2. Page 714 #1-10 ANSWERS

4. Lesson Reflection for Chapter 14 Section 7

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

6. 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) Learn to find and use Hamiltonian circuits (14-7)

7. Today’s Learning Goal Assignment Learn to find and use Hamiltonian circuits

8. Pre-Algebra HW Page 718 #1-6

9. 14-7 Hamiltonian Circuits Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation Pre-Algebra

10. Warm Up 1. Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain. Yes; all vertices have even degrees. 14-7 Hamiltonian Circuits Pre-Algebra A B D C

11. Problem of the Day Find the missing polynomial that makes the equation true. a 3 + 2a 2 + 3 Pre-Algebra (3a 3 – a 2 – 3) + = 4a 3 + a 2 Pre-Algebra 14-7 Hamiltonian Circuits

12. Today’s Learning Goal Assignment Learn to find and use Hamiltonian circuits. Pre-Algebra 14-7 Hamiltonian Circuits

13. Vocabulary Hamiltonian circuit Insert Lesson Title Here Pre-Algebra 14-7 Hamiltonian Circuits

14. Pre-Algebra Roger and his friends are planning a trip to four Major League baseball parks. They will start their trip at Busch Stadium, in St. Louis, visit each stadium once, and then return to Busch Stadium. Use the graph on the next slide to find a path that Roger and his friends can travel. Note that the lengths shown are distances, not side lengths of geometric figures. Reading Math Pre-Algebra 14-7 Hamiltonian Circuits

15. Pre-Algebra Wrigley Field (Chicago) Busch Stadium (St. Louis) Comisky Park (Chicago) Miller Park (Milwaukee) 92 mi 374 mi 301 mi 10 mi 99 mi 269 mi A Hamiltonian circuit is a path that ends at the beginning vertex and passes through each of the other vertices in the graph exactly once. In a Hamiltonian circuit, the path is not required to traverse each edge. Pre-Algebra 14-7 Hamiltonian Circuits

16. Find a Hamiltonian circuit in the graph. Additional Example 1: Finding Hamiltonian Circuits Pre-Algebra Start/ Finish Visit each vertex once, but do not necessarily traverse each edge. Pre-Algebra 14-7 Hamiltonian Circuits

17. Try This: Example 1 Find a Hamiltonian circuit in the graph. Insert Lesson Title Here Pre-Algebra Start/ Finish Visit each vertex once, but do not necessarily traverse each edge. Pre-Algebra 14-7 Hamiltonian Circuits

18. Use the information in the graph to determine the shortest Hamiltonian circuit beginning at Boston. Additional Example 2: Problem Solving Application Pre-Algebra Cleveland Miami Boston Baltimore New York 522 mi 1,086 mi 955 mi 360 mi 190 mi 408 mi 1088 mi 170 mi Pre-Algebra 14-7 Hamiltonian Circuits

19. Pre-Algebra 1 Analyze the Problem Find the shortest path between the cities. In the graph the vertices represent different cities. 2 Choose a Strategy Find all the Hamiltonian circuits beginning and ending at Boston. Additional Example 2 Continued Solve 3 Find the length of each path. The first letter of each city’s name is used to represent the city. Pre-Algebra 14-7 Hamiltonian Circuits

20. Pre-Algebra Solve 3 B 190 N 170 B 955 M 1086 C 522 B = 2923 B 190 N 408 C 1086 M 955 B 360 B = 2999 B 522 C 1086 M 1088 N 170 B 360 B = 3226 B 360 B 170 N 1088 M 1086 C 522 B = 3196 B 522 C 1086 M 955 B 170 N 190 B = 2923 B 360 B 955 M 1088 N 408 C 522 B = 3333 B 360 B 955 M 1086 C 408 N 190 B = 2999 B 522 C 408 N 1088 M 955 B 360 B = 3333 Additional Example 2 Continued The shortest circuit is B N B M C B or B C M B N B. Pre-Algebra 14-7 Hamiltonian Circuits

21. Pre-Algebra Look Back 4 Be sure that you have all possible paths. There are 8 paths that meet the Hamiltonian criteria of only passing through each vertex once. Additional Example 2 Continued Pre-Algebra 14-7 Hamiltonian Circuits

22. Pre-Algebra Use the information in the graph to determine the shortest Hamiltonian circuit beginning and ending at B. Try This: Example 2 4 in. 6 in. 4 in. 3 in. 7 in. 5 in. 2 in. B A C D E Pre-Algebra 14-7 Hamiltonian Circuits

23. Insert Lesson Title Here Pre-Algebra 1 Analyze the Problem Find the shortest path between locations. In the graph the vertices represent different locations. 2 Choose a Strategy Find all the Hamiltonian circuits beginning and ending at B. Try This: Example 2 Pre-Algebra 14-7 Hamiltonian Circuits

24. Pre-Algebra Solve 3 Find the length of each path. The vertex locations are represented by letters. B 6 A 4 E 3 D 7 C 4 B = 24 in. B 4 C 2 E 3 D 5 A 6 B = 20 in. B 6 A 5 D 3 E 2 C 4 B = 20 in. B 4 C 7 D 3 E 4 A 6 B = 24 in. The shortest circuit is B C E D A B or B A D E C B. Try This: Example 2 Pre-Algebra 14-7 Hamiltonian Circuits

25. Look Back4 Be sure that you have all possible paths. There are 4 paths that meet the Hamiltonian criteria of only passing through each vertex once and beginning and ending at B. Pre-Algebra Try This: Example 2 Pre-Algebra 14-7 Hamiltonian Circuits

26. Lesson Quiz Use the graph for problems 1-2. Insert Lesson Title Here Pre-Algebra 12 in. 6 in. 13 in. 10 in. B A C D 4 in 4 in. 1. Find a Hamiltonian circuit in the graph. Possible answer: A-B-C-D-A 2. Determine the shortest Hamiltonian circuit beginning and ending at A, and find its length. A-B-D-C-A or A-C-D-B-A; 32 in. Pre-Algebra 14-7 Hamiltonian Circuits

27. Page 718 #1-6 ANSWERS