Transcript Number Patterns in Nature and Games

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?
– 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 (-1) 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 7
13
Year 6
8
Year 5
5
Year 4
3
Year 3
Year 2
Year 1
2
1
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?