Logo Symmetry Learning Task. Unit 5

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to solve simple radical equations. In working with symmetry of graphs, students will apply the concepts of similarity and transformations of geometric figures inherent in the Grade 7 standards for geometry. The task offers students an in-depth discussion of even and odd symmetry of graphs of functions and transformations of graphs by reflection in the coordinate axes. These topics and the discussion of solving rational equations and quadratic equations of the form x² c = 0, c 0, reinforce topics from geometry, especially the topics of symmetry and transformation of geometric figures, similar triangles, and the Pythagorean Theorem. Key Standards MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. MM1A3. Students will solve simple equations. a. Solve quadratic equations in the form ax² + bx + c = 0 where a = 1 by using factorization and finding square roots where applicable. d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1. Possible Materials o Miras o Patty paper o Rulers/ Straightedges o TI Navigator System o Graphing calculators

2 Task In middle school, students learned about line and rotational symmetry. Students can be asked to remember that a figure has line symmetry if there is a line that divides the figure into two parts that are mirror images of each other and that a figure has rotational symmetry if, when rotated by an angle of 180 degrees or less about its center, the figure aligns with itself. A figure can also have point symmetry. A figure is symmetric about a single point if when rotated about that point 180 degrees it aligns with itself. So, rotational symmetry of 180 degrees is also symmetry about the center. Sample Questions/Solutions 1.We all see many company logos everyday. These logos often have symmetry. For each logo shown below, identify and explain any symmetries you see. Solution: Except for the Logo 4, Logo 14 and Logo 16, each logo above has line symmetry with respect to the vertical that cuts the figure in half. If we omit the year 1785, the remaining figure in Logo 14 has symmetry with respect to the vertical that divides the figure in half. Logo 4 has rotational symmetry of 180 ; this symmetry is also described as point symmetry about the center of the design. Logos 9, 10, and 17 have rotational symmetry of 120 in the both the clockwise and counterclockwise directions. Logos 9 and 10 also have line symmetry about the lines obtained by rotating the vertical line through the center 120 clockwise and counterclockwise. Logo 11 has rotational symmetry of 72 and 144 in both the clockwise and counterclockwise directions. If we ignore the colors, Logo 16 has rotational symmetry of 72 and 144 in both the clockwise and counterclockwise directions. Logo 2 has rotational symmetry of 20, 40, 60, 80, 100, 120, 140, 160, and 180 in both the clockwise and counterclockwise directions. The logo also has line symmetry across any line obtained by rotating the vertical line through the center through a multiple of 10. These lines are the diagonal lines that go from one of the outer points through the center to another outer point and the lines half way between any two of these diagonals.

3 2. A textile company called Uniform Universe has been hired to manufacture some military uniforms. To complete the order, they need embroidered patches with the military insignia for a sergeant in the United States Army. To save on costs, Uniform Universe subcontracted a portion of their work to a foreign company. The machines that embroider the insignia design require a mathematical description. The foreign company incorrectly used the design at the right, which is the insignia for a British Sergeant. They sent the following description for the portion of the design to be embroidered in black. a. Match the lines in the design to the functions indicated in the table. Comments: Students may use graphing calculators or other technology to reproduce the design. In order to restrict the domain in a graph drawn by technology, divide the expression for the function by the Boolean expression ( 7 x) ( x 7). When the Boolean expression is false, we have division by 0 and no graph is drawn. When the Boolean expression is true, the graph is drawn. Solution: The graphs match the functions listed in the table in order from top to bottom. The absolute value graph is shifted down 1 unit, 3 units, 5 units, and 7 units respectively. b. When the design is stitched on a machine, wide black stitching is centered along the lines given by the equations above. The British Sergeant s insignia has a light-colored embroidery between the lines of black. Write a description for the lines on which the light-colored stitching will be centered. Use a table format similar to that shown above.

4 3. Jessica, a manager at Uniform Universe, immediately noticed the design error when she saw some of the prototype uniforms. The sergeant s insignia was upside down from the correct insignia for a U.S. sergeant, which is shown at the right. Jessica checked the description that had been sent by the foreign contractor. She immediately realized how to fix the insignia. So, she ed the foreign supplier to point out the mistake and to inform the company that the error could be corrected the by reflecting each of the functions in the x-axis. Ankit, an employee at the foreign textile company, ed Jessica back and included the graph below to verify that Uniform Universe would be satisfied with the new formulas. a. What type of symmetry does the incorrect insignia have? Line symmetry b. If it is symmetric about a point, line, or lines, write the associated coordinates of the point or equation(s) for the lines of symmetry. The line of symmetry is the y-axis, which is the vertical line with equaiton x = 0. c. Does the corrected insignia have the same symmetry? The correct insignia is symmetric with respect to the y-axis, that is, the vertical line x = 0. d. Write the mathematical description of the design for the U.S. sergeant insignia, as shown in the graph below. Verify that your mathematical description yields the graphs shown.

5 The correct insignia is described by reflecting each of the equations through the x-axis. To reflect a graph through the x-axis, we set the y-coordinate equal to the negative of the previous y-coordinate. So, to reflect y = x 7 through the x-axis, we form the equation y = ( x 7) = x + 7. We can see that the graphs are the graph of y = x shifted up 1 unit, 3 units, 5 units, and 7 units, respectively. e. Let f denote any one of the functions graphed in the British sergeant s insigniz or the U.S. sergeant s insignia. Compare f(1) and f( 1), f(2.5) and f( 2.5), f(3.7) and f( 3.7). If x is a number such that 0 x 7, how do f(x) and f( x) compare? On each of the graphs, f(1) = f( 1), f(2.5) = f( 2.5), f(3.7) = f( 3.7). If x is a number such that 0 x 7, then f(x) = f( x). f. Let a be a constant other than the number 0 and let g denote the function whose formula is given by g(x) = ax 2. You studied the shapes of these graphs in Unit 1. Look back at some examples for particular choices of a. What type of symmetry do these graphs have? If x is a positive number, how do g(x) and g( x) compare? Graphs of some of the functions of the form g(x) = ax 2 considered in Unit 1, Painted Cubes Learning Task, are shown below. Each of the graphs has line symmetry with respect to the y-axis. For any positive number x, the value of g(x) is the same as the value of g( x).

6 4. We call a function f an even function if, for any number x in the domain of f, x is also in the domain and f( x) = f(x). a. Suppose f is an even function and the point (3, 5) is on the graph of f. What other point do you know must be on the graph of f? Explain. Comments: Since (3, 5) is on the graph, 3 is in the domain and f(3) = 5. Then, by the definition of an even function, 3 is also in the domain and f( 3) = f(3) = 5. Solution: ( 3, 5) is also on the graph of f b. Suppose f is an even function and the point ( 2, 4) is on the graph of f. What other point do you know must be on the graph of f? Explain. Comments: Since ( 2, 4) is on the graph, 2 is in the domain and f( 2) = 4. Then, by the definition of an even function, ( 2) = 2 is also in the domain and f(2) = f( 2) = 4. Solution: (2, 4) is also on the graph of f c. If (a, b) is a point on the graph of an even function f, what other point is also on the graph of f? Comments: Since ( a, b) is on the graph, a is in the domain and f(a) = b. Then, by the definition of an even function, a is also in the domain and f( a) = f( a) = b. Solution: ( a, b) d. What symmetry does the graph of an even function have? Explain why.

7 Solution: The graph of an even function has line symmetry with respect to the y-axis. For any number b, the points (x, b) and ( x, b) are at the same height on the grid and are equidistant from the y-axis; thus, they represent line symmetry with respect to the y-axis. Since f(x) and f( x) are the same number, the points (x, f(x)) and ( x, f( x)) are of the form (x, b) and ( x, b), and thus, are symmetric with respect to the y-axis. See the diagram above. e. Consider the function k, which is an even function. Part of the graph of k is shown at the right. Using the information that k is an even function, complete the graph for the rest of the domain. Comments: Since the graph of an even function is symmetric with respect to the y- axis, we reflect the part of the graph shown in the y-axis to obtain the rest of the graph, as shown.

8 5. When Jessica s supervisor, Malcom, saw the revised design, he told Jessica that he did not think that the U.S. Army would be satisfied. He pointed out that, while the revision did turn the design right-side up, it did not account for the slight curve in the lines in the real U.S. sergeant s insignia. He suggested that a square root function might be a better choice than an absolute value function and told Jessica to work with the foreign contractor to get a more accurate design. Jessica ed Ankit to let him know that the design needed to be revised again to have lines with a curve similar to the picture from the U.S. Army website, as shown above, and suggested that he try the square root function. a. Ankit graphed the square root function, y = x, and decided to limit the domain to 0 x 4. Set up a grid using a scale of ½ -inch for each unit, and graph the square root function on this limited domain. For accuracy, plot points for the following domain values: : 0, 1/4, 1, 25/16, 9/4, 49/16, 4. Comments: The graph, using the scale of ½ -inch for each unit, is shown in blue below, after part b. b. Ankit saw that his graph of the square root function (on the domain 0 x 4 ) looked like the curve that forms the lower right edge of the British sergeant s insignia. He knew that he could reflect the graph through the x-axis to turn the curve over for the U.S. sergeant s insignia, but first he needed to reflect the graph through the y-axis in order to get the full curve to form the lower edge of the British sergeant s insignia. He knew that he needed to input a negative number and then get the square root of the corresponding positive number, so he tried graphing the function y = x on the domain -4 x 0. Reproduce Ankit s graph using the scale of ½-inch for each unit and using the opposites of the domain values in part a as some accurate points on the graph: 0, 1/4, 1, -25/16, -9/4, -49/16, 4. Comments: This graph is shown below in red on the same axes with the graph for part a.

9 c. To see an additional example of reflecting a graph through the axes, graph each of the following and compare the graphs. d. Given an equation of the form y = f(x), there is a specific process for creating a second equation whose graph is obtained by reflecting the graph of y = f(x) through the x-axis, and there is different process for creating an equation whose graph is obtained by reflecting the graph of of y = f(x) through the y-axis. Expalin these processes in your own words. Comments: This item begins a process of helping students understand the similarities and differences in the two types of reflection. Reflecting the graph of a function through the x-axis is the simpler idea; it was introduced in Unit 1. Understanding reflection through the y-axis requires more sophisticated understanding of function notation and, hence, was delayed until Unit 5. One of the difficulties in helping students at this level understand the distinction is that, for three of the six basic functions studied in Mathematics I, the two types of reflection produce the same result. In particular, for the function f(x) = x Given an equation of the form y = f(x).: to reflect through the x-axis, form the equation y = -f(x).. to reflect through the y-axis, form the equation y = f(-x)..

10 e. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part a in the x-axis and then shifting it up 8 units. Call this function f1, read f sub one. f. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part b in the x-axis and then shifting it up 8 units. Call this function f2, read f sub two. When graphed on the same axes, the graphs of the functions f1 and f2 give the curve shown above. Solution: f 2 (x) = - -x + 8

11 g. Ankit completed his mathematical definition for the U.S. Army sergeant s insignia and sent it to Jessica along with the graph shown at the right. Based on the graph and your work above, write Ankit s specifications for black embroidery of the insignia as shown in the graph. Supporting Lessons Available :

The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

Section 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

To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

Algebra 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

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations

Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

Clovis Community College Core Competencies Assessment 2014 2015 Area II: Mathematics Algebra

Core Assessment 2014 2015 Area II: Mathematics Algebra Class: Math 110 College Algebra Faculty: Erin Akhtar (Learning Outcomes Being Measured) 1. Students will construct and analyze graphs and/or data

For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

Algebra 2: Q1 & Q2 Review

Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).

CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

Accelerated Mathematics I Frameworks Student Edition. Unit 7 Algebraic Investigations: Quadratics and More, Part 3

Accelerated Mathematics I Frameworks Student Edition Unit 7 Algebraic Investigations: Quadratics and More, Part 3 nd Edition March, 011 Accelerated Mathematics I Unit 7 nd Edition Table of Contents INTRODUCTION:...

High School Functions Interpreting Functions Understand the concept of a function and use function notation.

Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

0.4 FACTORING POLYNOMIALS

36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

Chapter 11 Equilibrium

11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

Warm-Up Oct. 22. Daily Agenda:

Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

Administrative - 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

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

MATH 110 College Algebra Online Families of Functions Transformations

MATH 110 College Algebra Online Families of Functions Transformations Functions are important in mathematics. Being able to tell what family a function comes from, its domain and range and finding a function

Estimating the Average Value of a Function

Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

Algebra I Credit Recovery

Algebra I Credit Recovery 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,

Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS

THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE. Sample Solutions for Specimen Test 2

THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE Sample Solutions for Specimen Test. A. The vector from P (, ) to Q (8, ) is PQ =(6, 6). If the point R lies on PQ such

Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

Quickstart for Desktop Version

Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easy-to-use package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions These materials are for nonprofit educational purposes only. Any

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

Rolle s Theorem. q( x) = 1

Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

Gouvernement du Québec Ministère de l Éducation, 2004 04-00815 ISBN 2-550-43543-5

Gouvernement du Québec Ministère de l Éducation, 004 04-00815 ISBN -550-43543-5 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

Study Guide 2 Solutions MATH 111

Study Guide 2 Solutions MATH 111 Having read through the sample test, I wanted to warn everyone, that I might consider asking questions involving inequalities, the absolute value function (as in the suggested

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

3.6 The Real Zeros of a Polynomial Function

SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

04 Mathematics CO-SG-FLD004-03. Program for Licensing Assessments for Colorado Educators

04 Mathematics CO-SG-FLD004-03 Program for Licensing Assessments for Colorado Educators Readers should be advised that this study guide, including many of the excerpts used herein, is protected by federal

Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

GeoGebra. 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

Mathematics as Problem Solving The students will demonstrate the ability to gather information from a graphical representation of an equation.

Title: Another Way of Factoring Brief Overview: Students will find factors for quadratic equations with a leading coefficient of one. The students will then graph these equations using a graphing calculator

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

College Readiness Math MOOC Instructor Information: Dr. Jennifer Kosiak, jkosiak@uwlax.edu General email: mathmooc@uwlax.edu Mathematics Department, University of Wisconsin- La Crosse Description: The

2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

Polynomials and Factoring

Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

Appendix A Designing High School Mathematics Courses Based on the Common Core Standards

Overview: The Common Core State Standards (CCSS) for Mathematics are organized by grade level in Grades K 8. At the high school level, the standards are organized by strand, showing a logical progression

Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 )

Quiz: Factoring by Graphing Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 ) (x-3)(x-6), (x-6)(x-3), (1x-3)(1x-6), (1x-6)(1x-3), (x-3)*(x-6), (x-6)*(x-3), (1x- 3)*(1x-6),

Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

Guide for Texas Instruments TI-83, TI-83 Plus, or TI-84 Plus Graphing Calculator

Guide for Texas Instruments TI-83, TI-83 Plus, or TI-84 Plus Graphing Calculator This Guide is designed to offer step-by-step instruction for using your TI-83, TI-83 Plus, or TI-84 Plus graphing calculator

8th Grade Texas Mathematics: Unpacked Content

8th Grade Texas Mathematics: Unpacked Content What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards mean a student must

Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior

The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

http://school-maths.com Gerrit Stols

For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

Factoring, Solving. Equations, and Problem Solving REVISED PAGES

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

Functional Math II. Information CourseTitle. Types of Instruction

Functional Math II Course Outcome Summary Riverdale School District Information CourseTitle Functional Math II Credits 0 Contact Hours 135 Instructional Area Middle School Instructional Level 8th Grade

COLLEGE 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

9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

March 2013 Mathcrnatics MATH 92 College Algebra Kerin Keys. Dcnnis. David Yec' Lscture: 5 we ekly (87.5 total)

City College of San Irrancisco Course Outline of Itecord I. GENERAI- DESCRIPI'ION A. Approval Date B. Departrnent C. Course Number D. Course Title E. Course Outline Preparer(s) March 2013 Mathcrnatics

Performance Assessment Task Which Shape? Grade 3. Common Core State Standards Math - Content Standards

Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to

Common Core Standards Practice Week 8

Common Core Standards Practice Week 8 Selected Response 1. Describe the end behavior of the polynomial f(x) 5 x 8 8x 1 6x. A down and down B down and up C up and down D up and up Constructed Response.

MAT 113-701 College Algebra

MAT 113-701 College Algebra Instructor: Dr. Kamal Hennayake E-mail: kamalhennayake@skipjack.chesapeake.edu I check my email regularly. You may use the above email or Course Email. If you have a question

MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

Teaching & Learning Plans. Complex Number Operations. Leaving Certificate Syllabus

Teaching & Learning Plans Complex Number Operations Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

Common Tools for Displaying and Communicating Data for Process Improvement

Common Tools for Displaying and Communicating Data for Process Improvement Packet includes: Tool Use Page # Box and Whisker Plot Check Sheet Control Chart Histogram Pareto Diagram Run Chart Scatter Plot

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

1.3. Maximum or Minimum of a Quadratic Function. Investigate A

< P1-6 photo of a large arched bridge, similar to the one on page 292 or p 360-361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

The Quadratic Formula Target Audience: Introduction to Algebra Class at Bel Air High School, 90 minutes Lesson Topic: The Quadratic Formula NCTM Standards: (Algebra Standard for Grades 9-12) Understand

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Contents by Grade Level: Overview of Exam Grade Level Results Cut Score and Grade History

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

Don't Forget the Differential Equations: Finishing 2005 BC4

connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College

Homework #1 Solutions

Homework #1 Solutions Problems Section 1.1: 8, 10, 12, 14, 16 Section 1.2: 2, 8, 10, 12, 16, 24, 26 Extra Problems #1 and #2 1.1.8. Find f (5) if f (x) = 10x x 2. Solution: Setting x = 5, f (5) = 10(5)