Playground Networks

Playground Networks

Background:

Graph theory is a branch of mathematics focused on networks. A network is a collection of straight or curved lines that intersect at various points. For example, a hopscotch court, and a foursquare court, are both examples of networks. In graph theory, networks are called graphs, the points of intersection are called points, and the straight or curved lines between the points are called edges. Leonard Euler (1707-1783) was a Swiss mathematician known for solving a famous mathematics problem called the Seven Bridges of Konigsberg using graph theory.

Here are three activities involving networks:

Traveling Networks

Cooperative Network Logic

Map Coloring with Movement

Traveling Networks

NOTE: THIS ACTIVITY WAS THE MOST DIFFICULT TO EXPLAINAND MOST DIFFICULT FOR PEOPLE TO DO AT THE MARCH 20 WORKSHOP. THIS IS A CANDIDATE FOR REMOVAL

Try to discover the rule of traveling networks.

Materials:

Playground with painted lines (playing courts, hopscotch, etc.)

And/or paved surface

Chalk

Try This:

1. Identify several networks on the playground, such as a hopscotch court, a foursquare court, or portions of a larger court (e.g. everything inside the three point line on a basketball court). The networks do not need to be closed figures. You can also use chalk to draw networks on the paved surface. Make sure you have a number of networks to try.

2. Just by looking at the network, decide if it can be “traveled”. Successfully traveling a network is to walk on every line (or edge) without ever walking on the same edge more than once. You may cross any point of intersection more than once.

3. After you have thought about it, choose a starting location, and stand there. Think about how you might keep track of whether or not you have traveled an edge. Depending on the figure, you may or may not need to end up where you started.

4. Now try to travel the network. If you don’t succeed at first, try again. How many times do you need to try to travel a network before you decide if it is impossible? Not all networks can be traveled.

5. Try another network, repeating steps two through four.

6. As you try different networks, deciding if they can or cannot be traveled, look for a pattern or a rule that will tell you which networks can be traveled, and which networks cannot be traveled.

7. Copy those that you drew, noting whether you thought they were or were not possible. Note that a network that you found not possible could be possible –provided that there is a solution, although you did not discover it. Share your drawings of networks and see if the “not possible” networks can actually be traveled.

What’s Going On?

Points in the networks can be described as odd or even. Odd points have an odd number of edges. Even points have an even number of edges. A network can be traveled if it has only zero or two odd points. For paths that don’t make a complete circuit (e.g. the starting point and the ending point are not the same), the first and last points will be odd – hence the two odd points.

Graph theory, and traveling networks, have a number of practical applications: people who figure out the routes for garbage/recycling pickup, mail delivery, and street cleaning, use this kind of geometry to find the most efficient routes for the drivers.

Cooperative Network Logic

This activity was adapted from a collection of activities called “More Games on Graphs” from The Los Alamos National Laboratory Mega-Math Program.

Background:

In this game, each person starts on a different point of the network, and must get back to his/her “home” point. The object is for all travelers to get home, so everyone wins by cooperation.

Materials:

Playground with painted lines (playing courts, hopscotch, etc.)

And/or paved surface

Chalk

Small ball or other object to toss

Other people

Blank stickers, or 3 x 5 cards with holes and string, to make labels for people to wear

Marking pens

Try This:

2. Choose or draw a network that has at least one more point than the number of people in your group. It will be easier if the network has two or three points more than the number of people in your group. For example, on a foursquare court, there are nine points. Eight people could play on this network, but the game would be quite challenging. Start with a group of six.

3. Use the chalk to label each point on the network with letters: A, B, C, etc. until each point is labeled.

4. Use the stickers (or the index cards) to make labels for each of the players, using the same letters as those on the network labeled in step three.

5. Each person should start at a point that is not the one that matches the label she is wearing.

6. Give the ball to one of the players.

7. Review the rules:

The object of the game is for all the travelers get to their home – the point whose label matches the one they are wearing.

Players can only run down an edge to an adjacent point if 1) the point is empty, and 2) they are holding the ball.

The person who is holding the ball can throw it can throw it to any other player.

What’s Going On?

This game is easiest if the number of players is two or three less than the number of points in the network. Through cooperation and logical thinking, all the travelers can get to their home points.

This game illustrates routing and deadlock in networks.

Map Coloring With Movement

This activity was adapted from a collection of activities called “More Games on Graphs” from The Los Alamos National Laboratory Mega-Math Program.

Materials:

Playground with painted lines (playing courts, hopscotch, etc.)

And/or paved surface

Chalk

Other people

Try This:

1. Identify several networks on the playground, such as a hopscotch court, a foursquare court, or portions of a larger court (e.g. everything inside the three point line on a basketball court). The networks need to be closed figures. You can also use chalk to draw large networks on the paved surface.

2. Use chalk to divide large regions into smaller subregions, so that you have the same number of regions (or subregions) as you have people. Each region (or subregion) needs to be large enough to stand or sit in.

3. Choose and rehearse a number of different body postures that are not uncomfortable to hold for several minutes. Try to come up with at least six. Here’s a sample list: Stand, Crouch, Lean, Sit Crosslegged, Stand with Arms Akimbo, Stand with Arms Teapot.

4. Each person chooses a region, and sits crosslegged.

5. Review the rules:

Each player must assume a position that is different from any of her neighbors.

A neighbor is someone in a region that shares a side or edge with your region.

6. Players assume different positions until no one is in the same position as any of her neighbors.

7. How many different positions are necessary? What is the minimum number of different positions needed to solve this problem?

What’s Going On?

This activity is based on map coloring problems. Maps are typically divided into regions, and no two neighboring regions are the same color. What is the fewest number of colors needed so that no two neighboring regions are the same color? It is impossible to construct a map (or playground network) that would require more than four colors (or four different positions). However, it’s helpful to start out with more than four options. Many maps can be colored with two or three different colors.

1Exploratorium Teacher Institute