review jigsaw

8/9/2019 Review Jigsaw

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o Created consistenttheoremsy Natural uses forPythagorean Theoremhelped cultures independently discovertheoremy Euclid and Pythagoreans worked on discovering theplatonic solids

Big Names:

y Euclidy Pythagorasy Euler

Relation to Other Topics:

y Relates to Algebrathrough coordinate geometry

NON-EUCLIDEAN GEOMETRY

Therearethreebrands of geometry: Euclidean (exactly 180 degrees), Non-Euclidean

(Lobachevskian)(less than 180 degrees), and Riemannan geometry (greaterthan 180 degrees)

Euclidean geometry is on the surface ofaplaneand non-Euclidean is not on the surface ofa

plane.

Def: geometrybased upon one ormorepostulates that differ fromthose of Euclid, esp. fromthe

postulatethat only one linemaybe drawn througha givenpointparallel to a given line.

Lines thatare less than or greaterthan 180 degrees.

Big Names:

y Riemanny Euclid- Euclids Postulatey Lobachevskyy Gauss

Big Ideas:

y The geometries are distinguished byhow theytreatparallel lines.Vocabulary:

y Parallel Linesy Euclids Parallel Postulate


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y Playfairs form oftheParallel Postulate- Throughapoint not on a linethere is exactlyone lineparallel to the given line.

y GreatCirclesHistorical Development:

y Bythemid 19thCentury, there werethree distinguished strands of geometry.y Gauss (1810) Explored can therebea system ofplane geometry in whichthroughapoint

not on the linethere is morethan one lineparallel to the givenpoint.

o (1829) This investigation was published by Lobachevsky.o This came fromtrying to provethattheparallelpostulate was atheoremand nota

postulate.

Class Activities:

y The world/globeactivity- west wing videoRelationship:

y Euclidean Geometry was used to relateto Non-Euclidean Geometry.

ALGEBRA: LINEAR AND QUADRATIC

Big Ideas and Themes:

y Functions in the firstand second powersy Linear importantbecause whathappens naturally in the real worldy Quadratic is importantbecause it is the firstthing that considers bothpositiveand

negative

y Uses arithmetic-algebra is a generalization ofarithmeticCommon Vocabulary:

y Symbolicy Rhetoricy Falseposition methody Quadratic equation

Historical Development

y Rhetoric to symbolic style


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y Withthe invention of zero-betterableto solvequadratic equations, recognized as easierto set itequal to zero

Class Activities

yLinearmodeling activity

Big Names

y Pythagorasy Al-Khwarizmiy Harrioty Descartes

Relates to other strands

y Led to cubic and quartic equationsy Imaginary numbersy Basis for other strands ofmathematics-Calculus, Geometry

ALGEBRA- CUBIC AND QUARTIC AND TRIGONOMETRY

Big ideas

y Algebray Formulato solve cubic and quartic equations.y 14 different kinds of cubic equations.

Common Vocabulary

y Cubicy Quarticy Siney Cosy Tangent

TheHistorical Development

y It was competitive, caused mathematicians to be secretive.y Still didnt use zero or negative numbers

Class activities


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y Seeing someoneelses eyes in themirror:trigy Ladderagainstthe wall activity

Big Names in Your Strand

y Algebray Al-Khayammiproduced a solution method to the cubic equationy Rafael Bombelli showed that it ispossibleto work with square roots of negative

numbers and still get reasonableanswers.

How This Strand Relates To Other Strands

y Historicallyand generally

STATISTICS

Big Ideas

y Logic and chancey Statistics andprobability developed togetheras two closely related fields

Common vocabulary for statistics

y Theory ofprobabilityy Standard deviationy Logicy Datay Margin oferror

Historical Development of statistics:

y Finding outprobability in gambling gamesy Statistics comes from state: it was it was coined in the 18th centuryto mean the scientific

study ofthe stateand quickly shifted to focus on political and demographic data of

interestto the government

Class activities for statistics

y Lets Makea dealy Buffon Needle Activitya


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Big Names in statistics

y Bernoulli-Sharpened Cardanos Ideas ofthe law of large numbersy Cardano- Law of LargeNumbersy Pascal and Fermat-Theory ofProbabilityy De Morgan- Truthtables

Statistics related to the other strands, historically or generally?

y One ofthe later developed strands ofmathematics.y Statistics utilizes graphing

ARITHMETIC

Big Ideas

y NumberSystemsy Algorithms ofarithmetic

o Egyptian Multiplicationo Naplers Boneso Roman Multiplicationo Russian Multiplication

y Base systemsy

Positional structurey Zeroy Fractionsy Types ofNumbersy NegativeNumbersy ImaginaryNumbers

Vocabulary

y Types ofNumbero Naturalo Wholeo Integerso Rationalo Realo Complexo Irrational Imaginary

y Basevaluey Negative


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Historical Development

y Fibonacci broughtthe number systemto the westy Groups fromaround the word madehugeadvances in mathematics.

Class Activities

y Differentmethods ofarithmetic algorithmsy Hops on the number line

Big Names

y Brahmaguptaand Francis Maseres impacted knowledgeabout zeroy Fibonacci broughtthe number systemto the west

How does it relate to other strands

y Recognize zero as a numbermadeand madethe field ofalgebrapossible which isintegrated within all other fields

y You need itto do most other fields in math. It is thebuilding blocks.

CALCULUS

Big Ideas:

y Areas underthe curvey Approaching Infinity with limitsy Integration bybreaking up into parts

Vocabulary:

y Differentiationy Integraly Derivativey Limitsy Slopey Riemann Sumy Method of Exhaustiony Maximumand Minimumy Convergenceand Divergencey Continuous and non-continuousy Method of Equilibrium


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Historical Development:

y Astronomy withexploring planet orbitsy Motion and Gravity FromNewton

Class Activities:

y Toothpickactivityy Activity similarto theCalculus Packet in class

Big Names:

y Newton- wrotePrincipiay Liebniz- developed theRules of Calculusy Cauchy-theFundamental theorem of Calculusy

Johann and JacobBernoulli- developedF

ou

ndation of Calcu

lus of Variations

Relates to other Strands:

y Historically, analytical geometrypaved the way for calculus.y Generally, geometry (areaand volume), algebra (slopeand changing rates), and trig (sine

and cosine)

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