GO IN DEPTH

# Finding Satisfactory Solutions

### Purpose

To explore the creative aspects of problem-solving and practice creative problem-solving strategies in the context of a story problem.

### Context

In this activity, students decide where to locate ice cream stands in a town so that no one has to travel too far to buy a treat. The problem-solving strategies for this problem give students a chance to grapple with the notion of proof and to decide what makes a solution satisfactory. Students should be encouraged to state their own criteria for what is a satisfactory result and to discuss their judgments in terms of their purposes.

Science for All Americans (p.16, pp.19-22) and Benchmarks for Science Literacy (p.37) take the approach that mathematics can be characterized as a cycle of investigation that is intended to lead to the development of mathematical ideas. Students should have the opportunity to use this cycle of investigation in their own work.

Before doing The Ice Cream Stands Problem in the Development, you should read the Mathematics in Maps and Planning and the Materials and Preparation sections on the MegaMathematics site.

### Motivation

Ask students to write a brief description of what they think mathematicians do. Then, have students read Mathematics is What Mathematicians Do which you can print out from the MegaMathematics website.

After students have discussed their ideas, ask them to revise their description of what mathematicians do based on the discussion.

### Development

Begin the lesson by saying to students: We will act as mathematicians to solve a problem. Our problem is to decide where to locate ice cream stands in a town so everyone in town can easily get to an ice cream stand. Our goal is to build the minimum number of ice cream stands so that no one will have to walk more than two blocks to get ice cream.

Continue by following the instructions for carrying out the activity called Algorithms and Ice Cream for All

Divide the class into groups of two or three. Each student group will need several copies of the map of Iceberg, which can be found on the website. Tell students to use the different copies of the map to try multiple solutions. You will also need to make overhead transparencies of the map as well as of the Secret Solution.  This overhead shows that it is possible to solve the problem by building only six ice cream stands.

After students have come up with their own solutions, you can lay the map of Iceberg over the Secret Solution on the overhead projector. But before you do this, allow time for students to display and discuss their own solutions. Once students understand the relationship between the Secret Solution overlay and the map, have them try to make their own ice cream puzzles following the instructions in the activity.

After students have made their ice cream puzzles and you have demonstrated the Secret Solution, discuss the activity using the discussion questions 1, 2, 3, and 5 listed on the bottom of the page. Students who are interested in finding out more about discrete mathematics can explore question #4 that discusses one-way functions. However, in order to focus student attention on the ideas contained in the central benchmarks for this lesson, it is best to concentrate on the questions that discuss problem-solving strategies.

### Assessment

The experience of looking for an answer to the Ice Cream Stands Problem and devising strategies to come up with an answer are more important than finding the right answer.

See the Perspectives for Evaluation section of the Algorithms and Ice Cream for All lesson for specific assessmnet strategies to use in this lesson.

You can also use some of the Extensions listed below to further assess students' problem-solving strategies.

### Extensions

Continue to work on the rest of the Algorithms and Ice Cream for All  module. In the Brute Force and Other Algorithms, students take a closer look at two algorithms that can be used to solve The Ice Cream Stands Problem and evaluate their effectiveness and efficiency. In Proving You Have Found the Minimum, students explore whether or not they can be certain that a better solution to the Ice Cream Stands Problem cannot be found.

Check out more stories on the Mega-Mathematics website. These can be done as class work or independent projects.

The Discrete Mathematics Project,  from the University of Colorado, Boulder, contains a large collection of discrete mathematics activities, many of which are appropriate for grades 6-8. The following are particularly relevant:

• Getting There From Here extends the ideas in the Ice Cream Stands Problem lesson by asking students to determine the number of different routes of shortest length that connect two points in a city with streets arranged in a square grid.
• In Bus Routes, students explore the concept of route design and finding the shortest route as it pertains to graph theory.
• What's the Shortest Route is an activity in which students work in small groups to arrive at the shortest route solution and be able to explain, and justify, how they arrived at this solution.

For an additional Nature of Mathematics lesson for grades 6-8, go to The Fibonacci Series

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