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Number Patterns in Nature and Math

Clarkson Summer Math Institute: Applications and Technology. Number Patterns in Nature and Math. Peter Turner & Katie Fowler. 1. Arithmetic Progression. Rules Start with a “given” number Add a constant quantity repeatedly Questions What happens? Can we find a formula?

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Number Patterns in Nature and Math

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  1. Clarkson Summer Math Institute: Applications and Technology Number Patternsin Nature and Math Peter Turner & Katie Fowler

  2. 1. Arithmetic Progression • Rules • Start with a “given” number • Add a constant quantity repeatedly • Questions • What happens? • Can we find a formula? • Compare rates of growth

  3. 1. Arithmetic Progression • More critical thinking • Which will grow faster? • What happens to ratios? • What about relative size of two progressions? • Advanced number topic • Use fraction or decimal “differences” • When will you be half you Dad’s age? • Do negative differences make sense?

  4. 2. Sums of Arithmetic Progression • Rules • Add the terms of the progression together • Questions • What happens? • Will the graphs be straight lines? • How do they increase? • Can we find a formula?

  5. 2. Sums of Arithmetic Progression • More critical thinking • Try to predict faster/ slower growth • What happens to differences? • Differences of differences? • Can you think of faster growth? • Advanced number topic • Squares of numbers (and variations) • Compare sums, squares, and combinations

  6. 3. Doubling – A Geometric Progression • Rules • Start with a “given” number • Double it repeatedly • Questions • What are the first few terms in this progression? • What is the tenth one? • How did you get it? Recursively or by repeated multiplication each time? • What about sums? • What’s the pattern here? • Can you explain it?

  7. 3. Geometric Progressions • More critical thinking • Other common ratios • What happens to the formulas/patterns • For the terms? • For the sums? • Which grows faster? • Advanced number topic • What about fractional or negative ratios? • Try (-1) and its sums • What do you know about the ratios of the sums?

  8. 4. Tree growth – Fibonacci Numbers • Also applies to other “growth” problems • Population of rabbits, for example • Pineapple skin • Sunflower seeds • Rules • Tree can add branches to branches that are at least two years old • Other branches persist

  9. 4. Tree growth – Fibonacci Numbers • Questions • How many branches are there after one, two, three years? • Four, five, …, ten years? • What about n years? • Can you spot a pattern? • What about ratios?

  10. 4. Tree growth – Fibonacci Numbers 13 Year 7 8 Year 6 5 Year 5 3 Year 4 2 Year 3 Year 2 1 1 Year 1

  11. 4. Tree growth – Fibonacci Numbers • More critical thinking • Patterns in the numbers • Will it persist forever? • Why, or why not? • What about different starting conditions? • Advanced number topic • “Exponential growth” • Ratio is not a fraction – introduce irrational numbers • Differences do not have a simple pattern – or do they?

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