Problem Solving in Mathematics

Fall 2015

Last modified on December 3, 2015.     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


Quadratic Equations in One Variable  FALL
Deriving the Quadratic Formula  FALL
Alternative Quadratic Formula

Equations of a Parabola FALL

Prove All Parabolas are Similar Fall

Triangle Constructions  (revised)  FALL
Triangle Constructions with Altitudes
Triangle Built Outward (12 problems) FALL
Triangle with Median Equal to the Base and Sides of Length 1 and 2 (revised) FALL
Bottles and Cans (Revised) FALL

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

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

Circle Tangent to Circumcircle and Two Sides of a Given Triangle FALL
Inverse Geometry FALL

Conic Section Images FALL

Recreations  FALL
Geometric Demonstration of  the Product of Two Negative Numbers FALL
Alternative Demonstrations of the Product of Two Negative Numbers FALL

Locus Problems (revised) FALL

Conchoid Construction FALL

Construct Three Congruent Segments in a Fixed Angle FALL
Locus of the Vertex of an Angle Subtending a Line Segment FALL
Two (Mythical) Investment Strategies FALL
Right Angle in a Configuration (SSM 3936)




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  FALL

Environment Problems


Distance survey  FALL

Volume of a curved trench, Trapezoid Cross-section

Volume of an Irregular Solid FALL

The Three Point Problem from Geology  


   Geometry Problems

20-30-130 Triangle   FALL

100 degree isosceles triangle FALL

Angle Bisector of two line non-intersecting line segments FALL

Angle of View

Arc Length equal to a given segment

Arctan Sum

Area of a Segment of a Circle  FALL

Area and Side of a Rhombus Given its Diagonals

Area Bounded by Quarter Circle Arcs in a Square FALL

Areas of a Rectangle  

Areas of Lunes I FALL

Areas of Lunes II FALL

Areas of Lunes III

Bisector of an angle of a triangle FALL

Bisectors Problem in 120 degree Obtuse Triangle   Fall

Bouncing Barney FALL

Brahmagupta's Formula FALL

 Carl's Cone FALL

Ceva's Theorem  FALL

Checkerboard Problems

Circle, given Endpoints of a Diameter FALL

Circle Theorems Review

Circles of Apollonius for a Triangle ABC  FALL

Circular Window FALL

Color a Circle

Concurrency Theorems

Cone Half Full FALL

Count the Triangles -- I  

Count the Triangles -- II  

Count the Triangles -- III  

Construct Equilateral Triangle with Vertices on Three Given Parallel Lines FALL

Cutting the Cake  FALL

Cyclic Quadrilateral

Dissect a Square into a Set of Acute Triangles    FALL

Dissect an Equilateral Triangle a form a Square FALL

Divide a Circle into Five Equal Areas FALL

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

Equilateral Triangle Altitude Theorem  FALL

Equiperimetric Areas FALL

Excircle Problems    

FlowingStream FALL

Fly and Spider  

Folding a sheet of paper into equal areas  FALL

Four dogs

Goat Tethered in a Field   FALL

Goat Tethered to the Edge of a Square Field     FALL

Golden Ratio Calipers FALL

Golden Ratio Construction   FALL

Golden Triangle  (same as Sublime Triangle)     FALL

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 FALL

Heron's Formula: Geometric Proof  FALL

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 FALL

Inscribed Square for Given Triangle

Inscribed Trapezoids 

Inscribed Triangle in a given triangle  

Island Treasure  FALL 

Isosceles Right Triangles--Path of the Mid-Point

Isosceles Right Triangles With a Common Vertex

Kite   FALL

Lines of Symmetry in a Polygon FALL

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

Lighthouse Problem: Three lighthouses FALL

Locus of intersection of two secants

Locus Problems  Revised FALL

Maximum Quadrilateral  

Medians of a Triangle  FALL

Menelaus's Theorem

Minimum Path    FALL

Networks of Minimal Length   FALL

Notch for Felling a Tree

Obtuse Triangle Relationship

Octagon Construction and Formulas FALL

Orthogonal Parabolas

Overlap Two Circles  FALL

Pairs of Segments with the Same Sum of Lengths

Paper Knot Pentagon FALL

Parallelogram with Integer Sides and Integer Diagonals

Parallel Chords

Partition Square into Acute Triangles FALL

Patterns Constructed in Circles FALL

Patterns of Circles and Squares; Comparision of Areas 

Pappus Areas

Path Problems  FALL

Perfect Triangles FALL

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    FALL 

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

Segment Parallel to Bases of a Trapezoid through Intersection of Diagonals FALL

Septagon  FALL

 Spheres FALL

 Square Inscribed along a base of any Triangle (Revised) 

 Squares Inscribed in a Right Triangle (Revised)   FALL

Square Inscribed in a Semicircle -- find a ratio FALL

 Squares on the Sides of a Parallelogram  

Sublime Triangle(Same as Golden Triangle)   FALL

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) FALL

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 FALL

Trapezoid with Parallel through the Intersection of the Diagonals FALL

Trapezoid Inscribed in a Semicircle

 Triangle areas/Steinhaus  FALL

 Triangle areas/Triangle Built on Outside of a Given Triangle FALL

 Triangle Area and the Circumcircle

 Triangle Constructions  FALL

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. FALL

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

 Triangle Loci

 Triangle Mid-Segment Theorem    

 Triangles and Squares  

Triangles with Integer Area and Integer Sides FALL

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

Trirectangular Vertex Problem FALL

Two triangle problems  

Trisection of the Area of a Triangle

Two Squares

Volume of Holes Left by Tree Spade  FALL

Algebra Problems



7-11 Problem    Fall

A Tangled Tale Problem  FALL

Area bounded by Quarter-Circles in a Square FALL

ASTROID Equations

Big Tires FALL

Calculator sequence

Candy ProblemFALL

Check Cashing Problem   FALL  

Combining Percentages   FALL

Combining Rates      (Danielsville travel problem . . . ) FALL

Comparision of Two Radical Expressions

Equiperimetric Areas     FALL

Exponents Equation

Golden Ratio and Fibonacci Sequence I

Golf Ball Flight  FALL

Heart to Bell

Iron Ball Floating in Vat of Mercury FALL

Jordan's Inequality

Linear functions tangent to their product function   

Maximum Volume of a Cone FALL

McNuggets  FALL

Minimal area triangle

Multiple Solutions FALL

Orthogonal Parabolas  

Phonograph groove  FALL

Powers of the Golden Ratio

Pythagorean Triples 

Quotient of Two Linear Functions

Rational Equation

s(s-c) = (s-a)(s-b)   FALL

 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  FALL

Two Stamp Problem FALL

  Volume of a Frustum  

Conversion Problems

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

Cryptarithm Problems


Cryptarithms  FALL

Division Problem with 8 in the Quotient FALL

Nine digit number FALL


"Mean" Problems


AM-GM Problems   FALL

Arithmetic Mean -- Geometric Mean Inequality Fall

Arithmetic Mean -- Geometric Mean -- Harmonic Mean Inequality FALL

Areas in square of sides a+b  Fall

Box Problem FALL

Combining Rates     (Danielsville travel problem . . .) FALL

Comparing Segments in two circles  FALL 

Comparison of altitude and median in a right triangleFall 

Construct Square with Same Area as a Given Rectangle FALL

Cost of Fencing a Field FALL

Equiperimetric Areas    See problem max area with partitions     FALL

Harmonic Mean FALL

Harmonic Mean Problems  FALL

Inequalities FALL

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

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

Maximum Area of Rectangular Pen with Diagonal Partition FALL

Maximum area -- rectangle   FALL

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

Maximum Area of Trapezoid, Base twice the Lateral Side FALL

Maximum area -- triangle with Fixed Perimeter FALL

Maximum and Minimum of   FALL

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

Maximum Volume of a Cylinder Inscribed in a Sphere

Mean Problem

Minimum distance from (0,0) to FALL

Minimum of x + 1/x  FALL

Minimum of  FALL

Minimum Surface area of a can of fixed volume FALL

Phonograph groove  FALL

Product =  1; Sum ≥ n FALL 


Segment Parallel to Bases of a Trapezoid through Intersection of Diagonals FALL

Segments in a Circle FALL

Square Roots and Cube Roots FALL

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


      Trigonometry Problems


Arctan Sum

Area of a Segment of a Circle  FALL

Barn in a square field FALL

Bridge Expansion Problem FALL

Closed Form for Some Values of Cosine and Sine

Cos 36 degrees

Golf Ball Flight FALL

Navigation Problems

Navigation Problems 2

Overlap Two Circles FALL

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

Prove (The equation is a link.)

Ratio of Sines 

Segments in a Trig Drawing FALL

Square Inscribed in a Semicircle -- find a ratio   FALL

Sum of Two Sines or Sum of Two Cosines

Three Mile Roadway -- Two line segments and an arc  FALL


Mixture of all 

7-11 Problem  FALL

Area of Texas FALL

A Sequence from Eudoxus

A Will to be Interpreted      Fall

Area of Golfing Greens FALL

Arithmetic Mean Sequence

Bottles and Cans  FALL

Census Taker Problem FALL

Change for a Dollar FALL

Check Cashing Problem FALL

Coins FALL

Compare Radicals 

David Rock's House Number

Distance to Nearest Road FALL

Division Problem with 8 in the Quotient FALL

Evaluate MENTALLY and describe your work.

Flowing Stream FALL


Friday 13 FALL

Four Number Challenge Problem FALL

Grass Consumption by Oxen FALL

How Wide is the Alley?FALL

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

Ladder and Box   FALL

Magic Square  

Maximum Volume of a Cone FALL

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

Mirror Image  FALL

Oil Tank ProblemsFALL

Patterns in a Whole Number Array

Perfect Shuffle  FALL

Rational or Irrational

Rational Equation

Revolutions of a Rolling Circle FALL

Rolling Circle I -- Cycloids and Trochoids FALL

Rolling Circle II -- Number of Revolutions   FALL

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

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

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

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

Square Roots

The Salesperson's Journey

Triangular Numbers FALL

Triangular Number Problems FALL

Sum of Cubes FALL

Sum of Squares FALL

Sums of Powers of Integers -- Derivations from summations FALL

Finite Differences Instructions (Clay Kitchings) FALL

Finite Differences Comments, Examples, and Problems FALL

Where is the store? FALL

Final Project


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