Problem Solving in Mathematics

Fall 2015

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

Resources

References Fall

EMAT 4600/6600 Class Members, Fall 2015

Go to the Class list for individual e-mail addresses.

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.    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.

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.

PROBLEMS

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

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

Project INTERMATH

Environment Problems

Volume of a curved trench, Trapezoid Cross-section

Geometry Problems

Angle of View

Arc Length equal to a given segment

Arctan Sum

Area and Side of a Rhombus Given its Diagonals

FALL

Areas of Lunes I FALL

Areas of Lunes III

Bouncing Barney FALL

Carl's Cone FALL

Ceva's Theorem  FALL

Checkerboard Problems

Color a Circle

Cone Half Full FALL

FALL

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

FlowingStream FALL

Golden Triangle  (same as Sublime Triangle)     FALL

Heron's Formula FALL

Inscribed Equilateral Triangle in a Square -- Construction

Inscribed Equilateral Triangle in a Square -- Problem

Inscribed Square for Given Triangle

Inscribed Trapezoids

Inscribed Triangle in a given triangle

Island Treasure  FALL

Kite   FALL

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

Locus Problems  Revised FALL

Medians of a Triangle  FALL

Minimum Path    FALL

Notch for Felling a Tree

Obtuse Triangle Relationship

Orthogonal Parabolas

Pairs of Segments with the Same Sum of Lengths

Parallel Chords

Pappus Areas

Path Problems  FALL

Perpendicular Chords in a Circle

Projections: Homothetic Similarity

Quadrilateral Determined by Intersecting Orthogonal Parabolas

Ratio on a line segment -- Something Golden    FALL

Septagon  FALL

Spheres FALL

Squares Inscribed in a Right Triangle (Revised)   FALL

Sublime Triangle(Same as Golden Triangle)   FALL

Tangent Circles and Spheres

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

Three-Color Circle Patterns

Three Intersecting Circles with Collinear Intersection Points

Trapezoid Inscribed in a Semicircle

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

Trisection of the Area of a Triangle

Two Squares

Volume of Holes Left by Tree Spade  FALL

Algebra Problems

FALL

7-11 Problem    Fall

ASTROID Equations

Big Tires FALL

Calculator sequence

Candy ProblemFALL

Combining Rates      (Danielsville travel problem . . . ) FALL

Exponents Equation

Golden Ratio and Fibonacci Sequence I

Golf Ball Flight  FALL

Jordan's Inequality

McNuggets  FALL

Powers of the Golden Ratio

Quotient of Two Linear Functions

Rational Equation

Volume of a Frustum

Cryptarithm Problems

Cryptarithms  FALL

"Mean" Problems

AM-GM Problems   FALL

FALL

Combining Rates     (Danielsville travel problem . . .) FALL

Comparing Segments in two circles  FALL

Equiperimetric Areas    See problem max area with partitions     FALL

Harmonic Mean FALL

Inequalities FALL

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

Maximum area -- triangle with Fixed Perimeter FALL

Maximum Volume of a Cylinder Inscribed in a Sphere

Mean Problem

Minimum of  FALL

Prove

Trigonometry Problems

Arctan Sum

Closed Form for Some Values of Cosine and Sine

FALL

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

Prove (The equation is a link.)

Mixture of all

7-11 Problem  FALL

Area of Texas FALL

A Sequence from Eudoxus

Arithmetic Mean Sequence

Coins FALL

David Rock's House Number

Evaluate MENTALLY and describe your work.

Flowing Stream FALL

Friday 13 FALL

Magic Square

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

FALL

Patterns in a Whole Number Array

Perfect Shuffle  FALL

Rational Equation

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

The Salesperson's Journey

Sum of Cubes FALL

Sum of Squares FALL