Transcript Chapter 11

Chapter
11
Introductory Geometry
© 2010 Pearson Education, Inc.
All rights reserved
February 5th was Super bowl XLVI (46)
Know your Roman Numerals for Math and for
Outlines in other Classes and for the super bowl! (3 min)
http://www.youtube.com/watch?v=9nUmjX8EpI8&feature=related
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Now that you understand a bit about the Roman
Numeral system of counting…
Let us keep our brains working with rearranging letters:
In 1930, Mars introduced Snickers, named after the
favorite horse of the Mars family
In three minutes try to write down as many words as you
can you make with using only the letters in the phrase :
happy birthday snickers
Examples: third or sad
Slide 11.5- 3
Before we get to networks
let us warm up our synaptic connections with
the nine dot problem.
Given the nine stars(points) can you connect them using
only straight segments and never lifting your pencil?
Set timer for three minutes: GO!
http://www.youtube.com/watch?v=Rq3ta6SvlTo
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11-5 Networks
Teaching thinking and reasoning skills.
A teacher is one who makes themselves
progressively unnecessary.
- Thomas Carruthers
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Networks
Vertices – the points in a network
Arcs – the curves in a network (also called edges)
Traversable network – a network having a path
beginning at some vertex and ending at the
same or another vertex such that each arc is
traversed exactly once.
Euler circuit – a traversable network having the
same starting point and stopping point.
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Königsberg Bridge Problem
Is it possible to walk across all the bridges so that
each bridge is crossed exactly once on the same
walk?
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Euler Paths and Circuits
http://www.youtube.com/watch?v=KhW4I0D4_mQ&feature=related
Watch clip.
Set timer for 5 minutes:
GO!
Then proceed to next
slide…
a pioneering Swiss
mathematician.Euler made
important discoveries in logical
reasoning and graph theory.
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Königsberg Bridge Problem
The solution is represented as a network.
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Networks
This is an Euler circuit.
Slide 11.5- 10
Traversable Networks
Traversable network – a network having a path beginning at
some vertex and ending at the same or another vertex
such that each arc is traversed exactly once.
(a), (b), and (c) are traversable.
(d) is not traversable.
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Network Vertices
Even vertex – a vertex where the number of
arcs that meet is even.
Odd vertex – a vertex where the number of arcs
that meet is odd.
If a network is traversable,
then each vertex that is not a starting point or
a stopping point must be even.
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Properties of a Network
 If a network has all even vertices, it is
traversable. Any vertex can be a starting point,
and the same vertex must be the stopping
point. Thus, the network is an Euler circuit.
 If a network has two odd vertices, it is
traversable. One odd vertex must be the
starting point, and the other odd vertex must be
the stopping point.
Slide 11.5- 13
Which of these networks are traversable?
Which of these networks are Euler circuits?
Euler circuit – a traversable network having the
same starting point and stopping point.
Traversable networks:
(i), (ii), (iii), and (v)
Euler circuits:
(ii) and (v)
Slide 11.5- 14
Let us try a puzzle. Ready or Not! Here it is!
http://www.youtube.com/watch?v=CQNH2ifOAQE&feature=related
Start the video
Pause when necessary
Let us try 5 minutes on this puzzle too…
Then finish the video.
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The network is not traversable,
and it is impossible to go through all the rooms and pass through
each door exactly once.
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Paths and walks
A path P = (v1,v2,v3,...,vk) is an ordered list of vertices such that
From each of its vertices there is an edge to the next vertex in the list
No vertex occurs in the list more than once.
The first vertex of a path is called the origin and the last vertex is called
the destination. Both origin and destination are called endpoints of
the path.
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Example:
In the graph below, there are many paths from node 6 to
node 1. One such path is P1 = (6, 4, 3, 2, 5, 1); another
path from 6 to 1 is P2 = (6, 4, 5, 1).
Any node all by itself makes a trivial path. For example, (3)
is the path that begins and ends at node 3. Another way
to see (3) is as an ordered list with exactly one element:
the node 3.
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Gas, Water and Electricity Puzzle
http://www.youtube.com/watch?v=hjAP8Fy5WhE&feature=related
Albert, Betty and Chris are not getting along the best today,
But all three want to play
Basketball, Hopscotch and Jump rope at recess.
Can that happen without running into each other?
I am sure that they will be friends again tomorrow.
Not everyone gets along ALL THE TIME!
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Social bookmarking,
bipartite graphs, and structural equivalence
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Mathematical or
Reasoning and Logic
Trees
Chain of Command
Or
Emergency Contact Tree
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Sound Familiar?
Proofs?
Mathematical Induction
A mathematical proof is a rigorous explanation of such a claim.
(Example claim: "MapQuest finds the shortest path between
any two addresses in the United States.")
A proof starts with statements that are accepted as true (called
axioms)
and uses formal logical arguments to show that the desired
claim is a necessary consequence of the accepted statements.
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http://greatmathsgames.com/number/item/33-brainteasers/32-traversable-networks-part-1.html
Traversable network grade school puzzles
Logical Reasoning in the movies:
http://www.math.harvard.edu/~knill/mathmovies/swf/shrek3_lies.html
The Pinocchio puppet which cannot lie
is trying not tell the truth…
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He who dares to teach must never cease to learn.
- Anonymous
Next class we will review Chapter 11
Then you will be ready for the MathXL test!
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