2. Lesson Reflection for Chapter 14 Section 6

3. Pre-Algebra Learning Goal Students will understand collecting, displaying, & analyzing data.

4. Students will understand collecting, displaying & analyzing data by being able to do the following: Learn to identify populations & recognize biased samples (4-1) Learn to organize data in tables and stem-and-leaf plots (4-2) Learn to find Euler circuits (14-6)

5. Today’s Learning Goal Assignment Learn to find Euler circuits.

6. Pre-Algebra HW Page 714 #1-10

7. 14-6 Networks and Euler Circuits Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation Pre-Algebra

8. Warm Up Fill in each blank. 1. A ________ is perfectly straight and extends forever in both directions. 2. A ________ is the part of a line between two points. 3. A ________ names a location. line line segment point 14-6 Networks and Euler Circuits Pre-Algebra ? ? ?

9. Problem of the Day There are 8 classes of 27 student each in the eighth grade. If Mr. Allen buys boxes of pencils that contain 36 pencils each, how many boxes will he need to buy so that he can give each eighth- grader 2 pencils? 12 boxes Pre-Algebra 14-6 Networks and Euler Circuits

10. Learn to find Euler circuits Pre-Algebra 14-6 Networks and Euler Circuits

11. Vocabulary graphcircuit networkEuler circuit vertex edge path connected graph degree (of a vertex) Insert Lesson Title Here Pre-Algebra 14-6 Networks and Euler Circuits

12. Pre-Algebra Kansas City Minneapolis Pittsburgh Denver Houston A new airline may begin by offering service to only a few cities. Suppose a small airline has flights between only the cities shown. In mathematics, there are graphs of equations, bar graphs, and various other types of graphs. The representation of the airline’s routes is a type of graph. 14-6 Networks and Euler Circuits

13. Pre-Algebra In a branch of mathematics called graph theory, a graph is a network of points and line segments or arcs that connect the points. The points are called vertices. The line segments or arcs joining the vertices are called edges. A path is a way to get from one vertex to another along one or more edges. A graph is a connected graph if there is a path between every vertex and every other vertex. The degree of a vertex is the number of edges touching that vertex. 14-6 Networks and Euler Circuits

14. Find the degree of each vertex, and determine whether each graph is connected. Additional Example 1: Identifying the Degree of a Vertex and Determining Connectedness Pre-Algebra 2 3 3 2 4 2 The graph is connected. There is a path between every vertex and every other vertex. A B C D EF VertexDegree A B C D E F 14-6 Networks and Euler Circuits

15. Find the degree of each vertex, and determine whether each graph is connected. Pre-Algebra The graph is connected. There is a path between every vertex and every other vertex. Try This: Example 1 A BC D E G F H 2 3 3 3 2 2 VertexDegree A B C D E F G H 5 2 14-6 Networks and Euler Circuits

16. Pre-Algebra A circuit is a path that ends at the same vertex at which it began and doesn’t go through any edge more than once. An Euler circuit (pronounced oiler) is a circuit that goes through every edge of a connected graph. 14-6 Networks and Euler Circuits

17. Pre-Algebra Every vertex in an Euler circuit has an even degree. To understand why this is true, suppose a vertex has an odd degree. In an Euler circuit, two edges are required each time a path enters and exits the vertex. A vertex with an odd degree would have an edge that would be traveled twice or not at all. 14-6 Networks and Euler Circuits

18. Pre-Algebra One famous problem in graph theory is the Konigsberg Bridge problem. The goal is to find a path that crosses every bridge only once and returns to the starting point. Solving the Konigsberg Bridge problem is equivalent to finding an Euler circuit in the graph. ¨ ¨ 14-6 Networks and Euler Circuits

19. Additional Example 2: Application Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain. A B C D The graph is an Euler circuit because it is connected, and all vertices have even degrees. Pre-Algebra 14-6 Networks and Euler Circuits

20. Try This: Example 2 Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain. Insert Lesson Title Here Pre-Algebra The graph is a Euler circuit because each of the vertices has an even number of degrees. A B C D E F 14-6 Networks and Euler Circuits

21. Lesson Quiz Use the graph for problems 1-2. No; vertices C and D each have an odd degree. A: 2, B: 4, C: 3, D:3; yes Insert Lesson Title Here Pre-Algebra A B C D 1. Find the degree of each vertex, and determine whether the graph is connected. 2. Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain. 14-6 Networks and Euler Circuits