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Transcript of Page 714 #1-10 ANSWERS
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Page 714 #1-10 ANSWERS
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Lesson Reflection for
Chapter 14 Section 7
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Pre-Algebra Learning GoalStudents will
understand collecting, displaying, & analyzing
data.
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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)
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Today’s Learning Goal Assignment
Learn to find and use Hamiltonian circuits
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Pre-Algebra HW
Page 718
#1-6
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14-7 Hamiltonian Circuits
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
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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
BD
C
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Problem of the Day
Find the missing polynomial that makes the equation true.
a3 + 2a2 + 3
Pre-Algebra
(3a3 – a2 – 3) + = 4a3 + a2
Pre-Algebra
14-7 Hamiltonian Circuits
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Today’s Learning Goal Assignment
Learn to find and use Hamiltonian circuits.
Pre-AlgebraPre-Algebra
14-7 Hamiltonian Circuits
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Vocabulary
Hamiltonian circuit
Insert Lesson Title Here
Pre-AlgebraPre-Algebra
14-7 Hamiltonian Circuits
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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
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Pre-Algebra
Wrigley Field (Chicago)
Busch Stadium(St. Louis)
Comisky Park(Chicago)Miller Park
(Milwaukee)
92 mi374 mi 301 mi
10 mi99 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
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Find a Hamiltonian circuit in the graph.
Additional Example 1: Finding Hamiltonian Circuits
Pre-Algebra
Start/Finish
Visit each vertex once, but do not necessarilytraverse each edge.
Pre-Algebra
14-7 Hamiltonian Circuits
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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 necessarilytraverse each edge.
Pre-Algebra
14-7 Hamiltonian Circuits
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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 mi170 mi
Pre-Algebra
14-7 Hamiltonian Circuits
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Pre-Algebra
11 Analyze the Problem
Find the shortest path between the cities. In the graph the vertices represent different cities.
22 Choose a Strategy
Find all the Hamiltonian circuits beginning and ending at Boston.
Additional Example 2 Continued
Solve33Find 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
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Pre-Algebra
Solve33
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
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Pre-Algebra
Look Back44
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
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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
AC
D
E
Pre-Algebra
14-7 Hamiltonian Circuits
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Insert Lesson Title Here
Pre-Algebra
11 Analyze the Problem
Find the shortest path between locations. In the graph the vertices represent different locations.
22 Choose a Strategy
Find all the Hamiltonian circuits beginning and ending at B.
Try This: Example 2
Pre-Algebra
14-7 Hamiltonian Circuits
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Pre-Algebra
Solve33
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
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Look Back44
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
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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 D4 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
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Page 718 #1-6 ANSWERS