The Ice Cream Stands Problem

Laura Drew

Topic:

One Way Functions, Problem Solving, Critical Thinking

Grade Level:

4 or 5

Objective: Students should

be able to solve the problem,
which is an example of a one way function. Therefore, the students
should be able

to

solve and identify
one way functions.

– A map of the Town of Iceberg for each student

– Overhead transparencies of the Iceberg map and the Secret
Solution

– A copy of the scenario for each student

– About 20 Unifix Cubes available for each student

Anticipatory Set – Pass out a copy of map and the scenario
to the students.

– Read it with them, making sure everyone understands what

Ivan and Ivana are trying to accomplish and why.

Concept Development – All the students need to do is figure
out where to put the ice

cream stands, so they are placed in positions most

advantageous to Ivan and Ivana.. In other words, using as few

stands as possible, how can Ivan and Ivana build enough stands

so that at the very most the farthest anyone will have

to

walk

is down one street to get to a stand.

– Do an example using the overhead. Start with an inappropriate

solution, and then show a better choice so the students can

model after your example.

Practice – The students should work with each other and
begin to try to

find the solution.

– The students should use the unifix cubes to organize
their

thoughts. For example, put one color on a corner where you

put a stand. Then put markers of a second color on all the

corners that are one street away from that ice cream stand.

– All the people who live in those houses will walk one
street

to

get

to

the ice cream stand on the next corner.

– Give plenty of time for experimentation. Remind students
that

part of the goal is to have as few stands as possible, because

stands are expensive to build.

– If you want to, you can tell students that it is possible

to

build only six stands and still
serve the whole town, or you

can let them discover this on their own.

– After the students have had a chance to work on the puzzle

and solve it, show them how it was made.

– Display the Transparency of the Solution.

Explain how you can make a puzzle like the ice Cream Puzzle

by drawing the solution first. This looks easy.

Lay the map of the town over the Secret Solution, so the

students can see the "easy" solution disappear.

Closure

Have the students make a simple puzzle of their own, using the

solution first method.

Explain to students that this puzzle is an example of what

mathematicians call a

one – way function.

If you start
with the

solution and create the puzzle, it’s easy. If you start with the

puzzle and have to find the solution, it’s not.

Ask the students to think of other situations in real life that

may present themselves as a puzzle like this. Some ideas could

include hospitals, hamburger restaurants, firestations,

restrooms, etc.

Explain that trying to find strategies for solving puzzles like

this is important when you need to plan in real life.

Assessment

1. Observation of individuals as they participate in groups. Note
any lack of participation. Each group should submit an example
of a correctly completed puzzle, with at least fewer than ten
stands. You may want

to

reward
those with fewer stands with a better grade, since their solution
is more desirable. However, this is up

to

the teacher, as there really is no incorrect answer to
this problem.

Bibliography:

– Internet

http ://www.coe.missouri.edu/~kyle/edu.html

http ://www.c3.1anl.gov/mega – math *

– Magazine

Teaching Children Mathematics Oct 1996, and Dec 1996.

Click

here

to view a sample map. Click

here

to view a sample solution to that map.

What you have in your hands is a
map of the town of Iceberg. It's a somewhat unusual way to draw
a map. The lines on this map represent streets and the dots are
street corners. The map doesn't have any houses on it, but we
so know that there is at least one house at each corner.

Iceberg would be a nice place to
live, except for one problem: you can't get ice cream anywhere
in town. So Ivan and Ivana Icicle have founded the Icicle and
IceBerg Ice Cream Company in order to do something about that.
Ivan and Ivana want to do something good for their town, so they
are going to build ice cream stands all over town where people
can go to buy ice cream. They also want to make money.

At first, Ivan and Ivana had hoped
to put an ice cream stand on every corner, knowing how, in the
summertime, they would just rake in the money, But ice cream stands
are expensive to build: you have to but all that lumber, and nails,
and windows, etc. Then you have to put big freezers inside them,
and pay people to work in them all day, and so forth. It didn't
seem possible to sell enough ice cream to pay for ice cream stands
on every corner.

They figured, however, that people
would still eat lots of ice cream if they only had to walk down
the street to get it. Their second plan was to build the ice cream
stands so that people could get ice cream either right there on
the corner where they live, or at the very most, have to walk
down only one street to find a corner where there was an ice cream
stand.

Now, all they have to do is figure
out where to put the ice cream stands. Where should they put them?
How many do they have to build?