The Reason for Numbers
Lesson 1 box with 15 sets of boxes (work in groups of two
The following script goes page by page on the slideshow called "The Reason for Numbers." Many of the math concepts are studied in a pre-algebra class. The connection with science helps students to appreciate mathematical reasoning and how it is used in science.
Historically, who required an understanding of numbers? Traders who may have not been able to speak to each other because they came from different areas of the world, needed to communicate through numbers. Common system is needed if you want to trade.
This area of the Arabic Peninsula (present-day Iraq), which was known as Mesopotamia, was the heart of trade. It was an ancient civilization that started at least 5000 BC. Several of the cultures learned to produce more crops and then sell them for other resources. Numbers were important to buy and sell throughout the region.
The Arabian numeral system was developed by ancient Indian mathematicians in which sequence of digits is read as a single number. In ancient times the two regions Arabia and Indian traded together and traditionally thought to have adopted the practice and passed it to other trading areas.
Numbers need to mean something.
Click on the worm. When you add one worm and one worm, can you get a two-headed worm? No, not on this planet. But what would happen if we were on a different universe and that is what happened? Would numbers be different? Yes. So numbers were invented to be useful on our world .
What happens if we live in a world where is it only water. One drop plus one drop is really one big drop. Our math would be different if all we could compare it to was water. Math is used relative to the world we live in. Sir Isaac Newton invented calculus in order to investigate the universe. Even today's mathematicians are exploring different ways of looking at things, especially since we are able to look farther out into the universe.
The Babylonians were a civilization that traded in Mesopotamia and used the base 60 system. They used the system developed by the ancient Sumerians in about 2000 BC, and used it in their trade. Today it is still used in a modified form in measuring time, angles, and geographic coordinates. It was a flexible system with many divisible units. The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, of which 2, 3, and 5 are prime. It was easy to divide into sellable groups.
A dozen is from the base 60 system, as well as 60 seconds, 60 minutes, and 24 hours.
Today in many stores eggs are sold in containers that reflect a similar size which are sorted by the egg manufacturer. We can easily see if there are any broken eggs. In ancient times eggs were sold in containers that sometimes you could not see if there were broken or malformed eggs. A seller has an advantage because they could hide bad eggs.
Are there any other examples today where the manufacturer tries to make us think we are buying more than what we actually get? Let's do an experiment.
Look at the different packages, which one do you think holds the most. Use a hand count to determine which one the students think will hold the most. Use the caliper to measure the width or diameter of each of the items. Show them how to use the caliper.
Then give them a package of red "candy." The red chips could represent any product. Remind students that the lid must be on. Have the students count how many fit in each. Make sure the students shape the box a little to get as many as they can. Discuss if the students predicted the correct one. Most will think the cup, but because it has a top that cuts into the amount, it loses!
After everything is put away, focus students back to whether numbers were invented or discovered. So let us discuss more "rules."
Before we can discuss if numbers were invented or discovered, we need to discuss some definitions.
number refers to an actual item, so the picture shows three flowers.
A cardinal number refers to an exact amount of natural items, for example there is 8 saber tooth cats.
An integer is any whole number, whether positive or negative and can be shown on a number line.
An ordinate number refers to the ordering of items like first, second and third.
A ratio is when you compare one number to another number. We can symbolically use a fraction. The numerator is on the number on the top and the denominator is the number on the bottom.
A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. In other words, a proportion is a way to compare numbers that will reduce to the same comparison.
Pythagoras (582-507 BC) was a philosopher that studied properties of numbers similar to number theory of today. He spent most of his life in the Mediterranean area in Europe including Greece, Egypt, and Italy. He was fascinated by mathematics developed from Mesopotamians and felt that numbers held the answers to many questions. He looked at even and odd numbers, triangular numbers, perfect numbers, and tried to make understand of them.
Pythagoras and his followings would look into whether numbers has masculine or feminine traits including whether they were beautiful or ugly. For example, ten was a beautiful number because if you add the first 4 digits it equals 10 [1 + 2 + 3 + 4 = 10]. Pythagoreans observed vibrating strings (in musical instrumentals) produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and these ratios were applied to other instruments. The Pythagoreans studied different subjects from astronomy and music, always looking for relationships. Their conclusion was that the world should be reduced to numbers and are related to each other.
The Pythagoreans helped to develop formulas after looking at data collected from their observations. Their most famous observation is the Pythagorean Theorem. This theorem states that for a right angle triangle the square of the hypotenuse is equal to the sum of the squares on the other sides (a2 + b2 = c2). Right angles are important in our environment, especially when constructing. Have students look at the room and see how many right angles there are.
Make two right angle triangles. May have refresh them memory of what is a right angle (90 degrees). (answers: red, blue, purple and orange, yellow green
The Pythagorean Theorem is a good example of the problem of inventing a number system. Sometimes the formula works great and provides a whole number, but sometimes it produces a number that has not end. For example if "a" and "b" are both 1, then c = the square root of 2, which is 1.41421356237..... with no end. To many people that makes no sense, according to the math you cannot measure "c" but we know we can. There is problems with the "invention" of numbers. We call that "irrational."
Numbers are grouped into irrational and rational. Rational numbers can be expressed as a fraction or ratio. Irrational numbers continue to infinity and cannot be expressed as a fraction. We will investigate some very famous irrational numbers in a later chapter.
Your students may wonder why we have irrational numbers. There are some problems with the invention of numbers, but it allows the system of our numbers to work.
Slide 10. Other famous irrational numbers.
The circle goes back before recorded history with the discovery of the wheel. Circles are closed curves that are equidistant from a fixed point called the center. The circumference (c) is the outside perimeter of a circle. If you draw a line through the center it is called the diameter (d). The center to the edge it is called the radius (r) or half the diameter (1/2d). A circle is a highly symmetrical shape
If you divide the circumference of any circle by its diameter it yields a number that never ends or repeats. This number is referred to as pi (B). The circumference to diameter ratio (c/d) is a constant, no matter how large or small the circle. This ratio produces is the number pi (22/7 or 3.1419…). As early as the 2000 BC Babylonian mathematicians used the ratio 25/8.
The Pythagoreans were fascinated with the number 5, and saw the pentagram as a mathematical perfection. A pentagram is the shape of a 5-pointed star which in many ancient civilizations is associated with magical properties. The pentagram is related to a regular pentagon.
The Pythagoreans studied the Golden Ratio often referred to as phi M (and sometimes tau J). Euclid also makes mention of the Golden Ratio in the developing a ratios of “extreme and mean ratio.” This term was used to describe the Golden Ratio from 3 BC to the 18th Century.
What it is the Golden Ratio? If you connect the vertices of the pentagon by straight lines you obtain a pentagram. The lines form smaller pentagons at the center which can continue to infinity, making the pentagons and pentagrams smaller and smaller. The property of all these figures was discovered, that no matter what size you start with, if you create a ratio of a to b in the figure, and b to c, and c to d, and so on you always get phi. Phi turns out to be an irrational number of 1.6180339887……., a never ending number.
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. This grid shows the placement of the prime numbers. Notice that there seems to be a pattern of prime numbers. Mathematicians always look for patterns as well as scientists. When we can see the relationship of one thing to another, it can help make connections. Patterning can also help students learn how to memorize sequences.