Problem Solving in Mathematics

Fall 2015

Last modified on February 15, 2018.     Send e-mail to Back to Jim Wilson's Home Page

EMAT 4600/6600 Documentation

EMAT 4600/6600 Syllabus   - General information about the objectives and operation of the course.   (This is our syllabus but it is a work in progress.)

Introductory Remarks


References Fall


EMAT 4600/6600 Class Members, Fall 2015

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

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.  FALL 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.     (Scroll down to discussion of fixed perimeter triangle with maximum area and proof that the maximum is equilateral).

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  

Using the Arithmetic Mean-Geometric Mean Inequality in Problem Solving FALL. 

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a very useful algebraic tool for reasoning about many maximization and minization problems without calculus.   Yet, this tool gets very little attention in modern mathematics curriculum.   I have presented basic information and definitions about the Arithmetic Mean-Geometric Mean Inequality, showed a range of proofs and demonstrations for the cases with 2 positive numbers and a range of problem solving episodes where the AM-GM inequality is used.

A presentation for the Annual Meeting of the School Science and Mathematics Association, Birmingham, November 8-10, 2012, was prepared using parts of this paper.



Recently Added







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



        Algebra -- Functions and Relations -- Recommended -- Multiple Solutions 

Environment Problems


Distance survey  

Volume of a curved trench, Trapezoid Cross-section

Volume of an Irregular Solid

The Three Point Problem from Geology  


   Geometry Problems

20-30-130 Triangle   

100 degree isosceles triangle 

Angle Bisector of two line non-intersecting line segments 

Angle of View

Arc Length equal to a given segment

Arctan Sum

Area of a Segment of a Circle 

Area and Side of a Rhombus Given its Diagonals

Area Bounded by Quarter Circle Arcs in a Square 

Areas of a Rectangle  

Areas of Lunes I 

Areas of Lunes II 

Areas of Lunes III

Bisector of an angle of a triangle

Bisectors Problem in 120 degree Obtuse Triangle 

Bouncing Barney

Brahmagupta's Formula

 Carl's Cone

Ceva's Theorem 

Checkerboard Problems

Circle, given Endpoints of a Diameter

Circle Tangent to Circumcircle and Two Sides of a Given Triangle

Circle Theorems Review

Circles of Apollonius for a Triangle ABC  

Circular Window

Color a Circle

Concurrency Theorems

Cone Half Full

Count the Triangles -- I  

Count the Triangles -- II  

Count the Triangles -- III  

Construct Equilateral Triangle with Vertices on Three Given Parallel Lines

Construct Three Congruent Segments in a Fixed Angle

Cutting the Cake 

Cyclic Quadrilateral

Dissect a Square into a Set of Acute Triangles    

Dissect an Equilateral Triangle a form a Square

Divide a Circle into Five Equal Areas 

Dividing a Line Segment for Perimeter of a Square and Circumference of a circle

Equilateral Triangle Altitude Theorem 

Equiperimetric Areas

Excircle Problems    

Flowing Stream 

Fly and Spider  

Folding a sheet of paper into equal areas  

Four dogs

Goat Tethered in a Field   

Goat Tethered to the Edge of a Square Field    

Golden Ratio Calipers

Golden Ratio Construction   

Golden Triangle  (same as Sublime Triangle)    

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

Heron's Formula: Geometric Proof 

Incircle Problems  

Inequality in a triangle

Inscribed Equilateral Triangle in a Square -- Construction  

Inscribed Equilateral Triangle in a Square -- Problem  

Inscribed Quadrilateral

Inscribed Rectangle with Maximum Area for Given Triangle

Inscribed Square for Given Triangle

Inscribed Trapezoids 

Inscribed Triangle in a given triangle

Inverse Geometry  

Island Treasure   

Isosceles Right Triangles--Path of the Mid-Point

Isosceles Right Triangles With a Common Vertex


Lines of Symmetry in a Polygon

Lines Parallel to the Bases of a Trapezoid -- HM-GM-AM-RMS Inequalities 

Lighthouse Problem: Three lighthouses

Locus of intersection of two secants

Locus of the Vertex of an Angle Subtending a Line Segment

Locus Problems  Revised

Maximum Quadrilateral  

Medians of a Triangle  

Menelaus's Theorem

Minimum Path    

Networks of Minimal Length   

Notch for Felling a Tree

Obtuse Triangle Relationship

Octagon Construction and Formulas

Orthogonal Parabolas

Overlap Two Circles  

Pairs of Segments with the Same Sum of Lengths

Paper Knot Pentagon

Parallelogram with Integer Sides and Integer Diagonals

Parallel Chords

Partition Square into Acute Triangles 

Patterns Constructed in Circles 

Patterns of Circles and Squares; Comparision of Areas 

Pappus Areas

Path Problems  

Perfect Triangles

Perpendicular Chords in a Circle

Points closer to the Centroid

Points on a Circle

Problem of Apollonius

Projections: Homothetic Similarity

 Ptolemy's Theorem

 Pyramids in a Prism

Quadrilateral Determined by Intersecting Orthogonal Parabolas

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 Angle in a Configuration (SSM 393) 

Right Triangle Constructions

Right Triangle Relationships

 Right Triangle with perimeter 60, altitude to hypotenuse 12

 Rotating Triangle

Segment Parallel to Bases of a Trapezoid through Intersection of Diagonals 



 Square Inscribed along a base of any Triangle (Revised) 

 Squares Inscribed in a Right Triangle (Revised)   

Square Inscribed in a Semicircle -- find a ratio

 Squares on the Sides of a Parallelogram  

Sublime Triangle(Same as Golden Triangle)   

Sum of the Reciprocals of Powers of 2 

Tangent and Secant Problem

Tangent Circles and Spheres

Tangent Lines Common to Two Given Circles

Three Adjacent Squares (THIS IS A GEOMETRY PROBLEM -- not trigonometry)

Three Circles Problem

Three-Color Circle Patterns

Three Intersecting Circles with Collinear Intersection Points

Trapezoid Divided into Equal Areas by a Line Parallel to the Base 

Trapezoid with Parallel through the Intersection of the Diagonals

Trapezoid Inscribed in a Semicircle

 Triangle areas/Steinhaus  

 Triangle areas/Triangle Built on Outside of a Given Triangle

 Triangle Area and the Circumcircle

Triangle Built Outward (12 problems)

 Triangle Constructions

Triangle Constructions  (revised) 

Triangle Constructions with Altitudes

Triangle Inscribed in a Rectangle Leaving 3 Right Triangles of Equal Area

Triangle Inscribed in a Rectangle: Right triangle APQ in Rectangle ABCD with PQ fixed.

Triangle Inscribed in a Rectangle: Right triangle APQ in Rectangle ABCD with PQ fixed. (Geometry Problem -- Revised)

Triangle Inscribed in a Rectangle: Right triangle APQ in Rectangle ABCD with PQ fixed. (Algebra Problem --  New)

Triangle Loci

 Triangle Mid-Segment Theorem    

 Triangles and Squares  

Triangles with Integer Area and Integer Sides 

Triangle with Median Equal to the Base and Sides of Length 1 and 2

Triangle with Median Equal to the Base and Sides of Length 1 and 2 (revised)

Trirectangular Vertex Problem

Two triangle problems  

Trisection of the Area of a Triangle

Two Squares

Volume of Holes Left by Tree Spade  

Algebra Problems


7-11 Problem

Alternative Quadratic Formula   

A Tangled Tale Problem  

Area bounded by Quarter-Circles in a Square

ASTROID Equations

Big Tires

Calculator sequence

Candy Problem

Check Cashing Problem    

Combining Percentages  

Combining Rates      (Danielsville travel problem . . . )

Comparision of Two Radical Expressions

Deriving the Quadratic Formula 

Equations of a Parabola

Equiperimetric Areas    

Exponents Equation

Golden Ratio and Fibonacci Sequence I

Golf Ball Flight 

Heart to Bell

Iron Ball Floating in Vat of Mercury

Jordan's Inequality

Linear functions tangent to their product function   

Maximum Volume of a Cone


Minimal area triangle

Multiple Solutions 

Orthogonal Parabolas  

Phonograph groove  

Powers of the Golden Ratio

Prove All Parabolas are Similar

Pythagorean Triples 

Quadratic Equations in One Variable

Quotient of Two Linear Functions

Rational Equation

s(s-c) = (s-a)(s-b)  

 Simultaneous Quadratics  

 Square Root Equation

Sum of the Reciprocals of Powers of 2   

  Sum of Unit Fractions

Three Mile Roadway -- Two line segments and an arc 

Two Stamp Problem

  Volume of a Frustum  

Conversion Problems

Big Tires
Cubic Foot 
Million Drops of Water 
Volume of a 12 ounce can

Cryptarithm Problems



Division Problem with 8 in the Quotient

Nine digit number


"Mean" Problems


AM-GM Problems   

Arithmetic Mean -- Geometric Mean Inequality 

Arithmetic Mean -- Geometric Mean -- Harmonic Mean Inequality

Areas in square of sides a+b 

Box Problem 

Combining Rates     (Danielsville travel problem . . .)

Comparing Segments in two circles   

Comparison of altitude and median in a right triangle 

Construct Square with Same Area as a Given Rectangle 

Cost of Fencing a Field 

Equiperimetric Areas    See problem max area with partitions    

Harmonic Mean

Harmonic Mean Problems 


Lines Parallel to the Bases of a Trapezoid -- HM-GM-AM-RMS Inequalities 

Maximize the Area of a Triangle with sides  9, 40x, and 41x

Maximum Area of Rectangular Pen with Diagonal Partition

Maximum area -- rectangle  

Maximum Area of a Sector of a Circle -- fixed perimeter of the sector

Maximum Area of Trapezoid, Base twice the Lateral Side 

Maximum area -- triangle with Fixed Perimeter

Maximum and Minimum of  

Maximum of f(x) = (1-x)(1+x)(1+x) 

Maximum Volume of a Cylinder Inscribed in a Sphere

Mean Problem

Minimum distance from (0,0) to

Minimum of x + 1/x  

Minimum of  

Minimum Surface area of a can of fixed volume

Phonograph groove  

Product =  1; Sum ≥ n 


Segment Parallel to Bases of a Trapezoid through Intersection of Diagonals 

Segments in a Circle

Square Roots and Cube Roots

Trapezoid Divided into Equal Areas by a Line Parallel to the Base -- RMS

      Trigonometry Problems


Arctan Sum

Area of a Segment of a Circle 

Barn in a square field

Bridge Expansion Problem

Closed Form for Some Values of Cosine and Sine

Conic Section Images

Cos 36 degrees

Golf Ball Flight

Navigation Problems

Navigation Problems 2

Overlap Two Circles

Show, without tables or a calculator, (The equation is a link.)

Prove (The equation is a link.)

Ratio of Sines 

Segments in a Trig Drawing

Square Inscribed in a Semicircle -- find a ratio  

Sum of Two Sines or Sum of Two Cosines

Three Mile Roadway -- Two line segments and an arc 


Mixture of all 

7-11 Problem  

Area of Texas

A Sequence from Eudoxus

A Will to be Interpreted      

Area of Golfing Greens 

Arithmetic Mean Sequence

Bottles and Cans

Bottles and Cans (Revised) 

Census Taker Problem

Change for a Dollar

Check Cashing Problem 


Compare Radicals

Conchoid Construction 

David Rock's House Number

Distance to Nearest Road

Division Problem with 8 in the Quotient

Evaluate MENTALLY and describe your work.

Flowing Stream


Friday 13

Four Number Challenge Problem

Geometric Demonstration of  the Product of Two Negative Numbers

Alternative Demonstrations of the Product of Two Negative Numbers

Grass Consumption by Oxen

How Wide is the Alley?

Integer that is a Perfect Square, Cube, and Fifth Power

Ladder and Box  

Magic Square  

Maximum Volume of a Cone

Minimum Integer Resulting from Dividing a 5-Digit Whole number by the Sum of its Digits

Mirror Image  

Oil Tank Problems

Patterns in a Whole Number Array

Perfect Shuffle

Rational or Irrational

Rational Equation


Revolutions of a Rolling Circle

Rolling Circle I -- Cycloids and Trochoids

Rolling Circle II -- Number of Revolutions   

Rolling Circle III -- Cycloids, epitrochoids, hypocycloids, hypotrochoids

Rolling Circle IV -- When the ratio of circles is 2:1

Rolling Circle V -- When the ratio of circles is 3:2  

Show some square root of 2 and square root of 15 are irrational

Square Roots

The Salesperson's Journey

Triangular Numbers

Triangular Number Problems

Two (Mythical) Investment Strategies

Sum of Cubes

Sum of Squares

Sums of Powers of Integers -- Derivations from summations

Finite Differences Instructions (Clay Kitchings)

Finite Differences Comments, Examples, and Problems

Where is the store?

Final Project


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