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^1`1`1a1a1a12a1a1a1a1a1b1b0>+ 1Ld0#1nd1dEvery math student studies sequences and series: arithmetic sequences, geometric sequences, Fibonacci sequences, and arithmetic series. Students can learn about these in their prealgebra days and we can exploit these topics later when we teach about functions, graphing, and so forth. For students to get the idea of iteration, there s no substitute for actually doing some of these calculations by hand, but student understanding is deepened by also learning to specify sequences at a more abstract level, thinking about the seed and the operations performed on the seed. Here are a couple of example activities that use Sketchpad to encourage this step of abstraction.
Activities: Arithmetic Sequence, Geometric Sequence, Fibonacci sequence, possibly arithmetic series.GeometryRepetition in geometry, applying the same rule repeatedly, produces striking results. We can iterate in many places in geometry, from constructing a regular polygon to tesselating the plane to drawing an object known as a complete graph. Let s just look at a couple of the applications of iteration in geometry.
Activities: Construct a regular polygon, do a tesselation, show a complete graph.FractalsIteration is so important in geometry that it s recently spawned an entirely new area of study: the1d1d14&1x&1&1&1J1d1
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.Iteration through the Math Curriculum:
Sketchpad 4 Does It Again and Again
2002 NCTM Annual Meeting Session 491
Scott Steketee
KCP Technologies, Emeryville, California
There are a number of big ideas in mathematics that form unifying themes across different branches of mathematics. Such ideas include functions, transformations, data, proof, and my topic today: iteration.
Function: The concept of function is fundamental to much of mathematics. The idea that applying a particular operation to a particular input determines a unique output is at the heart of algebra, trigonometry, calculus, and just about anything else we want to do in mathematics.
Transformation: Heres a related concept The concept of function is fundamental to much of mathematics. The idea that applying a particular operation to a particular input determines a unique output is at the heart of algebra, trigonometry, calculus, and just about anything else we want to do in mathematics.
is session will demonstrate activities involving iteration and recursion in
a wide range of mathematical disciplines, including algebra (e.g.,
sequences, series, optimization), geometry (e.g., symmetry, fractals) and
calculus (e.g., limits, Eulers method, Riemann sums).
Marco Polo 806; Venetian (capacity 70)
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{6B*OJQJnH B*OJQJnH HH@R(XXd'CjV, LatkaHH@R(XXd'CjV, Latkaossible explorations include pand threebody sandwill beuseful to thinkas theyfigure out how to specifyS of that data,,F ` (XXd'CjVgeneratescan be used to explore using iterative methodsossible explorations include pand threebody sandwill beuseful to thinkas theyfigure out how to specifyS of that data,,F ` (XXd'CjVgeneratescan be used to explore using iterative methods,,F ` (XXd'CjV[This document is the handout for my talk. There are also two Sketchpad files, before and after: before contains the sketches as they appeared when I began the talk, and after contains the sketc