Gear Math
Patterns, Groups, and Relative Size
We are going to investigate gears and how gears of different sizes mesh together along a beam. Before we do that, let's first decide how many different possible combinations there are. There are mainly four different size gears in the simple machines set. (There are specialty gears such as the crown gear and the bevel gear, but let's ignore those for now). With these four gears, how many different combinations of two gears can be made? (include same-gear pairs)

A further thing to investigate is the relative sizes of the gears. See that two small gears meshed together equal the distance across of the second smallest gear. If we call the smallest gear a "one," the other gears are a "two", "three", and a "five".

After the number of combinations and the relative sizes have been figured out, draw beams on the blackboard (one for every combination), asking the students to draw the gear combinations onto the beams. Students need to put axles into the different gears and investigate how many holes to leave between gears so that they will mesh before going to the board.
 

Gear Addition
Once the combinations are drawn on the board, students should start looking for patterns. In particular, try and find combinations of gears that have the same number of holes between them. One such example is the "five" and the "one" and two "three" gears - they both have two empty holes between the centers of the two gears. If we look at the gears from end to end we also see that they are the same length. What is going on? Students notice that 5 + 1 is the same as 3 + 3. What other combinations are the same?

The next fun thing to investigate is how many different ways to create the distance of "10". Two fives, five twos, ten ones, ... how many others?
 

Gear Division
The next thing to investigate is the relationship in speed between the meshed gears. If you put the yellow catch onto the axle, it is easy to count the number of times that the gears go around. For every time the big gear goes around once, how many times does the small gear go around?

was on
http://www.ceeo.tufts.edu/ldaps/htdocs/curriculum/gearmath.html
 
 

Gear Multiplication
The next question to ask is "can I get any better than five times faster/slower?" Using the biggest and smallest gear meshed together there is a relationship of 5 to 1. Can I get any better than that? The first thing that people think of is of adding more gears in between the two gears. Eventually students begin to see that no matter how many gears are between - the relationship in the speeds is the same! This is why the gears in between the "driver" and the "follower" are called "idlers".

What about putting more than one gear on the same axle? Sometimes students ask this on their own, sometimes not... In any case, that is the next thing to try out. With three axles, and two gears of different sizes on the middle axle, it is possible to get better than a 5 to 1 ratio.

The idea that putting multiple gears on an axle has a multiplicative effect is not an easy concept to grasp. A good idea is to try to have students that understand it explain it to others, for there are many different ways in which to think about it. Here Jenny explains how her gear train of 25 to 1 works.
 
 

I get a lot of LEGO curriculum ideas from the isles of TARGET Greatland :) . During the summer of 1997 I got the idea of taking apart the diet scale. This summer I took apart the bubble blower shaped like a helicopter myself and showed the teachers. With each turn of the helicopter's wheels, the fan/flywheel that blows the bubbles turns many times. It accomplishes this with gears of different sizes on the same axle.