Mathematics

Problem Solving Workshop

 

Mastery Project

Mathematics and Science Teacher Education

in Yemen

May 16 - 20, 2009

 

 

 

Jim Wilson

Workshop Leader


This is the web site page is for use in workshop, in Sana'a Yemen, for the Mastery Project, Mathematics and Science Teacher Education in Yemen.

 

The material is taken from the EMAT 6600 Problem Solving in Mathematics course that is taught at the University of Georgia, as organized and led by Jim Wilson. Other instructors at the University of Georgia and elsewhere use some of the material in similar courses but no links to other courses is provided here. The material for the workshop draws heavily from EMAT 6600. Click here to go to the EMAT 6600 web site if that is desired.

 

EMAT 6600 Problem Solving in Mathematics is taught in the Department of Science and Mathematics Education. At other universities a similar course is often taught in a Department of Mathematics.


Special Web instruction: Use the browser back key to return to this workshop page. Links provided in the problem sets will return you to the EMAT 6600 page rather than the workshop page. I have placed a link on the EMAT 6600 page to return here if you inadvertantly go that route.


Last modified on April 13,  2009


Send e-mail to jwilson@uga.edu


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Goals for the Workshop in Yemen

 

-- To explore what a course on mathematics problem solving for secondary school teachers might be.

 

-- To explore the objectives and assessment of a mathematics problems solving course.

 

-- Doing mathematics requires doing problem solving.

 

-- Using technology when appropriate

 

-- Other descriptions of Problem Solving -- inquiry, exploration, reasoning, proving

 

References:

 

• Polya, G. (1945) How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press. Second edition 1956.

 

• Polya, G. (1965) Mathematical Discovery, Vol I & II. New York: Wiley.

Chapters 1-4 of Volume 1 contain extensive problem sets that Polya developed for The Stanford-Sylvania Mathematics Institutes for Teachers.

• Polya, G. (1954) Mathematics and Plausible Reasoning. Vol.1 Induction and Analogy in Mathematics; Vol. II Patterns of Plausible Inference. Princeton, NJ: Princeton University Press.

 

•Polya, G., & Kilpatrick, J. (1974) The Stanford Mathematics Problem Book. New York: Teachers College Press.

 

•Brown, S. I., & Walter, M. I. (1990) The Art of Problem Posing. 2nd edition. Hillsdale, NJ: Erlbaum.

 

• Courant, R., & Robbins, H. (1941) What is Mathematics? An Elementary Approach to Ideas and Methods. New York: Oxford University Press. (Revised Edition prepared by Ian Stewart, 1996)

 

•Taylor, H. & Taylor, L. (1993) George Polya: Master of Discovery. Palo Alto, CA: Dale Seymour Publications.

 

•Steinhaus, H. (1950) Mathematical Snapshops. New York: Oxford University Press.

 

Other Resources

Polya, G. (1964) Let us Teach Guessing. Film/Video. Washington, DC: Mathematical Association of America

 

Math Forum (alias Geometry Forum). Formerly at Swarthmore. Now at Drexel University. (HIGHLY RECOMMENDED)
Mathematical Association of America
Mathematical Sciences Education Board
National Council of Teachers of Mathematics
A Visual Dictionary of Special Plane Curves
MacTutor History of Mathematics Site -- St. Andrew's University
SIAM


 

Schedule

EMAT 6600 Documentation

         EMAT 6600 Syllabus   - General information about the objectives and operation of the course.
 

         Introductory Remarks  
 

         Resources


PAPERS:

         Synthesis of Research on Problem Solving. This paper was published as Chapter 4 in

Wilson, P. S. (Ed.) (1993). Research Ideas for the Classroom: High School Mathematics. New York: MacMillan.

The book was part of the National Council of Teacher of Mathematics Research Interpretation Project, directed by Sigrid Wagner.

The bibliographic reference for the published version is

Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research Ideas for the Classroom: High School Mathematics (pp. 57-78). New York: MacMillan.

         Squares. X What is the ratio of areas of the two squares?  This is a discussion of some exploration and extensions of this problem.

         Roots 2 and 5. This paper examines sets of equations that have graphs crossing the x-axis only at 2 and 5. For a preview on one family of such graphs, click here.

 

         Problem Solving with Heron's Formula.  X  This is a paper on the development and demonstration of Heron's formula for the area of a triangle given the lengths of its three sides. Problems and explorations are included for using Heron's formula.
 

         An Investigation with Parametric Equations.  This paper examines the movement of triangles when one vertex is moved along the x-asis and another is moved along the y-axis. We trace trace the movement of the third vertex.
 

         Extended Concurrencies of the Triangle


PROBLEMS:

 

Project InterMath  

Project InterMath has a web site with many investigations deemed appropriate for Middle School mathematics teachers. I believe secondary mathematics teachers may find some challenges here as well. Please access the site: Project INTERMATH

 

Environment Problems

         Volume of an Irregular Solid   X

         The Three Point Problem from Geology

         Volume of a curved trench, Trapezoid Cross-section

         Distance survey  X

 

Geometry Problems

         A Mean Problem  X

         Angle Bisector of two line non-intersecting line segments  X

         Angle of View  

         Arc Length equal to a given segment  

         Areas of a Rectangle X

         Bisector of an angle of a triangle X

         Bisectors Problem in 120 degree Obtuse Triangle  X

Bouncing Barney X *new*

         Brahmagupta's Formula X

         Carl's Cone  X

       Ceva's Theorem X *new*

  Checkerboard Problems  

         Circle Theorems Review X

         Circles of Apollonius for a Triangle ABC  X

         Circular Window

         Color a Circle   

         Coloring Circles   

         Concurrency Theorems   X

         Cone Half Full    X

         Count the Triangles -- I  X

         Count the Triangles -- II   X

         Count the Triangles -- III  X

Construct Equilateral Triangle with vertices on three given parallel lines X

         Cutting the Cake   X

Divide a Square into a Set of Acute Triangles X *new*

         Equilateral Triangle Altitude Theorem  

         Equiperimetric Areas  X

         Excircle Problems  X

         Find Inscribed Square for Given Triangle  X

         Find Inscribed Rectangle with Maximum Area for Given Triangle X

         FlowingStream  X

         Fly and Spider  X

         Folding a sheet of paper into equal areas  X

         Four dogs  

         Half the Area of a Triangle: A Line Parallel to a Side

         Half the Area of a Triangle: A line Through a Point on the Side

         Heron's Formula   X

         Incircle Problems  X

         Inequality in a triangle

         Inscribed Equilateral Triangle in a Square – Construction  X

         Inscribed Equilateral Triangle in a Square – Problem  X

         Inscribed Quadrilateral  X

         Inscribed Triangle in a given triangle  X

         Island Treasure   X

         Isosceles Right Triangles--Path of the Mid-Point

         Isosceles Right Triangles With a Common Vertex

         Isosceles trapezoid   X

         Isosceles Trapezoid -- Equal areas  X

         Kite  X

         Lines of Symmetry in a Polygon  X

         Locus of intersection of two secants

         Locus Problems  

         Maximum Quadrilateral  

         Medians of a Triangle  

         Menelaus's Theorem

         Mid-Segment Theorem

Minimum Path  X
Networks of Minimal Length   X

         Notch for Felling a Tree

         Obtuse Triangle Relationship

         Pairs of Segments with the Same Sum of Lengths

         Parallelogram with Integer Sides and Integer Diagonals

         Partition Square into Acute Triangles   X

         Perfect Triangles X  

         Pappus Areas

         Perpendicular Chords in a Circle

         Points closer to the Centroid than the Sides

         Polya's Problem 2.35.1

Projections: Homothetic Similarity *new*

         Ptolemy's Theorem

         Pyramids in a Prism

         Quadrilateral Inscribed in a Semicircle  

         Quadrilateral with Maximum area  

         Quadrilateral with Squares on the Sides  

         Ratio Sums from Orthocenter   

         Ratio on a line segment -- Something Golden  

         Ratio: Segments cut off by an angle bisector to the adjacent sides   

         Right Triangle Constructions

         Right Triangle Relationships

         Right Triangle with perimeter 60, altitude to hypotenuse 12

         Rotating Triangle  

         Spheres

         Square Inscribed along a base of any Triangle  X

         Squares Inscribed in a Right Triangle  X

         Squares on the Sides of a Parallelogram  

         Sum of Ratios

         Tangent and Secant Problem

         Tangent Lines Common to Two Given Circles X

         The Problem of Apollonius  X

         Three Circles Problem
Three Intersecting Circles X

         Trapezoid Inscribed in a Semicircle

         Triangle areas/Steinhaus  X

         Triangle areas/Triangle Built on Outside of a Given Triangle

         Triangle Area and the Circumcircle

         Triangle Constructions  

         Triangle Loci

         Triangle Mid-Segment Theorem    

         Triangle/Square  

         Triangles and Squares  

         Triangles with Integer Area and Integer Sides  X

         Two triangle problems  X

         Two Squares

         Volume of Holes Left by Tree Spade   X


Algebra Problems

         Solve by iteration.

         7-11 Problem  X

         A Tangled Tale Problem  

         Average Rate   X 

         Big Tires  X

         Calculator sequence  

         Candy Problem X

         Comparision of Two Radical Expressions X

         Equiperimetric Areas  X

         Heart to Bell

         Jordan's Inequality X

         Linear functions tangent to their product function   

         Maximum Volume of a Cone  X

         McNuggets  X

         Minimal area triangle  

         Phonograph groove  X

         s(s-c) = (s-a)(s-b)  X

         Simultaneous Quadratics  

         Square Root Equation  

         Sum of Unit Fractions

         Volume of a Frustum  

         Solve by iteration

Conversion Problems

         Big Tires   X

         Cubic Foot  X

         Million Drops of Water   X

         Volume of a 12 ounce can  X


Cryptarhythm Problems

         Cryptorithms  

         Nine digit number

         OCTOBER Primes


"Mean" Problems

         AM-GM Problems   X

         Arithmetic Mean -- Geometric Mean Inequality   X

         Areas in square of sides a+b  X

         Area of a Sector of a Circle  X

         Average Rate  X

         Box Problem   

         Comparing Segments in two circles  

         Comparison of altitude and median in a right triangle  X

         Equiperimetric Areas  X

         Harmonic Mean Problems X

         Maximum area -- rectangle  X

         Maximum area -- triangle  X

         Maximum and Minimum of (x/(1+x^2))

         Maximum of f(x) = (1-x)(1+x)(1+x)

         Minimum of x + 1/x  X

         Minimum Surface area of a can of fixed volume  X

         Phonograph groove  X

Trigonometry Problems

         Barn in a square field    X

Bridge Expansion Problems

         Cos 36 degrees

         Navigation Problems

           

         Sum of Two Sines or Sum of Two Cosines


Mixture of all

         7-11 Problem   X

         Area of Texas   X

         A sequence from Eudoxus  

         A Will to be Interpreted   X

         Area Golfing Greens  X

         Arithmetic Mean Sequence

         Bottles and Cans  

         Census Taker Problem  

         Change for a Dollar  

         Coins  

         Compare Radicals X

         Distance to Nearest Road  X

         Fractions   

         Friday 13   X

         FlowingStream X

         Four Number Challenge Problem  X

         Grass Consumption by Oxen X

         How Wide is the Alley?  X

         Ladder and Box  

         Magic Square  

         Maximum Volume of a Cone  X

         Mirror Image  X

         Oil Tank Problems  X

         Rational or Irrational

         Square Root of 2  

         The Salesperson's Journey X

         Triangular Numbers X

         Where is the store? X


 

 


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