Bluffton’s Explore and Explain Mathematics Courses for Middle

School Teachers

Donald E. HooleyBluffton University

Bluffton, Ohiohooleyd@bluffton.edu

2007 MAA/AMS Joint Meetings

Outline

• Bluffton’s Program• Goals• Teaching Strategies• Pedagogical Approaches• Number Theory and Algebra course• Geometry course• Results• Examples

Bluffton’s Program

CBMS MET BLUFFTONNumbers & Oper. Fund. for El.Tea.Data, Prob. & Stat. Under.Num.DataCalculus CalculusDiscrete Math Discrete MathMeasure and Geom. Geom. for MC*Algebra & Functions Alg.&Func.for

MC* *special Bluffton MC courses

Goals

• Organized exploration

• Inductive reasoning

• Verbal explanation

• Writing mathematics

• Group processes

• Enjoyment of mathematics

Teaching Strategies

• Summary reports

• Written reports

• Question introduction

• Group work

exploration, conjecture, explanation

Pedagogical Approaches

• Guided constructivism

• Generalization

Algebra and Functions

• Partitions• Palindromes• Primes and factors• Figurate numbers• Diophantine equations• Congruences• Pythagorean triples• Pell’s equation• Continued fractions• Prop. of binary operations

• TSP• PERT• Bin packing• Conflict graphs• Linear programming• Loans and annuities• Function labs• Symmetric polynomials• Factoring models• Cubics and quartics

Geometry for MC

• Geom. software• Coord. geometry• Quad. and midpoints• Taxicab geometry• Pythagorean Ther.• Triangle trig• Cyclic quadrilaterals• Circles

• Fibonacci and GR• Symmetry groups• Strip patterns• Trans. Matrices• Tesselation• Regular polyhedra• 3D symmetry• Fractals

Results

• High interest

• High involvement

• Good summaries

• Average scores

Algebra Examples

How many ways are there to distribute 8 ice cream bars among 5 persons?

How many n-digit palindromes are there?

Is every positive even integer the sum of two primes?

Which positive integers are abundant?

Can you find positive integers a, b, c so that T(a) + T(b) = T(c)?

Algebra Examples 2

How many beetles might be in a box of beetles and spiders containing 46 legs?

What happens to a+b mod 9?Can you find a Pythagorean triangle whose area

is equal to its perimeter?For which positive integers D does Pell’s

equation x2 – Dy2 = 1 have solutions?Can you find a continued fraction equal to the

square root of 2?

Algebra Examples 3

Investigate commutativity, associativity, and existence of identity and inverses for:

a*b = b a*b = a+b+1

a*b = a+b+ab a*b = min(a,b)

a*b = gcd(a,b) a*b = lcm(a,b)

a*b = (a+b)/2 a*b = ABS(a-b)

a*b = 1/(a+b) a*b = (a+b)/(1+ab)

Geometry Examples 4

What length relationships do you find when constructing midlines of a triangle?

What length relationships do you find when constructing medians of a triangle?

What shapes are made when midpoints of sides of an arbitrary quadrilateral are connected?

What points are equidistant from two given points in Taxicab geometry?

Geometry Examples 5

What happens when you form sinT/t for each angle T and side t of a triangle?

How are the lengths of the sides and the product of the diagonals of a cyclic quadrilateral related?

How do the products of the lengths of the segments created by intersecting chords of a circle compare?

Geometry Examples 6

What are the vertex angle measures in a regular n-gon?

Which regular polygons tesselate the plane?What are possible semi-regular tesselations

of the plane?Which polygons are faces of regular

polyhedra?What are the axes of symmetry for regular

polyhedra?

Credits

Kay, David C., College Geometry: A Discovery Approach, 2nd ed., Addison-Wesley Pub. Co., Boston, 2001.

Masingilla, Joanna O., and Lester, Frank K., Mathematics for Elementary Teachers via Problem Solving, Prentice-Hall, Inc., Upper Saddle River, NJ, 1998

Otto, Al, personal correspondence, Illinois State University, Normal-Bloomington, IL, 1994.

O’Daffer, Phares G., and Clemens, Stanley R., Geometry: An Investigative Approach, 2nd ed., Addison-Wesley Pub. Co., Reading, MA, 1992.

CMJ 513

Let

p(x) = x3 – 87x2 + 181x + c

For which integers c does this cubic have three integer roots?