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Puzzles, Games, and Problem Solving (Major and Minor)
I.
Course description
Puzzles
Games, and Problem Solving is a course focusing on the techniques used in both
mathematical and non-mathematical problem solving. In this course students
will learn how many of the same strategies and techniques used in solving puzzles
and playing games transfer to more traditional problem solving applications
and, conversely, how techniques and strategies used in solving problems can
be used to solve puzzles and develop winning strategies in playing games. Students
may expect to discover and develop problem solving methods through working with
puzzles, games and non-traditional problems in such areas as number theory,
geometry, probability, logic, and statistics. Non-mathematical problems explored
in the class may come from varied disciplines. Emphasis in this class will
be on developing methods for finding solutions and discovering and proving why
some shortcut methods work rather than simply finding answers or learning “tricks.”
Students will also be given the opportunity to appreciate the elegance and beauty
often found in these solutions and the paths leading to them.
II. Instructor
Don Arni
- B.S. Physics, University
of Missouri-Columbia, 1973
- M.Ed., Curriculum and
Instruction, University of Missouri-Columbia
- Mathematics and physics
instructor
- Glasgow R-II High School
- Glasgow, Missouri 65254
- darni@glasgow.k12.mo.us
III. Rationale for
inclusion in a program for gifted students
Puzzles, games, and problem
solving serve as both a vehicle and an end in this course. As a vehicle, puzzles,
games, and problem solving will be used to introduce students to a broad range
of mathematical topics, many of which are not customarily included in traditional
high school courses, and, as an end, students will be more experienced in solving
problems with a wider variety of methods.
IV. Major topics covered
- Defining problems, puzzles,
and games
- Polya’s strategies
for problem solving
- Classic problems in
history—solved and unsolved
- Sequences and series
- Limits and derivatives
- Logarithms and analytical
geometry
- Binary numbers and
other base systems
- Patterns
- Pascal’s triangle
- Combinations, permutations,
and probability
- Fibonacci numbers
- Golden ratio
- Networks
- Inductive and deductive
reasoning
- Logic and logic games
- Lateral thinking puzzles
- Word play and word
puzzles
- Manipulative puzzles
and topology
- Empirical problem solving
- Modeling and simulation
- Number theory
- Game theory
- Mathematics and problem
solving contests
- Graphing and solving
systems of equations
- Trigonometry
- Strategy games
- Cooperative problem
solving—team approach
- Geometric constructions
- Physics-related applications
V. Prerequisite knowledge:
Students should have successfully
completed introductory courses in Algebra I and geometry and possess a desire
to explore a variety of problem solving methods while investigating a broad
range of mathematical and non-mathematical topics. Students should also be
prepared to be persistent in meeting the challenge of a mixture of puzzles and
games.
VI. Learning objectives
The student will
- develop working definitions
for problems, puzzles, and games
- identify which of Polya’s
problem solving strategies they use when solving a problem
- restate some of the
classic solved and unsolved problems in history
- define sequences recursively
and by a general term formula
- use the method of finite
differences to find general term formula
- identify arithmetic
and geometric sequences
- find sums of finite
and infinite series
- find limits of sequences
and functions
- use derivatives to
solve maximum and minimum problems
- solve problems by using
logarithmic and exponential equations
- use binary numbers
and other base systems to solve puzzles, develop game strategies, and solve
problems
- use patterns in Pascal’s
triangle to solve problems
- calculate combinations,
permutations, and probability
- list the sequence of
Fibonacci numbers and give examples of their occurrences in nature
- calculate the golden
ratio and cite its significance in mathematics, art, and architecture
- solve networking problems
of various types
- distinguish between
inductive and deductive reasoning and explain the importance of each
- develop strategies
to solve logic puzzles
- use Boolean algebra
to solve logic problems
- solve lateral thinking
puzzles
- solve and design word
puzzles
- construct all 12 pentominoes
- use pentominoes and
tangrams to replicate challenge patterns
- use topology to solve
puzzles and problems
- solve manipulative
puzzles
- use the empirical method
to solve problems
- use simulations and
models to solve problems
- use geometric constructions
to solve problems
- use number theory concepts
to solve problems
- use game theory to
make decisions
- compete in mathematical
and problem solving contests
- solve systems of equations
- sketch curves of quadratic,
cubic, and higher degree functions
- use trigonometric functions
to solve right triangle problems
- use the law of sines
and the law of cosines to solve problems
- develop winning strategies
for strategy games
- solve problems cooperatively,
using a team approach
- solve problems using
basic physics laws and principles
VII. Primary source
material
- Books
- Art and Techniques
of Simulation. Quantitative Literacy Series. Dale Seymour Publications.
1987.
- Bergamini, David.
Mathematics. Time-Life Books, Inc. 1980.
- Britton and Seymour.
Introduction to Tesselations. Dale Seymour Publications. 1989.
- Camilli, Thomas.
A Case of Red Herrings Book B1. Critical Thinking Press and Software.
1992.
- ______. A Case
of Red Herrings Book B2. Critical Thinking Press and Software. 1993.
- Carlson, Roger.
“Random Digits and Some of Their Uses.” Statistics by Example: Weighing
Chances. Addison-Wesley Publishing Company. 1973.
- Conrad and Flegler.
The 1st High School Math League Problem Book. Math
League Press. 1989.
- ______. Math
Contests! Volume 2. Math League Press. 1992.
- Coxford, Arthur F.
and Joseph N. Payne. Advanced Mathematics: A Preparation for Calculus.
Harcourt, Brace, Jovanovich. 1978.
- Eves, Howard. In
Mathematical Circles—Series. Prindle Weber and Schmidt, Inc.
- Exploring Probability.
Quantitative Literacy Series. Dale Seymour Publications. 1987.
- Fixx, James F. Games
for the Superintelligent. Popular Library. 1972.
- Gardner, Martin.
Aha! Gotcha: Paradoxes to Puzzle and Delight. W. H. Freeman and
Company. 1982.
- ______. Aha!
Insight. W. H. Freeman and Company. 1978.
- ______. Mathematical
Carnival. Vintage Books. 1975.
- ______. Mathematics
Magic and Mystery. Dover Publications, Inc. 1956.
- Garland, Trudi Hammel.
Fascinating Fibonaccis, Mystery and Magic in Numbers. Dale Seymour
Publications. 1987.
- Green and Hamberg.
Pascal’s Triangle. Dale Seymour Publications. 1986.
- Loyd, Sam. Mathematical
Puzzles of Sam Loyd. Dover Publications. 1959
- ______. More
Mathematical Puzzles of Sam Loyd. Dover Publications. 1960.
- Mottershead, Lorraine.
Metamorphosis, A Source Book of Mathematical Discovery. Dale Seymour
Publications. 1977.
- Pappas, Theoni.
The Joy of Mathematics. Wide World Publishing. 1989.
- Polya, G. How
to Solve It. Princeton University Press. 1973.
- Problem Solving
in School Mathematics—NCTM 1980 Yearbook. National Council of Teachers
of Mathematics. 1980.
- Runion, Garth E.
The Golden Section. Dale Seymour Publications. 1990.
- Seymour and Shedd.
Finite Differences. Dale Seymour Publications. 1973.
- Sloane, Paul. Lateral
Thinking Puzzlers. Sterling Publishing Co., Inc. 1992.
- Sloane and MacHale.
Challenging Lateral Thinking Puzzles. Sterling Publishing Co.,
Inc. 1993.
- ______. Great
Lateral Thinking Puzzles. Sterling Publishing Co., Inc. 1994.
- A Sourcebook of
Applications of School Mathematics. National Council of Teachers
of Mathematics. 1980.
- Wujec, Tom. Pumping
Ions. Doubleday. 1988.
- Journals Games Magazine
- The Mathematics Teacher
- Quantum
- Student Math Notes, MCTM
Bulletin
VIII. Supplementary source
material
- AHSME Contests
- AIME Contests
- Pentominoes
- Tangrams
- Manipulative and Strategy
Puzzles such as
- Rubik’s Cubes
- Tavern Puzzles—horseshoe
and other interlocking chain type puzzles
- Wooden Barrel
and other 3D interlocking puzzles
- Twisted Nail
type puzzles, etc.
- Strategy games such as
Mastermind and other logic and strategy games will be used.