Transcript The Elevator Problem - North Carolina School of Science and

THE ELEVATOR
PROBLEM
Christine Belledin
The North Carolina School of
Science and Mathematics
Teaching Contemporary
Mathematics Conference 2014
What is mathematical modeling?
According to the Common Core Standards for High School:
Modeling is the process of choosing andusing appropriate
Mathematics and statistics to analyze empirical situations,
to understand them better, and to improve decisions.
What did students have to say about
the Elevator Problem?
“Never before, in all my math experiences, had I seen a
problem as open ended and varying as this one. Working
on a problem like this with no obvious answer and many
different options was a wholly new experience for me. This
problem helped me visualize the role math could and
most likely will play in my future.”
The Elevator Problem
In some buildings, all of the elevators can travel to all of the
floors, while in others, the elevators are restricted to stopping
only on certain floors. Why? What might be the advantage of
having elevators that travel only between certain floors?
Suppose a building has 5 floors (1-5) that are occupied. The
ground floor (0) is not used for business. Each floor has 60
people working on it. There are three elevators (A, B and C)
available to take these employees to their offices in the morning.
Everyone arrives at approximately the same time and enters the
elevators on the ground floor. Each elevator holds 10 people
and takes approximately 25 seconds to fill on the ground floor.
The elevators then take 5 seconds between floors and 15
seconds on each floor on which it stops.
Simplifying Questions
1)
Suppose it is a holiday and only 5 people come to work
today. Each person works on a different floor, and they
all ride the same elevator. How long will it take for
everyone to get to work?
2) Now suppose that 5 people come to work and these five
people do not all work on different floors. How long will
it take for everyone to get to work?
Why was question (2) harder to answer than question (1)?
What assumptions will you need to make in order to
simplify the problem?
Assumptions my students made to
simplify the problem
• The elevator is filled to capacity (10 people) for each trip.
• The elevator must stop at every floor during each trip.
• No one takes the stairs.
• No one uses the elevator to go down during this time (or if
they do, it does not impact the time for the elevator to
complete it’s trip).
• The elevator doors don’t have to re-open on any floors.
Updated Questions
• If all elevators go to all floors, how long will it take
everyone to get to work?
• If 80 people were late using the unrestricted elevators,
approximately what time did the employees begin arriving
at the ground floor?
• Reassign the elevators to transport the employees to their
offices as quickly as possible. What arrangement
produces the shortest time? If this arrangement had been
used today, would everyone have arrived at their floor on
time?
What strategies did we use that we can apply to
future mathematical modeling adventures?
• We used a simple case to understand the structure of the
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problem.
We drew a diagram to help us visualize the scenario.
We thought about what made the problem hard to help us
figure out simplifying assumptions.
We considered the worst case scenario and solved this
rather than trying to think about all of the different
possibilities.
We found a solution that worked, then we modified it to see if
we could improve it.
We had to make sure our solution was realistic. (Sometimes
“mathematically optimal” is not optimal in the real world.)
Extensions
• If a fourth elevator is added to the building, how should
the elevators be organized to get employees to their
offices as quickly as possible?
• Consider our assumption that each elevator stops at each
floor on each trip. How realistic is this? In other words,
how likely is it that an elevator could skip one floor in a
given trip? Two floors?
• Suppose the number of workers on each floor is not
equal. How does this distribution of workers change your
elevator assignment?
Floor
0
1
2
3
4
5
# employees
0
100
100
70
100
50
Things I’ve learned…
• Think about what information you give to students and
when you give them that information.
• Check in often. Struggle is important, but it needs to be
productive.
• Communicate with students how they will be graded on
this assignment.
• If this is their first experience, consider grading only on effort (or not
at all).
• If you want to assess their ability to communicate their findings,
consider giving students the chance to revise their work.
• You want to encourage creativity. What grade do you assign to an
incorrect answer that is based on a really good idea versus a
correct answer that was very straightforward?
It’s hard for us to let go of details.
“I loved how realistic it was, aside from no one taking the
stairs.”
What else did students have to say?
We see the power of collaboration.
“In my pod, I felt like none of us could have solved the
problem on our own but we pooled together our knowledge
and we found that it was possible to solve it together.”
We struggled. And we learned that we
can persist through the struggle.
“It was also a very complex problem, much more complex
than I had ever done before. We had to set assumptions to
reduce the amount of variables and make the project
manageable. Even with the assumptions the problem was
daunting. We had to break it down logically instead of just
trying to plug it into a memorized equation. This thought
process is very common in this class, and while I found it
confusing and hard, I end with deeper understanding of
how to do the problems.”
“The elevator problem was probably the first time in the
class that I felt like I was trying to comprehend something
completely beyond my intelligence, but eventually I figured
out what we are doing.”
We liked it!
“I absolutely loved the elevator problem because it was so
intricate and complex. I also liked that it might actually be
helpful one day and have real world application. I liked that
in order to find the one of many possible final solutions, you
must first solve for one tiny section, how long it takes to get
to one floor, and then apply it to the whole process. I think
this was also one of my favorite problems because it was a
reasoning problem instead of a computation problem. I
wish we would do more problems like these more often.”
We can see the relevance of this
problem and this process to our lives.
“I don't know what a profession that focuses on efficiency
(workplace or public) is called, but I would love to work out
things like this for a living.”
“In most of my other math classes, the concepts were
mostly superficial; in the sense that we only learned the
basics and processes of a certain idea without working on
how it could be used in real life. Of course, this was often
nice and easy, bit if I'm looking to work in a STEM field one
day, it is crucial to understand the applications of the
different things we learn.”
Request for feedback
• What kinds of resources could we provide that would help
you use this kind of problem with your students?
• What kind of resources would your colleagues want to
see?
All feedback is welcome!
[email protected] or [email protected]
References
• Common Core State Standards Initiative, Common Core State Standards:
Mathematics, http://www.corestandards.org/the-standards/mathematics
• Compton, Helen and Dan Teague. “The Elevator Problem.” Everybody’s
Problems, Consortium, Number 57, COMAP, Inc. Lexington, Massachusetts,
1986.
• Mueller, Claudia M., and Carol S. Dweck. "Praise for Intelligence Can
Undermine Children's Motivation and Performance." Journal of Personality
and Social Psychology 75.1 (1998): 33-52.
• PISA 2012 Mathematics Framework Draft,
http://www.oecd.org/pisa/pisaproducts/46961598.pd
• President’s Council of Advisors on Science and Technology, PCAST, Engage
to Excel: Producing One Million Additional College Graduates with Degrees in
Science, Technology, Engineering, and Mathematics, Office of Science and
Technology Policy, Washington DC, February 2012,
http://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-engage-toexcel-final_2-25-12.pd