Math Saturday Series

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Math Saturday Series

• Presented by West Virginia University
• Institute for Mathematics Learning
• Mathematics Department
• Dr. Robert Mayes
• rmayes_at_math.wvu.edu
• 293-2011 ext. 2304

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Math Saturday Series Kickoff

• Purpose
• To explore mathematics you may not have seen
• To have FUN.
• Audience
• Any 6-8 grade students in Mon County
• No cost, no obligation
• No grade or assessment

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Math Saturday Series Kickoff

• What topics would you like to see in your Math
  Saturday Series
• Probability and Statistics
• Geometry non-Euclidean, transformational, and
  Euclidean
• Trigonometry
• Mathematical Modeling
• Discrete Math graph theory, combinatorics
• Other topics of interest?

4
Arithmetic-24 Game

• Lets warm up with a game from China.
• http//www-personal.umich.edu/huahaiy/arith24/ind
  ex.html

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What is math?

• Pair up with a fellow participant and spend a few
  minutes discussing what you believe mathematics
  to be.
• Math can have a number of definitions. For today
  we will take the following view.
• Math is a search for patterns.

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Integer Patterns on a Circle

• Step 1 Place 4 integers on a circle.
• Step 2 Compute all differences of pairs of
  consecutive numbers.
• Step 3 Replace the original numbers with the
  absolute value of these differences.
• Step 4 Repeat steps 2 and 3 until you see a
  pattern.

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Integer Patterns on a Circle

• What pattern did you discover for 4 numbers on
  the circle?
• Solution Differences go to zero.
• What if we have 5 numbers? 6 numbers? Can you
  determine a pattern for these cases? Use the
  following applet to explore.
• http//www.cut-the-knot.com/SimpleGames/IntIter.sh
  tml

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Patterns in Sequences

• Sequence an infinite ordered list (we will limit
  the terms in the sequence to integers).
• Sum of the first n positive integers
• What is the sum of 1 2?
• What is the sum of 1 2 3?
• What is the sum of
• for any positive integer n?

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Patterns in Sequences

• Gausss Solution
• http//www-groups.dcs.st-and.ac.uk/history
• Sequence of Sums Solution
• How do we determine the pattern in a sequence?

10
Method of Finite Differences

• Based on subtracting successive terms of the
  sequence.
• If the sequence has a linear pattern, what will
  happen as we take successive differences?
• Compare this to what happens in the quadratic
  pattern for adding the first n integers.

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Method of Finite Differences

• Apply the Method of Finite Differences to find
  the pattern for the following
• Sum of the squares of the first n integers.
• Sum of the cubes of the first n integers.

12
Geometric Patterns

• How many diagonals are there in a regular polygon
  with n sides?

13
Geometric Patterns

• How many diagonals are there in a regular polygon
  with n sides?
• Sum of Integer Solution
• Finite Differences Sol.

14
Venn Diagram

• Picture of all the possible intersections of n
  sets. For two sets how many regions are there?

B
A
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Venn Diagram

• U 1, 2, 3, 4, 5, 6,
  7, 8
• A 1, 2, 6, 7
• B 2, 3, 4, 7
• C 4, 5, 6, 7

A 1, 2, 6, 7
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Venn Diagram for Order n

• Construct Venn Diagrams of the following order
  (for the given number of sets). Remember that
  all possible intersections between sets must be
  represented.
• Order 1 A will have two regions A and A
• Order 2 A, B will have 4 regions A, B, A ? B,
  and A ? B

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Venn Diagram of Order n

• Order 3 A,B,C
• Number of regions?
• Order 4 A,B,C,D
• Number of regions?

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Venn Diagram for four sets A,B,C,D
Identify the region representing the intersection
of all 4 sets.
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Existence of Venn Diagrams

• The largest Venn diagram that can be created
  with circles consists of 3 circles...you've
  probably seen it. Can you prove that no Venn
  diagram can be created with 4 circles?
• The largest Venn diagram that can be created with
  ellipses has 5 ellipses and was discovered only
  recently, and was thought for a long time not to
  exist!

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Venn Diagram showing the intersection of five sets
OPEN PROBLEM Is it possible to create a Venn
diagram with six Triangles?
21
Tower of Hanoi

• Ancient puzzle that predicts when the world will
  end.
• Try to solve the Tower for cases where n
  1 through 7. Predict how many moves for n discs.
  Use the Java applet below to assist in
  collecting data.
• http//www.adifferentplace.org/math48.htm

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Tower of Hanoi

• Inductive Reasoning Solution working forward
  making a conjecture based on cases
• Recursive Reasoning Solution working backwards
• Move n-1 small disks from A to B (say mn moves)
• Move largest disk from A to C
• Move n-1 small disks from B to C (mn more moves)
• Total moves 2mn 1
• Moving from Recursive to Inductive Form

23
Sequence Puzzles

• Sequences can have patterns other than those
  found using finite differences or recursive
  reasoning.
• Explore the Puzzle Sequences Site for more
  challenges
• http//www.research.att.com/njas/sequences/Spuzzl
  e.html