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For these investigations you will be using a spreadsheet to examine patterns, make conjectures, and test your conjectures.
Work on all of these problems then choose two you will use for your write up.
1) Friday the 13th
The typical year is 365 days, but make sure you take under consideration leap years when investigating these questions.
Will there be at least one Friday the 13th each year?
What is the maximum number of Friday the 13th's you can have in any given year?
It is possible to have two consecutive months with Friday the 13th. When does this happen?
2) Fibonacci
Generate the Fibonacci sequence in a spread sheet using f(0) = 1 and f(1) = 1 and f(n) = f(n-1) + f(n-2)
Construct the ratio of consecutive Fibonacci numbers. What do you observe as the numbers increase?
How does squaring each term in the Fibonacci sequence change the ratio? What about cubing?
Explore the sequence for any arbitrary values for f(0) and f(1). For example: Let f(0) = 1 and f(1) = 4. Discuss some of the things you notice.
Is it possible to go backwards in the Fibonacci sequence, ie f(-1), f(-2)? Discuss any properties discovered.
3) The 7-11 problem
A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.
What are the exact costs of each item?
Use a spreadsheet to help inform you how to solve this problem.
4) Change for a dollar
What is the largest amount of money you can have (using pennies, nickels, dimes, and quarters) so that you can not make change for a dollar?
How does this amount change if you include half dollars?
How does this problem change if you only include pennies, nickels, and dimes?
5) The absolute value problem
Place four numbers in the first row as follows
A B C D
For each successive row replace the entries by the absolute value of the difference of the entry just above and the entry just to the right in the previous row. In the fourth position use the absolute value of the difference of the fourth and the first (i.e. cycle)
|A-B| |B-C| |C-D| |D-A|
Will the process lead to a 0 in all 4 entries for some row?
Determine properties of numbers that produce 0's after 3 iterations. Explain why the properties always produce 0's after 3 iterations.
Determine properties of numbers that produce 0's after 4 iterations. Explain why the properties always produce 0's after 4 iterations.
Determine properties of numbers that produce 0's after 5 iterations. Explain why the properties always produce 0's after 5 iterations.
What is the largest number of rows before a zero row is generated? Explain how you were able to come up with the numbers that produced this cycle.
6) Using data to model, explain, and make predictions
Part 1. Price of Gas (1990 - 2012)
The price of gasoline has been rising for the last few years. Explore the gasoline prices spreadsheet (Data 1 tab, column B) and answer these questions:
Hint: One approach you can take with this giant data set is to only use the average of regular gasoline prices at the beginning of each year. Another approach is to examine the average gasoline prices of each month.
Is there such a thing as the "rate of increase" for the price of gas? If so what is it? If not, why?
Use a model to predict when the price of gasoline will reach $5 per gallon. Explain how you determined this date.
Many people believe that the price of gasoline rises quicker during the winter (because of heating oil) than any other month. Use the data to determine if there is there any truth to this statement.
Part 2. Stamp prices
The price of a first class stamp has also been increasing for the last few years, actually since 1933. Explore the stamp prices spreadsheet and answer these questions:
Is there such a thing as a "rate of increase for the price of a first class stamp? If so what is it? If not, why?
Use a model to predict when the price of a first class stamp will be 60 cents. Explain how you determined this date.
Many people believe that the price of stamps have been going up more rapidly in the last two decades. Use the data to determine the truth to this statement.
7) The Money Bags Post
In October 2010, postings on social media sites and emails went viral with the claim of a"Money Bag Month" and that they were very rare. The email/post stated that August 2010 was a money bag month since there were 5 Sundays, Mondays, and Tuesdays. Months that are considered to be a money bag month when they have 3 consecutive days that occur 5 times. The post goes on to explain that such an event only happens every 800+ years.
Part 1) Use a spreadsheet to examine the claim that money bag months occur every 800 years. Explain why the claim is true or false.
Part 2) Is it possible to have a month with 4 or more consecutive days that occur 5 times?
Part 3) How often does a month have 2 consecutive days that occur 5 times?
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