Week 11 Notes

8/9/2019 Week 11 Notes

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2/25

Itinerary

Announcements Biographies

Timeline

Sharing out of linear systems activity

Discussion of Readings

Break

Slide Rule Activity

Closing


3/25

Portfolio Check

The next binder check will be in 2 weeks. Portfolioswill be due at the start of class on Week 13(03/13/10)

If you want us to re-check for lost points from lasttime, pick up a pinksheet, circle which parts youwish to be re-checked, and attach it to the gradedsheet you got when your binders were returned toyou. This should be the FIRST thing in your binder.


4/25

Contents of Binder Biography Facts

Apollonius, Eudoxus, Plato, Hypatia, Mandelbrot, Sierpiski, Lobachevsky, al-

Khwrizm

, Diophantus, Gauss, Galois, Grothendieck, Abel, Jacobi, Bernoulli(Jacob), Germain, Bernoulli (Johann), Newton, Leibniz, Cauchy

Activities

Origami Maps Linear Systems Slide Rule Jigsaw

Geometry and Arithmetic Journal Entries Notes from Class (Week 7-12 - Biographies & Timelines you may have done)

03/02/10

SME 430 Solidswith ModularOrigami

Thepurposeof thisactivityis to beable to constructsolidsbyusing modularorigamiand

to describepropertiesof solidswe haveconstructed byrelating to theproperties provided for

PlatonicSolids.

Regularpolyhedronsarenamed asPlatonicsolids.

Task

In thisactivitywewillconstructtwo solidsbyusing modularorigami.

In Japanese, theword orimeans to foldand theword kamimeans paper. So,

origamimeans to fold paper1. Modularorigami, or unitorigami, is a paperfoldingtechniquewhich usesmultiplesheetsofpaper to createa largerand morecomplex structurethan

would bepossibleusing single-pieceorigamitechniques. Each individualsheetofpaper isfolded

into amodule, orunit, and then modulesareassembled into an integrated flatshape orthree-

dimensionalstructure by inserting flapsinto pocketscreated by the folding process. These

insertionscreatetension orfriction thatholdsthe modeltogether2.

Wellusethe sameunitstructurefor both ofthe solidswellconstructtogether. In orderto

havesomefamiliaritywith this techniquewepicked the firstone to be an easierone. Well

constructacube in thefirstpart ofthisactivity.

Allwe need in thisactivityis 6 setsquarepieces ofpaper -forthefirst task and 12squarepiece

ofpaper-for thesecond task.

Descriptionof BasicModule (Unit)

You will need a square piece of paper. Fold and unfold the paper in half Fold and unfold the two sides intowards the center crease.

!"!#$$%&''((()*+,-./,0+12*3+410415$1+)4*/',5617)#$/8!9!#$$%&''15)(,:,%16,.)*+-'(,:,';*638.+


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7/25

Arabic MathematiciansAl-Nasawi (1010-1075)

Khayyam (1048-1122)

Abraham (1070-1130)

Ezra (1092-1167)

Geber (1100-1160)

Gherard (1114-1187)

al-Samawal (1130-1180)

al-Tusi, Sharaf (1135-1213)

Indian MathematiciansSripati (1019-1066)

Brahmadeva (1060-1130)

Hemchandra (1089-1173)

Bhaskara II (1114-1185)

Chineese MathematicianJia Xian (1010-1070)

Shen (1031-1095)

European MathematiciansHermann of R. (1013-1054)

Adelard (1075-1160)

Grosseteste (1168-1253)

Fibonacci (1170-1250)

1000 A.D. 1050 A.D. 1100 A.D. 1150 A.D. 1200


8/25

Linear SystemsActivity

Last week each group completed one problem set.

This week each problem set will share out what theydid while other groups take notes.

Come to the board and write down the work youused to solve a problem in the set


9/25

Babylonian Algebra: Rhetorical Style

Rhetorical

[Given] 32 the sum;

252 the area

[Answer] 18 length, 14 width

One follows this method: Take half of 32 [this gives16]

16x16 = 256256 252 = 4

The square root of 4 is 2.

16 +2 = 18

16 2 = 14

[Check] I have multiplied 18 length by 1 width

18x14 = 252 area

Symbolic

Problem: Length, width. I have multiplied length and width, thus obtaininarea: 252. I have added length and width:32. Required: length and width

[cuneiform clay tablets, King Hammurabi, 1700 BC]


10/25

Greek Geometric Algebra

Greek algebra was geometric.

Formulated by Pythagoreans (540 BC) and Euclid (300 BC)

If a straight line be divided into any two parts, the square on the wholeline is equal to the squares on the two parts, together with twice the

rectangle contained by the parts. [That is, (a+b)2 = a2+ b2 + 2ab


11/25

Discussion(Quadratic Equations)

What is meant by quadratic equations?

What are different methods that have been used tosolve quadratic equations in the past?

What are some of the quadratic equations andsolutions (current method & al-Khwarizmis method)you came up with?

What are the advantages of setting all quadraticequations equal to zero


12/25

Discussion(Cubic Equations)

Describe the culture of mathematics in Europe in the15th and 16th centuries.

Historically, why did mathematicians believe that therewere 14 different ways of describing cubic equations (p.110). What are some of the ways you came up with?

The book mentions al-Khayammi producing a solutionmethod to the cubic equation. Describe the method andits advantages and limitations.

What do we know about the solutions to cubic, quartic,and quintic equations?


13/25


14/25

Slide Rule Activity

Begin working on this with a partner/small group

We will be coming back together at variouspoints during this activity

If you have a laptop, you will find under this weeksfolder on ANGEL a link to a digital slide rule.


15/25

Homework

On ANGEL, read the chapter from Mathematics andthe Imagination

In the Dropbox on ANGEL, submit a journal entry forAlgebra

Consider the following topics: algebra, math

history, and education. Write a couple ofparagraphs that describes your understanding of(and how your understanding has improved in)these three topics and how they intersect witheach other.


16/25

Sketch 10 Quadratic Equations

What is meant by quadratic equations?Dictionary.com equations containing a single variable to the second degree. (Its

general form is ax^2+bx+c=0, where a is not equal to zero.).

Wikipedia a polynomial of degree 2.

Comes from the latin word quadractum which means square.

What are different methods that have been used to solve quadratic equationsin the past?

Solving an equation finding where the graph of the equation passes through the x-

axis. This is also finding the values of x, such that when you plug them back into the

equation, the equation comes out to zero.

Different approaches graphic, algebraic (factoring, quadratic formula, completingthe square, difference of squares), geometric

What are some of the quadratic equations and solutions (current method &al-Khwarizmis method) you came up with?

X^2 + 8X = 20.

X^2+8X+16=20+16 (X+4)^2=36

x

x 8

x

x 4

4


17/25

X+4=6 X=2

X^2+8X=20

X^2+8X-20=0

(X+10)(X-2)=0

X= -10 OR X=2

=b^2-4ac

x=[-bsqrt()]/(2a)

What are the advantages of setting all quadratic equations equal to zero


18/25

Sketch 11 Solving Cubic Equations

Describe the culture of mathematics in Europe in the 15th and 16thcenturies.

Competitive. There were scholarly competitions with the winners getting support

by rich patrons and universities. This caused many mathematicians to be secretabout their work. Betrayals.

Del Ferrro found the solution of the cubic, but secret. On his deathbed, he gave it to

his student, Ferrari. Ferrari challenged Tartaglia to a competition. Tartaglia

discovered the solution to the cubic equation, which happened to be the same

method that Ferraro had used. Cardano, convinced Tartaglia to take him on as a

student, and so Cardano learned of the method, but was told to never reveal it.

Historically, why did mathematicians believe that there were 14 differentways of describing cubic equations (p. 110). What are some of the ways you

came up with?

Still didnt used zero and negative numbers, so they treated ax^3+bx^2=cx+d as

different from ax^3+bx^2+cx=d

The book mentions al-Khayammi producing a solution method to the cubicequation. Describe the method and its advantages and limitations.

Geometric solution. Produces a line segment, but didnt produce a number. This

shows the existence of positive roots.

Cardanos solution method produced negative roots. The culture of mathematics at

this time didnt accept negative roots (false roots).

What do we know about the solutions to cubic, quartic, and quinticequations?

Quartic equations also solutions, but quintic equations do not have solutions. By this

we mean there does not exist one formula that accounts for all quintic equation

roots.


19/25

Slide Rule Applet

https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3
https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3https://angel.msu.edu/section/content/default.asp?WCI=pgDisplay&WCU=CRSCNT&ENTRY_ID=E3C18A740E494704A312DDD1A97C77D3


20/25

Name:____________________________

Slide Rule Activity

A Brief History of the Slide Rule

William Oughtred and others developed the slide rule in the 1600s based on theemerging work on logarithms by John Napier. Before the advent of the pocketcalculator, it was the most commonly used calculation tool in science and engineering.

The use of slide rules continued to grow through the 1950s and 1960s even as digitalcomputing devices were being gradually introduced; but around 1974 the electronicscientific calculator made it largely obsolete and most suppliers exited the business.

- from wikipedia.com (http://en.wikipedia.org/wiki/Slide_rule)

______________________________________________________________________

A Review of Some AlgebraWrite the solution to the following problems as a prime number raised to a power.

1. 24 x 23 = ________

2. 35 x 36 = ________

3. 7100 x 710 = ________

The definition for logarithms is that loga(c)=b means ab = c. Use this fact to solve for theunknown x in the following equations

Write a general rule for multiplying twoexponential numbers that have the same bases:

____ x ____ = _____

4. log2(4) = x x=________

5. log2(8) = x x=________

6. log2(32) = x x=________

7. log10(x) = 7 x=________

8. log10(x) = 4 x=________

9. log10(x) = 3 x=________

Use your answers to illustrate the rule logz(x)+logz(y)=log(x*y).

SME430: History of Mathematics 03/30/10


21/25

Use the the rule logz(x)+logz(y)=log(x*y) to explain the following progression:

log10(4)=0.60206log10(40)=1.60206log10(400)=2.60206

...log10(4*10n)=n.60206

______________________________________________________________________

Using a Slide Rule

The clear plastic slider is called the hairline. This is used for keeping track in more

difficult problems. The 1 and 10 on the moveable scale are called the left and right indices, respectively We keep track of powers of 10 when using the C/D scale. This scale treats 16 the

same as 160 and 1.6

Instructions for multiplying:1. To multiply the numbers aand b, slide the middle piece so one of the indices is overthe number aon the fixed part of the rule.2. Look on the middle piece for your number b. This other number of the problem shouldbe directly over the solution to the problem.3. If the number b is hanging outside the rule, put the other index over the first number a

and try step 2 again. This will give you your solution.

Try multiplying the following problems with the slide rule:a) 2 x 4b) 3 x 2

c) 1.5 x 6d) 2.4 x 3.5


C & D Scales

Index


22/25

Try the following problems on the slide rule: 3 x 5. What is different about this problem,than problems (a)-(d) above?

If we wanted to multiply 35 x 28, how would we do that on a slide rule?

What do users of a slide rule have to keep track of as they re doing multiplicationproblems?

How are the numbers laid out on a slide rule? How does a slide rule maker know where

to put each number on the slide rule?

Explain the connection between multiplying numbers, and adding the distances on theslide rule.



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Week 11 Notes

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