Week 09 Notes

8/9/2019 Week 09 Notes

1/20


2/20

Itinerary

Announcements

History of Mathematics Jigsaw

Biographies

Timeline

Discussion on Trigonometry

Homework


3/20

Announcements

Grading is up-to-date. If youre missing grades, letus know.

Discussion posts

Broken links - let us know when theyre

updated. Difference between Euclidean & Non-

Euclidean Geometries


4/20

History of

Mathematics Jigsaw

For homework you were assigned to read the remaining sections of the beginning chapter History of Mathematics in a Large Nutshell

Get together with groups of your same color (15 min) Become experts on an assigned chapter. Talk about the main ideas, and prepare a 3-minute

presentation Get together with groups of the same number (25 min)

Give three minute presentations on the big ideasfrom your chapters


5/20


6/20


7/20


8/20

Math Timeline

Brahmagupta (598-670) - Math & Astro & Ujjain, Brahmasphutasiddhanta , operations with zeroBhaskara (~600-~680) - Commentaries on Aryabhatiya , and improved pi approximationsLi Chunfeng - (602-670) - Astronomy & calendar reform

600 A.D. 650 A.D. 700 A.D.


9/20

Math Timeline

ero

Lalla (~720-~790) - Application of math to astronomy, and on the sphere

700 A.D. 750 A.D. 800 A.D.

Al-Khwarizmi (~780-~850) - House of Wisdom scholar, Hisabal-jabr w'al-muqabalacompletion & balancing

~775 -House of Wisdomfounded inBaghdad


10/20

Trigonometry


11/20


12/20

Ladder Against a Wall

Suppose that you had a ladder that

was 10 feet in length.

If the ladder isnt rated forinclinations greater than 65degrees from horizontal, whatsthe tallest height the ladder canreach?

If theres a 3 foot wide ower bedat the bottom of a wall that youcant put your ladder in, whatsthe tallest height ladder canreach?


13/20


14/20


15/20

Trig Discussion

Get together with those around you and discuss thefollowing questions...

What is trigonometry? Where did trigonometry come from? What

problems did it help solve? What is the origin of the word sine and cosine? What are common uses of trigonometry? What are the different ways of visualizing

trigonometry and trigonometric ratios?


16/20

Trig Applet

http://catcode.com/trig/trig01.html
http://catcode.com/trig/trig01.htmlhttp://catcode.com/trig/trig01.html


17/20

Homework

Read the following sketches...

8 - The Cossic Art (Writing Algebra withSymbols) - p. 95-100

9 - Linear Thinking (Solving First DegreeEquations) - p. 101-104

Respond to the discussion forum online.


18/20

Math Through the AgesA Gentle History for Teachers and Others

William P. Berlinghoff Fernando Q. Gouvea

Meanwhile, in India

(Page 24)

After 8 th century math started (not much before) Influence by Babylonian astronomy Math problems inspired by astronomical questions

Decimal number system most important Indian discovery 1-9, and 0 was a place holder Quadratic formula used but done in words Did give proof for their math

Arabic Mathematics

(Page 28)

Focus on positive numbers Huge part of math has Algebra

o Did it all in words (not numbers) Focused on Euclids 5 th postulate Discovered 3 rows of Pascals triangle

Medieval Europe

(Page 32)

2 schools developed out of the cathedralo Tribium focused on grammar and logico Qudribium more advanced and focused on music and astronomy

Trade opened allowing lost of math to spread 15 th /16 th century Maya hand sophisticated structures for problems

o Little trade of math between groups, base 23 and an elaborate calendar


19/20

Algebra Comes of Age

(Page 37)

16

th

/17th

century expressed with words Simple and non complex problems interested in connection between algebra and

geometry Theory of polynomial/roots evaluated Discort of Format found the coordinate plane by connecting geometry and

Algebra Format came up with questions about ratios and algebra

Calculus and Applied Mathematics

(Page 42)

Calc came about because they wanted to understand the universe Cavalier worked on principal of indivisibles

o Helped to get area and volume of shapes that it was once harder to get Euler was an important mathematician of this time

o His textbooks establish style and notation of calculus Hard for them to prove it

Rigor and Professionalism

(Page 47)

Physical problems led to more rigor of problems Mathematics became more professional because they had to learn how to teach Developed MC French schools

o Metric system was created and standardized Cauchy taught calculus and redid the foundations of calculus

o He wanted to do calc right and developed a derivative and integral Computations of algebraic side of calc Moved to find rigor for Algebra Move from only allowing to think of some ways to considering ALL ways A push towards abstraction

o Geometry Euclids Albert Einstein developed new line of thinkingo Theoretical Physics Joseph Forner Fourner series

Renan hypothesis written when something still needs to be proven Linking non-Euclidean with algebra Rigor and abstraction were key to success to progress in math


20/20

Abstraction, Computers, and New Applications

(Page 53)

Golden Age of math Increase in math professionals and specifications

o Formalist systematic matho Institutionalist not all math ideas are well founded

Push towards abstractions Made them look at old problems differently French revolutionized Algebra, but discrepancies over who wrote the book Math then moved more towards application Computer systems allowed larger computations to be done Questions over what we teach Governments looked to mathematics to solve problems

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