# Number Theory and Algebra Working Group

## Monday, July 9

The people

We began by introducing ourselves and discovered that we had some "interesting" career paths. Francisco, a native of Portugal, who began his career in designing and marketing molds for plastics. He had always enjoyed teaching and became a Newark teacher. Victoria, a native of the Philippines, has taught in the Philippines and for eight years in Nigeria. After moving to the U.S., she is now teaching in Newark. Steve ended his formal education at 17, returned as a college first year student at 31 and completed his studies at 40 and is presently teaching at the University of Louisville. John worked for the phone company as an engineer. . He volunteered to substitute for teachers twice a month so that the teachers could attend workshops at the Educational Development Center (EDC) in Newton, Mass. John became a full time teacher in Wayland, Mass. The group leader of the Algebra/Number Theory group, Al Cuoco, heads the curriculum in mathematics education. Al was a high school teacher for 24 years, took two years leave and obtained a Ph.D. in Algebraic Number Theory.

What we would like to accomplish

We would like to engage our students in interesting and motivating mathematics. At the workshop we hope to create activities that lead students to engage in the same processes that mathematicians use to solve problems.

A main theme is given a set of data, several values of a function, find functions that fit the data. For example, given such a data, is a linear function an appropriate function the data?

Here is an example. (Actually we were not given the Delta's but observed as a group that a pattern emerged with the Delta's.)

Would a linear function be appropriate to model the data given below? If so, what linear function do you think works best? Explain.

Values of a certain function f(n)

 n f(n) Delta = f(n+1)- f(n) 0 3 1 8 5 2 13 5 3 18 5 4 23 5 5 28 5 6 33 5

We worked this one out.   Then we were given a new set of data. (Again we were not given the Delta’s but observed as a group that a pattern emerged with the “second differences”,  Delta(Delta).)

 n G(n) Delta Delta (Delta) 0 1 1 -2 -3 2 1 3 6 3 10 9 6 4 25 15 6 5 46 21 6 6 73 27 6

We agreed that a linear function would not be the best way to model the data.  We came to the conclusion that the fact that the second differences were constant indicated that

a quadratic function might be good to model the data.  We discussed and discovered methods to determine the appropriate a,b,c  of g(x) = ax^2 +bx +c.   We concluded that if the second differences were constant that a quadratic was indeed a good choice to model and we found a technique to determine from the chart the coefficients of the appropriate quadratic.

We're hungry now so we shall conclude here.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.