Page 714 #1-10 ANSWERS
Lesson Reflection for
Chapter 14 Section 7
Pre-Algebra Learning Goal
Students will
understand
collecting, displaying,
& analyzing data.
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)
Today’s Learning Goal Assignment
Learn to find
and use
Hamiltonian
circuits
Pre-Algebra HW
Page 718
#1-6
14-7 Hamiltonian Circuits
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
14-7 Hamiltonian Circuits
Warm Up
1. Determine whether the graph can be
traversed (traveled) through an Euler circuit.
A
Explain.
D
B
C
Yes; all vertices have even degrees.
Pre-Algebra
14-7 Hamiltonian Circuits
Problem of the Day
Find the missing polynomial that makes
the equation true.
(3a3 – a2 – 3) +
= 4a3 + a2
a3 + 2a2 + 3
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Today’s Learning Goal Assignment
Learn to find and
use Hamiltonian
circuits.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian
Insert Lesson
Title Here
Circuits
Vocabulary
Hamiltonian circuit
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
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.
Reading Math
Note that the lengths shown are distances, not side
lengths of geometric figures.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Wrigley Field (Chicago)
269 mi
10 mi
99 mi
Miller Park
(Milwaukee)
92 mi
Busch Stadium
(St. Louis)
Comisky Park
(Chicago)
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
Pre-Algebra
14-7 Hamiltonian Circuits
Additional Example 1: Finding Hamiltonian Circuits
Find a Hamiltonian circuit in the graph.
Start/
Finish
Visit each vertex once, but do not necessarily
traverse each edge.
Pre-Algebra
Pre-Algebra
14-7 Insert
Lesson
Title Here
Hamiltonian
Circuits
Try This: Example 1
Find a Hamiltonian circuit in the graph.
Start/
Finish
Visit each vertex once, but do not necessarily
traverse each edge.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Additional Example 2: Problem Solving Application
Use the information in the graph to
determine the shortest Hamiltonian
circuit beginning at Boston.
Boston
522 mi
Cleveland
360 mi
New York 1,086 mi
Baltimore
Pre-Algebra
Pre-Algebra
955 mi
Miami
14-7 Hamiltonian Circuits
Additional Example 2 Continued
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.
3
Solve
Find the length of each path. The first letter of
each city’s name is used to represent the city.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Additional Example 2 Continued
3
B
B
B
B
Solve
190
190
522
360
N 170 B 955 M 1086 C 522 B = 2923
N 408 C 1086 M 955 B 360 B = 2999
C 1086 M 1088 N 170 B 360 B = 3226
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
The shortest circuit is B N B M C B or
B C M B N B.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Additional Example 2 Continued
4
Look Back
Be sure that you have all possible paths.
There are 8 paths that meet the Hamiltonian
criteria of only passing through each vertex
once.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Try This: Example 2
Use the information in the graph to
determine the shortest Hamiltonian
circuit beginning and ending at B.
B
6 in.
A
4 in. E 2 in.
3 in.
5 in.
D
Pre-Algebra
Pre-Algebra
4 in.
7 in.
C
14-7 Insert
Lesson
Title Here
Hamiltonian
Circuits
Try This: Example 2
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.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Try This: Example 2
3
Solve
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.
Pre-Algebra
Pre-Algebra
14-7 Hamiltonian Circuits
Try This: Example 2
4 Look Back
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
Pre-Algebra
14-7 Hamiltonian
Insert Lesson
Circuits
Title Here
Lesson Quiz
Use the graph for problems 1-2.
A
12 in. 4 in.
C
10 in.
D
6 in.
4 in
13 in.
B
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
Pre-Algebra
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