Vocabulary and Theorems: Section 1
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Notes……Unit 2.1 Modeling with Direct and Inverse Variation
Objective: To introduce how scientists use direct and inverse variation to find important patterns in science.
[pic] [pic]Archimedes
From ancient times, people have tried to make sense of the world around them. Nomadic Native Americans made observations about the behavior of the buffalo herds in the central plains of North America. These observations led them to make conjectures about the movement of the herd, which was essential to the tribe’s survival. If they successfully followed the herd, then food was plentiful. If the herd left them behind, then they starved.
Some civilizations began to use numbers to describe relationships. In ancient Egypt, people began writing down numerals to keep count of things. They began to write simple equations to perform calculations.
These numeric observations prompted early scientists to want to learn more. They noticed that certain quantities are related, such as the length of day and the location of certain stars, and began to use mathematics to describe these relationships. These observations were the early stages of what we now call mathematical modeling.
In this unit, we will explore scientists who have used modeling to make discoveries. Archimedes was a Greek scientist who lived in Syracuse, Sicily, during the third century BC. Archimedes wrote many books about geometry, mathematics, and physical science and was a good friend of King Hiero of Syracuse. Archimedes was instrumental in defending his home city against the Roman siege in 212 BC, before his death that same year.
Archimedes is perhaps best known for what is today called Archimedes’ Principle. This idea describes the forces that interact between a fluid, such as water, and an object that is submerged in that fluid. We will explore this farther today.
Other scientists like Robert Hooke (invented Hooke’s Law) and Robert Boyle (invented Boyle’s Law) worked on mathematical and scientific relationships with their work on springs and air pressure. Scientists continue to use mathematical modeling to describe natural events and to make predictions. Astrophysicists use mathematical modeling to chart the paths of stars and solar systems. Aeronautical engineers use mathematical modeling to build better airplanes and space vehicles. Social scientists use mathematical modeling to make predictions about populations and natural resource management.
Throughout this course, you will study how different people use mathematical modeling to make decisions about everyday life. You will also see how mathematics appears in unlikely places and is used by people from a variety of backgrounds. In this unit, you will explore some important ideas in science using mathematical modeling.
2.1 Archimedes and the Crown
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In the third century BC lived a famous mathematician and scientist named Archimedes. Archimedes is famous for many discoveries including an irrigation device, the law of the lever, the formula for the volume of a sphere, and possibly even the odometer. One of his discoveries is the Archimedes Principle, which describes the concept of buoyancy. Vitruvius, a Roman architect, tells a famous story about Archimedes and King Hiero of Syracuse.
King Hiero hired a craftsman to make a crown of gold. The king measured out the exact amount of gold for the craftsman to use. The craftsman later delivered to the king a beautiful crown. The crown weighed the same as the measured amount of gold. Rumors began floating around Syracuse that the crown was not made of pure gold. It was suggested that the craftsman had replaced part of the gold with an equal mass of silver. King Hiero asked Archimedes to prove or disprove the rumors without damaging the crown in any way. (Today this is called nondestructive testing.) As Archimedes began to sit down in a bath pondering the problem, he noticed the water level in the bath rising as he submerged more and more of his body into the bath. Realizing he had found a way to solve the problem he ran down the street, still naked, shouting, “Eureka!” (“I have found it!”) The craftsman admitted he was guilty of stealing part of the gold and replacing it with silver.
1. What do you think Archimedes’ solution might have been?
2. Gold has greater mass than silver. If a gold crown and a silver crown have the same weight, would they have the same volume? Why?
3. How can the volume of an object be determined by placing it in water?
4. Suppose your teacher places 15 pennies in a film canister and 20 pennies in another. Which canister has greater mass?
5. Which canister, the one with 15 pennies or the one with 20 pennies, would displace more water when submerged?
6. Does mass or volume cause displacement?
Assignment: Unit 2.1 Classwork (All problems)
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