|
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
…Lewis Carroll, Alice in Wonderland Sometimes we have problems that seem too big or too complicated. Sometime they seem impossible. We will investigate a couple of methods of problem solving and a new tool for you to use to get a handle on some of these big problems. When you have one of these problems to solve, there are a few different things that you can try before you get frustrated. Larger problems can often be broken down into smaller problems that are easier to figure out. Look for patterns in the smaller problems. If you find a pattern, and are convinced that it is a good pattern, then you can predict from it without doing some of the complicated math. That sounds easier than it really is, but you also will have a new tool to use to break down problems and look for patterns. That tool is the spreadsheet. Introduction to Spreadsheets  A spreadsheet is a computer software program that is used to organize, calculate and graph numbers and formulas. It works like a calculator but allows you to see all of you work. We will use Microsoft Excel to set up, solve and graph the solutions to the following problems: Problem 1
Congratulations! You have just been offered a new job. You will work at this job for exactly 30 days and then comfortably retire! But first you must make a choice. You can choose to be paid in one of two ways. Salary A. You earn $10,000 on the first day and get a $10,000 raise each day you work after that (that means you earn $20,000 on the second day, $30,000 on the third day and so on …that’s a pretty good salary!) Salary B. You earn one cent on the first day and double your salary each day that you work (hmm, that means you earn 1 cent on the first day, 2 cents on the second day, 4 cents on the third day, 8 cents on the fourth day and so on …doesn’t sound so good!) 1. Which salary gives you more money? By how much?
2. How much would you lose if you took just one day off? Does it matter which day you take off? Why?
3. What if the job only lasted for 20 days? Would your answer change? Why?
4. When are the two salaries exactly equal? How can you see this on your graph? Problem 2
A very famous mathematician named Carl Gauss (click on the link to see a picture of him …a pretty good looking guy!) solved the following problem almost immediately when he was 7 years old!.
What is the sum of the first 100 counting numbers?
or
What is the sum of 1 + 2 + 3 + 4 + 5 … + 97 + 98 + 99 + 100?He said:
"If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries".
He solved this problem by finding a pattern. Can you use a pattern to solve this problem quickly? Remember, Gauss did not have the use of a calculator in the 1700's. You can probably do this one pretty easily with a calculator, and it's a really easy one to do with a spreadsheet, but what would you do if you needed to find the sum of the first 1,000 counting numbers? How will you know you have the right answer? Problem 3 The King's Chessboard Use the right-click on your mouse and "Open in New Window". You will open a second browser window. Just close the window when you are done with the story and you'll come back to our SchoolNotes investigation.
 The King's Chessboard link is one version ...here is another.
In ancient China, a peasant (who just so happened to be a mathematician) saved the emperor's daughter from drowning in the river. For his heroic action, the emperor wanted to give the man a reward. He asked the man what he wanted for payment for this brave deed. The man looked around the room and noticed a chess board. He asked that the emperor place 1 grain of rice on the first square of the chess board, 2 grains on the second, 4 on the third, and to keep continuing the pattern of doubling the grains of rice for all 64 squares of the board. He would take all of the rice as his reward. The emperor (who was not a mathematician) thought this was a bargain and immediately agreed. How many grains of rice did the man earn? How much rice is this? What size container would he need to take his reward home? How many grains of rice fit inside this room? How can you describe the number of grains of rice? To investigate the size of this large number, investigate the following:
How long does it take to count to a thousand? A million (which is a thousand thousands)? A billion (which is a thousand millions)? A trillion (which is a million millions …wow, that‘s big!)? Our country’s national debt is over $6 trillion! What is national debt? How long would it take us to pay it back if we paid $1 every second? $100 every second? $1,000,000 every second? And that doesn’t even take into account interest! What is interest on a loan or debt? How much interest do we pay on our national debt every year? Every day? Problem 4. The Handshake Problem
 How many handshakes would it take for everyone in this classroom (including the teachers) to shake hands exactly once with everyone else? How can you use a spreadsheet to solve this problem? The golden rule is:
If the problem seems too long, and if the numbers seem too many, LOOK FOR A PATTERN to see if you can predict what will happen.
If you can find a pattern, often times it's easy to find a formula. Formulas, or functions (or the really fancy math term algorithms) are one reason why spreadsheets are such an important part of learning Algebra (and you didn't even know you were doing Algebra, did you?). :-). | 
