6366)
- Go to this
- Note the number that is
constantly changing. Does it seem to be getting
closer to a well known constant? It's Pi -
- How did Pi get into the
the number of your individual throws and divide it
by the number of crosses. What do you get? Check
with other groups. What can you conclude? (Your
answer is an approximation for Pi.)
- Why does this happen? (See
George Reese's explanation.)
- Here's alternative lesson
- Watch this simulation.
1:17 min video.
Pi 2:07 min video. Teacher TV. Buffon's needle
experiment done using french bread and taped lines!
- Imagine throwing a green
dart onto an xy plane (dartboard) when a toothpick
crosses a line and a red dart when the toothpick
lands between the lines. After over 56,000 throws
here's what the dartboard looks like.
- To see the actual simulation done in Scratch go to here.
- In the simulation above, the number of throws was
56926 (35703 + 21223). So if we double the number of
throws (113825) and divide by the number of crosses
(green dart hits: 35703) you get 3.18... which is a
reasonably close approximation to š.