Page 714 #1-10 ANSWERS Lesson Reflection for Chapter 14 Section 7.

#1-10 ANSWERS

Lesson Reflection for

Chapter 14 Section 7

Pre-Algebra Learning GoalStudents 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

18

#1-6

14-7 Hamiltonian Circuits

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

Warm Up

1. Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain.

Yes; all vertices have even degrees.


Pre-Algebra

A

BD

C

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


Today’s Learning Goal Assignment

Learn to find and use Hamiltonian circuits.

Pre-AlgebraPre-Algebra


Vocabulary

Hamiltonian circuit

Insert Lesson Title Here

Pre-AlgebraPre-Algebra


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


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


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


Try This: Example 1

Find a Hamiltonian circuit in the graph.


Pre-Algebra

Start/Finish

Visit each vertex once, but do not necessarilytraverse each edge.

Pre-Algebra


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


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


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


The shortest circuit is B N B M C B or B C M B N B.

Pre-Algebra


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.


Pre-Algebra


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



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


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


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


Lesson Quiz

Use the graph for problems 1-2.


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


#1-6 ANSWERS

Page 714 #1-10 ANSWERS Lesson Reflection for Chapter 14 Section 7.

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