How did mathematics begin?
Human beings from our earliest beginnings have searched for ____ to basic problems, building homes, measuring space, keeping track of ____ and counting objects. Over 30,000 years ago, Early Paleolithic people kept track of the passing seasons and the changes of ____ for planting. To represent the passing of time, they carve tally marks on cave walls or slash tallies on bones, __or stone. Each tally stood for ones. But this system was awkward when it came to large amounts. So ____ were eventually created that stood for groups of objects. Sumerian clay stones have been found that date to the fourth millennium BC. A small clay cone was used for one. A clay ball was used for 10. And a large cone stood for 60. Written records from around 3,300 BC showed that Babylonians inscribed amounts on clay tablets with a read. They used a ____ shape for ones, and a v on its side for tens, combining these symbols to write other numbers. For example, Babylonians wrote the ____19 as the ancient Egyptians used objects from their everyday life as symbols - - a rod stood for one, a cattle hobble was 10, a coiled ____ was 100, a lotus flower was a 1000 and so on. The number 19 was a cattle hobble and nine rods.
The Early Romans
The early Romans created a number system that we still see today. Along with other symbols, they used an X for 10 and an i for 1. By the ____ Ages, Romans were putting the I to the right of the X for 11, and to the left for nine. So they wrote 19 as XIX. All these creative number systems show groups of objects as well as individual objects.
Human Counting
Some of the oldest human counting systems rely on ____ and __. So they were based on ones, fives, 10 and 20s. The Zulu word for six means to take the ____ of the right hand meaning that all the fingers on the left ____ had been added up and the other thumb was needed. Other systems evolved from Commerce, the Uruba in Nigeria used cowry shells as currency and developed an amazingly complex number system. It was based on 20s and on the operations of ____, ____ and ____. For example, they thought of 45 as ________ . Knots tied in chords and strings were used for recording amounts by many cultures like the Persians. The Incans used a more refined version called the kipu - - a thick cord held horizontally from which hung knotted string. The kind of ____ the Incans used along with the length and color of the cord represented ones, tens and hundreds. In today’s world, almost every industrial culture uses the numeral 0 through 9 but these symbols weren’t invented until the ____ Century BC in India and it took another ____ years for the idea of zero with place value to be constructed. This big idea dramatically changed the ____ of mathematics. We humans have always shared with one another when early cultures shared their food and water or wanted to divide their land in ways that were fair and equal. ____ gradually emerged as symbols for these fair share situations.
Ancient Egyptians
The ancient Egyptians used unit fractions. Fractions where the numerator is one like 1/2 1/3 and 1/5 and would add and have these fractions. If they wanted to divide three loaves of bread equally among five family members, they’d first divide the first and second loaves into thirds, then they divide the third loaf into fifths, finally they take the remaining 1/3 from the second Loaf and divide that into five pieces. They wrote this as 1/3 1/5 1/15. Today we would represent this sharing with the fraction 3/5. 3/5 of a loaf for each person or three loaves divided by five people.
Numbers System of Fractions
The samarians and early Babylonians invented a number system of fractions based on ____ that we still use 4,000 years later. Our days have 60-minute in hours and 60-second in minutes and our circles Encompass 360°. Chinese societies used an abacus with a system based on tens although it had no __. An early form of decimal fractions came from the abacus. For example, ____ would be ____ out of ____ on an abacus. The Chinese lovingly named the numerator the Son and the denominator the mother. It wasn’t until the 12th century that common fractions with the bar notation that we use today were invented even then these fractions weren’t widely used until the Renaissance period only 500 years ago. Throughout history, every culture around the globe has created inventive ways to calculate to solve a problem say ____.
Doubling and Halving
Early Russian peasants used a system of doubling and halving. When an ____ number had resulted in a fraction, they rounded down. Then they added the ____ associated with the odd __. Ancient Egyptians relied on a doubling procedure until they produced enough groups. Then they added these groups to find the ____. Across Europe and Asia during the Middle Ages, the Abacus was the handheld calculator of its day but only very few people knew how to use it, usually wealthy merchants and money lenders. By simply moving beads that each had place value, an abacus was a highly efficient way to compute. Then the great Arab mathematician Al quisi introduced the Hindu Arabic numerals 0 through 9 into North America and ____ and created new procedures for computation. These algorithms could be written on the ____. Over the centuries learning the algorithms became the Hallmark of an education, as students were taught to compute long Columns of ____, borrow and carry and do long division efficiently and reliably. They could now keep records of these procedures and check results. Today, complex ____ are done with a handheld ____. This means students need the ability to check the reasonableness of the answer and to have a rich repertoire of mental math strategies to do that. Most simpler computations like ____ can be solved mentally using a variety of strategies.
