Teaching Mathematics and its
Applications: An Approach
for the Middle School
Presented by
Jessica Alfano
Mentored by Dr. Robert Mayans
Fairleigh Dickinson University
Math Can Be Fun
 Use puzzles, games, and real-life problems
 Utilize the Internet and technology to enhance the
learning environment of mathematics
Build on
curiosity,
puzzlement,
encouragement,
pleasure,
elation,
and s a t i s f a c t i o n
Purpose

Difficulty in including “trendy” methods into
math education. (i.e. constructivism, fuzzy
math, etc.)

Lack of focus on math in education classes.

Lack of useful material.

Math teaching should focus on grasp of
content while using traditional and fun
techniques (i.e. games, puzzles, hands-on
activities, technology).
Topics
Bipartite Graph Theory
Probability and Statistics
Number Theory
Bipartite graph
1
2
3
1
2
4
3
4
7
5
8
6
5
6
7
8
9
9
Graph Importance To Middle
School Mathematics
New Jersey Core Content Standards:
Standard 4.4 (Data Analysis, Probability,
and Discrete Mathematics):
All students will develop an understanding
of the concepts and techniques of analysis,
probability, and discrete mathematics,
and will use them to model situations, solve
problems, and analyze and draw
appropriate inferences from data.
Some Definitions
 A graph is a finite set of dots called vertices (or
nodes) connected by links called edges.
 A directed graph is a graph in which the edges
have an orientation, denoted by arrowheads.
 A weighted graph is a graph in which each
edge has a value. The value can represent
distance, time, cost, etc.
 A graph is Bipartite if its vertices can be
separated into two groups, so that each edge
joins a vertex in one group with a vertex in the
other group. Every bipartite graph can be
colored with two colors.
Definitions Cont’d
 A Matching M is a set of edges of a graph
in which no two edges share a vertex. A
match is between vertices. Matched vertices
are connected by an edge.
 A Maximum Matching is one which
matches as many vertices as possible.
 A Perfect Matching is one which matches
every vertex; this can happen only if the
number of vertices are even.
“The Domino Effect”
Can you cover the checkerboard with 7
dominoes (each covering 2 squares) so that
two squares of the same color are left
uncovered?
Joe’s Game
This game can be played using any graph. Player 1 and Player 2
alternate picking vertices in the graph, subject to the
following rules:
1.
2.
Neither player can pick a vertex that has been picked
earlier in the game.
Every vertex must be adjacent to the vertex picked just
before.
The winner is the last player who can choose a vertex.
A.
1
2
B.
Player 1
is the
WINNER!
1
2
1
2
2
1
1
Player 2
is the
WINNER!
The Secret of Joe’s Game
Find a maximum matching in the graph.
If the maximum matching is perfect, pick to be Player 2.
Perfect
Maximum
Matching
Matching
If the maximum matching is not perfect (if the number
of vertices are odd), pick to be Player 1.
Carpoolers Dilemma
The basketball team of Hillside Middle School,
the Hillside Hawks, are in the state
championships. Laura, Christopher, Jason,
Amy, and Tommy want to attend the games
to cheer on their fellow classmates. Their
parents can only drive specific days. Can we
assign a day for each parent so that no one
needs to drive more than once?
Laura’s Mom
Monday
Wednesday
Christopher’s
Dad
Monday
Thursday
Jason’s Dad
Tuesday
Friday
Amy’s Dad
Tuesday
Friday
Tommy’s Mom
Wednesday
Thursday
The Carpooler’s Compatibility
Graph
Laura’s Mom
Monday
Christopher’s Dad
Tuesday
Jason’s Dad
Wednesday
Amy’s Dad
Thursday
Tommy’s Mom
Friday
Two Parallel Problems
“The Sweetheart’s Dance” – The Hawks are
holding their annual Valentine’s Day
Dance. Each student provides a list of
mutually acceptable dates of the opposite
sex for the dance. Can we pair each
student with an acceptable date?
“Field Day!” – On your mark, get set, GO!
The Hawks field day consists of many
competitions and games. For the threelegged race, the students must pair up.
Each student provides a list of mutually
acceptable partners. Can we pair each
student with an acceptable partner?
To solve the Sweetheart Dance Problem…
After graphing the problems in Inspiration, the
students will see that the graph for the dance
problem is bipartite. For a bipartite graph there
is a method of finding a perfect matching.
To solve the Field Day Problem…
After graphing the problems in Inspiration, the
students will see that the graph for the field day
problem is not a bipartite graph. If there is a
perfect matching, there is no defined method
that will always find it.
Probability and Statistics
New Jersey Core Content Standards:
Standard 4.4 (Data Analysis, Probability,
and Discrete Mathematics):
All students will develop an understanding
of the concepts and techniques of analysis,
probability, and discrete mathematics, and
will use them to model situations, solve
problems, and analyze and draw
appropriate inferences from data.
Some Definitions
 The mode is the value that occurs most
frequently in a given series of numbers.
 The mode of the set {13, 5, 9, 11, 11, 8, 10} is 11
 The mean is the average obtained by dividing
the sum of two or more quantities by the
number of these quantities.
 The mean of the set {1, 3, 5} = (1+3+5)/3 = 3
 The median is the middle number in a set of
ordered data.
 The median of the set {1, 1, 1, 2, 4, 6, 6} is 2 since 2
is the middle number when all of the numbers are
placed in order
Definitions Cont’d
 Probability is a quantitative description of the
possible likelihood of a particular event. It is the
ratio of the number of outcomes favoring an event
to the total number of possible outcomes.
 Statistics is a part of mathematics that deals with
collecting, organizing, and analyzing data with a
probability measure defining the likelihood of those
values.
 A scatterplot is a graph used to visually display
and compare two sets of related quantitative or
numerical data by displaying only finitely many
points, each having a coordinate on a horizontal
and a vertical axis.
Survey Scatterplot
 T.V vs. AIM – Do Middle School students spend
more time watching TV or talking on AOL Instant
Messanger?
 Students conduct a survey on the amount of time 30
students spend on the specified activities.
 They will then construct 1 handwritten scatterplot
and 1 scatterplot in Excel.
BURGER AND FRENCH FRIES
 Each letter of BURGER AND FRENCH FRIES
is written on a piece of paper and placed in a
bag. What is the probability of pulling out an
F? An R? A letter that appears in the first
half of the alphabet? A vowel? A consonant?
 In teams students will pick out letters and
record data.
Spinner Winner
 If you were to spin the wheel, it is
equally likely to stop at any point. You
win if it stops on a space that is 6 or
higher. What is the probability of
winning?
 Students manually figure out actual
probability.
 Just for fun:
 The class then plays with the wheel. If a
student wins 5 or more times they receive a
prize.
What Will I Wear Tomorrow?
 Students write down 5 shirts, 5 pants, and 2 pairs of
shoes they own and may wear tomorrow.
 They figure out the possible outfit combinations.
Dice
 Using a pair of regular dice, what is the
probability of rolling a 2, 4, 5, 6, 7, 8, 9, 10, 11, or
12?
 Roll dice and record data.
Number Theory Importance to
Middle School
New Jersey Core Content Standards:
Standard 4.3 (Patterns and Algebra)
All students will represent and analyze
relationships among variable quantities and
solve problems involving patterns,
functions, and algebraic concepts and
processes.
Fountain of Knowledge
Is it possible to use only unmarked 6- and 10ounce glasses to produce exactly 8 ounces in a
10 ounce glass?
First Step
Second Step
Final Step
The 3x + 1 Problem
 Start with any number.
 If the number is even, divide it by 2.
 If the number is odd, triple it then add1.
 Repeat the process with this new number.
 You win (and stop) if you get to 1.
Aliquot Game
 Aliquot part is another name for a proper
divisor, i.e. any divisor of a given number other
than the number itself.
 For example: 1, 2, 3, 4, 6 are all aliquot parts of
12.
 The number 1 does not have aliquot parts.
Aliquot Game Cont’d
 In the Aliquot game, players take turns
subtracting an aliquot part of the number
left by their opponent.
 The winner is the last player able to
perform such a subtraction.
 The loser is the player left with a number
that has no aliquot parts - 1.
What’s Next?
Students will visit http://www.cutthe-knot.com and play the interactive
game What’s Next?
They surmise the pattern and guess
what would be the next term.
Fibonacci Sequence
 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, …
 Many plants produce new branches in
quantities that are based on
Fibonacci numbers.
 Plants illustrate the Fibonacci series
in the numbers and arrangements of
petals, leaves, sections and seeds.
Golden Ratio
 The Golden Ratio is a ratio based on a phi.
Phi = 1.618033988749895...
 The ratio of two successive numbers in the
Fibonacci's series approach phi.
 It is said that things that are appealing to the
eye are composed of the golden ratio.
 Greek architecture.
 It’s said that the
Renaissance artists knew
it as the Divine Proportion
and used it for beauty and
balance in the design of
architecture and in the
design of art.
 Design of Notre Dame in
Paris.
 Found in art, architecture,
and design and physical
proportions in nature,
humans, and many other
aspects of life and the
universe.
Activities
1.
Students measure the
length and width of items
at home (i.e. TV, credit
card, etc.) and calculate
ratio -- Specify items with
golden ratio.
2.
Measure pointer finger
from knuckle to knuckle,
and from top of finger to
third knuckle – calculate
ratios.
3.
Students examine the seed
sections of fruit (i.e.
banana, apple, etc.) to
determine if sections are of
Fibonacci sequence.
Item
Measured
Length
(cm)
Width
(cm)
Ratio
(L:W)
Recap
Topics Covered:
 Bipartite Graphs
 Probability & Statistics
 Number Theory
We applied these topics to puzzles, games,
and real-world problems with the
incorporation of technology. Using these
techniques, higher level mathematics can
be brought into the Middle School level
while staying within the NJCC Standards.