Week 10 Notes

8/9/2019 Week 10 Notes

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SME 430: History of MathematicsWeek 10 - Linear Equations


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Itinerary

AnnouncementsBiographies

Timeline

Proof Discussion

Discussion of Reading- Writing Algebra

Break

Discussion of Reading- Solving LinearEquations

Activ ity - LinearSystems

Closing


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Biographies

Alexander

Grothendieck Niels Henrik Abel

Carl Gustav Jaco

Jacobi


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Timeline


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Al-Jawhari (800 - 860)

Mahavira (800 - 870)

Govindasvami (800 - 860)al-Kindi (805 - 873)

Hunayn (808 - 873)

Banu Musa, Ahmad & al-Hasan & Muhammad (810-873)

Al-Mahani (820 - 880)

Prthudakasvami (830 - 890)

Ahmed (835 - 912)

Thabit (836 - 901)

Sankara Narayana (840 - 900)

Abu Kamil (850 - 930)

al-Battani (850 - 929)

Sridhara (870 - 930)

Sinan (880 - 943)

Al-Nayrizi (875 - 940)

Al-Khazin (900 - 971 )

Ibrahim (908 - 946)

al-Uqlidisi (920 - 980)

Aryabhata II (920 -1000)

Abu'l-Wafa (940 - 998)

al-Quhi (940 -1000)

Al-Khujandi (940 -1000)

Vijayanandi (940 -1010)

al-Sijzi (945 -1020)

Yunus (950 -1009)

A.D. 850 A.D. 900 A.D. 950 A.D.


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Proof

In small groups discuss the following questions..What is proof?How is proof used in mathematics? In matheducation? In real life?

How are these the same and how are they

different?Why do we use proof in these differentcontexts?

What is the difference bet ween a deductiveproof and an inductive proof?


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Deduction vs. Induction

Begin with premisesthat are assumed to betrue. Then determinewhat else would haveto be true if thepremises are true.

Suppose for any line kand any point p not on

that line there existsonly one line j such

that j is parallel to kand passes through p...

Begin with some dataand determine whatgeneral conclusionscan be logicallyderi ved from thatdata.

We know that the sumof the first n integersis equal to n*(n+1)/2.Prove this for n+1numbers.


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Geometric vs.Algorithmic

(a+b)2 = a2+2ab+b2

(a+b)2 = (a+b)*(a+b)= (a+b)*a+(a+b)*b= a*(a+b)+b*(a+b)

= a*a+a*b+b*a+b*b= a*a+a*b+a*b+b*b= a*a+2*a*b+b*b= a2+2ab+b2

a

a

b

b b2

a2 ab

ab


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Sketchpad & Proof

Diagonals of a Rhombus


10/31

Technology & Proof4-Color Theorem

1974 - Kenneth Appel & Wolfgang Hakenat the University of Illinois633 fundamental configurations of maps

Each is reducible to a simplerconfiguration

1200 hours of computer timeComputer performed tens of millions ofcalculations

1975 article contained...50 pages of text and diagrams85 pages of almost 2500 additionaldiagrams400 microfiche pages of furtherdiagrams and verificat ions of claims inthe main text.


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Discussion of Reading(Writing Algebra)

What is your own definit ion/description of algebra?

Is there a formal definition of algebra? (If so, what

What are the characteristics of algebra thatdist inguish it from other branches of mathematics?

How is a symbolic style different from a rhetoricalstyle in algebra?

Give an example of each, and state one advantage each approach.


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Break(Back in 10 minutes)


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Discussion of Reading(Solving Linear Equations)What is the false position method?

Give some examples of the method.Consider the problem, a quantit y and its fifthbecomes 24.A third of a quantity is taken from half of a

quantity and becomes 18.Solve these problems with our traditional algorithm

How is the mathematics used in these two solutionthe same? Different? Are there any similaritiesbetween this method other methods we use today.


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Discussion of Reading(Solving Linear Equations)

What are the different approaches we use to solvethese problems today?

Will the false position method method work for allfirst degree algebraic problems? How do you know?


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Activity

Representations of Linear EquationsBreak into groups of three

Complete one problem set

Prepare to explain how you solvedeach problem

Take notes on how other groupssolved their problems


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Homework

Read the following sketches...10 - A Square and Things (QuadraticEquations) p. 105-108

11 - Intrigue in Renaissance Italy

(Solving Cubic Equat ions) p. 109-112Respond to the prompts on thediscussion forum.


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What is proof?Proof (provides/is) evidence that something has to be true (or not true). Provides a

reason that something is true (or not true).

I proved the Pythagorean Theorem means

This means I took a given, and used supporting facts in a step by step (sequential)

manner which can be followed to supports a conclusion, and arrived at a conclusion.

How is proof used in mathematics? In math education? In real life?o How are these the same and how are they different?

Mathematics tend to think its something new, uses theorems and theories

Math Education tend to think its something already done, use manipulatives (not

just words), proof can also mean convincing someone, but not necessarily a formal

proof. Visual signals can be convincing. Understanding a proof can lead to greaterunderstanding of what was proved.

Real life the scope (one instance vs. all instances), trying to get evidence to be

convinved. Visual signals can be convincing.

All of these areas tend to be about convincing someone that something is true.

o Why do we use proof in these different contexts?

What is the difference between a deductive proof and an inductive proof?


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Discussion of Reading (Writing Algebra)

What is your own definition/description of algebra?o Is there a formal definition of algebra? (If so, what?)

Algebra (colloquial) equations and expressions that include symbols and variables

(that represent quantities). Can use algebra to maipulate equations to isolate a

variable. Solving for an unknown. Manipualtion of plynomials that can also

represent graphs.

Algebra (formal) A part of mathematics in which letters and other general symbols,

are used to represent numbers and quantities in formulae and equations.

Algebra problems (formal) regardless of how it is written, it is a questions about

numerical operations and relations in which an unknown quantity must be deduced

from known ones.

What are the characteristics of algebra that distinguish it from otherbranches of mathematics?

Equations are already set equal to something, as opposed to being a variable.

Algebra uses arithmetic to find an answer, but arithmetic doesnt use algebra to find

an answer.

Algebra is a generalization of arithmetic.

Geometry involves shapes/pictures.

Algebra may be used in geometry solutions.

How is a symbolic style different from a rhetorical style in algebra?o Give an example of each, and state one advantage of each approach.

X^2+5x+6=0


19/31

Discussion of Reading (Solving Linear Equations)

What is the false position method?o Give some examples of the method.o Consider the problem, a quantity and its fifth becomes 24.o A third of a quantity is taken from half of a quantity and becomes 18.

Solve these problems with our traditional algorithm.o How is the mathematics used in these two solutions the same?

Different? Are there any similarities between this method other

methods we use today.

What are the different approaches we use to solve these problems today?

Will the false position method method work for all first degree algebraicproblems? How do you know?


20/31

Linear Modeling Activity 3/23/10

SME430: History of Mathematics

Name_____________________________

Problem Set A

Try to solve the problem below first by guessing. Then, solve the problem using three

different methods: using pictures, rhetorically, and symbolically.

1. An automobile mechanic is called to a huge parking lot where severe weather hasdamaged the vehicles. In the parking lot there are only motorcycles (that have 2wheels each) and cars (that have 4 wheels each). If the mechanics supplies can onlyrepair 100 tires, how many vehicles can they repair? (Assume the mechanic fixes allthe tires on each vehicle before moving on to another vehicle)


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Problem Set A (Continued)

Solve the problems below using any method you would like.

2. If the mechanic makes $12 for every car he fixes and $6 for every bike he fixes, what

combination of car and bike repairs will make the mechanic the most money?

3. Each vehicle also has one license-plate holder, which was also damaged in thestorm. The mechanic only has enough supplies to fix 40 license-plate holders. If themechanic is fixing the license-plate holders while fixing each vehicles wheels, howmany vehicles could the mechanic repair?


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Problem Set B

Try to solve the problem below first by guessing. Then, solve the problem using threedifferent methods: using pictures, rhetorically, and symbolically.

1. A toy manufacturer is trying to use up his excess inventory. The manufacturer makesbicycles (that have 2 wheels each) and tricycles (that have 3 wheels each). If themanufacture has 150 wheels to use, how many toys can the manufacturer make?


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Problem Set B (Continued)


2. If the manufacturer earns $9 for every bicycle they make and $14 for every tricycle

they make, what combination of bicycles and tricycles will earn the manufacturer themost money?

3. Each toy also has one handle bar. The manufacturer only has 60 handle bars instock. If the manufacturer is only paying attention to the number of wheels andhandlebars in stock, how many toys could the manufacturer make?


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Problem Set C

Try to solve the problem below first by guessing. Then, solve the problem using threedifferent methods: using pictures, rhetorically, and symbolically.

1. A circus clown is preparing to make balloon animals for their guests. The clownspecializes in making spiders (that have 8 legs each) and jumbo ants (that have 6legs each). If each leg requires its own balloon and the clown has 456 balloons touse, how many balloon animals can the clown make?


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Problem Set C (Continued)


2. If the clown earns $5 for every spider they make and $3 for every jumbo ant they

make, what combination of spider balloon animals and jumbo ant balloon animals willearn the clown the most money?

3. Each balloon animal also contains one specialty balloon that is only used for the bodyof each animal. The clown only has 72 of these specialty balloons. Considering thenumber of regular balloons and specialty balloons that the clown has, how manyballoon animals could the clown make?


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