# Quotient Calculations – Using Intuition When Using tuples in Calculus Examples

Triples is a language which is the language of triangles. It is easy to understand. The problem is that it seems difficult to translate into English, which is what we are doing when we say things like “We found nine red boats”. What we actually mean is “We found nine identical ships (with the names of three of their owners)”. It does not say “We found nine red boats, these are identical red boats we found”.

In order for you to understand this example you need to know something about the subject of mathematics. You can learn all sorts of things about this subject, though most people only study triangles. You probably know there are forty basic geometric shapes, right? And then you have things like the cone, cylinder and sphere. So those are the basic geometric shapes.

Now if you study algebra then you will find this example easier to understand. All people study triangles. So, if you have a set of people whose names start with the letter A, such as UPS, TRW, or JK it would be easy to translate it into English. For instance UPS is the name of the company that has just sent us ten million dollars in cash. That is the name of one of their pilots. Then TRW and JK are two people whose names start with the letter R, for Ray Wilson who is the United States Air Force’s aerial fighter.

This example uses the words like “rays” and “wing” to refer to planes. It uses “rays” to refer to a part of the plane. It uses “wing” to describe the shape of the airplane. These words are easy to translate and mean almost the same thing.

Another example is lawyers. You know all types of lawyers, from barristers to criminal lawyers to patent attorneys. So let’s say there is a patent attorney named Paul Smith. He has done a patent search and finds one patent by an inventor called Robert Kaplan.

Paul finds out that Robert makes goggles for hockey players. He talks to some people at his law firm and explains what he did. One of the people tells him about a conference that he attended where he met Kaplan. He thanks him and he tells him his business over lunch. So we have two people and their names.

Another example uses words like “work” and “drive” to refer to a job. It could also use words like “labor” or “organize”. The person with the “work” job might have organized the resources for the other people to use so they can work on their project. The example uses the word “organize” to describe the process where people put things together to make something useful and “labor” to describe the work people do.

The intuition behind these examples is that a topic is important and it has practical value. The intuition fails, however, when we use our common sense. A very common sense method is to say “Aha!” when we see something we have not seen before. These examples are prime examples of “Aha!” moments in everyday life.

Another example is this: Two carpenters are working on a new addition to the library. One man moves a large book shelf with a plank between them. The other man measures the height of the shelf and uses that to install new legs to it. The one who moved the shelf did not realize that the legs needed to be attached so that they would not tip over. The intuition here is that this is a dangerous project and that the man who moved the shelf was too careless. The library was closed for a day to allow people to gather to watch the work.

The intuition fails again in the following example. A young girl has painted her bedroom a beautiful color. She thinks that the color is so bright and beautiful that she will love to decorate her room. Her mother tells her not to paint it because she might injure herself. The intuition here is that the color is so bright and brilliant that it may hurt her if she touches it.

The intuition is most often used in the second degree courses, when students have little experience with higher mathematics. Students use their intuition most frequently when solving a problem using a real life example. If you ask a child to draw a car, most of them will just do the simplest possible thing without considering their intuition. That’s why it’s called the intuitive faculty. After all, what kid wouldn’t want to do the simplest thing?

The intuition can also be used in the first two years of calculus courses when students need an easy example to understand the concepts of higher algebra. It doesn’t have to be a complicated one. Sometimes a simple event can be used as a teaching tool to help the student understand the concepts behind harder problems. These examples often come from the student’s textbook or the instructor’s lecture. There are plenty of these examples online if you search for “intuition” or “instant”.