Teaching Mathematics and its Applications: An Approach for the Middle School

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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 - PowerPoint PPT Presentation

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Teaching Mathematics and its Applications: An

Approach for the Middle School

Presented byJessica Alfano

Mentored by Dr. Robert MayansFairleigh 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 curiositycuriosity,

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

21 3

4 5 6

7 8 9

21

34

5 6

7 8

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.

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. Neither player can pick a vertex that has been picked earlier in the game.

2. Every vertex must be adjacent to the vertex picked just before.

The winner is the last player who can choose a vertex.

B.1

2

12

Player 1 is the

WINNER!

12

1

2

1

Player 2 is the

WINNER!

A.

The Secret of Joe’s GameFind a maximum matching in the graph.

If the maximum matching is perfect, pick to be Player 2.

If the maximum matching is not perfect (if the number of vertices are odd), pick to be Player 1.

Perfect Maximum Matching

Matching

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 MondayWednesday

Christopher’s Dad

MondayThursday

Jason’s Dad TuesdayFriday

Amy’s Dad TuesdayFriday

Tommy’s Mom WednesdayThursday

The Carpooler’s Compatibility Graph

Laura’s MomMonday

Christopher’s DadTuesday

Jason’s DadWednesday

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 three-legged 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 DefinitionsThe 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 10- ounce 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.cut-the-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.

Activities1. 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.

ItemMeasured

Length

(cm)

Width(cm)

Ratio(L:W)

Recap

Topics Covered:Bipartite Graphs Probability & StatisticsNumber 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.

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