Problem Solving in Mathematics

Spring 2014

Last modified on April 14, 2014.     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.

Introductory Remarks


EMAT 4600/6600 Class Members, Spring 2014.

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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.          Sp 2014

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.  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.    Sp 2014  (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   Sp 2014


Recently Added


ASTROID Equations Sp 2014
Circle, given Endpoints of a Diameter  Sp 2014
Dividing a Line Segment for Perimeter of a Square and Circumference of a circle Sp 2014
Revolutions of a Rolling Circle Sp 2014

  Sp 2014




Sum of the Reciprocals of Powers of 2  Sp 2014

Tangent Circles and Spheres Sp 2014

Golf Ball Flight Sp 2014

Points on a Circle Sp 2014


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 Sp 2014


Sp 2014 Project INTERMATH

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

Environment Problems


Distance survey  Sp 2014

Volume of a curved trench, Trapezoid Cross-section Sp 2014

Volume of an Irregular Solid Sp 2014

The Three Point Problem from Geology  


   Geometry Problems

20-30-130 Triangle    Sp 2014

100 degree isosceles triangle  Sp 2014

Angle Bisector of two line non-intersecting line segments  Sp 2014

Angle of View Sp 2014

Arc Length equal to a given segment Sp 2014

Arctan Sum

Area of a Segment of a Circle Sp 2014

Area and Side of a Rhombus Given its Diagonals

Area Bounded by Quarter Circle Arcs in a Square 

Areas of a Rectangle   SPRING 2014, Examined APRIL 10

Areas of Lunes I  Sp 2014

Areas of Lunes II  Sp 2014

Areas of Lunes III

Bisector of an angle of a triangle SPRING 2014, Examined APRIL 10

Bisectors Problem in 120 degree Obtuse Triangle   Sp 2014

Bouncing Barney Sp 2014

Brahmagupta's Formula

 Carl's Cone SPRING 2014, Examined APRIL 10

Ceva's Theorem

Checkerboard Problems  

Circle Theorems Review

Circles of Apollonius for a Triangle ABC  

Circular Window  Sp 2014

Color a Circle

Concurrency Theorems

Cone Half Full Sp 2014

Count the Triangles -- I  

Count the Triangles -- II   Sp 2014

Count the Triangles -- III  

Construct Equilateral Triangle with Vertices on Three Given Parallel Lines Sp 2014

Cutting the Cake Sp 2014

Cyclic Quadrilateral

Dissect a Square into a Set of Acute Triangles  Sp 2014

Dissect an Equilateral Triangle a form a Square

Divide a Circle into Five Equal Areas   Sp 2014

Equilateral Triangle Altitude Theorem

Equiperimetric Areas Sp 2014

Excircle Problems  

Find Inscribed Square for Given Triangle  

Find Inscribed Rectangle with Maximum Area for Given Triangle

FlowingStream Sp 2014 

Fly and Spider  

Folding a sheet of paper into equal areas  Sp 2014

Four dogs

Goat Tethered in a Field    SPRING 2014, Examined APRIL 10

Goat Tethered to the Edge of a Square Field Sp 2014

Golden Ratio Calipers Sp 2014

Golden Ratio Construction Sp 2014

Golden Triangle  (same as Sublime Triangle) Sp 2014

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 Trapezoids 

Inscribed Triangle in a given triangle  

Island Treasure    Sp 2014

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  Sp 2014

Lighthouse Problem: Three lighthouses

Locus of intersection of two secants

Locus Problems  

Maximum Quadrilateral  

Medians of a Triangle  

Menelaus's Theorem

Minimum Path    Sp 2014

Networks of Minimal Length 

Notch for Felling a Tree

Obtuse Triangle Relationship

Octagon Construction and Formulas Sp 2014

Orthogonal Parabolas

Overlap Two Circles Sp 2014

Pairs of Segments with the Same Sum of Lengths

Paper Knot Pentagon Sp 2014

Parallelogram with Integer Sides and Integer Diagonals

Parallel Chords

Partition Square into Acute Triangles Sp 2014 

Patterns Constructed in Circles  Sp 2014

Patterns of Circles and Squares; Comparision of Areas 

Pappus Areas

Path Problems Sp 2014

Perfect Triangles Sp 2014

Perpendicular Chords in a Circle

Points closer to the Centroid

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   Sp 2014

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  Sp 2014

Septagon  Sp 2014


 Square Inscribed along a base of any Triangle   Sp 2014

 Squares Inscribed in a Right Triangle   Sp 2014

Square Inscribed in a Semicircle -- find a ratio Sp 2014

 Squares on the Sides of a Parallelogram  

Sublime Triangle(Same as Golden Triangle) Sp 2014

Tangent and Secant Problem

Tangent Lines Common to Two Given Circles

The Problem of Apollonius

Three Adjacent Squares (THIS IS A GEOMETRY PROBLEM -- not trigonometry) Sp 2014

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  Sp 2014

Trapezoid with Parallel through the Intersection of the Diagonals Sp 2014

Trapezoid Inscribed in a Semicircle

 Triangle areas/Steinhaus   Sp 2014

 Triangle areas/Triangle Built on Outside of a Given Triangle

 Triangle Area and the Circumcircle

 Triangle Constructions

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 Loci

 Triangle Mid-Segment Theorem    

 Triangles and Squares  

Triangles with Integer Area and Integer Sides  Sp 2014

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

Trirectangular Vertex Problem

Two triangle problems  

Trisection of the Area of a Triangle

Two Squares

Volume of Holes Left by Tree Spade  Sp 2014

Algebra Problems


Sp 2014

7-11 Problem    Sp 2014

A Tangled Tale Problem    Sp 2014

Area bounded by Quarter-Circles in a Square

Big Tires Sp 2014

Calculator sequence

Candy Problem Sp 2014

Check Cashing Problem   SPRING 2014, Examined APRIL 10    

Combining Percentages SPRING 2014, Examined APRIL 10

Combining Rates    Sp 2014  (Danielsville travel problem . . . )

Comparision of Two Radical Expressions

Equiperimetric Areas Sp 2014

Exponents Equation SPRING 2014, Examined APRIL 10

Golden Ratio and Fibonacci Sequence I

Heart to Bell

Iron Ball Floating in Vat of Mercury SPRING 2014, Examined APRIL 10

Jordan's Inequality

Linear functions tangent to their product function   

Maximum Volume of a Cone


Minimal area triangle

Multiple Solutions  Sp 2014

Orthogonal Parabolas  

Phonograph groove SPRING 2014, Examined APRIL 10

Powers of the Golden Ratio

Pythagorean Triples  Sp 2014

Quotient of Two Linear Functions

Rational Equation

s(s-c) = (s-a)(s-b)  

 Simultaneous Quadratics  

 Square Root Equation  

  Sum of Unit Fractions

Three Mile Roadway -- Two line segments and an arc

Two Stamp Problem

  Volume of a Frustum  

Conversion Problems

Big Tires Sp 2014
Cubic Foot 
Million Drops of Water  SPRING 2014, Examined APRIL 10
Volume of a 12 ounce can

Cryptarithm Problems



Division Problem with 8 in the Quotient 

Nine digit number


"Mean" Problems


AM-GM Problems   Sp 2014

Arithmetic Mean -- Geometric Mean Inequality    Sp 2014

Arithmetic Mean -- Geometric Mean -- Harmonic Mean Inequality

Areas in square of sides a+b  Sp 2014

Box Problem 

Combining Rates   Sp 2014    (Danielsville travel problem . . .)

Comparing Segments in two circles   Sp 2014 

Comparison of altitude and median in a right triangle Sp 2014 

Construct Square with Same Area as a Given Rectangle  Sp 2014

Cost of Fencing a Field Sp 2014

Equiperimetric Areas  Sp 2014   See problem max area with partitions

Harmonic Mean Problems


Lines Parallel to the Bases of a Trapezoid -- HM-GM-AM-RMS Inequalities    Sp 2014

Maximize the Area of a Triangle with sides  9, 40x, and 41x SPRING 2014, Examined APRIL 10

Maximum Area of Rectangular Pen with Diagonal Partition Sp 2014

Maximum area -- rectangle  Sp 2014

Maximum Area of a Sector of a Circle -- fixed perimeter of the sector Sp 2014

Maximum Area of Trapezoid, Base twice the Lateral Side 

Maximum area -- triangle Sp 2014 

Maximum and Minimum of   Sp 2014

Maximum of f(x) = (1-x)(1+x)(1+x)  Sp 2014

Maximum Volume of a Cylinder Inscribed in a Sphere Sp 2014

Mean Problem Sp 2014

Minimum distance from (0,0) to Sp 2014

Minimum of x + 1/x   Sp 2014

Minimum of  Sp 2014

Minimum Surface area of a can of fixed volume Sp 2014

Phonograph groove

Product =  1; Sum ≥ n   Sp 2014


Segment Parallel to Bases of a Trapezoid through Intersection of Diagonals  Sp 2014

Segments in a Circle

Square Roots and Cube Roots Sp 2014

Trapezoid Divided into Equal Areas by a Line Parallel to the Base -- RMS Sp 2014


      Trigonometry Problems


Arctan Sum

Area of a Segment of a Circle Sp 2014

Barn in a square field

Bridge Expansion Problem Sp 2014

Closed Form for Some Values of Cosine and Sine

Cos 36 degrees

Navigation Problems

Navigation Problems 2

Overlap Two Circles Sp 2014

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   Sp 2014

Area of Texas Sp 2014

A Sequence from Eudoxus

A Will to be Interpreted Sp 2014

Area of Golfing Greens  Sp 2014

Arithmetic Mean Sequence

Bottles and Cans Sp 2014

Census Taker Problem Sp 2014

Change for a Dollar Sp 2014

Check Cashing Problem  Sp 2014

Coins  Sp 2014

Compare Radicals  Sp 2014

David Rock's House Number

Distance to Nearest Road

Division Problem with 8 in the Quotient

Flowing Stream Sp 2014


Friday 13

Four Number Challenge Problem Sp 2014

Grass Consumption by Oxen

How Wide is the Alley?Sp 2014

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

Ladder and Box   Sp 2014

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   Sp 2014

Oil Tank Problems

Patterns in a Whole Number Array

Perfect Shuffle  SPRING 2014, Examined APRIL 10

Rational or Irrational

Rational Equation

Rolling Circle I -- Cycloids and Trochoids

Rolling Circle II -- Number of Revolutions   Sp 2014

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 Sp 2014

Square Roots Sp 2014 

The Salesperson's Journey Sp 2014

Triangular Numbers Sp 2014

Triangular Number Problems Sp 2014

Sum of Cubes Sp 2014

Sum of Squares Sp 2014

Sums of Powers of Integers -- Derivations from summations Sp 2014

Finite Differences Instructions (Clay Kitchings) Sp 2014

Finite Differences Comments, Examples, and Problems Sp 2014

Where is the store?

Final Project


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