South Carolina ETV

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Network This (Grades 5-6)

Master Teacher

Linda Brown

Time Allotment

Two 90-minute blocks

Overview

Through the activities presented in this lesson, students will define, construct and analyze networks. By using a school map to discover the shortest paths to all of their classes, the students will find networks at their school. Then they will use networks to determine the best place to build ice cream shops in a make believe neighborhood on the Internet.

Subject Matter

Mathematics

Learning Objectives

Students will be able to:

determine the number of paths in a network with up to six vertices
construct a network to solve a problem situation.
describe and defend the solution to a network problem

South Carolina Standards

Visit the South Carolina Department of Education for the South Carolina Mathematics Standards.

Geometry—Grades 6_8

I. Use visualization, spatial reasoning and geometry modeling to solve problems.

A. Use visual tools such as networks to represent and solve problems.

Media Components

Video

Math Vantage: Patterns, Lesson 5: Networks, Paths and Knots

Web Site

Algorithms and Ice Cream for All. Scroll down to the big box named "The Big Picture." Double click on "Activities" and then choose "The Ice Cream Stands Problem." This activity asks: "Where shall we locate ice cream stands in our town so that no one has to travel too far to buy a treat?" The problem-solving strategies for this exercise apply to many other situations that require planning facilities. Students will have a chance to grapple with the notion of proof and decide what makes a solution satisfactory. They are also exposed to an important unsolved problem in mathematics.

Materials

Per student:

map of the school
Activity Sheet 1
colored pencils
rulers
map of "Iceberg" from Web problem
unifix cubes

Prep for Teachers

Run off copies of a map of your school for each student.
Go to the Web page: Algorithms and Ice Cream for All and preview "The Ice Cream Stands Problem." Run off copies of the "map of Iceberg" for each student and make overhead transparencies of the "Iceberg Map" and the "Secret Solution."
Bookmark the Web site on the computer for the students.

Introductory Activity: Setting the Stage

After the warm-up activity, the students will explore networks already set up at their school. Then they will create a new underground network that will link the cafeteria to all parts of the school.

Step 1: Warm-up Activity

Find enough space for the class to stand in a circle, shoulder to shoulder. You might want to do this in two groups if you have a large number of students.
Ask students to grab hold of somebody's right hand with their right hand.
Then have them do the same thing with their left hand.
Now, ask them to untangle the circle without letting go.

The students started off in a tangled network. As they untangled themselves, they straightened out their network and made it easier to see.

Step 2: School Map Activity

Pass out maps of your school and colored pencils to each student.
Ask the students to draw a path to all of their classes.
Then ask them to draw alternate routes using a different colored pencil.
Discuss which paths are shorter. Longer. They may need to measure the paths with rulers.
Then pretend that the students want to build an underground path to the cafeteria from all of the different buildings in the school, but they want to use the fewest tunnels. Tunnels have to be straight. Have them work in groups to come up with the best paths with the fewest tunnels.
Ask the different groups to present their solutions to the class.

Learning Activities

Step 1

The "Don't Lift That Pencil" activity (Activity Sheet 1) will allow students to explore different networks. They will discover how paths are put together to create a working network. Pass out Activity Sheet 1.

Ask students to try to trace each figure without lifting their pencil or going over a line segment twice. Can they do this on all of the shapes? If not, why?

If the students don't figure it out, tell them that only the networks with no more than two odd-numbered intersections will work. Odd-numbered intersections are points where an odd number of line segments meet. Each figure has exactly two odd-numbered intersections except the third one. It has four odd-numbered intersections. Therefore, it will be impossible to trace without lifting or tracing over a line twice.

Step 2

Focus for Media Interaction: Have students raise their hands when they think they know the definition of networks as they watch the video you are about to show. (They will be talking about the different networks used to deliver the US Mail.) The students might have an idea here about what networks are. If not, the girl in the video gives the mathematical definition of networks when she is looking at a circuit board. The definition she gives is "a series of points on a surface connected by lines." As soon as she gives this definition, pause the video and ask your students to name some other uses of networks. They might mention a sprinkler system, telephone lines, cable TV, highways, cellular phones, class schedules, and tournament schedules. The human brain, circuit boards, and the US Mail have already been mentioned in the video.

START the video at the beginning after the Math Vantage sign flashes on the screen and the girl is looking into the mailbox. PAUSE the video after the girl give the definition of networks.

RESTART the video and listen to the girl list other networks. STOP the video when the girl is in the army tank and it starts rolling away with Math Vantage written under it.

Focus for Media Interaction: Compare the networks the students named with the list of networks mentioned in the video. Ask the students if they can think of any more networks not mentioned in the video or that they already have named.

Step 3

Have students do the Ice Cream Stands Problem. The problem that is explained in the story is easy to understand, but, surprisingly, there is no simple, straightforward way to solve it. Students will find themselves experimenting with a variety of approaches.

1. Pass out copies of the map of Iceberg (which is a graph) and present the following to the students:

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 do 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 The Icicle & 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 want it to be easy for the people to get 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 buy 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?

2. Show the students how to use the unifix cubes to mark the places where they think that the ice cream stands should go. Place a marker of one color on a corner where you would put an ice cream 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.

3. Have the students experiment with their maps and decide where they think the ice cream stands should go. As students find configurations of ice cream stands that will serve all the houses, remind them that the ice cream stands are expensive to build, and that Ivan and Ivana want to build as few as possible. Ask if there's any way to rearrange their configuration of ice cream stands so that one or more can be eliminated.

4. You can tell students that it is possible to build only 6 ice creams stands and serve all of the houses in this town, or you can let them try to discover that this is the minimum on their own.

5. After students have had a chance to work on the puzzle and solve it, show them how it was made.

Display the transparency of the Secret Solution.

Explain that you can make a puzzle like the Ice Cream Stands Puzzle by drawing the solution first. It is easy to see what the solution to the puzzle is when looking at the Secret Solution.

Then lay the map of Iceberg over the Secret Solution on the overhead. Almost by magic, the easy-to-see solution disappears. After drawing the solution to the puzzle, the puzzle-maker draws just enough disguising lines to make the solution hard to find.

Invite each student to make a similar puzzle of his/her own. It is not necessary to make puzzles with ice cream stands only. Perhaps the students would like to make a puzzle about something that is more interesting to them than ice cream. One way to begin is by asking, "What kinds of things do you think a town should have many of so that there is one close to everyone's house?" Students' ideas may include: convenience stores, video rentals, swimming pools, shopping malls, parks, etc.

Step 4: Discussion

Generate a discussion with the following questions.

1. What strategies did you use to try to figure out where to put the ice cream stands? How did you check to make sure that no house was too far away from an ice cream stand? Do you think it is possible to arrange the ice cream stands in a different way so that Ivan and Ivana won't need to build so many of them?

2. How did you make your own puzzle? If you didn't use ice cream stands in the puzzle that you made, how did you decide what to use instead? What happened when you showed your puzzle to other people? Were they able to solve it?

3. Do you think it would be possible to make a puzzle like this, and then come up with a solution that is even better (i.e., takes fewer ice cream stands) than the one you built into the puzzle in the first place? Try to make a puzzle where that happens.

4. 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. One way it's easy, one way it's hard. A place that one-way functions are useful is for inventing secret codes. When you have a good scheme for a secret code, it is easy to encode a secret message, but difficult to decode it. Invite students to find other examples of one-way functions.

5. Ask students to think of other situations in real life that might present themselves as a puzzle like this. Some of their ideas might include: finding locations for warehouses, main and branch offices, electrical switching stations, telecommunication centers, airports, hamburger joints, hospitals, fire stations, and restrooms. Remind students that when people are faced with problems like this in real life, someone hasn't made the puzzle ahead of time. Trying to find strategies for solving puzzles like this is important when you can't just say, "I give up. What's the solution?"

Culmination Activity/Assessment

Step 1

Present this problem to the students to solve and ask them to defend their answers: There are seven friends who want to build their own business with individual offices. They want to connect the offices with underground tunnels. How can they arrange their offices so that each person can connect with everybody else in the fewest number of tunnels and in the shortest distance? (This should work best if everyone is in a circle except one who is in the middle. The person in the middle is connected to everyone else. Then each person should only have to go through two tunnels at the most to get to someone else.)

Step 2

Ask each student to draw his/her conclusion (network) on paper and to tell why his/her answer has the fewest possible tunnels.

Cross-Curricular Extensions

History: Research Euler and how he proved the theory on networks with the famous "Seven Bridge Problem." This is explained in the video in the segment right after you stopped the video where the tank was rolling away with Math Vantage written under it.

Find out how networks have been used throughout history.

Science: Experiment with different objects by dipping them in a bubble solution and lifting them out to discover the shortest networks to the points on the objects. When you dip a cube or tetrahedron into a bubble solution and lift it out, there will be a network of lines made by the bubbles.

Community Connections

Have somebody from the city come in and discuss the way the city decides to network sewers and streets.
Ask someone from the telephone company to come speak about how the telephone lines are networked.
Visit the post office and go on a tour to see how the mail is networked

Activity Sheets (PDF)

Activity Sheet 1