SituationThree green frogs and three brown frogs sit on their own lily pads. There is one spare pad between the two sets of frogs. No pad can support more than one frog at a time. (See the diagram below.)

Goal & Rules The green frogs are going to change places with the brown frogs, but must observe the following rules.

• A frog can hop to an adjacent pad.
• A frog can jump over another frog to an unused pad.
• A frog cannot move backward. For example, once a frog moves to the right, it can only continue to move to the right.

Question What is the minimum number of moves needed to have the green frogs switch places with the brown frogs?

The Locker Investigation

Imagine you are at a school that still has student lockers. There are 1000 lockers, all shut and unlocked, and 1000 students.

1. Suppose the first student goes along the row and opens every locker.

2. The second student then goes along and shuts every other locker beginning with the 2nd locker.

3. The third student changes the state of every third locker beginning with locker number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)

4. The fourth student changes the state of every fourth locker beginning with locker number 4.

5. Imagine that this continues until all of the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed? Why?

Find the sum of all the multiples of 3 from 1 through 300. Do you see a pattern developing? Find the sum of the multiples of 7 from 1 through 700. Have you found the pattern? Next suppose we let N be any whole number. Find the sum of all the multiples of N from 1 through 100*N.

Two elevators leave the nth floor at 2:00 P.M. The faster elevator takes one minute to travel between floors and theslower elevator takes two minutes to travel between floors. The first elevator to reach a floor must stop for threeminutes to take on passengers. If both elevators arrive at a floor at the exact same time, they becomeconfused and do not stop for passengers.

If the final stop for an elevator is the lobby (1st floor), then describe n if the faster elevator arrives at the lobby first. Describe n if the slower elevator arrives at the lobby first.

Compare the variables (time, floor) for the fast elevator.

You've got to see this!! Click here to see the "Safety Rider Video"

You're sitting in math class, minding your own business, when in walks a Bill Gates kind of guy ­ the real success story of your school. He's made it big, and now he has a job offer for you.

He doesn't give too many details, mumbles something about the possibility of danger. He's going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?) And you've got to make sure your passport's current. (Get real, Bill, this ain't Paris). But do you ever sit up at the next thing he says.

You'll have your choice of two payment options:

1.One cent on the first day, two cents on the second day, and double your salary every day thereafter for the thirty days;

OR

2.Exactly $1,000,000. (That's one million dollars!)

Which option is the better choice??

• How long would it take to make $10 million dollars if your pay was tripling every day instead of doubling? What would be the formula you could use to get this answer? (Hint: How would you adjust the formula found on this page?)
• Imagine the pay isn't doubled, but it's 1 and 1/2 times each previous day's pay (1.5 times). Or maybe it's 1 and 1/3 times (1.333... times). Now here's the question. At what rate should you say, no, I'll take the million. In other words, at what rate would the 30 days Total Pay be less than $1 million?

The Tower of Hanoi (sometimes referred to as the Tower of Brahma or the End of the World Puzzle) was invented by the French mathematician, Edouard Lucas, in 1883. He was inspired by a legend that tells of a Hindu temple where the pyramid puzzle might have been used for the mental discipline of young priests. Legend says that at the beginning of time the priests in the temple were given a stack of 64 gold disks, each one a little smaller than the one beneath it. Their assignment was to transfer the 64 disks from one of the three poles to another, with one important proviso­a large disk could never be placed on top of a smaller one. The priests worked very efficiently, day and night. When they finished their work, the myth said, the temple would crumble into dust and the world would vanish.

Now let's try to solve the problem that Lord Brahma presented to his priests. How long would it take to move the tower made up of 64 discs? Assume that one move would be made every second.) I suggest that you start with a small numbers of discs and look for a pattern(s)

Once you've discovered a pattern(s), can you use it to figure out when Lord Brahma predicted the world to end?

PS See these wonderful websites: http://www.lhs.berkeley.edu/Java/Tower/towerhistory.html

http://chemeng.p.lodz.pl/zylla/games/hanoi5e.html

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Chirping Crickets Investigation

Scientists have noticed that crickets move their wings faster in warm temperatures than in cold temperatures. Therefore, by listening to the pitch of the chirp of crickets, it is possible to tell (reasonably accurate) the temperature of the air?

PS For all you could ever want to know about chirpping crickets... click here

You have been hired by a company that makes all kinds of cups - foam hot cups, plastic drinking cups, paper cups, and more - of different sizes. For each of the kinds of cups it makes, the company needs to know the measurements of cartons that can hold 50 cups. Your task is to provide this information. Also provide the company with a gerealized formula they can use with any type of cup.

PS Check out this website - click here And you won't believe this one! - click here

Squares Cubed Investigation

Investigate the functional relationships between the length of the sides of a square and its perimeter, area and volume as the sides increase in length