3.7"Perfecting"My"Quads" A"Practice"Understanding"Task"
|
|
|
- Shanon Summers
- 5 years ago
- Views:
Transcription
1 42 3.7PerfectingMyQuads APracticeUnderstandingTask CarlosandClarita,TiaandTehani,andtheirbest friendzacarealldiscussingtheirfavoritemethods forsolvingquadraticequationsoftheform ax 2 + bx + c = 0.Eachstudentthinksaboutthe relatedquadraticfunction y = ax 2 + bx + c aspart 2013www.flickr.com/photos/soldiersmediacenter ofhisorherstrategy. Carlos: Iliketomakeatableofvaluesforxandfindthesolutionsbyinspectingthetable. Clarita: Iliketowritetheequationinfactoredform,andthenusethefactorstofindthe solutions. Tia: IliketotreatitlikeaquadraticfunctionthatIamtryingtoputinvertexformby completingthesquare.icanthenuseasquareroottoundothesquaredexpression. Tehani: Ialsoliketotreatitlikeaquadraticfunction,butIusethequadraticformulato findthesolutions. Zac: Iliketographtherelatedquadraticfunctionandusemygraphtofindthesolutions. Demonstratehoweachstudentmightsolveeachofthefollowingquadraticequations. Solve: Carlos Strategy Zac sstrategy x 2 2x 15 = 0 Clarita sstrategy Tia sstrategy Tehani sstrategy 2013MathematicsVisionProject M VP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
2 43 Solve: 2x 2 + 5x 12 = 0 Clarita sstrategy Solve: x 2 + 4x 8 = 0 Clarita sstrategy Carlos Strategy Tia sstrategy Carlos Strategy Tia sstrategy Zac sstrategy Tehani sstrategy Zac sstrategy Tehani sstrategy 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
3 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Solve: 8x 2 + 2x = 3 Carlos Strategy Zac sstrategy Clarita sstrategy Tia sstrategy Tehani sstrategy Describewhyeachstrategyworks. Asthestudentscontinuetotryouttheirstrategies,theynoticethatsometimesonestrategyworks betterthananother.explainhowyouwoulddecidewhentouseeachstrategy. 44
4 45 Hereisanextrachallenge.Howmighteachstudentsolvethefollowingsystemofequations? Solvethesystem: y 1 = x 2 4x +1 y 2 = x 3 Carlos Strategy Zac sstrategy Clarita sstrategy Tia sstrategy Tehani sstrategy 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
5 46 SolvingQuadraticandOtherEquations 3.7 Name: Ready,'Set,'Go' 2013www.flickr.com/photos/soldiersmediacenter Ready' Topic:SymmetryandDistance Thegivenfunctionsprovidetheconnectionbetweenpossibleareas,Ax),thatcanbecreatedbya rectangleforagivensidelength,x,andasetamountofperimeter.youcouldthinkofitasthe differentamountsofareayoucancloseinwithagivenamountoffencingaslongasyoualways createarectangularenclosure. 1. = 10 ) 2. = 50 ) Findthefollowing: Findthefollowing: a.a3) =b.a4) = a.a 10 =b.a20) = c.a6) =d.) = 0 c.a30) =d.ax) = 0 e.whenisax)atitsmaximum?explainor e.whenisax)atitsmaximum?explainor showhowyouknow. showhowyouknow. 3.) = 75 ) 4.) = 48 ) Findthefollowing: Findthefollowing: a.20) =b.35) = a.a10) =b.a20) = c.40) =d.) = 0 c.a28) =d.ax) = 0 e.whenisax)atitsmaximum?explainor e.whenisax)atitsmaximum?explainor showhowyouknow. showhowyouknow. 2013MATHEMATICSVISIONPROJECT M V P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
6 SolvingQuadraticandOtherEquations 3.7 Set' Topic:SolvingQuadraticEquationsEfficiently For'each'of'the'given'quadratic'equations'find'the'solutions'using'an'efficient'method.''State' the'method'you'are'using'as'well'as'the'solutions.'you'must'use'at'least'three'different' methods.' = = = = = = 7 Summarize'the'process'for'solving'a'quadratic'by'the'indicated'strategy.''Give'examples' along'with'written'explanation,'also'indicate'when'it'is'best'to'use'this'strategy.' 11.CompletingtheSquare 12.Factoring 13.QuadraticFormula Go' Topic:Graphingquadraticsandfindingessentialfeaturesofthegraph.Solvingsystemsof equations. Graph'the'quadratic'function'and'supply'the'desired'information'about'the'graph.' 14.) = a.lineofsymmetry: b.x4intercepts: c.y4intercept: d.vertex: 2013MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
7 SolvingQuadraticandOtherEquations ) = 4 1 a.lineofsymmetry: b.x4intercepts: c.y4intercept: d.vertex: Solve'each'system'of'equations'using'an'algebraic'method'and'check'your'work' = = 6 = = = = 10 = 24 = MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
3.10"iNumbers" A"Practice"Understanding"Task"
62 3.10iNumbers APracticeUnderstandingTask Inordertofindsolutionstoallquadraticequations,wehavehadto extendthenumbersystemtoincludecomplexnumbers. 2013www.flickr.com/photos/milongadas Dothefollowingforeachoftheproblemsbelow:
x(t)*=*2t"+*1* y(t)*=*3t* *4**
47 7.8HParametrically8DefinedCurves A"Solidify"and"Practice"Understanding"Task" " Aparametriccurveisdefinedbyasetofequationsthatgivethe coordinatesofpointsonthecurveintermsofanothervariablecalleda parameter.
4.7$Graphing$Rational$Functions$$ A"Practice!Understanding+Task!
35 4.7$Graphing$Rational$Functions$$ A"PracticeUnderstanding+Task PartI:SeeingStructure Foreachfunction,determineintercepts,domain,asymptotes,andcompleteasignline.Usethis informationtosketchthegraph. 1.
7.2"Circle"Dilations" A"Develop"Understanding"Task"
9 7.2CircleDilations ADevelopUnderstandingTask Thestatement allcirclesaresimilar mayseemintuitivelyobvious,sinceall circleshavethesameshapeeventhoughtheymaybedifferentsizes. However,wecanlearnalotaboutthepropertiesofcirclesbyworkingonthe
3.2"Half"Interested" A"Solidify"Understanding"Task"
3.2HalfInterested ASolidifyUnderstandingTask CarlosandClarita,theMartineztwins,haveruna summerbusinesseveryyearforthepastfiveyears. Theirfirstbusiness,aneighborhoodlemonadestand, earnedasmallprofitthattheirfatherinsistedthey
3.8"To"Be"Determined"."."." A"Develop"Understanding"Task"
49 3.8ToBeDetermined... ADevelopUnderstandingTask IsraelandMiriamareworkingtogetheronahomework assignment.theyneedtowritetheequationsofquadratic functionsfromtheinformationgiveninatableoragraph.at first,thisworkseemedreallyeasy.however,astheycontinued
4.5$Watch$Your$ Behavior $ A"Develop!Understanding+Task!
23 4.5$Watch$Your$ Behavior $ A"DevelopUnderstanding+Task Inthistask,youwilldevelopyourunderstandingoftheendbehaviorofrationalfunctionsaswellas discoverthebehaviorofevenandoddfunctions. PartI:Endbehaviorofrationalfunctions
Advanced Mathematics II
1 Advanced Mathematics II Module 7 Circles a Geometric Perspective By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
5.3 It s All In Your Head A Solidify Understanding Task
16 5.3 It s All In Your Head A Solidify Understanding Task In the previous task you were asked to justify some claims by writing paragraphs explaining how various figures were constructed and how those
7.1*Function*Family*Reunion* A"Solidify"Understanding"Task"
2 7.1FunctionFamilyReunion A"Solidify"Understanding"Task" Duringthepastfewyearsofmathclassesyouhavestudieda varietyoffunctions:linear,exponential,quadratic, polynomial,rational,radical,absolutevalue,logarithmicand
1.5 Look Out Below! A Solidify Understanding Task
22 1.5 Look Out Below A Solidify Understanding Task What happens when you drop a ball? It falls to the ground. That question sounds as silly as Why did the chicken cross the road? (To get to the other
7.4*Composing*and*Decomposing* A"Develop"Understanding"Task"
23 7.4ComposingandDecomposing A"Develop"Understanding"Task" Asthedaygetwarmer,youandyourfriendsdecidetocooloffby takingarideontheturbulent"waters"dive.asyouarewaitingin lineyourtourguideexplainsthemathematicsbehinddesigning
1.6 The Tortoise and the Hare A Solidify Understanding Task
29 1.6 The Tortoise and the Hare A Solidify Understanding Task In the children s story of the tortoise and the hare, the hare mocks the tortoise for being slow. The tortoise replies, Slow and steady wins
8.2 Getting Centered A Solidify Understanding Task
8.2 Getting Centered A Solidify Understanding Task Malik s family has decided to put in a new sprinkling system in their yard. Malik has volunteered to lay the system out. Sprinklers are available at the
Quadratic)Functions) 1.6)
Name: Ready,Set,Go Quadratic)Functions) 1.6) Ready 2013www.flickr.com/photos/88394234@N04/8139271342 Topic:Recognizingfunctions Identifywhichofthefollowingrepresentationsarefunctions.IfitisNOTafunctionstate
3.9"My"Irrational"and" Imaginary"Friends" A"Solidify"Understanding"Task"
55 3.9MyIrrationaland ImaginaryFriends ASolidifyUnderstandingTask Part1:Irrationalnumbers Findtheperimeterofeachofthefollowingfigures. Expressyouranswerassimplyaspossible. 2013www.flickr.com/photos/lel4nd
Secondary Mathematics III: An Integrated Approach Module 5 Modeling with Geometry
1 : An Integrated Approach Module 5 Modeling with Geometry By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics
1 Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =
Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals
Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis
WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations
SECONDARY MATH TWO An Integrated Approach WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2017 Original
3.6 Curbside Rivalry. A Solidify Understanding Task
Carlos and Clarita have a brilliant idea for how they will earn money this summer. Since the community in which they live includes many high schools, a couple of universities, and even some professional
Up a Little, Down a Little
Up a Little, Down a Little A Solidif Understanding Task One of the most common applications of eponential growth is compound interest. For eample, Mama Bigbucks puts $20,000 in a bank savings account that
Math%2%Final%Exam% %Skills%Review% %Answer%Key% Explicit!Rule!!! = 2! + 9! 3,! = 0!! 1 + 3,! > 0
% 1) a.recursiverule = b.recursiverule = ) a.explicitrule = b.recursiverule = Math : Algebra, Geometry and Statistics Math%%Final%Exam% %Skills%Review% %Answer%Key% 9, = 0 1 +, > 0 ExplicitRule = + 9 3,
( a ) m "they"are" If"n"is"a"positive"integer"greater"than"1"and"both"a"and"b"are"positive"real"numbers"then,"
22 3.4RadicalIdeas APracticeUderstadigTask NowthatTiaadTehaikowthat a m = a ) m theyare woderigwhichform,radicalformorexpoetialform,is besttousewheworkigwithumericaladalgebraic expressios. 2013www.flickr.com/photos/zjootsuite
Secondary One Mathematics: An Integrated Approach Module 3 Linear and Exponential Functions
1 Secondary One Mathematics: An Integrated Approach Module 3 Linear and Exponential Functions By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
6.10!Finding&the&Value&of&a&Relationship! A"Solidify"Understanding"Task!
6.10Finding&the&Value&of&a&Relationship A"Solidify"Understanding"Task Part%I:%What s%your%angle?% AndreaandBonitaaregoingforawalkstraightupthesideofahill.Andreadecidedto stretchbeforeheadingupthehillwhilebonitathoughtthiswouldbeagoodtimetogeta
5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
3.4 Pascal s Pride. A Solidify Understanding Task
3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very
Lesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
E k E k 1 E 2 E 1 A = B
Theorem.5. suggests that reducing a matrix A to (reduced) row echelon form is tha same as multiplying A from left by the appropriate elementary matrices. Hence if B is a matrix obtained from a matrix A
4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe ECE133A (Winter 2018) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
Solutions to Chapter Review Questions, Chapter 0
Instructor s Solutions Manual, Chapter 0 Review Question 1 Solutions to Chapter Review Questions, Chapter 0 1. Explain how the points on the real line correspond to the set of real numbers. solution Start
a t =Bx a v s 4 c. at +2ax Exam 1--PHYS 101--F15 Name: Class: Date:
Name: Class: Date: Exam --PHYS 0--F5 Multiple Choice Identify the choice that best completes the statement or answers the question.. Which of these best approximates the volume of a person s head? a. 8
Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring
Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis
When they compared their results, they had an interesting discussion:
27 2.5 Making My Point A Solidify Understanding Task Zac and Sione were working on predicting the number of quilt blocks in this pattern: CC BY Camille King https://flic.kr/p/hrfp When they compared their
Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal
Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make
Review : Powers of a matrix
Review : Powers of a matrix Given a square matrix A and a positive integer k, we define A k = AA A } {{ } k times Note that the multiplications AA, AAA,... make sense. Example. Suppose A=. Then A 0 2 =
The greatest common factor, or GCF, is the largest factor that two or more terms share.
Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of
Mar 10, Calculus with Algebra and Trigonometry II Lecture 14Undoing the Marproduct 10, 2015 rule: integration 1 / 18
Calculus with Algebra and Trigonometry II Lecture 14 Undoing the product rule: integration by parts Mar 10, 2015 Calculus with Algebra and Trigonometry II Lecture 14Undoing the Marproduct 10, 2015 rule:
6.3 More Sine Language
6.3 More Sine Language A Solidify Understanding Task Clarita is helping Carlos calculate his height at different locations around a Ferris wheel. They have noticed when they use their formula h(t) = 30
3.4 Pascal s Pride. A Solidify Understanding Task
3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very
Matrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
Permutations and Polynomials Sarah Kitchen February 7, 2006
Permutations and Polynomials Sarah Kitchen February 7, 2006 Suppose you are given the equations x + y + z = a and 1 x + 1 y + 1 z = 1 a, and are asked to prove that one of x,y, and z is equal to a. We
6.3 More Sine Language
6.3 More Sine Language A Solidify Understanding Task Clarita is helping Carlos calculate his height at different locations around a Ferris wheel. They have noticed when they use their formula h(t) = 30
Computer Assignment 2
Computer Assignment 2 Chapter 2, section, problem 3 Apes = [7 5;3 6] Diet = [25 4 2;5 3 25] Result = Apes * Diet Apes = 7 5 3 6 Diet = 25 4 2 5 3 25 Result = 25 43 265 65 3 2 The meaning of the calculation:
Find two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
Math 4377/6308 Advanced Linear Algebra
2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377
5. 4 Pampering and Feeding Time
14 5. 4 Pampering and Feeding Time A Practice Understanding Task Carlos and Clarita have been worried about space and start-up costs for their pet sitters business, but they realize they also have a limit
9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.
Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1
Math 308 Practice Final Exam Page and vector y =
Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution
Lecture 27. Quadratic Formula
Lecture 7 Quadratic Formula Goal: to solve any quadratic equation Quadratic formula 3 Plan Completing the square 5 Completing the square 6 Proving quadratic formula 7 Proving quadratic formula 8 Proving
Section 19 Integral domains
Section 19 Integral domains Instructor: Yifan Yang Spring 2007 Observation and motivation There are rings in which ab = 0 implies a = 0 or b = 0 For examples, Z, Q, R, C, and Z[x] are all such rings There
Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1-10.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2
Matrix Differentiation
Matrix Differentiation CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Matrix Differentiation
3.4 Elementary Matrices and Matrix Inverse
Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary
2. Two binary operations (addition, denoted + and multiplication, denoted
Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between
Polynomial expression
1 Polynomial expression Polynomial expression A expression S(x) in one variable x is an algebraic expression in x term as Where an,an-1,,a,a0 are constant and real numbers and an is not equal to zero Some
3.2 Subspace. Definition: If S is a non-empty subset of a vector space V, and S satisfies the following conditions: (i).
. ubspace Given a vector spacev, it is possible to form another vector space by taking a subset of V and using the same operations (addition and multiplication) of V. For a set to be a vector space, it
Logarithmic and Exponential Equations and Change-of-Base
Logarithmic and Exponential Equations and Change-of-Base MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to solve exponential equations
How to write proofs - II
How to write proofs - II Problem (Ullman and Hopcroft, Page 104, Exercise 4.8). Let G be the grammar Prove that S ab ba A a as baa B b bs abb L(G) = { x {a, b} x contains equal number of a s and b s }.
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
Lecture Notes: Matrix Inverse. 1 Inverse Definition. 2 Inverse Existence and Uniqueness
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
9-8 Completing the Square
In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When
STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part I. 1 st Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part I 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
Jim Lambers MAT 419/519 Summer Session Lecture 11 Notes
Jim Lambers MAT 49/59 Summer Session 20-2 Lecture Notes These notes correspond to Section 34 in the text Broyden s Method One of the drawbacks of using Newton s Method to solve a system of nonlinear equations
Watch Your Step Proofs in Algebra
LESSON 9 Watch Your Step Proofs in Algebra LEARNING OBJECTIVES Today I am: explaining each step in a solution. So that I can: practice using the equality properties. I ll know I have it when I can: write
Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x
11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of
Solving Quadratic Equations
Concepts: Solving Quadratic Equations, Completing the Square, The Quadratic Formula, Sketching Quadratics Solving Quadratic Equations Completing the Square ax + bx + c = a x + ba ) x + c Factor so the
Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*
Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course
Integration - Past Edexcel Exam Questions
Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point
PLC Papers Created For:
PLC Papers Created For: Quadratics intervention Deduce quadratic roots algebraically 1 Grade 6 Objective: Deduce roots algebraically. Question 1. Factorise and solve the equation x 2 8x + 15 = 0 Question
Algebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
Math 1B03/1ZC3 - Tutorial 2. Jan. 21st/24th, 2014
Math 1B03/1ZC3 - Tutorial 2 Jan. 21st/24th, 2014 Tutorial Info: Website: http://ms.mcmaster.ca/ dedieula. Math Help Centre: Wednesdays 2:30-5:30pm. Email: dedieula@math.mcmaster.ca. Does the Commutative
GROUP THEORY BY THE SPMPS 2013 GROUP THEORY CLASS
GROUP THEORY BY THE SPMPS 2013 GROUP THEORY CLASS Abstract. This paper presents theorems proven by the Group Theory class of the 2013 Summer Program in Mathematical Problem Solving. It also includes the
Vectors. February 1, 2010
Vectors Febrary 1, 2010 Motivation Location of projector from crrent position and orientation: direction of projector distance to projector (direction, distance=magnitde) vector Examples: Force Velocity
Equations in Quadratic Form
Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written
Inverses and Elementary Matrices
Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose a 11 x 1 +a 12 x 2 + +a 1n x n = b 1 a 21 x 1 +a 22 x 2 + +a 2n x n = b 2 a n1 x
Chapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
MS 2001: Test 1 B Solutions
MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question
Lesson 18: Problem Set Sample Solutions
Problem Set Sample Solutions Problems 5 7 serve to review the process of computing f(g(x)) for given functions f and g in preparation for work with inverses of functions in Lesson 19. 1. Sketch the graphs
For comments, corrections, etc Please contact Ahnaf Abbas: Sharjah Institute of Technology. Matrices Handout #8.
Matrices Handout #8 Topic Matrix Definition A matrix is an array of numbers: a a2... a n a2 a22... a 2n A =.... am am2... amn Matrices are denoted by capital letters : A,B,C,.. Matrix size or rank is determined
Solutions to Exam I MATH 304, section 6
Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0
JUST THE MATHS UNIT NUMBER 1.5. ALGEBRA 5 (Manipulation of algebraic expressions) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.5 ALGEBRA 5 (Manipulation of algebraic expressions) by A.J.Hobson 1.5.1 Simplification of expressions 1.5.2 Factorisation 1.5.3 Completing the square in a quadratic expression
review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17
1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient
Homework 3/ Solutions
MTH 310-3 Abstract Algebra I and Number Theory S17 Homework 3/ Solutions Exercise 1. Prove the following Theorem: Theorem Let R and S be rings. Define an addition and multiplication on R S by for all r,
Regional Solution of Constrained LQ Optimal Control
Regional Solution of Constrained LQ Optimal Control José DeDoná September 2004 Outline 1 Recap on the Solution for N = 2 2 Regional Explicit Solution Comparison with the Maximal Output Admissible Set 3
Announcements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
Can there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either
MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring
NORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
Algebra. x + a 0. x 2 + a 1. are constants and a n. x n 1,..., a 0. x n, a n 1
2 Algebra Introduction We have studied about a particular type of algebraic expression, called polynomial, and the terminology related to it. Now, we shall study the factorisation of polynomials. In addition,
A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
![A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].](/thumbs/78/77913594.jpg "A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].")
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
Lesson 7: Algebraic Expressions The Commutative and Associative Properties
: Algebraic Expressions The Commutative and Associative Properties Four Properties of Arithmetic: The Commutative Property of Addition: If a and b are real numbers, then a + b = b + a. The Associative
A mathematical statement that asserts that two quantities are equal is called an equation. Examples: x Mathematics Division, IMSP, UPLB
EQUATIONS A mathematical statement that asserts that two quantities are equal is called an equation. Examples: 1.. 3. 1 9 1 x 3 11 x 4xy y 0 Consider the following equations: 1. 1 + 9 = 1. x + 9 = 1 Equation
Synchronous Sequential Logic
1 IT 201 DIGITAL SYSTEMS DESIGN MODULE4 NOTES Synchronous Sequential Logic Sequential Circuits - A sequential circuit consists of a combinational circuit and a feedback through the storage elements in
Class 10 Quadratic Equations
ID : in-10-quadratic-equations [1] Class 10 Quadratic Equations For more such worksheets visit www.edugain.com Answer t he quest ions (1) Had Sunil scored 8 more marks in his mathematics test out of 30
Math 24 Spring 2012 Questions (mostly) from the Textbook
Math 24 Spring 2012 Questions (mostly) from the Textbook 1. TRUE OR FALSE? (a) The zero vector space has no basis. (F) (b) Every vector space that is generated by a finite set has a basis. (c) Every vector
CM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation
CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics