Bluffton’s Explore and Explain Mathematics Courses for Middle School Teachers Donald E. Hooley Bluffton University Bluffton, Ohio 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 METBLUFFTON Numbers & Oper.Fund. for El.Tea. Data, Prob. & Stat.Under.Num.DataCalculusDiscrete Math Measure and Geom.Geom. for MC* Algebra & FunctionsAlg.&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 x 2 – Dy 2 = 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 = ba*b = a+b+1 a*b = a+b+aba*b = min(a,b) a*b = gcd(a,b)a*b = lcm(a,b) a*b = (a+b)/2a*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, 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, 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) = x 3 – 87x x + c For which integers c does this cubic have three integer roots?