Making Math: A Hands on History Beth Powell
My City School, San Francisco, CA [email protected] 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 Where – East Bay, CA Where – Syria to Saudi Arabia, Similar ones in Peru Purpose – Calendar? Purpose – Unknown Purpose – Ritual? Age- 75,000 Years Old - Oldest Man Made Structure? Age – Unknown, Native Americans didn’t build them? Age – Unknown - Prehistoric to 2000 years old
6 4 5 Where – Scotland Where – Southern Africa Where - Jordan Purpose – Decoration? Purpose – Energy Grid? Purpose – Unknown Age - Prehistoric Age – 180,000 Years Old 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 Small Circle:
Circle Time!! 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 Wine and Beer Grains: Wheat, Barley, Amaranth Stored in Jars and Amphora Stored in Amphora Stored in Silos Oldest Known: 5800 BC Oldest Known: 6000 BC 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. 100
10
1 1 What Numbers Are These?!? Are Numbers What What numbers can you find? Name that number! Notes What numbers can you write using hieroglyphs? 9:
47:
437: Another Number System!!!
• How many symbols are there? Does the position of the symbols matter? 1,573:
13,727: • What did the Egyptians use to write or record math?
234,567:
• What type of problems and benefits do you see with this number system? 1,273,645: 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 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 21 didn’t?
46
• What did the Maya use to write or record math? 127
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.