Making Math: A Hands on History Beth Powell My City School, San Francisco, CA bethciis@yahoo.com
Why Study the History of Math Full of Epic Failures Creates a Sense of Wonder Connections, Integration, and Creativity
Keep in Mind Inquiry and Exploration over Lecture Hands-on and Creative Engagement and Critical Thinking over Memorization
Ideas for Study People, Places, Mathematical Objects, History of a Number Students Can Pick Topic and Design a Lesson History of Math Fair Integrate with Current History Topics
Materials Babylonia: Clay and a Stylus or Reed Egypt: Papyrus and Soap Bars Guatemala: Sticks and Stones & Book Making Materials
Prehistoric Math Unit 1
Bruniquel Cave France 176,000 years old
Lebombo Africa 43,000 Years Old
Ishango Bone Africa 20,000 years old
Sketch of Lebombo Bone Sketch of Ishango Bone
Tally Systems Make Up Your Own!
Notes France What shapes do you notice? What do you think they used to make those shapes? Is this above ground or below ground? What is significant about where this was made? What do you think they did here? Africa What do you think these were made of? Why? What did you notice about the markings? What do you think they were used for? Timeline: What year (approximately) were each of these from?
Prehistoric Math Unit 2
1 2 3 Where South Africa Purpose Calendar? Age- 75,000 Years Old - Oldest Man Made Structure? Where East Bay, CA Purpose Unknown Age Unknown, Native Americans didn t build them? Where Syria to Saudi Arabia, Similar ones in Peru Purpose Ritual? Age Unknown - Prehistoric to 2000 years old 4 5 6 Where Scotland Purpose Decoration? Age - Prehistoric Where Southern Africa Purpose Energy Grid? Age 180,000 Years Old Where - Jordan Purpose Unknown Age Prehistoric?
Sketch of Syrian Wheels Sketch of Jordan Circle
Notes Shapes What shapes do you notice? What do you think they used to make those shapes? Which locations would be easy to figure out the shape in person? Which shapes would be harder to notice from the ground? Purpose Why don t we know what all of these were used for? Why would people need a calendar? Any ideas what else these might be used for?
Prehistoric Math Unit 3
çatalhöyük Mound, Turkey Approximate Date: 7000 BC Lascaux Cave, France Approximate Date: 15,000 BC Chauvet Cave, France Approximate Date: 32,000 BC
Sketch of a Cave Drawing
Notes Drawings What animals do you see? Why do you think people drew these? Which cave has the oldest drawings? Origin of Numbers Which do you think came first paintings or numbers? Why? How would you describe the number of animals you see without using numbers? Without language? Guess which numbers are hard-wired into our brains.
Ancient Math: Babylonia Unit 1
Ancient Math: Civilization What do you think this is? What do you think it is made of? What common shapes do you see? What do you think they did about it?
Ancient Math Unit 1
Circle Time!! Small Circle: Diameter (D): Circumference (C): C divided by D: Medium Circle: Diameter (D): Circumference (C): C divided by D: Big Circle: Diameter (D): Circumference (C): C divided by D:
Find the Average of Your C/D Measurement Small Circle: C divided by D Medium Circle: C divided by D Big Circle: C divided by D Add those together Divide that by 3 You just found an approximate value of...
Approximate Values of Pi add your name to the list! 1900 1680 BC Babylonia 3.125 1650 BC Egypt 3.1605 250 BC Greece 3.1429 to 3.1408 Archimedes 287 212 BC China 3.1415926 and 3.1415927 Zu Chongzhi
Notes Circle Time! What two parts of the circle did you use to find pi? What did you do with those parts to find pi? What is the number that is always the same no matter how big or small a circle is? Pi Which place came up with the first calculation of pi that we know of? Who came up with the best approximation (the closest to the actual number) of pi? How many years did it take to get from the first approximation to the best approximation?
Ancient Math: Babylonia Unit 2
Ancient Math: Farming Once people settled down and starting farming, they had extra food that they needed to store. They built places where everyone stored their food together. Play the food storage game. What kind of problem did people need to solve? What do you think they did about it?
Olive Oil Stored in Jars and Amphora Oldest Known: 5800 BC Wine and Beer Stored in Amphora Oldest Known: 6000 BC Grains: Wheat, Barley, Amaranth Stored in Silos Oldest Known: 9500 BC
Notes Growing Food How did farming create the need for math? What type of information did people need to record? Why was it important to keep track of extra food? What was one kind of storage system people used?
Ancient Math: Babylonia Unit 3
Ancient Math: Clay Tokens What do you think these pictures represent? Give students time to discuss. Picture 1: Clay tokens each token represents a specific crop. For instance, if you stored 1 bushel of wheat, you d get 1 triangular token. We don t know for sure what each token represented. Picture 2: What would happen if you d tried to keep track of all these clay tokens? So people made an envelope. Picture 3: They finally figured out that they could make impressions of the tokens on the outside of the envelope so that they didn t have to break the envelope open to see what was inside. Guess how long it took to figure that out? About two thousand years! And they still put the tokens inside. Picture 4: Eventually, they realized they could just make markings in clay and get rid of the tokens. This eventually led to the first number system. What problem did they solve?
Notes Clay Tokens, Envelopes, and Tablets What were the tokens in Picture 1 used for? What does Picture 2 show? What was the big leap from Picture 2 to Picture 3? What does Picture 4 show?
Ancient Math: Babylonia Unit 4
Ancient Math: Base Sixty Ancient Babylonians were the first to divide the circle into 360 degrees They also gave us our 60 minutes and 60 seconds for time. What incredibly useful tool is made from a circle? They may have been among the first to use the wheel. Circular divisions were an important tool for sea explorations What shape was critical for the development of civilization?
What Numbers Are These?!?
Notes Write the numbers in Cuneiform on the tablets: 4 15 35 64 127 632 First Number System!!! How many symbols are there? Does the position of the symbols matter? What did the Babylonian s use to write math? What type of problems and benefits do you see with this number system?
Ancient Math: Egypt Unit 1
Ancient Math: Egypt What do you know about Egypt? Pyramids, hieroglyphs, papyrus Let s see if you can figure out their numbers! Small group, then large group discussion. Use papyrus to create a project. Discuss what it is if needed. Use soap (instead of stone) to carve a project.
What Numbers Are These?!? 11 10 100
What numbers can you find? Name that number!
Notes What numbers can you write using hieroglyphs? 9: 47: 437: 1,573: Another Number System!!! How many symbols are there? Does the position of the symbols matter? 13,727: What did the Egyptians use to write or record math? 234,567: 1,273,645: What type of problems and benefits do you see with this number system?
Ancient Math: Egypt Unit 2
. Ancient Math: Egypt To find the product of 13 x 11, in one column, start from 1, doubling in each row until there are enough numbers in that column to add up to the number Then double the second number. Give students time to discover what to do from here. Add only the numbers in the left column that are needed and then add up the corresponding right column Example: 8, 4, and 1 add up to 13 so (11 + 44 + 88) is the answer. 13 x 11 1 11 2 22 4 44 8 88 13 X 11=143
One of the better preserved ancient scrolls from Egypt is the Ahmes Papyrus also known as the Rhind papyrus is from the 17 century BC. Its scribe (known only as Ahmes) had copied it from a school text which, he reported, had been a standard for nearly 400 years before his own time. It has tables to help a student with multiplication and division, showing methods that are very different from ours, but fascinating and (dare I say it?) useful even today. http://classicalschool.blogspot.com/search/label/math "One of the many puzzles on the Rhind papyrus: Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats of wheat. What is the total of all of these?" http://www.bitwisemag.com/copy/wilf/wilf3.html
Multiplying Egyptian Style 11 X 15 13 X 15 15 X 15
Multiplying Egyptian Style 13 X 29 23 X 29 29 X 29
Ancient Math: Guatemala Unit 1
. Ancient Math: Guatemala Use sticks, rocks, and cacao beans to create a number system They had a very complex calendar that used another number system. Much of the history of the Maya was lost because a Catholic priest burned their manuscripts thinking it showed demons. See if you can figure out the numbers in the picture? Can you find any numbers? Teacher helps students discover the numbering system. For older students, introduce numbers above 20.
What numbers can you find?
MAYAN BOOKS Just as with modern books, paper was the most common material out of which codices (books) were made. The Maya made paper from the inner bark of fig trees. The large codices were folded like screens, covered with layer of starch, and then with a thin, white, paste. Subjects varied from religion, astronomy, agricultural cycles and history to prophecies. One or more themes occupied each page and in all cases, the contents related to the spiritual world.
Notes LARGE NUMBERS Rewrite using Mayan numbers: 18 21 One More Number System!!! How many symbols are there? Does the position of the symbols matter? What number did the Maya used that the Egyptians and Babylonians didn t? 46 127 What did the Maya use to write or record math? 368 What type of problems and benefits do you see with this number system? 567
Ancient Math: Babylonia Extra Info
YBC 7289 There are a number of remarkable facts about the tablet, which is one of the very oldest mathematical diagrams extant. Given our vast ignorance about the era, speculation is inevitable... We [may be] looking here at the very origins of mathematical reasoning. The Babylonians, unlike the early Greeks much later on, interpreted ratios of lengths as numbers. They weren't just finding a good (very good) approximation to the ratio of a diagonal to a side of a square. They knew that the ratio of the diagonal of a square to a side was a number whose square was 2. They possessed an algorithm for finding approximations to the square root of 2. http://www.math.ubc.ca/people/faculty/cass/euclid/ybc/comments.html
Mathematics Exercise Tablet Geometric Patterns Language: Akkadian Babylonian 1700 BC This large fragment is from a tablet containing mathematics exercises and questions, written in Akkadian. It dates back to around 1700 BC. The text in the lower right corner says: "The side of the square equals one. I have drawn four triangles in it. What is the surface area? Babylonian schools would train young scribes to learn geometry because they were required to draw up accurate deeds and calculate agricultural yields. This Tablet contained the student s geometry lesson, the measure of weight, and the medical tract that offered remedies for a variety of illnesses. www.bible-history.com/babylonia/babyloniamathematics_tablet.htm
"This is one of the first clear examples of multiplication known to man," says Robert K. Englund, co-principal investigator at the Cuneiform Digital Library Initiative at University of California at Los Angeles. Clay tablet from ~3,100 BC showing how Babylonian landowners kept accounts. The face of the tablet is divided into five fields, each referring to a single parcel of agricultural land. Inside each field are symbols giving surface measurements of the parcels. http://www.fas.org/main/pu_content_printable.jsp?formaction=156&contentid=322
Ancient Math: Egypt Extra Info
. Ancient Math: Fractions Story of the Eye of Horus Egyptians only wrote fractions using unit fractions: 1/2, 1/4, 1/8, 1/16, 1/32 What do you notice about these fractions? How can you write 5/8 using fractions?
The Eye of Horus The Eye of Horus represents the combination of a human eye, with the cheek markings of a falcon. It was considered a powerful symbol for imparting protection and life. 2 parts 4 parts 8 parts Can you divide each circle into equal parts?
The Eye of Horus Divide each circle into equal parts. Shade in some of the parts in each circle. Label the fraction you created in each circle.