Math Model, Series Model for Third of Square or Triangle
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What is one third? What do most kids think of when you ask them about simple fractions such as one half, one third, etc.? I have used series models to stimulate discussions about fractions and some of the beautiful ideas behind them. There are numerous nice models or processes. Here are two of them for one third (1/3). The side length is about 50mm; the wall is 1mm thick and 5mm or 10mm high. Question: What is the sum of all the squares/triangles with bottoms, if we allow the pattern to go on forever and ignore the walls between them? In general, let R be a real number with R > 1. Then, The sum of an infinite series of fractions is equal to one over the difference between R and one. But what does it look like at least in some cases?
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