# Learning Objectives 9.2. Media Run Times 9.3

Size: px
Start display at page:

Transcription

1 Unit 9 Table of Contents Unit 9: Factoring Video Overview Learning Objectives 9.2 Media Run Times 9.3 Instructor Notes 9.4 The Mathematics of Factoring Polynomials Teaching Tips: Conceptual Challenges and Approaches Teaching Tips: Algorithmic Challenges and Approaches Instructor Overview 9.8 Tutor Simulation: Perfecting the Long Kick in Soccer Instructor Overview 9.9 Puzzle: Match Factors Instructor Overview 9.11 Project: A Cool Million Glossary 9.18 Common Core Standards 9.20 Some rights reserved. Monterey Institute for Technology and Education 2011 V1.1 "#\$

2 Unit 9 Learning Objectives Unit 9: Factoring Table of Contents Lesson 1: Factoring Monomials and Polynomials Topic 1: Factoring and the Distributive Property Learning Objectives Use the distributive property to factor a monomial out of a polynomial. Topic 2: Factoring Trinomials by Grouping 1 Learning Objectives Factor polynomials of the form x 2 + bx + c by grouping. Topic 3: Factoring Trinomials by Grouping 2 Learning Objectives Factor polynomials of the form ax 2 + bx + c by grouping. L e ar ni n g O bj e ct iv e s Lesson 2: Factoring Special Products of Polynomials Topic 1: Factoring Special Products Learning Objectives Identify and factor special products of binomials. Topic 2: Solving Quadratic Equations by Factoring Learning Objectives Solve quadratic equations using factoring techniques and express the solution(s) as a set. "#%

3 Unit 9 Media Run Times Unit 9 Lesson 1 Topic 1, Presentation 4.9 minutes Topic 1, Worked Example 1 3 minutes Topic 1, Worked Example minutes Topic 1, Worked Example minutes Topic 2, Presentation 5.4 minutes Topic 2, Worked Example minutes Topic 2, Worked Example 2 5 minutes Topic 2, Worked Example 3 3 minutes Topic 3, Presentation 4.8 minutes Topic 3, Worked Example minutes Topic 3, Worked Example minutes Topic 3, Worked Example minutes Lesson 2 Topic 1, Presentation 5 minutes Topic 1, Worked Example minutes Topic 1, Worked Example minutes Topic 1, Worked Example minutes Topic 2, Presentation 4.3 minutes Topic 2, Worked Example minutes Topic 2, Worked Example minutes Topic 2, Worked Example minutes "#&

4 Unit 9 Instructor Notes Unit 9: Factoring Instructor Notes The Mathematics of Factoring Polynomials This unit builds upon students' knowledge of factoring to teach them how to pull apart polynomials. They will learn how to use the distributive property and greatest common factors to find the factored form of binomials and how to factor trinomials with grouping. Students will also learn how to recognize and quickly factor special products (perfect square trinomials and difference of squares binomials). Finally, they'll get experience combining these techniques by using them to solve quadratic equations. Teaching Tips: Conceptual Challenges and Approaches This unit on factoring is probably one of the most difficult students will spend a lot of time carrying out multi-step, complex procedures for what will often seem to be obscure purposes. At this stage in algebra, factoring polynomials may feel like busy work rather than a means to a useful end. To combat the high level of abstraction, we begin by connecting the new mathematics to ideas students are familiar with, prime and composite numbers. Visual models and manipulatives help extend these concepts to polynomials. Topic 1 reviews the mathematics involved in finding greatest common factors before demonstrating how to factor expressions by using the distributive property in reverse to pull out the greatest common monomial from each term in a polynomial. If students are struggling with these ideas then it may be valuable to revisit the area model of multiplication. A graphical representation of prime and composite numbers can help students understand how to apply the same ideas to trinomials. "#'

5 Example The number six, which is a composite number, can be represented as the area of a rectangle like this: Students can clearly see that 3 and 2 are factors of 6. In contrast, it is impossible to make a rectangle out of seven tiles (other than the case of a rectangle with sides of 7 and 1) because seven is a prime number: This visual representation, used in Unit 8, can be continued to support the work with factoring trinomials. The example above used the area model as a basis for understanding prime and composite numbers. Students can use similar manipulatives to model the area representation of trinomials. An excellent example can be found here [MAC users will need to copy/paste url into browser]: Having students work with a manipulative like this as they begin to build their understanding, either individually or in small groups, could help all students, including English Learners, build their conceptual understanding. "#(

6 Example One key mathematical point to raise concerns building rectangles to represent trinomials such as x 2 x 6. In this case we have an x 2 *+ x, and 6. x 2 x 6 Notice that it is impossible to build a rectangle from these pieces, even though this is a factorable trinomial, with factors (x 3) and (x + 2). Explain to students that we can fill in the rectangle by adding more pieces, as long as they don't change the value of the expression. For example, pairs of +x and x will help build the rectangle but they cancel each other out in the polynomial. So let s add a positive x and a negative x to the existing tiles: 2 x 2x + x 6 We re not there yet, but you can see we re getting closer, so let s add another pair, +x and x, And now we have a rectangle. x 2 3x + 2x 6 Teaching Tips: Algorithmic Challenges and Approaches The process shown in the example above exactly models factoring by grouping. This is the technique taught in Topics 2 and 3. Many textbooks do not use factoring by grouping for trinomials, and instead use essentially a guess and check method. While factoring by grouping may initially be a more complex procedure, it has many significant advantages in the long term for students and is used in this course. "#)

7 Factoring by grouping has the great advantage of working for all trinomials, from the most basic all the way to simple cubic polynomials, for example: x 3 + 3x 2 4x 12 x 2 (x + 3) 4(x + 3) (x 2 4)(x + 3) (x 2)(x + 2)(x + 3) Students are now equipped with a powerful technique that will transfer into Algebra 2 and other mathematics courses. Summary Factoring trinomials and solving quadratic equations by factoring are some of the most abstract mathematics in Algebra 1, and students will struggle to learn the techniques because they don't see the point. Connecting these procedures to prior ideas of prime and composite numbers, and using a powerful technique like factoring by grouping, which works in all cases, will help students confidently work with this complex mathematics. "#,

8 Unit 9 Tutor Simulation Unit 9: Factoring Instructor Overview Tutor Simulation: Perfecting the Long Kick in Soccer Purpose This simulation is designed to challenge a student s understanding of factoring polynomials. Students will be asked to apply what they have learned to solve a real world problem by demonstrating understanding of the following areas: Visualizing Problems Factoring Polynomials Trajectories Problem Students are given the following problem: Your school's soccer coach spent a lot of money on a computer system that would analyze his team's kicking technique, but the system broke down before he got the results for his star player, Renaldo. It's up to you to take over and finish the trajectory calculations. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, make sure they have completed all other unit material. explain the mechanics of tutor simulations o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. emphasize that this is an exploration, not an exam. "#-

9 Unit 9 Puzzle Unit 9: Factoring Instructor Overview Puzzle: Match Factors Objective Match Factors is a puzzle that tests a player's ability to factor trinomials and provides a fun means of practicing grouping. It reinforces the technique of factoring a trinomial in the form ax 2 + bx + c by finding two integers, r and s, whose sum is b and whose product is ac. Puzzle play, especially when done by eye rather than with pencil and paper, will help students learn to quickly identify the components of factors. Figure 1. Match Factors players choose the factors of the polynomial in the center from a rotating ring of possibilities. "#"

10 Description Each Match Factors game consists of a sequence of 4 polynomials orbited by 8 possible factors. As each polynomial is displayed, players are asked to pick the matching pair of factors. If they choose correctly, the next polynomial appears. If not, they must try again before play advances. There are three levels of play, each containing 10 games. In Level 1, polynomials have the form x 2 + bx + c. Level 2 polynomials have the form ax 2 + bx + c. Players in Level 3 must factor ax 2 + bxy + cy 2 polynomials. Match Factors is suitable for individual or group play. It could also be used in a classroom setting with the whole group taking turns calling out the two factors of the expression. "#\$.

15 w(2w + 8) = w + 8w 960 = 0 2 2( w + 4w 480) = 0 2( w 20)( w + 24) = 0 w = 20, 24 Problem 4 The cost for the remaining area will vary based on the material chosen. Material Total Square Cost per Total Cost Footage square foot Adult Area Pool 8400 \$75 \$630,000 Concrete 6600 \$3 \$19,800 Kiddie Area Pool 4000 \$75 \$300,000 Concrete 1000 \$3 \$3000 Landscape Landscape 960 \$4 \$3840 Remaining 9040 Area TOTAL \$956,640 Recommendations: have students work in teams to encourage brainstorming and cooperative learning. assign a specific timeline for completion of the project that includes milestone dates. provide students feedback as they complete each milestone. ensure that each member of student groups has a specific job. Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The following are other examples of free Internet resources that can be used to support this project: Sketchup is a free graphics download that students may wish to use to draw the swimming pool complex. It may be helpful to demonstrate this or similar software to the entire class, so students can focus most of their time on the algebra. An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. "#\$(

17 Unit 9: Algebra Factoring Unit 9 Glossary Glossary binomial a sum of two monomials, such as 3x coefficient Distributive Property factor factored form of a polynomial factoring greatest common factor (GCF) grouping technique monomial a number that multiplies a variable states that the product of a number and a sum equals the sum of the individual products of the number and the addends: for all real numbers a, b, and c, a(b + c) = ab + ac for any number x, the numbers that can be evenly divided into x are called factors of x. For example, the number 20 has the factors 1, 2, 4, 5, 10, and 20. a polynomial written as a product of factors, and each nonmonomial factor has no common factors in its terms the process of breaking a number down into its multiplicative factors. Every number x has at least the numbers 1 and x as factors. the largest factor that two numbers have in common a factoring technique involving finding common factors among groups of terms rather than among all of terms a number, a variable, or a product of a number and one or more variables with whole number exponents, such as -5, x, and 8xy3 perfect square any of the squares of the integers. Since 12 = 1, 22 = 4, 32 = 9, etc., 1, 4, and 9 are perfect squares perfect square trinomial a trinomial that is the product of a binomial times itself, such as r2 + 2rs + s2 (from (r + s)2), and r2 2rs + s2 (from (r s)2) polynomial a monomial or sum of monomials, like 4x2 + 3x 10 prime factor prime factorization prime trinomial a factor that has no factors but 1 and itself. For example, 2 is a prime factor of 12 because its only factors are 1 and 2, while 6 is not a prime factor of 12 because it has more factors than 1 and 6 (i.e. 2 and 3). the process of breaking a number down into its prime factors a trinomial that cannot be factored using integers quadratic equation an equation that can be written in the form ax2 + bx + c = 0 where a 0. When written as y = ax2 + bx+ c the expression becomes a quadratic function. root of an equation trinomial any number that makes the equation true when the variable is equal to that number. That is, a solution of the equation. a three-term polynomial Zero Product Property states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0 "#\$,

18 Unit 9 Common Core NROC Algebra 1--An Open Course Unit 9 Mapped to Common Core State Standards, Mathematics Algebra 1 Factoring Factoring Monomials and Polynomials Factoring and the Distributive Property Grade: 7 - Adopted 2010 STRAND / DOMAIN CC.7.EE. Expressions and Equations CATEGORY / CLUSTER Use properties of operations to generate equivalent expressions. STANDARD 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Algebra 1 Factoring Factoring Monomials and Polynomials Factoring Trinomials by Grouping 1 Grade: Adopted 2010 STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Write expressions in equivalent forms to solve problems. EXPECTATION A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. GRADE EXPECTATION A-SSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines. Algebra 1 Factoring Factoring Monomials and Polynomials Factoring Trinomials by Grouping 2 Grade: Adopted 2010 STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Write expressions in equivalent forms to solve problems. EXPECTATION A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. "#\$-

### Unit 12: Introduction to Factoring. Learning Objectives 12.2

Unit 1 Table of Contents Unit 1: Introduction to Factoring Learning Objectives 1. Instructor Notes The Mathematics of Factoring Teaching Tips: Challenges and Approaches Additional Resources Instructor

### Learning Objectives 8.2. Media Run Times 8.3. Instructor Overview 8.8 Tutor Simulation: Roman Numerals and Polynomials

Unit 8 Table of Contents Unit 8: Polynomials Video Overview Learning Objectives 8.2 Media Run Times 8.3 Instructor Notes 8.4 The Mathematics of Monomials and Polynomials Teaching Tips: Conceptual Challenges

### Unit 4: Analyze and Graph Linear Equations, Functions, and Relations

Unit 4 Table of Contents Unit 4: Analyze and Graph Linear Equations, Functions and Relations Video Overview Learning Objectives 4.2 Media Run Times 4.3 Instructor Notes 4.4 The Mathematics of Analyzing

### Unit 5: Analyze, Solve, and Graph Linear Inequalities

Unit 5 Table of Contents Unit 5: Analyze, Solve, and Graph Linear Inequalities Video Overview Learning Objectives 5.2 Media Run Times 5.3 Instructor Notes 5.4 The Mathematics of Linear Inequalities Writing,

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### 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.

### Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results

Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At

### 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

### Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

### 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

### 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

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### Unit 10: Solving Equations and Inequalities. Learning Objectives 10.2

Unit 10 Table of Contents Unit 10: Solving Equations and Inequalities Learning Objectives 10.2 Instructor Notes The Mathematics of Solving Equations and Inequalities Teaching Tips: Challenges and Approaches

### 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:

### Unit 14: Systems of Equations and Inequalities. Learning Objectives 14.2

Unit 14 Table of Contents Unit 14: Systems of Equations and Inequalities Learning Objectives 14.2 Instructor Notes The Mathematics of Systems of Equations and Inequalities Teaching Tips: Challenges and

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

### 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,

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Curriculum Alignment Project

Curriculum Alignment Project Math Unit Date: Unit Details Title: Solving Linear Equations Level: Developmental Algebra Team Members: Michael Guy Mathematics, Queensborough Community College, CUNY Jonathan

### Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

### Factors and Products

CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

### called and explain why it cannot be factored with algebra tiles? and explain why it cannot be factored with algebra tiles?

Factoring Reporting Category Topic Expressions and Operations Factoring polynomials Primary SOL A.2c The student will perform operations on polynomials, including factoring completely first- and second-degree

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Factor Polynomials Completely

9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

### 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

### Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

### Factoring Polynomials

Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

### 6.3 FACTORING ax 2 bx c WITH a 1

290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### FACTORING OUT COMMON FACTORS

278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

### 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

### ( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

### Algebra 1. Curriculum Map

Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Tennessee Department of Education

Tennessee Department of Education Task: Pool Patio Problem Algebra I A hotel is remodeling their grounds and plans to improve the area around a 20 foot by 40 foot rectangular pool. The owner wants to use

### 4/27/2010 ALGEBRA 1 CHAPTER 8 FACTORING. PAR Activities Cara Boening

4/27/2010 ALGEBRA 1 CHAPTER 8 FACTORING PAR Activities Cara Boening Table of Contents Preparation Strategies... 3 Scavenger Hunt... 4 Graphic Organizer Chapter 8.... 6 Word Inventory... 8 TEASE... 12 Anticipation

### 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

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Pearson Algebra 1 Common Core 2015

A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).

### How To Solve Factoring Problems

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

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

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

### Factoring Algebra- Chapter 8B Assignment Sheet

Name: Factoring Algebra- Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.

### 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

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

### Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

### Multiplying and Factoring Notes

Multiplying/Factoring 3 Multiplying and Factoring Notes I. Content: This lesson is going to focus on wrapping up and solidifying concepts that we have been discovering and working with. The students have

### 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,...}

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

### Successful completion of Math 7 or Algebra Readiness along with teacher recommendation.

MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 8-11 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION

### Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.

5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological

### Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

### A Systematic Approach to Factoring

A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

### In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials

Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials

Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:

### Polynomial Expressions and Equations

Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### Polynomials and Polynomial Functions

Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions Overview Number of instruction days: 13-15 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Prove

### AIP Factoring Practice/Help

The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

### Polynomial Equations and Factoring

7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

### Algebra II New Summit School High School Diploma Program

Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

### Factoring Polynomials

Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,

### SOL Warm-Up Graphing Calculator Active

A.2a (a) Using laws of exponents to simplify monomial expressions and ratios of monomial expressions 1. Which expression is equivalent to (5x 2 )(4x 5 )? A 9x 7 B 9x 10 C 20x 7 D 20x 10 2. Which expression

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### Florida Math for College Readiness

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

### Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials

visit us at www.cpm.org Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials For more information about the materials presented, contact Chris Mikles mikles@cpm.org From CCA

### A. Factoring out the Greatest Common Factor.

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

### Algebra II A Final Exam

Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### 15.1 Factoring Polynomials

LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE