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Report from the 2007 MAA MathFest Meeting

San Jose, California
August 2007

This convivial meeting was the first Sketchpad User Group to convene at a MathFest, and based on the interest, we'll try to reconvene at MathFests in the future. Sketchpad enthusiasts shared work and sketch ideas in ad hoc presentations hosted by Martin Flashman, Viva Hathaway, Nick Jackiw, Kendra Lockman, and Scott Steketee. Sketches from some of these presentations are available below.

Sketchpad User Group

Iterating Despite Dissimilarity

Abstract: Sketchpad's Iteration command gives a powerful means for generating sequences or series of geometric shapes or numeric values, by constructing and then generalizing the relationship between the first and second term (the "pre-image" and "first iterated image") in the sequence. However, iteration requires a "strong similarity" between first and second terms: a triangle must always map to a triangle, a single value must always map to some other single value, etc. However, in some iterative contexts, one wants to express some idea of choice or variation in the rule by which one term in the sequence is replaced by the next. For example, in the Chaos Game played inside a triangle, a single point (the first term) maps to one of three possible next points depending on the roll of a die. In a Penrose tiling, kite-shaped quadrilaterals decompose into both kite-shaped and dart-shaped quadrilaterals (and vice versa). This talk describes methods for encoding this idea of choice or decision within a single image, so that iteration's "strong similarity" requirement can be satisfied, while still expressing the variation needed by the mathematical production rule, which one is attempting to iterate. Examples include the Chaos Game, Conway's pinwheel tessellation, and Penrose tilings.

Presenters: Nick Jackiw and Kendra Lockman, KCP Technologies

Antenaresis & Other Algorithms

Abstract: This talk surveys ideas about constructing algorithms in Sketchpad using a combination of techniques centered on iteration. We look at binary search in a data set, the Babylonian square root, and the geometric foundation of Euclid's greatest common divisor algorithm, exploring the irrational ramifications of that last method when the two numbers have no GCD—when, in the Euclidean setting (the so-called "antenaresis"), the two lengths are incommensurate.

Presenter: Nick Jackiw, KCP Technologies

Bring Calculus Alive with Sketchpad

Abstract: Calculus can be described as the mathematics of change, and many calculus topics have striking visual representations. What better way to explore these topics than with a program designed for dynamic mathematical visualization? This presentation includes three examples: Tracing the Slope Function, Probing the Antiderivative, and Slope Fields. The presentation materials include classroom-ready worksheets, sketches, and additional support materials.

Presenter: Scott Steketee, Key Curriculum Press

Circles by Tangencies

Apollonius of Perga (ca. 262–ca. 190 BC) posed and solved the following problem: Given any three objects chosen from a point, line, or circle, find a way to construct a circle tangent to all three. Unfortunately, although each case (of 10 possible combinations) is possible, few students of mathematics have carried out each construction by hand. Many of the constructions have several steps, and there is a loss of precision at each step of constructing by hand, often leading to unsatisfying results. Moreover, once a construction is made, the geometer who made them is still left with the solution to only one case of a posed problem. In this talk we explored the cases involving points and lines in the dynamic context of Sketchpad.

Presenter: Kendra Lockman, KCP Technologies