Number Patterns in Nature and Math Peter Turner & Katie Fowler Clarkson Summer Math Institute: Applications and Technology

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

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?

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?

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

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?

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  and its sums –What do you know about the ratios of the sums?

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

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?

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

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?