Classroom Tips and Techniques: The Student Precalculus Package  Commands and Tutors. Content of the Precalculus Subpackage


 Dulcie McCoy
 2 years ago
 Views:
Transcription
1 Classroom Tips and Techniques: The Student Precalculus Package  Commands and Tutors Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This article provides a systematic exposition of the functionalities available in the Student Precalculus package. The commands and tutors comprising the package are exhaustively listed and detailed. Content of the Precalculus Subpackage Table 1 lists the mathematical functionalities addressed in the Precalculus subpackage of the Student package. The first column describes the functionality; the second lists the available commands; the third indicates if the commands draw graphs ("V" for visualization), or output mathematical calculations ("C" for computational); the fourth column lists the interactive tutors that implement the relevant functionality in a Maplet format; and the last column provides the commnd with which the tutor can be launched from the keyboard. Table 2 in the next section (Launching Tutors) provides alternatives for these commands. Column four in Table 1 uses for the name of the tutor, the term seen in the Tools/Tutors/Precalculus menu. Unfortunately, these terms do not match the titles displayed within the tutors themselves, and these titles often do not match the name embedded in the command that launches the tutor from the keyboard. Hence, Table 1 has its fifth column, no pun intended. Mathematical Task Command TypeInteractive Tutor Keyboard Command Composition of functions CompositionPlot V Function Composition CompositionTutor Graphing conics Conic Sections ConicsTutor Secant line becoming tangent line FunctionSlopePlotV Slopes FunctionSlopeTutor
2 Numeric intuition about limits Equation and graph of a straight line Graphing linear inequalities LimitPlot V Limits LimitTutor Line V, C Lines LineTutor Linear Inequalities LinearInequalitiesTu tor Graph and zeros of polynomials Polynomials PolynomialTutor Graph of rational function and its asymptotes RationalFunctionPV lot Rational Functions RationalFunctionTut or Graphs of elementary functions Standard Functions StandardFunctionsT utor Weighted average CenterOfMass C Distance between points Midpoint of line segment Distance C Midpoint C Slope of a line Slope C Complete the square CompleteSquare C Table 1 Content of the Student Precalculus subpackage. Command types: V = visualization; C = computational The Line command outputs either the equation of a line, or a graph of the line. Launching Tutors Table 2 lists three ways any of the Precalculus tutors can be launched. 1. Launch tutor and type equation 2. Copy equation (even in 2D
3 math) Launch tutor and paste equation 3. Load Student[Precalculus] package Launch context menu on tutor input Select Tutors/Precalculus/... Table 2 Options for launching a tutor A tutor can be launched interactively from the Tools/Tutors menu, or can be launched by the command listed in the last column of Table 1. Executing the command with empty an argument launches the tutor with its default data. Alternatively, provide the input as argument to the command. This is another way to avoid having to enter data in text mode. This is the import of Table 2, namely, ways to supply to the tutors data typed in math mode. If the tutor is launched first from the Tools/Tutors menu, then any data that is entered into the tutor must by typed in text mode. Thus, multiplications must be made explicit, must be entered as Pi, etc. If input to a tutor already exists in math mode, it can be copied and pasted into the tutor. Because the tutor is primal, the copy must be made before the tutor is launched. If the Student[Precalculus] package has been loaded, then the context menu for an appropriate mathematical expression will contain the Tutor option from which all relevant tutors can be accessed and launched. This is usually the most convenient way to launch a tutor. For selected tutors, there are Task Templates that provide access to the tutor and some of its related calculations. These tutors are the Conic Sections Tutor, the Linear Inequalities Tutor, the Rational Functions Tutor, and the Line Tutor. Initializations The SetColors command is available for all the Student subpackages. It sets a color sequence for commands and tutors that use default colors to distinguish various mathematical objects.
4 Composition of Functions To study the composition of two functions such as use the Function Composition tutor. The easiest way to launch this tutor with the functions and already embedded, is via the context menu. As per Table 2, if the Student Precalculus package has already been loaded, then the context menu will provide access to all relevant tutors. Load the Student Precalculus package via the Tools/Load Package menu. The context menu for a sequence of two expressions such as will contain the options Tutors/Function Composition. Figure 1 shows the Function Composition tutor for this pair of expressions. Figure 1 Function Composition tutor for and As per the indicated colorcoding, the graph shows in red, in black, in green, and in blue.
5 The graph provided by the tutor shows both compositions simultaneously. The associated CompositionPlot command draws just one of the two possible compositions, with being the default. To obtain the graph of use the syntax Conic Sections The Conic Sections tutor will analyze and graph the conic section determined by its associated quadratic equation. The equation can be given in Cartesian coordinates (using and ) or in polar coordinates using or where or, with either or. In Cartesian coordinates, the quadratic can have an term, which rotates the conic. Figure 2 shows the Conic Sections tutor applied to the Cartesian equation Figure 2 Conic Sections tutor applied to
6 The totality of the information in the analysis window appears in Figure 3. class: ellipse eccentricity:.723 semimajor axis (a): 4.28 semiminor axis (b): 2.96 latus rectum: 4.09 angle: 3/8*Pi In the xyplane: vertices: [(8.60,6.57), (.689, 3.29)] foci: [(7.50,6.11), (1.79, 3.74)] center (h,k): (4.64,4.93) directrix: y = 2.41*x In the x'y'plane: vertices: [(2.78,10.5), ( 2.78,1.90)] foci: [(2.78,9.27), (2.78, 3.08)] center (h',k'): (2.78,6.18) directrix: y' = Figure 3 Contents of analysis window of the Conic Sections tutor The graph of the conic is drawn in the plane. The analysis window (see Figure 3) provides details for the conic as drawn in the plane, and as it would appear in the  plane where the conic assumes the standard form shown in Figure 2. The eccentricity, major and minor axes, and length of the latus rectum are the same for any orientation of the conic. The coordinates of the center, vertices and foci change with orientation, as does the equation of the directrix. The Task Template "Conic  Analysis and Graph" appears in Figure 4. Pressing the launch button after the equation is dragged, pasted, or typed into the template's math container launches the tutor with the equation embedded. Closing the tutor afterwards writes the graph to the plot window of the template. Analyze a Quadratic Equation Using the
7 Conics Tutor Enter a quadratic equation: Figure 4 Task Template for launching the Conic Sections tutor
8 Secant and Tangent Lines To superimpose secant and tangent lines on the graph of the function use the Function Slope tutor. Launching this tutor from the Context Menu yields Figure 5 in which the default point of contact for the tangent line is at. Figure 5 The Function Slope tutor applied to In general, ten secant lines are drawn, each passing through the point of contact where. The other point coincident with the graph of the function has coordinate The tangent line is drawn in green, and its equation is given on the right in the tutor. The table of values in the tutor lists the coordinate common to the curve and secant line, and the slope of the corresponding secant line. Unfortunately, there is no way to control the location or spacing of the secant lines. Figure 6 is created with the FunctionSlopePlot command, as per the display at the bottom of the tutor shown in Figure 5. This command defaults to an animation. Click on the graph to access the animation toolbar, with which the animation of the secant line with intersections at can be activated.
9 > Figure 6 Secant and tangent lines on a graph of > Intuitive Limits To obtain some sense of what the word "limit" means mathematically, apply the Limits tutor to a function such as Figure 7 shows the use of this tutor. The graph in the tutor defaults to an animation in which a point moves along the curve according to the values listed in the tables to its right. If the coordinate of the point at which the limit is being investigated is, then the neighboring points at which the function is sampled are
10 Figure 7 The Limits tutor applied to The animation in the Limits tutor can be generated by the LimitPlot command, as shown in Figure 8. >
11 Figure 8 Animation illustrating "limit" > Straight Lines To graph and otherwise analyze the line menu. The result is shown in Figure 9., launch the Line tutor via the context
12 Figure 9 The Line tutor applied to the equation The line is graphed, and its equation is rendered in pointslope, twopoint, slopeintercept, and general forms. The pointslope form uses the intercept for the point, while the twopoint form uses the intercept and the point As per the display at the bottom of the tutor, the Line command provides a graph of the line. In addition, the Line command will provide the equation, slope, intercept, and  intercept if given any of the data listed in Table 3. Table 4 illustrates these uses. point and slope (in any order) two points slope and  intercept (in that order) Table 3 Computational inputs to the Line command Table 4 Examples of the use of the Line command The Line tutor can also be launched by the LineTutor command with any of the arguments in Table 4, or with the equation of a line.
13 Linear Inequalities The LinearInequalitiesTutor shows the feasible region for a set of linear inequalities. It can be accessed via the context menu applied to a sequence, list, or set of up to six linear inequalities. It can also be launched by the LinearInequalitiesTutor command with argument either a set or list of no more than six linear inequalities. Figure 10 shows the default content of this tutor, the feasible region for six linear inequalities. Unchecking any of the inequalities removes them from the set whose feasible region is graphed when the Display button is pressed. Figure 10 Default content of the Linear Inequalities tutor, with feasible region shown in red The graph is actually drawn with the inequal command from the plots package. This command is not restricted in the number of linear inequalities it can resolve, and has numerous options for coloring the feasible and infeasible regions and their boundaries. The Linear Inequalities tutor is simply a more convenient frontend to this command. However, data entry into the tutor must be in text mode. To provide the Linear Inequalities tutor data in math mode, launch it from the context menu applied to inequalities entered in math mode, or use the Task Template Algebra/Graph Linear Inequalities, as per Figure 11. Graph Linear Inequalities
14 Enter up to six linear inequalities separated by commas: Figure 11 The Task Template Algebra/Graph Linear Inequalities
15 The inequalities are entered using math mode, and the tutor launched by pressing the obvious button. When the tutor is closed, the graph it generates is embedded in the box on the right, thus preserving a view of the inequalities and the feasible region. Polynomials The real zeros and graph of a polynomial are provided by the Polynomial tutor, which can be launched by any of the methods in Table 2. Figure 12 shows the Polynomial tutor applied to. When the real zeros are not simple integers, the tutor reverts to floats to express them. They (the intercepts on the graph) are displayed in the box labeled "Roots". Figure 12 The Polynomial tutor applied to Rational Functions
16 The Rational Functions tutor draws a graph of a rational function  complete with all its asymptotes  and provides the equations of the asymptotes. Figure 13 shows this tutor applied to the function. Figure 13 Rational Functions tutor applied to the function Direct entry of the rational function requires separating the numerator and denominator. Alternate methods of launching the tutor as per Table 2 do not require this separation. Closing the tutor returns just the graph, and the equations of the asymptotes are lost. To compensate, use the Task Template Algebra/Rational Function  Graph and Asymptotes, shown in Figure 14.
17 Rational Function Tutor Enter a rational function Asymptotes Horizontal Plot Oblique Vertical
18 Figure 14 The Task Template Algebra/Rational Function  Graph and Asymptotes In Figure 14, the Rational Function Tutor button was pressed, as was the Asymptotes button. The tutor was launched, and then closed. The graph is written to the Task Template, and the information about asymptotes is likewise preserved.
19 As shown at the bottom of the tutor, the graph is drawn with the RationalFunctionPlot command. Users familiar with the syntax of the plot command and the commands in the plots package will find the syntax of this, and the other visualizaiton commands in the Student package awkward. Before using any of these commands, the reader is advised to look up the syntax on the relevant help page. The paradigms in use for these Student visualization commands is radically different from Maple's older plot commands. For example, in these older commands, the plot range is given by an equation of the form, and the plot window in which the graph appears is trimmed by the view option. In the Student package, the plot range is not given separately, but rather, it is extracted from the view option! Moreover, for the older commands, there is no title by default. To obtain a title, syntax for it must be included. The Student package commands default to a title, and to eliminate the title, the syntax title = "" (the empty string) must be explicitly given! Elementary Functions Given an elementary function, the StandardFunctionsTutor will draw its graph and allow the user to experiment with transformations of the form. Figure 15 shows this tutor applied to the function Figure 15 The Standard Functions tutor applied to curve curve corresponding to, (in red), with the black The tutor defaults to. Changing one or more of these values and pressing the Display button adds the varied curve (in black) to the graph.
20 The graph is drawn with the basic Maple plot command, as shown in the Maple Command window at the bottom of the tutor. The dropdown listing in the window includes the six trig functions and their inverses, the six hyperbolic functions and their inverses, the natural and common logarithmic functions, and the exponential functions. Polynomial functions are not included, and no other functions can be entered into this tutor. Weighted Average The center of mass of a discrete system of particles is the weighted average of their Cartesian coordinates or position vectors. The CenterOfMass command in the Student Precalculus package computes this weighted average using lists (for coordinates) and/or vectors. To give a weight (i.e., mass), enclose the object and its weight in list brackets. If no weights are explicitly given, the weights are assumed to be 1. Table 5 contains examples. The center of mass for uniform masses at and is, computed to the right. The center of mass for masses, respectively located at,,, is, as computed at the right. Note how weights are included and not, and how coordinate and vector notation can be mixed. Table 5 Examples of the CenterOfMass command used to compute the center of mass for discrete systems. Distance between Points The (Euclidean) distance between Cartesian points is obtained with the Distance command, which accepts locations given as points and/or vectors in any number of dimensions. Several examples are listed in Table 6. The distance between and
21 The distance between and Table 6 Examples of the distance between two points computed by the Distance command Midpoint of a Line Segment The coordinates of the midpoint of the line segment connecting the points, are given by As with the CenterOfMass and Distance commands, Cartesian points can be described as lists or vectors. Table 7 illustrates the use of the Midpoint command. Midpoint of the segment connecting and Midpoint of the segment connecting and Table 7 Examples of the midpoint of a line segment computed by the Midpoint command Slope of a Line The slope of the line passing through the points and is given by
22 As with the CenterOfMass, Distance and Midpoint commands, points in the Cartesian plane can be described as lists or vectors. Table 8 gives examples of the use of the Slope command for computing the slope between two planar points. The slope of the line passing through the points is given by and The slope of the line passing through the points and is given by by any of the formalisms at the right Table 8 Computation of the slope by the Slope command Algebraic Completion of the Square The CompleteSquare command will write the quadratic expression. This command can be applied to expressions and equations, to one or more variables, and to functions such as or. Table 9 lists a number of examples of the functioning of this command. as
23 Table 9 The CompleteSquare command illustrated The iterated integrals in the last example in Table 9 appear in gray. This indicates they are the inert form of the integral, corresponding to the Maple Int command. The Maple int command would immediately evaluate the integral, so it is essential that the inert form be used. To set this inert form in math notation, either type Int and use command completion (e.g.,tools/complete Command) or enter the indefinite integral template from the Expression palette, and use the context menu to convert it to its inert form. Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for noncommercial, nonprofit use only. Contact the author for permission if you wish to use this application in forprofit activities.
24
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationMaple Quick Start. Introduction. Talking to Maple. Using [ENTER] 3 (2.1)
Introduction Maple Quick Start In this introductory course, you will become familiar with and comfortable in the Maple environment. You will learn how to use context menus, task assistants, and palettes
More informationCourse Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics
Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)
More informationClassroom Tips and Techniques: Circle Inscribed in a Parabola. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Circle Inscribed in a Parabola Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The following problem is #11 on page 300 (Problems
More informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationBirmingham City Schools
Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 PreAssessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:
More informationCurve Fitting with Maple
Curve Fitting with Maple Maplesoft, a division of Waterloo Maple Inc., 2007 Introduction Maple includes a number of assistants that allows a user to experiment and easily perform key tasks. This Tips and
More informationChapter 1 Quadratic Equations in One Unknown (I)
Tin Ka Ping Secondary School 015016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real
More informationComal Independent School District PreAP PreCalculus Scope and Sequence
Comal Independent School District Pre PreCalculus Scope and Sequence Third Quarter Assurances. The student will plot points in the Cartesian plane, use the distance formula to find the distance between
More informationAble Enrichment Centre  Prep Level Curriculum
Able Enrichment Centre  Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
More informationAlgebra I. Copyright 2014 Fuel Education LLC. All rights reserved.
Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
More informationAlgebra Course KUD. Green Highlight  Incorporate notation in class, with understanding that not tested on
Algebra Course KUD Yellow Highlight Need to address in Seminar Green Highlight  Incorporate notation in class, with understanding that not tested on Blue Highlight Be sure to teach in class Postive and
More informationQuickstart for Desktop Version
Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easytouse package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationClassroom Tips and Techniques: Real Distinct Roots of a Cubic. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Real Distinct Roots of a Cubic Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The real distinct roots of the cubic equation can
More informationExam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.
Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the
More informationTools of Algebra. Solving Equations. Solving Inequalities. Dimensional Analysis and Probability. Scope and Sequence. Algebra I
Scope and Sequence Algebra I Tools of Algebra CLE 3102.1.1, CFU 3102.1.10, CFU 3102.1.9, CFU 3102.2.1, CFU 3102.2.2, CFU 3102.2.7, CFU 3102.2.8, SPI 3102.1.3, SPI 3102.2.3, SPI 3102.4.1, 12 Using Variables,
More informationMath Handbook of Formulas, Processes and Tricks. Algebra and PreCalculus
Math Handbook of Formulas, Processes and Tricks (www.mathguy.us) Algebra and PreCalculus Prepared by: Earl L. Whitney, FSA, MAAA Version 2.8 April 19, 2016 Copyright 2008 16, Earl Whitney, Reno NV. All
More informationALGEBRA 1/ALGEBRA 1 HONORS
ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical
More informationPrep for Calculus. Curriculum
Prep for Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular
More informationStudy Guide and Review
For each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the 11. (x + 3) 2 = 12(y + 2) (x + 3) 2 = 12(y + 2) The equation is in standard form and the squared term is x,
More information( ) 2 = 9x 2 +12x + 4 or 8x 2 " y 2 +12x + 4 = 0; (b) Solution: (a) x 2 + y 2 = 3x + 2 " $ x 2 + y 2 = 1 2
Conic Sections (Conics) Conic sections are the curves formed when a plane intersects the surface of a right cylindrical doule cone. An example of a doule cone is the 3dimensional graph of the equation
More informationALGEBRA II Billings Public Schools Correlation and Pacing Guide Math  McDougal Littell High School Math 2007
ALGEBRA II Billings Public Schools Correlation and Guide Math  McDougal Littell High School Math 2007 (Chapter Order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 10) BILLINGS PUBLIC SCHOOLS II 2009 Eleventh GradeMcDougal
More informationThe Desmos Graphing Calculator. Desmos in On Core
The Desmos Graphing Calculator The Desmos online calculator allows users to graph mathematical functions/equations and to manipulate the graphs on a computer, tablet, or whiteboard. Users can enter functions
More informationAlgebrator Manual Softmath
Algebrator Manual I Algebrator Manual Table of Contents Foreword 0 Part I Quick introduction to the Algebrator software 1 1 What... is new in version 5.0 (NEW features) 2 2 Getting... technical support
More informationhttp://www.aleks.com Access Code: RVAE4EGKVN Financial Aid Code: 6A9DBDEE3B74F5157304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationGOALS AND OBJECTIVES. Goal: To provide students of Zane State College with instruction focusing on the following topics:
Phone: (740) 8243522 ext. 1249 COURSE SYLLABUS Course Title: MATH 1350 PreCalculus Credit Hours: 5 Instructor: Miss Megan Duke Email: megan.duke@rvbears.org Course Description: Broadens the algebra
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationMercer County Public Schools PRIORITIZED CURRICULUM. Mathematics Content Maps Algebra II Revised August 07
Mercer County Public Schools PRIORITIZED CURRICULUM Mathematics Content Maps Algebra II Revised August 07 Suggested Sequence: C O N C E P T M A P ALGEBRA I I 1. Solving Equations/Inequalities 2. Functions
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationChapter R  Basic Algebra Operations (69 topics, due on 05/01/12)
Course Name: College Algebra 001 Course Code: R3RK6CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics
More informationFlorida Math for College Readiness
Core provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness in math. This fullyear course is aligned with Florida's Postsecondary
More informationClassroom Tips and Techniques: Directional Derivatives in Maple. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Directional Derivatives in Maple Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The directional derivative of a scalar function,
More informationMicrosoft Mathematics for Educators:
Microsoft Mathematics for Educators: Familiarize yourself with the interface When you first open Microsoft Mathematics, you ll see the following elements displayed: 1. The Calculator Pad which includes
More informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationPrecalculus with Limits Larson Hostetler. `knill/mathmovies/ Assessment Unit 1 Test
Unit 1 Real Numbers and Their Properties 14 days: 45 minutes per day (1 st Nine Weeks) functions using graphs, tables, and symbols Representing & Classifying Real Numbers Ordering Real Numbers Absolute
More informationSECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.
(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationCommon Curriculum Map. Discipline: Math Course: College Algebra
Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationItems related to expected use of graphing technology appear in bold italics.
 1  Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing
More information4. Factor polynomials over complex numbers, describe geometrically, and apply to realworld situations. 5. Determine and apply relationships among syn
I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of
More informationCD 1 Real Numbers, Variables, and Algebraic Expressions
CD 1 Real Numbers, Variables, and Algebraic Expressions The Algebra I Interactive Series is designed to incorporate all modalities of learning into one easy to use learning tool; thereby reinforcing learning
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationPRECALCULUS Semester I Exam Review Sheet
PRECALCULUS Semester I Exam Review Sheet Chapter Topic P.1 Real Numbers {1, 2, 3, 4, } Natural (aka Counting) Numbers {0, 1, 2, 3, 4, } Whole Numbers {, 3, 2, 2, 0, 1, 2, 3, } Integers Can be expressed
More informationThis is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0).
This is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/ 3.0/)
More informationConic Sections in Cartesian and Polar Coordinates
Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane
More informationQuickstart for Web and Tablet App
Quickstart for Web and Tablet App What is GeoGebra? Dynamic Mathematic Software in one easytouse package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationIntro to Excel spreadsheets
Intro to Excel spreadsheets What are the objectives of this document? The objectives of document are: 1. Familiarize you with what a spreadsheet is, how it works, and what its capabilities are; 2. Using
More informationIntroduction to the TINspire CX
Introduction to the TINspire CX Activity Overview: In this activity, you will become familiar with the layout of the TINspire CX. Step 1: Locate the Touchpad. The Touchpad is used to navigate the cursor
More informationCENTRAL COLLEGE Department of Mathematics COURSE SYLLABUS
CENTRAL COLLEGE Department of Mathematics COURSE SYLLABUS MATH 1314: College Algebra Fall 2010 / TuesThurs 7:309:00 pm / Gay Hall Rm 151 / CRN: 47664 INSTRUCTOR: CONFERENCE TIMES: CONTACT INFORMATION:
More informationGradient  Activity 1 Gradient from the origin.
Name: Class: p 31 Maths Helper Plus Resource Set 1. Copyright 2002 Bruce A. Vaughan, Teachers Choice Software Gradient  Activity 1 Gradient from the origin. 1) On the graph below, there is a line ruled
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More informationTable of Contents. Montessori Algebra for the Adolescent Michael J. Waski"
Table of Contents I. Introduction II. Chapter of Signed Numbers B. Introduction and Zero Sum Game C. Adding Signed Numbers D. Subtracting Signed Numbers 1. Subtracting Signed Numbers 2. Rewriting as Addition
More informationMATHEMATICS GRADE LEVEL VOCABULARY DRAWN FROM SBAC ITEM SPECIFICATIONS VERSION 1.1 JUNE 18, 2014
VERSION 1.1 JUNE 18, 2014 MATHEMATICS GRADE LEVEL VOCABULARY DRAWN FROM SBAC ITEM SPECIFICATIONS PRESENTED BY: WASHINGTON STATE REGIONAL MATH COORDINATORS Smarter Balanced Vocabulary  From SBAC test/item
More informationTOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM
Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 34 weeks Trigonometry 3 weeks Graphs of Trig Functions 34 weeks Analytic Trigonometry 56 weeks
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationClear & Understandable Math
Chapter 1: Basic Algebra (Review) This chapter reviews many of the fundamental algebra skills that students should have mastered in Algebra 1. Students are encouraged to take the time to go over these
More informationPlotting: Customizing the Graph
Plotting: Customizing the Graph Data Plots: General Tips Making a Data Plot Active Within a graph layer, only one data plot can be active. A data plot must be set active before you can use the Data Selector
More informationUsing GeoGebra to create applets for visualization and exploration.
Handouts for ICTCM workshop on GeoGebra, March 2007 By Mike May, S.J. mikemaysj@gmail.com Using GeoGebra to create applets for visualization and exploration. Overview: I) We will start with a fast tour
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationAdministrative  Master Syllabus COVER SHEET
Administrative  Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationUnit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks
Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More informationLevel: High School: Geometry. Domain: Expressing Geometric Properties with Equations GGPE
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Translate between the geometric
More informationProcedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:
Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series
More information3D Scatter Plots. Chapter 170. Introduction
Chapter 170 Introduction The 3D scatter plot displays trivariate points plotted in an XYZ grid. It is particularly useful for investigating the relationships among these variables. The influence of a
More informationALGEBRA I / ALGEBRA I SUPPORT
Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationREVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
More informationPrecalculus Orientation and FAQ
Precalculus Orientation and FAQ MATH 1011 (Precalculus) is a four hour 3 credit course that prepares a student for Calculus. Topics covered include linear, quadratic, polynomial, rational, exponential,
More informationCOLLEGE ALGEBRA LEARNING COMMUNITY
COLLEGE ALGEBRA LEARNING COMMUNITY Tulsa Community College, West Campus Presenter Lori Mayberry, B.S., M.S. Associate Professor of Mathematics and Physics lmayberr@tulsacc.edu NACEP National Conference
More informationHow are the properties of real numbers and the order of operations used to simplify an expression and/or solve an algebraic equation or inequality?
Topic: Algebra I Review Key Learning: Review Algebraic concepts of singlevariable expressions and equations using the order of operations, sets of real numbers and the properties of real numbers. How
More information1. Students will demonstrate an understanding of the real number system as evidenced by classroom activities and objective tests
MATH 102/102L InterAlgebra/Lab Properties of the real number system, factoring, linear and quadratic equations polynomial and rational expressions, inequalities, systems of equations, exponents, radicals,
More informationTI92 GRAPHING CALCULATOR BASIC OPERATIONS
TI92 GRAPHING CALCULATOR BASIC OPERATIONS by Carolyn Meitler Concordia University Wisconsin B1 Getting Started Press ON to turn on the calculator. Press 2nd 6 to get the MEMORY screen (shown at the right).
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationSEEING IS BELIEVING VISUALIZING CALCULUS
SEEING IS BELIEVING VISUALIZING CALCULUS Adam O. Hausknecht and Robert E. Kowalczyk University of Massachusetts Dartmouth Mathematics Department, 285 Old Westport Road, N. Dartmouth, MA 027472300 rkowalczyk@umassd.edu
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationIn this section, we ll review plotting points, slope of a line and different forms of an equation of a line.
Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:
More informationSpreadsheet View and Basic Statistics Concepts
Spreadsheet View and Basic Statistics Concepts GeoGebra 3.2 Workshop Handout 9 Judith and Markus Hohenwarter www.geogebra.org Table of Contents 1. Introduction to GeoGebra s Spreadsheet View 2 2. Record
More informationMTH304: Honors Algebra II
MTH304: Honors Algebra II This course builds upon algebraic concepts covered in Algebra. Students extend their knowledge and understanding by solving openended problems and thinking critically. Topics
More informationHigher Education Math Placement
Higher Education Math Placement 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry (arith050) Subtraction with borrowing (arith006) Multiplication with carry
More information