m n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.


 Dayna Morgan
 1 years ago
 Views:
Transcription
1 TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like Bses d the Power of Power rule. Studets lso ivestigte the vlue of power whose expoet is zero or egtive. As optiol extesio, studets ivestigte the vlue of power whose expoet is frctio with uertor of oe. Topic: Polyoils Use techology to discover the rules for fors such s, vrious iteger vlues of d d fixed iteger vlue. Use techology to verify for vrious vlues of d tht iteger., d ( ) for = where is Evlute siple uericl expressios rised to itegrl expoets (icludig zero expoets). Use techology to evlute ore coplex uericl expressios ivolvig expoets. Techer Preprtio d Notes This ctivity is desiged to e used i Alger clssroo. It c lso e used i PreAlger clssroo, or y y studet lerig the rules for opertig with expoets. Studets should lredy e filir with sic powers, expoets, d ses, such s 2 3 = 8. While studets c use Scrtchpd t y tie, you y wish to review the positive powers of two efore egiig this ctivity (2, 4, 8, 6, 32, 64...). This ctivity is iteded to e ily studetled, with reks for the techer to itroduce cocepts or rig the clss together for group discussio. Ech studet should hve his or her ow hdheld. Notes for usig the TINspire Nvigtor Syste re icluded throughout the ctivity. The use of the Nvigtor Syste is ot ecessry for copletio of this ctivity. To dowlod the studet TINspire docuet (.ts file) d studet worksheet, go to eductio.ti.co/exchge d eter 988 i the keyword serch ox. Associted Mterils ExpoetRules_Studet.doc ExpoetRules.ts 202 Texs Istruets Icorported Techer Pge Expoet Rules
2 TIth.co Alger Prole Discoverig expoet rules O pge.2, hve studets evlute the expressio i the lower left pe y selectig vrious vlues for d y ipultig the correspodig sliders. (Pressig /+e will llow studets to ove etwee pplictios i split widow.) TINspire Nvigtor Opportuity: Scree Cpture See Note t the ed of this lesso. Hve studets work idepedetly o pge.3. Perfor Scree Cpture to check o studets progress. After copletig these two pges, ecourge studets to discuss their fidigs. Be sure tht studets oserve tht these rules pply oly whe the powers hve like ses. Verify tht studets hve checked the rule for pge.2, = +, d tht it is clled the Product of Powers rule. Propt studets to coclude tht the rule for pge.4, =, is the Quotiet of Powers rule. Note: For coveiece, the vriles d will e used whe expressig the Expoet Rules o the questio pges i this docuet. Becuse of the use of sliders i this ctivity, severl other vriles were used o the explortio pges. 202 Texs Istruets Icorported Pge Expoet Rules
3 TIth.co Alger Repet siilr process for the rule o pge.6, ( ) =, kow s the Power of Power rule. Studets cotiue workig idepedetly through pges.8 to., which llows the to explore egtive d zero powers, respectively. Be sure they drw the correct coclusios, 0 ely tht = d = (for 0 d positive iteger). Note: If studets re hvig difficulty tryig to fid frctio represettios of their decil results o pge.8, they y fid it helpful to chge their clcultor s settigs to llow for this. This c e doe y selectig c > Settigs > Docuet Settigs d chgig Clcultio Mode to Exct. Next, select Mke Defult. O pge.0, it is iportt tht studets 0 oserve 0 is cosidered udefied y the clcultor. TINspire Nvigtor Opportuity: Quick Poll See Note 2 t the ed of this lesso. 202 Texs Istruets Icorported Pge 2 Expoet Rules
4 TIth.co Alger Studets fiish the ctivity y copletig pges.2 through.5. Pge.2 hs studets explore the Power of Product rule: ( ) = wheres o pge.4 they explore the Power of Quotiet rule: 0. =, where Agi, o pge.4, studets y fid it helpful to hve their hdheld s Clcultio Mode set to Exct. Prole 2 Extesio: Rtiol Expoets with Nuertor of Pge 2. sks studets to evlute the five expressios show to ke cojecture for They will see tht x = x. You y eed to expli tht whe = 2, the 2 is usully oitted: 2 =. x. TINspire Nvigtor Opportuity: Quick Poll See Note 3 t the ed of this lesso. 202 Texs Istruets Icorported Pge 3 Expoet Rules
5 TIth.co Alger Note Prole, Scree Cpture This would e good tie to do scree cpture to verify studets re ipultig the sliders correctly d euverig through the docuet t resole pce. Note 2 Prole, Quick Poll You y choose to use Quick Poll to ssess studet uderstdig. You y sk the studets to evlute Note 3 Prole 2, Quick Poll You y choose to use Quick Poll to ssess studet uderstdig. You y sk the studets to evlute 3 25 or write equivlet expressio i rdicl for for 5 x. 202 Texs Istruets Icorported Pge 4 Expoet Rules
Chapter 3 Section 3 Lesson Additional Rules for Exponents
Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters
More informationEXPONENTS AND RADICALS
Expoets d Rdicls MODULE  EXPONENTS AND RADICALS We hve lert bout ultiplictio of two or ore rel ubers i the erlier lesso. You c very esily write the followig, d Thik of the situtio whe is to be ultiplied
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationWe will begin this chapter with a quick refresher of what an exponent is.
.1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for
More informationA black line master of Example 3 You Try is on provided on page 10 for duplication or use with a projection system.
Grde Level/Course: Algebr Lesso/Uit Pl Nme: Geometric Sequeces Rtiole/Lesso Abstrct: Wht mkes sequece geometric? This chrcteristic is ddressed i the defiitio of geometric sequece d will help derive the
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL  INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More informationSect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals:
70 Sect 11.  Simplifyig Rdicl Epressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of epoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( ) 1/ 1/ 1/ Multiplictio
More informationMATH 90 CHAPTER 5 Name:.
MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio. 3 4 3 4 4 ( ) Expoet Properties
More informationA function f whose domain is the set of positive integers is called a sequence. The values
EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is
More informationSummation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
More informationArithmetic Sequences
Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece
More informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationSTUDENT S COMPANIONS IN BASIC MATH: THE SECOND. Basic Identities in Algebra. Let us start with a basic identity in algebra:
STUDENT S COMPANIONS IN BASIC MATH: THE SECOND Bsic Idetities i Algebr Let us strt with bsic idetity i lgebr: 2 b 2 ( b( + b. (1 Ideed, multiplyig out the right hd side, we get 2 +b b b 2. Removig the
More informationSINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355  INTERMEDIATE ALGEBRA I (3 CREDIT HOURS)
SINCLAIR COMMUNITY COLLEGE DAYTON OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355  INTERMEDIATE ALGEBRA I (3 CREDIT HOURS) 1. COURSE DESCRIPTION: Ftorig; opertios with polyoils d rtiol expressios; solvig
More informationA Resource for Freestanding Mathematics Qualifications
A pie chrt shows how somethig is divided ito prts  it is good wy of showig the proportio (or frctio) of the dt tht is i ech ctegory. To drw pie chrt:. Fid the totl umer of items.. Fid how my degrees represet
More informationGaussian Elimination Autar Kaw
Gussi Elimitio Autr Kw After redig this chpter, you should be ble to:. solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect
More informationn Using the formula we get a confidence interval of 80±1.64
9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge
More informationBasic Arithmetic TERMINOLOGY
Bsic Arithmetic TERMINOLOGY Absolute vlue: The distce of umber from zero o the umber lie. Hece it is the mgitude or vlue of umber without the sig Directed umbers: The set of itegers or whole umbers f,,,
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2πperiodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationUNIT FIVE DETERMINANTS
UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationApplication: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
More informationRADICALS AND SOLVING QUADRATIC EQUATIONS
RADICALS AND SOLVING QUADRATIC EQUATIONS Evaluate Roots Overview of Objectives, studets should be able to:. Evaluate roots a. Siplify expressios of the for a b. Siplify expressios of the for a. Evaluate
More informationAlgebra Work Sheets. Contents
The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will
More informationShowing Recursive Sequences Converge
Showig Recursive Sequeces Coverge Itroductio My studets hve sked me bout how to prove tht recursively defied sequece coverges. Hopefully, fter redig these otes, you will be ble to tckle y such problem.
More informationSequences and Series
Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.
More informationTo multiply exponential expressions with the same base, keep the base, add the exponents.
RULES FOR EXPONENTS Epoets re used to write repeted multiplictio of the sme fctor. I the epoetil epressio, the epoet tells us how m times the bse is used s fctor:. Similrl, i the epressio, the epoet tells
More informationGeometric Sequences. Definition: A geometric sequence is a sequence of the form
Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece
More informationIntroductory Explorations of the Fourier Series by
page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ 7764898 tzielis@momouth.edu Copyright
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationMath Bowl 2009 Written Test Solutions. 2 8i
Mth owl 009 Writte Test Solutios i? i i i i i ( i)( i ( i )( i ) ) 8i i i (i ) 9i 8 9i 9 i How my pirs of turl umers ( m, ) stisfy the equtio? m We hve to hve m d d, the m ; d, the 0 m m Tryig these umers,
More informationName: Period GL SSS~ Dates, assignments, and quizzes subject to change without advance notice. Monday Tuesday Block Day Friday
Ne: Period GL UNIT 5: SIMILRITY I c defie, idetify d illustrte te followig ters: Siilr Cross products Scle Fctor Siilr Polygos Siilrity Rtio Idirect esureet Rtio Siilrity Stteet ~ Proportio Geoetric Me
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationG r a d e. 5 M a t h e M a t i c s. Patterns and relations
G r a d e 5 M a t h e M a t i c s Patters ad relatios Grade 5: Patters ad Relatios (Patters) (5.PR.1) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome:
More informationUnderstanding Rational Exponents and Radicals
x Locker LESSON. Uderstadig Ratioal Expoets ad Radicals Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical
More informationSequences, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationParents Guide to helping your child with Higher Maths
Prets Guide to helpig your child with Higher Mths The essece of mthemtics is ot to mke simple thigs complicted, but to mke complicted thigs simple. S. Gudder Arithmetic is beig ble to cout up to twety
More informationhp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases
Numbers i Differet Bases Practice Workig with Numbers i Differet Bases Numbers i differet bases Our umber system (called HiduArabic) is a decimal system (it s also sometimes referred to as deary system)
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationModule 4: Dividing Radical Expressions
Her MTH 9 Secio IV: Rdicl Epressios, Equios, d Fucios Module 4: Dividig Rdicl Epressios Recll he propery of epoes h ses h oi logous propery for rdicls:. We c use his propery o 1 1 1 (usig he propery of
More informationChapter 13 Volumetric analysis (acid base titrations)
Chpter 1 Volumetric lysis (cid se titrtios) Ope the tp d ru out some of the liquid util the tp coectio is full of cid d o ir remis (ir ules would led to iccurte result s they will proly dislodge durig
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More informationHermitian Operators. Eigenvectors of a Hermitian operator. Definition: an operator is said to be Hermitian if it satisfies: A =A
Heriti Opertors Defiitio: opertor is sid to be Heriti if it stisfies: A A Altertively clled self doit I QM we will see tht ll observble properties st be represeted by Heriti opertors Theore: ll eigevles
More informationFourier Series (Lecture 13)
Fourier Series (Lecture 3) ody s Objectives: Studets will be ble to: ) Determie the Fourier Coefficiets for periodic sigl b) Fid the stedystte respose for system forced with geerl periodic forcig Rrely
More informationCHAPTER10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i
More informationHow to set up your GMC Online account
How to set up your GMC Olie accout Mai title Itroductio GMC Olie is a secure part of our website that allows you to maage your registratio with us. Over 100,000 doctors already use GMC Olie. We wat every
More informationChapter 04.05 System of Equations
hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationRoots, Radicals, and Complex Numbers
Chpter 8 Roots, Rils, Comple Numbers Agel, Itermeite Algebr, 7e Lerig Objetives Workig with squre roots Higherorer roots; ris tht oti vribles Simplifig ril epressios Agel, Itermeite Algebr, 7e Squre Roots
More informationSequences and Series Using the TI89 Calculator
RIT Calculator Site Sequeces ad Series Usig the TI89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informations = 1 2 at2 + v 0 t + s 0
Mth A UCB, Sprig A. Ogus Solutios for Problem Set 4.9 # 5 The grph of the velocity fuctio of prticle is show i the figure. Sketch the grph of the positio fuctio. Assume s) =. A sketch is give below. Note
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationSection 3.3: Geometric Sequences and Series
ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationBridging Units: Resource Pocket 4
Bridgig Uits: Resource Pocket 4 Iterative methods for solvig equatios umerically This pocket itroduces the cocepts of usig iterative methods to solve equatios umerically i cases where a algebraic approach
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationDefinition. Definition. 72 Estimating a Population Proportion. Definition. Definition
7 stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationEven and Odd Functions
Eve d Odd Fuctios Beore lookig t urther emples o Fourier series it is useul to distiguish two clsses o uctios or which the Euler Fourier ormuls or the coeiciets c be simpliied. The two clsses re eve d
More informationSolving equations. Pretest. Warmup
Solvig equatios 8 Pretest Warmup We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationSolving Inequalities
Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK12
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationReleased Assessment Questions, 2015 QUESTIONS
Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this
More informationOnestep equations. Vocabulary
Review solvig oestep equatios with itegers, fractios, ad decimals. Oestep equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property
More information17.3 ALGEBRA. Find Unknown Side Lengths. Are You Ready? Lesson Opener Making Connections. Resources. Essential Question
7. ALGEBRA Fid Ukow Side Leths? Essetil Questio How c you fid the ukow leth of side i polyo whe you kow its perimeter? Tes Essetil Kowlede d Skills Geometry d Mesuremet.7.B Determie the perimeter of polyo
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationHow Simple Is Your Rational Expression? Examples
How Siple Is Your Rtionl Epression? Eples. Rtionl epressions re lebric epressions whose nuertor nd denointor 5 re polynoils. The epression,, nd re eples of rtionl y 4 epressions or lebric frctions.. A
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More informationA. Description: A simple queueing system is shown in Fig. 161. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationThe Harmonic Series Diverges Again and Again
The Harmoic Series Diverges Agai ad Agai Steve J. Kifowit Prairie State College Terra A. Stamps Prairie State College The harmoic series, = = 3 4 5, is oe of the most celebrated ifiite series of mathematics.
More informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More informationLaws of Exponents Learning Strategies
Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More information23.3 Sampling Distributions
COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE SIC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through
More informationMultiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More information