FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10


 Coleen Summers
 2 years ago
 Views:
Transcription
1 FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies. [ME, PS, V] A2. Apply proportioal reasoig to problems that ivolve coversios betwee SI ad imperial uits of measure. [C, ME, PS] Geeral Outcome: Develop spatial sese ad proportioal reasoig. 1.1 Provide referets for liear measuremets, icludig millimetre, cetimetre, metre, kilometre, ich, foot, yard ad mile, ad explai the choices. 1.2 Compare SI ad imperial uits, usig referets. 1.3 Estimate a liear measure, usig a referet, ad explai the process used. 1.4 Justify the choice of uits used for determiig a measuremet i a problemsolvig cotext. 1.5 Solve problems that ivolve liear measure, usig istrumets such as rulers, calipers or tape measures. 1.6 Describe ad explai a persoal strategy used to determie a liear measuremet; e.g., circumferece of a bottle, legth of a curve, perimeter of the base of a irregular 3D object. 2.1 Explai how proportioal reasoig ca be used to covert a measuremet withi or betwee SI ad imperial systems. 2.2 Solve a problem that ivolves the coversio of uits withi or betwee SI ad imperial systems. 2.3 Verify, usig uit aalysis, a coversio withi or betwee SI ad imperial systems, ad explai the coversio. 2.4 Justify, usig metal mathematics, the reasoableess of a solutio to a coversio problem. WNCP Commo Curriculum Framework for Grades Mathematics Foudatios of Mathematics ad Precalculus (Grade 10) / 47
2 Measuremet (cotiued) A3. Solve problems, usig SI ad imperial uits, that ivolve the surface area ad volume of 3D objects, icludig: right coes right cyliders right prisms right pyramids spheres. [CN, PS, R, V] A4. Develop ad apply the primary trigoometric ratios (sie, cosie, taget) to solve problems that ivolve right triagles. [C, CN, PS, R, T, V] Geeral Outcome: Develop spatial sese ad proportioal reasoig. 3.1 Sketch a diagram to represet a problem that ivolves surface area or volume. 3.2 Determie the surface area of a right coe, right cylider, right prism, right pyramid or sphere, usig a object or its labelled diagram. 3.3 Determie the volume of a right coe, right cylider, right prism, right pyramid or sphere, usig a object or its labelled diagram. 3.4 Determie a ukow dimesio of a right coe, right cylider, right prism, right pyramid or sphere, give the object s surface area or volume ad the remaiig dimesios. 3.5 Solve a problem that ivolves surface area or volume, give a diagram of a composite 3D object. 3.6 Describe the relatioship betwee the volumes of: right coes ad right cyliders with the same base ad height right pyramids ad right prisms with the same base ad height. 4.1 Explai the relatioships betwee similar right triagles ad the defiitios of the primary trigoometric ratios. 4.2 Idetify the hypoteuse of a right triagle ad the opposite ad adjacet sides for a give acute agle i the triagle. 4.3 Solve right triagles, with or without techology. 4.4 Solve a problem that ivolves oe or more right triagles by applyig the primary trigoometric ratios or the Pythagorea theorem. 4.5 Solve a problem that ivolves idirect ad direct measuremet, usig the trigoometric ratios, the Pythagorea theorem ad measuremet istrumets such as a cliometer or metre stick. 48 / Foudatios of Mathematics ad Precalculus (Grade 10) WNCP Commo Curriculum Framework for Grades Mathematics
3 Algebra ad Number B1. Demostrate a uderstadig of factors of whole umbers by determiig the: prime factors greatest commo factor least commo multiple square root cube root. [CN, ME, R] B2. Demostrate a uderstadig of irratioal umbers by: represetig, idetifyig ad simplifyig irratioal umbers orderig irratioal umbers. [CN, ME, R, V] Geeral Outcome: Develop algebraic reasoig ad umber sese. 1.1 Determie the prime factors of a whole umber. 1.2 Explai why the umbers 0 ad 1 have o prime factors. 1.3 Determie, usig a variety of strategies, the greatest commo factor or least commo multiple of a set of whole umbers, ad explai the process. 1.4 Determie, cocretely, whether a give whole umber is a perfect square, a perfect cube or either. 1.5 Determie, usig a variety of strategies, the square root of a perfect square, ad explai the process. 1.6 Determie, usig a variety of strategies, the cube root of a perfect cube, ad explai the process. 1.7 Solve problems that ivolve prime factors, greatest commo factors, least commo multiples, square roots or cube roots. 2.1 Sort a set of umbers ito ratioal ad irratioal umbers. 2.2 Determie a approximate value of a give irratioal umber. 2.3 Approximate the locatios of irratioal umbers o a umber lie, usig a variety of strategies, ad explai the reasoig. 2.4 Order a set of irratioal umbers o a umber lie. 2.5 Express a radical as a mixed radical i simplest form (limited to umerical radicads). 2.6 Express a mixed radical as a etire radical (limited to umerical radicads). 2.7 Explai, usig examples, the meaig of the idex of a radical. 2.8 Represet, usig a graphic orgaizer, the relatioship amog the subsets of the real umbers (atural, whole, iteger, ratioal, irratioal). WNCP Commo Curriculum Framework for Grades Mathematics Foudatios of Mathematics ad Precalculus (Grade 10) / 49
4 Algebra ad Number (cotiued) B3. Demostrate a uderstadig of powers with itegral ad ratioal expoets. [C, CN, PS, R] Geeral Outcome: Develop algebraic reasoig ad umber sese. 1 a 3.1 Explai, usig patters, why a =, a Explai, usig patters, why a = a, > Apply the expoet laws: m m+ ( a )( a ) = a m m a a = a, a 0 m m ( ) a = a m ab = a b m m ( ) a b a b =, b 0 to expressios with ratioal ad variable bases ad itegral ad ratioal expoets, ad explai the reasoig. 3.4 Express powers with ratioal expoets as radicals ad vice versa. 3.5 Solve a problem that ivolves expoet laws or radicals. 3.6 Idetify ad correct errors i a simplificatio of a expressio that ivolves powers. 50 / Foudatios of Mathematics ad Precalculus (Grade 10) WNCP Commo Curriculum Framework for Grades Mathematics
5 Algebra ad Number (cotiued) Geeral Outcome: Develop algebraic reasoig ad umber sese. B4. Demostrate a uderstadig of the multiplicatio of polyomial expressios (limited to moomials, biomials ad triomials), cocretely, pictorially ad symbolically. [CN, R, V] B5. Demostrate a uderstadig of commo factors ad triomial factorig, cocretely, pictorially ad symbolically. [C, CN, R, V] (It is iteded that the emphasis of this outcome be o biomial by biomial multiplicatio, with extesio to polyomial by polyomial to establish a geeral patter for multiplicatio.) 4.1 Model the multiplicatio of two give biomials, cocretely or pictorially, ad record the process symbolically. 4.2 Relate the multiplicatio of two biomial expressios to a area model. 4.3 Explai, usig examples, the relatioship betwee the multiplicatio of biomials ad the multiplicatio of twodigit umbers. 4.4 Verify a polyomial product by substitutig umbers for the variables. 4.5 Multiply two polyomials symbolically, ad combie like terms i the product. 4.6 Geeralize ad explai a strategy for multiplicatio of polyomials. 4.7 Idetify ad explai errors i a solutio for a polyomial multiplicatio. 5.1 Determie the commo factors i the terms of a polyomial, ad express the polyomial i factored form. 5.2 Model the factorig of a triomial, cocretely or pictorially, ad record the process symbolically. 5.3 Factor a polyomial that is a differece of squares, ad explai why it is a special case of triomial factorig whereb = Idetify ad explai errors i a polyomial factorizatio. 5.5 Factor a polyomial, ad verify by multiplyig the factors. 5.6 Explai, usig examples, the relatioship betwee multiplicatio ad factorig of polyomials. 5.7 Geeralize ad explai strategies used to factor a triomial. 5.8 Express a polyomial as a product of its factors. WNCP Commo Curriculum Framework for Grades Mathematics Foudatios of Mathematics ad Precalculus (Grade 10) / 51
6 Relatios ad Fuctios Geeral Outcome: Develop algebraic ad graphical reasoig through the study of relatios. C1. Iterpret ad explai the relatioships amog data, graphs ad situatios. [C, CN, R, T, V] C2. Demostrate a uderstadig of relatios ad fuctios. [C, R, V] C3. Demostrate a uderstadig of slope with respect to: rise ad ru lie segmets ad lies rate of chage parallel lies perpedicular lies. [PS, R, V] 1.1 Graph, with or without techology, a set of data, ad determie the restrictios o the domai ad rage. 1.2 Explai why data poits should or should ot be coected o the graph for a situatio. 1.3 Describe a possible situatio for a give graph. 1.4 Sketch a possible graph for a give situatio. 1.5 Determie, ad express i a variety of ways, the domai ad rage of a graph, a set of ordered pairs or a table of values. 2.1 Explai, usig examples, why some relatios are ot fuctios but all fuctios are relatios. 2.2 Determie if a set of ordered pairs represets a fuctio. 2.3 Sort a set of graphs as fuctios or ofuctios. 2.4 Geeralize ad explai rules for determiig whether graphs ad sets of ordered pairs represet fuctios. 3.1 Determie the slope of a lie segmet by measurig or calculatig the rise ad ru. 3.2 Classify lies i a give set as havig positive or egative slopes. 3.3 Explai the meaig of the slope of a horizotal or vertical lie. 3.4 Explai why the slope of a lie ca be determied by usig ay two poits o that lie. 3.5 Explai, usig examples, slope as a rate of chage. 3.6 Draw a lie, give its slope ad a poit o the lie. 3.7 Determie aother poit o a lie, give the slope ad a poit o the lie. 3.8 Geeralize ad apply a rule for determiig whether two lies are parallel or perpedicular. 3.9 Solve a cotextual problem ivolvig slope. 52 / Foudatios of Mathematics ad Precalculus (Grade 10) WNCP Commo Curriculum Framework for Grades Mathematics
7 Relatios ad Fuctios (cotiued) Geeral Outcome: Develop algebraic ad graphical reasoig through the study of relatios. C4. Describe ad represet liear relatios, usig: words ordered pairs tables of values graphs equatios. [C, CN, R, V] C5. Determie the characteristics of the graphs of liear relatios, icludig the: itercepts slope domai rage. [CN, PS, R, V] 4.1 Idetify idepedet ad depedet variables i a give cotext. 4.2 Determie whether a situatio represets a liear relatio, ad explai why or why ot. 4.3 Determie whether a graph represets a liear relatio, ad explai why or why ot. 4.4 Determie whether a table of values or a set of ordered pairs represets a liear relatio, ad explai why or why ot. 4.5 Draw a graph from a set of ordered pairs withi a give situatio, ad determie whether the relatioship betwee the variables is liear. 4.6 Determie whether a equatio represets a liear relatio, ad explai why or why ot. 4.7 Match correspodig represetatios of liear relatios. 5.1 Determie the itercepts of the graph of a liear relatio, ad state the itercepts as values or ordered pairs. 5.2 Determie the slope of the graph of a liear relatio. 5.3 Determie the domai ad rage of the graph of a liear relatio. 5.4 Sketch a liear relatio that has oe itercept, two itercepts or a ifiite umber of itercepts. 5.5 Idetify the graph that correspods to a give slope ad yitercept. 5.6 Idetify the slope ad yitercept that correspod to a give graph. 5.7 Solve a cotextual problem that ivolves itercepts, slope, domai or rage of a liear relatio. WNCP Commo Curriculum Framework for Grades Mathematics Foudatios of Mathematics ad Precalculus (Grade 10) / 53
8 Relatios ad Fuctios (cotiued) Geeral Outcome: Develop algebraic ad graphical reasoig through the study of relatios. C6. Relate liear relatios expressed i: slope itercept form (y = mx + b) geeral form (Ax + By + C = 0) slope poit form (y y 1 = m(x x 1 )) to their graphs. [CN, R, T, V] C7. Determie the equatio of a liear relatio, give: a graph a poit ad the slope two poits a poit ad the equatio of a parallel or perpedicular lie to solve problems. [CN, PS, R, V] 6.1 Express a liear relatio i differet forms, ad compare the graphs. 6.2 Rewrite a liear relatio i either slope itercept or geeral form. 6.3 Geeralize ad explai strategies for graphig a liear relatio i slope itercept, geeral or slope poit form. 6.4 Graph, with ad without techology, a liear relatio give i slope itercept, geeral or slope poit form, ad explai the strategy used to create the graph. 6.5 Idetify equivalet liear relatios from a set of liear relatios. 6.6 Match a set of liear relatios to their graphs. 7.1 Determie the slope ad yitercept of a give liear relatio from its graph, ad write the equatio i the form y = mx + b. 7.2 Write the equatio of a liear relatio, give its slope ad the coordiates of a poit o the lie, ad explai the reasoig. 7.3 Write the equatio of a liear relatio, give the coordiates of two poits o the lie, ad explai the reasoig. 7.4 Write the equatio of a liear relatio, give the coordiates of a poit o the lie ad the equatio of a parallel or perpedicular lie, ad explai the reasoig. 7.5 Graph liear data geerated from a cotext, ad write the equatio of the resultig lie. 7.6 Solve a problem, usig the equatio of a liear relatio. 54 / Foudatios of Mathematics ad Precalculus (Grade 10) WNCP Commo Curriculum Framework for Grades Mathematics
9 Relatios ad Fuctios (cotiued) Geeral Outcome: Develop algebraic ad graphical reasoig through the study of relatios. C8. Represet a liear fuctio, usig fuctio otatio. [CN, ME, V] C9. Solve problems that ivolve systems of liear equatios i two variables, graphically ad algebraically. [CN, PS, R, T, V] 8.1 Express the equatio of a liear fuctio i two variables, usig fuctio otatio. 8.2 Express a equatio give i fuctio otatio as a liear fuctio i two variables. 8.3 Determie the related rage value, give a domai value for a liear fuctio; e.g., if f(x) = 3x 2, determie f( 1). 8.4 Determie the related domai value, give a rage value for a liear fuctio; e.g., if g(t) = 7 + t, determie t so that g(t) = Sketch the graph of a liear fuctio expressed i fuctio otatio. 9.1 Model a situatio, usig a system of liear equatios. 9.2 Relate a system of liear equatios to the cotext of a problem. 9.3 Determie ad verify the solutio of a system of liear equatios graphically, with ad without techology. 9.4 Explai the meaig of the poit of itersectio of a system of liear equatios. 9.5 Determie ad verify the solutio of a system of liear equatios algebraically. 9.6 Explai, usig examples, why a system of equatios may have o solutio, oe solutio or a ifiite umber of solutios. 9.7 Explai a strategy to solve a system of liear equatios. 9.8 Solve a problem that ivolves a system of liear equatios. WNCP Commo Curriculum Framework for Grades Mathematics Foudatios of Mathematics ad Precalculus (Grade 10) / 55
10 56 / WNCP Commo Curriculum Framework for Grades Mathematics
Grade 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 informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
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 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 informationAlgebra Vocabulary List (Definitions for Middle School Teachers)
Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf
More informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers
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 informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
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 information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
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 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 information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
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 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 informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationMath 114 Intermediate Algebra Integral Exponents & Fractional Exponents (10 )
Math 4 Math 4 Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over Mathematics/P DoE/November
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 informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
More informationMath 105: Review for Final Exam, Part II  SOLUTIONS
Math 5: Review for Fial Exam, Part II  SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x ad ycoordiates of ay ad all local extrema ad classify each as a local maximum or
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationLimits, Continuity and derivatives (Stewart Ch. 2) say: the limit of f(x) equals L
Limits, Cotiuity ad derivatives (Stewart Ch. 2) f(x) = L say: the it of f(x) equals L as x approaches a The values of f(x) ca be as close to L as we like by takig x sufficietly close to a, but x a. If
More informationME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision  A measure of agreement between repeated measurements (repeatability).
INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie
More informationFigure 40.1. Figure 40.2
40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called
More informationMATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12
Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationUnit 2 Sequences and Series
Mathematics IV Uit 1 st Editio Mathematics IV Frameworks Studet Editio Uit Sequeces ad Series 1 st Editio Kathy Cox, State Superitedet of Schools Uit : Page 1 of 35 Mathematics IV Uit 1 st Editio Table
More informationACCESS  MATH July 2003 Notes on Body Mass Index and actual national data
ACCESS  MATH July 2003 Notes o Body Mass Idex ad actual atioal data What is the Body Mass Idex? If you read ewspapers ad magazies it is likely that oce or twice a year you ru across a article about the
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationG r a d e. 5 M a t h e M a t i c s. shape and space
G r a d e 5 M a t h e M a t i c s shape ad space Grade 5: Shape ad Space (Measuremet) (5.SS.1) Edurig Uderstadigs: there is o direct relatioship betwee perimeter ad area. Geeral Outcome: Use direct or
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationSOLUTION & ANSWER FOR KCET2009 VERSION A2
SOLUTION & ANSWER FOR KCET9 VERSION A [MATHEMATICS]. cos ec( a) cos ecd si a [ si( a) cos ec] + C Sol. : si[ ( a) sicos(a) cos si( a) cos ec( a) cos ecd d si a si( a) [ cot( a) cot ] d si a si( a) +
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
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 informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
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 information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More informationNOTES AND FORMULAE SPM MATHEMATICS Cone
FORM 3 NOTES. SOLID GEOMETRY (a) Area ad perimeter Triagle NOTES AND FORMULAE SPM MATHEMATICS Coe V = 3 r h A = base height = bh Trapezium A = (sum of two parallel sides) height = (a + b) h Circle Area
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 informationReview of Fourier Series and Its Applications in Mechanical Engineering Analysis
ME 3 Applied Egieerig Aalysis Chapter 6 Review of Fourier Series ad Its Applicatios i Mechaical Egieerig Aalysis TaiRa Hsu, Professor Departmet of Mechaical ad Aerospace Egieerig Sa Jose State Uiversity
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a page formula sheet. Please tur over Mathematics/P DoE/Exemplar
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationAlternatives To Pearson s and Spearman s Correlation Coefficients
Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives
More informationThe Limit of a Sequence
3 The Limit of a Sequece 3. Defiitio of limit. I Chapter we discussed the limit of sequeces that were mootoe; this restrictio allowed some shortcuts ad gave a quick itroductio to the cocept. But may importat
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 informationFactoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationSUMS OF nth POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.
SUMS OF th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationGrade 7 Mathematics. Support Document for Teachers
Grade 7 Mathematics Support Documet for Teachers G r a d e 7 M a t h e m a t i c s Support Documet for Teachers 2012 Maitoba Educatio Maitoba Educatio Cataloguig i Publicatio Data Grade 7 mathematics
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 informationSENIOR CERTIFICATE EXAMINATIONS
SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over Mathematics/P1 DBE/016 INSTRUCTIONS AND INFORMATION
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
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 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 information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationG r a d e 6 M a t h e m a t i c s. Shape and Space
G r a d e 6 M a t h e m a t i c s Shape ad Space Grade 6: Shape ad Space (Measuremet) (6.SS.1, 6.SS.2) Edurig Uderstadig(s): All measuremets are comparisos. The uit of measure must be of the same ature
More informationM06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES
IB MATHEMATICS STANDARD LEVEL PAPER 2 DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 22067304 Thursday 4 May 2006 (morig) 1 hour 30 miutes INSTRUCTIONS TO CANDIDATES Do ot ope
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
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 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 informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
More informationSection 6.1. x n n! = 1 + x + x2. n=0
Differece Equatios to Differetial Equatios Sectio 6.1 The Expoetial Fuctio At this poit we have see all the major cocepts of calculus: erivatives, itegrals, a power series. For the rest of the book we
More informationRadicals and Fractional Exponents
Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.
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 informationMath 152 Final Exam Review
Math 5 Fial Eam Review Problems Math 5 Fial Eam Review Problems appearig o your iclass fial will be similar to those here but will have umbers ad fuctios chaged. Here is a eample of the way problems selected
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More informationG r a d e. 2 M a t h e M a t i c s. statistics and Probability
G r a d e 2 M a t h e M a t i c s statistics ad Probability Grade 2: Statistics (Data Aalysis) (2.SP.1, 2.SP.2) edurig uderstadigs: data ca be collected ad orgaized i a variety of ways. data ca be used
More informationConcept #1. Goals for Presentation. I m going to be a mathematics teacher: Where did this stuff come from? Why didn t I know this before?
I m goig to be a mathematics teacher: Why did t I kow this before? Steve Williams Associate Professor of Mathematics/ Coordiator of Secodary Mathematics Educatio Lock Have Uiversity of PA swillia@lhup.edu
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
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 informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 009() MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheet. Please tur over Mathematics/P DoE/November
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 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 informationNATIONAL SENIOR CERTIFICATE GRADE 12
NTIONL SENIOR ERTIFITE GRDE MTHEMTIS P EXEMPLR 04 MRKS: 50 TIME: 3 hours This questio paper cosists of pages, 3 diagram sheets ad iformatio sheet. Please tur over Mathematics/P DE/04 NS Grade Eemplar INSTRUTIONS
More informationMath 113 HW #11 Solutions
Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate
More informationFourier Analysis. f () t = + cos[5 t] + cos[10 t] + sin[5 t] + sin[10 t] x10 Pa
Fourier Aalysis I our Mathematics classes, we have bee taught that complicated uctios ca ote be represeted as a log series o terms whose sum closely approximates the actual uctio. aylor series is oe very
More information