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

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

Volume of an Irregular Solid

Geometry Problems

Angle of View

Arc Length equal to a given segment

Arctan Sum

Area and Side of a Rhombus Given its Diagonals

Areas of Lunes III

Brahmagupta's Formula

Carl's Cone

Checkerboard Problems

Color a Circle

Construct Equilateral Triangle with Vertices on Three Given Parallel Lines

Construct Three Congruent Segments in a Fixed Angle

Dissect an Equilateral Triangle a form a Square

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

Equiperimetric Areas

Golden Ratio Calipers

Golden Triangle  (same as Sublime Triangle)

Heron's Formula: Geometric Proof

Inscribed Equilateral Triangle in a Square -- Construction

Inscribed Equilateral Triangle in a Square -- Problem

Inscribed Rectangle with Maximum Area for Given Triangle

Inscribed Square for Given Triangle

Inscribed Trapezoids

Inscribed Triangle in a given triangle

Inverse Geometry

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

Lighthouse Problem: Three lighthouses

Locus Problems  Revised

Notch for Felling a Tree

Obtuse Triangle Relationship

Octagon Construction and Formulas

Orthogonal Parabolas

Pairs of Segments with the Same Sum of Lengths

Paper Knot Pentagon

Parallel Chords

Partition Square into Acute Triangles

Pappus Areas

Perfect Triangles

Perpendicular Chords in a Circle

Projections: Homothetic Similarity

Quadrilateral Determined by Intersecting Orthogonal Parabolas

Ratio on a line segment -- Something Golden

Right Angle in a Configuration (SSM 393

Septagon

Square Inscribed in a Semicircle -- find a ratio

Sublime Triangle(Same as Golden Triangle)

Tangent Circles and Spheres

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

Three-Color Circle Patterns

Three Intersecting Circles with Collinear Intersection Points

Trapezoid with Parallel through the Intersection of the Diagonals

Trapezoid Inscribed in a Semicircle

Triangle Built Outward (12 problems)

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 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

Trisection of the Area of a Triangle

Two Squares

Volume of Holes Left by Tree Spade

#### Algebra Problems

Area bounded by Quarter-Circles in a Square

ASTROID Equations

Big Tires

Calculator sequence

Candy Problem

Combining Rates      (Danielsville travel problem . . . )

Exponents Equation

Golden Ratio and Fibonacci Sequence I

Iron Ball Floating in Vat of Mercury

Jordan's Inequality

Powers of the Golden Ratio

Prove All Parabolas are Similar

Quotient of Two Linear Functions

Rational Equation

s(s-c) = (s-a)(s-b)

Two Stamp Problem

Volume of a Frustum

#### Cryptarithm Problems

Division Problem with 8 in the Quotient

Nine digit number

OCTOBER Primes

#### "Mean" Problems

Arithmetic Mean -- Geometric Mean -- Harmonic Mean Inequality

Combining Rates     (Danielsville travel problem . . .)

Comparing Segments in two circles

Cost of Fencing a Field

Equiperimetric Areas    See problem max area with partitions

Harmonic Mean

Inequalities

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 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 Volume of a Cylinder Inscribed in a Sphere

Mean Problem

Minimum of

Minimum Surface area of a can of fixed volume

Prove

Segments in a Circle

Square Roots and Cube Roots

#### Trigonometry Problems

Arctan Sum

Barn in a square field

Bridge Expansion Problem

Closed Form for Some Values of Cosine and Sine

Conic Section Images

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

Prove (The equation is a link.)

Mixture of all

Area of Texas

A Sequence from Eudoxus

Arithmetic Mean Sequence

Bottles and Cans

Bottles and Cans (Revised)

Census Taker Problem

Change for a Dollar

David Rock's House Number

Evaluate MENTALLY and describe your work.

Flowing Stream

Geometric Demonstration of  the Product of Two Negative Numbers

Alternative Demonstrations of the Product of Two Negative Numbers

Grass Consumption by Oxen

Magic Square

Maximum Volume of a Cone

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

Oil Tank Problems

Patterns in a Whole Number Array

Perfect Shuffle

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

The Salesperson's Journey

Triangular Number Problems

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