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

Size: px
Start display at page:

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

Transcription

1 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 and Approaches Teaching Tips: Algorithmic Challenges and Approaches Instructor Overview 8.8 Tutor Simulation: Roman Numerals and Polynomials Instructor Overview 8.9 Puzzle: Polynomial Poke Instructor Overview 8.11 Project: It's All Fun and Games Glossary 8.16 Common Core Standards 8.17 Some rights reserved. Monterey Institute for Technology and Education 2011 V1.1 "#$

2 Unit 8 Learning Objectives Unit 8: Polynomials Lesson 1: Operations on Monomials Learning Objectives Topic 1: Multiplying and Dividing Monomials Learning Objectives Multiply and divide monomials. Lesson 2: Operations on Polynomials Topic 1: Polynomials Learning Objectives Identify monomials, binomials and polynomials. Write polynomials to describe real world situations. Topic 2: Adding and Subtracting Polynomials Learning Objectives Add and subtract polynomials. Topic 3: Multiplying polynomials Learning Objectives Multiply polynomials and collect the like terms of the resulting sum of monomials. Topic 4: Special Products of Polynomials Learning Objectives Identify and multiply binomial products. L e ar ni n g O bj e ct iv e s "#%

3 Unit 8 Media Run Times Unit 8 Lesson 1 Topic 1, Presentation 4 minutes Topic 1, Worked Example minutes Topic 1, Worked Example 2 4 minutes Topic 1, Worked Example minutes Lesson 2 Topic 1, Presentation 3.8 minutes Topic 1, Worked Example minutes Topic 1, Worked Example minutes Topic 2, Presentation 4.4 minutes Topic 2, Worked Example 1 2 minutes Topic 2, Worked Example minutes Topic 2, Worked Example minutes Topic 3, Presentation minutes Topic 3, Worked Example minutes Topic 3, Worked Example minutes Topic 3, Worked Example minutes Topic 4, Presentation 5.3 minutes Topic 4, Worked Example minutes Topic 4, Worked Example 2 5 minutes Topic 4, Worked Example minutes "#&

4 Unit 8 Instructor Notes Unit 8: Polynomials Instructor Notes The Mathematics of Monomials and Polynomials Unit 8 introduces polynomials and teaches students how to work with them no matter how many terms they contain (in this course, monomials are included in the definition of polynomials). Students will learn how to carry out all the basic mathematical operations on polynomials. They ll also gain experience writing polynomials from verbal descriptions of real world situations. The ability to work fluently with polynomials will be critical for students who progress into higher math classes like Algebra 2 and beyond. Teaching Tips: Conceptual Challenges and Approaches Working with polynomials can present a significant challenge for many students. Most of the mathematics concerns symbolic manipulation, and if students don t build a meaningful conceptual understanding of how and why the techniques work, they will get lost as the terms become more numerous and complicated. Using a visual model can be very helpful when working with polynomials. Try beginning with terms that describe real objects, which can be sketched out and then manipulated and counted up. "#'

5 Example In the presentation for Lesson 2, Topic 3, the concept of multiplying two polynomials is introduced through the visual of planting a garden. Students see the result of multiplying polynomials as an understandable collection of objects instead of just symbols. They can count the various vegetables, compare them to the original terms, and by doing so, learn to appreciate what actually happens when polynomials are multiplied. Hands-On Opportunities The example above used the area model as a basis for understanding polynomial multiplication. Students can use virtual algebra tiles for further practice of this technique. One of the most useful virtual manipulative websites can be found here [MAC users will need to copy/paste url into browser]: Sketches and manipulatives are powerful tools that can help students build understanding and practice techniques. However, there are two very important ideas to keep in mind: 1. Students will need significant guidance to understand manipulatives, especially in the beginning. We suggest demonstrating any virtual tool in the classroom, either just by projecting the image and solving the problem on the computer, or better still, by using an interactive whiteboard and showing students how to solve this "#(

6 problem. Students could also use the board to demonstrate their ideas about how to take next steps. After seeing it done in the classroom, they ll be in a position to work with a tool like this either alone or in small groups. 2. Visual representation tools help students get started, but they are not an alternative method for ultimately doing the mathematics. Students still need to be fluent with the relevant symbolic manipulation. It is very important to discuss the connections between a visual model and its symbolic counterpart when working with polynomials. Teaching Tips: Algorithmic Challenges and Approaches It s tempting to teach students tricks for memorizing algebraic techniques. A lot of traditional teaching materials suggest using the acronym FOIL as a mnemonic device to remember how to multiply two binomials. While there is nothing inherently wrong with this memorization approach, it does have limitations. This mnemonic only works when multiplying two binomials when students who have grown comfortable with it are confronted by a more challenging situation like x + y + 2 ( )( 3 2x), they ll often struggle. As a result, it is more productive to teach a more general rule from the beginning: When multiplying two polynomials, multiply everything in the first parenthesis to everything in the second parenthesis. That approach is used exclusively in this course. If students are struggling to keep track of all of the terms when multiplying polynomials, they may find it useful to create a rectangular table (which is obviously connected to the visual area model) to diagram this operation. Example ( x + y + 2) ( 3 2x) x y 2 3 3x 3y 6 2x 2x 2 2xy 4x Once students have completed the multiplication, they can easily collect the like terms and find the answer. "#)

7 Summary This unit focuses on the addition, subtraction, multiplication, and division of polynomials. It uses general rules and visual models to explain the conceptual basis and the procedures involved in these operations. Students who struggle with these ideas may benefit from virtual or hands-on manipulatives, but they must learn how to carry out strictly symbolic manipulations. "#*

8 Unit 8 Tutor Simulation Unit 8: Polynomials Instructor Overview Tutor Simulation: Roman Numerals and Polynomials Purpose This simulation is designed to challenge a student s understanding of polynomials. Students will be asked to apply what they have learned to solve a real world problem by demonstrating understanding of the following areas: Polynomials Multiplying Polynomials The Distributive Property The Associative Property The Commutative Property Applying Properties to Polynomials Problem Students are given the following problem: You will take a look at Roman numerals and see how working with them is similar to working with polynomials. Once familiar with Roman numerals, you'll learn how to multiply them, then apply the same steps to multiply polynomials. 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 8 Puzzle Unit 8: Polynomials Instructor Overview Puzzle: Polynomial Poke Objective Polynomial Poke challenges students' familiarity with polynomial nomenclature. To play the game successfully, they must be able to distinguish between cubic, quadratic, and linear terms, and recognize monomials, binomials, and trinomials. Figure 1. Polynomial Poke asks players to pop balloons that contain specified types of polynomials. "#+

10 Description There are three levels in this puzzle, which each consist of 10 groups of floating balloons containing polynomials. In the first level, learners are challenged to pop balloons in order of degree of monomials, from cubic to quadratic to linear. In the second level, players must pop balloons depending on the number of terms in their polynomials. In the third level, players are asked to pop only those balloons that contain a specified degree of polynomial. Players earn points for correct answers and lose points for popping balloons out of sequence. The puzzle is primarily designed for a single player but in a classroom it could be played in a group with learners identifying the order or the degree and calling out the balloon for one to pop. "#$,

11 Unit 8 Project Unit 8: Polynomials Instructor Overview Project: It's All Fun and Games Student Instructions Introduction In algebra variables are used to represent unknowns. When first beginning algebra, the symbolic representation can be difficult. By now you should be quite comfortable with x and y, as symbols for unknowns, however, the roots of mathematics are engrained in complex symbol-based number systems. Get ready to explore these ancient systems and become an expert at interpreting and using the symbols found within Task Working together with your group, you will research one of four ancient number systems. Then, based upon what you have learned, you will design a team game based on that number system. The game needs to be complete with rules, scoring guidelines, and dimensions of the field based on the ancient number system. Finally, you will calculate the perimeter of your field using the number system. Instructions Solve each problem in order. Save your work along the way, as you will create a presentation at the conclusion of the project. Your audience will be the Mayan, Egyptian, Sumerian, or Roman people. You may use multi-media, make a movie, or create a website to highlight your game and how it connects to the number system. 1 First problem: With your group, choose one of the ancient number systems below to research. You may use the following links to begin, but there are multiple websites dedicated to the number systems. Egyptian: Mayan: Roman: Sumerian: "#$$

12 Use the following questions to guide your research: How do you write the following numbers: 1, 5, 10, 20, 50, & 100? What is the base of the number system? How has the number system impacted the base ten system that is used today? How do you perform basic addition using the number system? 2 Second problem: Once your group has a good understanding of the number system, begin thinking about games that are played today and how they might be adapted to fit with the number system. For instance, consider football. The field is based on 100 yards and advancing the ball is based on ten-yard increments. This game would fit well for the Egyptians and Romans, but not for the Sumerians and Mayan. Creativity and originality will make your presentation stand out. A good example of a made-up game is Quidditch from the Harry Potter book series. Information about the fictional game, including rules and field dimensions can be found at: First, decide the dimensions of your field. Be sure to keep in mind the foundation of your number system when making your decisions. Your dimensions should be written using the symbols from the system. You can draw a picture of your field using drafting software such as, Google Sketchup. The free download is available at If you have an artistic flair, a hand-drawn field or court is another option. 3 Third Problem: Now your team needs to work on developing rules and scoring guidelines for your game. What tools are necessary to play? How many points are various tasks worth? How many points are necessary to win? How many players are on the field at once? Make sure to consider how the answers to each of these questions would be impacted based on the number system that is studied. Hint: (Remember that you will be presenting the game at the end of the project. It may help to actually go outside and attempt to play the game in order to discover what works and what does not.) 4 Fourth problem: Your final task is to calculate the perimeter of the field, using the number system studied. Begin by learning to add small numbers within the system and then work your way to larger numbers. You will need to include the detailed process that was used to add the perimeter as part of your presentation. "#$%

13 Collaboration Get together with another group to discuss your game and how it is played. Discuss how the rules and field dimensions relate to what you have studied about the number system. Finally, work together to check each other s perimeter calculations. While reviewing the perimeter calculations, answer the following questions: How is adding within your number system related to adding polynomials? Do you see any other ties to algebra within your calculations? Conclusions Now you will get to present your game using modern technology to your classmates, who will be considered citizens of the ancient civilization that you researched. Some options for the presentation include using multi-media, making a movie, or creating a website to highlight your game and how it connects to the number system. Your presentation should include answers to each of the four problems above. Instructor Notes Assignment Procedures Problem 1 It is important for students to master basic addition and regrouping within the number system they have chosen before moving on. By the end, each group will calculate the perimeter of the playing field. Without a solid understanding of basic addition within their system, completing more difficult calculations will not be possible. Problem 4 If a group chooses to make their field LXV by XXV, the calculation for perimeter would be: By Addition: LXV + LXV + XXV + XXV By combining like terms: LLXXXXXXVVVV By simplifying: LL = C and VVVV = XX By substitution: CXXXXXXXX By simplifying: XXXXX=L By substitution: CLXXX By showing their work and justifying each step, the students should be able to see how addition within the number system relates very closely to adding polynomials in algebra. 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. "#$&

14 Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. The following are other examples of free Internet resources that can be used to support this project: 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. or Allows you create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work with parents. Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose. Rubric Score Content Presentation 4 Your project appropriately answers each of the problems. Background research is thorough. A detailed drawing of the game field is given, with dimensions labeled using the number system. Rules and scoring guidelines are complete and relate to the number system studied. A detailed calculation of perimeter, in the number system, is included. 3 Your project appropriately answers each of the problems. Background research is thorough. A detailed drawing of the game field is given, with dimensions labeled using the Your project contains information presented in a logical and interesting sequence that is easy to follow. Your project is professional looking with graphics and attractive use of color. Your project contains information presented in a logical sequence that is easy to follow. Your project is neat with graphics and attractive use of color. "#$'

15 number system. Minor errors may be noted. Rules and scoring guidelines are complete and relate to the number system studied. A detailed calculation of perimeter, in the number system, is included. Minor errors may be noted. 2 Your project attempts to answer each of the problems. Background research is present, but not complete. A drawing of the game field is given, with dimensions labeled using the number system. Major errors may be noted &/or some information is missing. Rules and scoring guidelines present and relate to the number system studied. The perimeter is given, but the detailed work used to obtain the answer is not given. Major errors may also be noted. 1 Your project attempts to answer some of the problems. Major errors are noted and information is missing. Your project has minimal information on rules and scoring guidelines. The perimeter calculation is missing. Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. Your project contains low quality graphics and colors that do not add interest to the project. Your project is difficult to understand because there is no sequence of information. Your project is missing graphics and uses little to no color. "#$(

16 Unit 8 Glossary Glossary Unit 8: Algebra - Polynomials area model a graphic representation of a multiplication problem, in which the length and width of a rectangle are the factors and the area is the product binomial a sum of two monomials, such as 3x coefficient like terms monomial a number that multiplies a variable two or more monomials that contain the same variables raised to the same powers, regardless of their coefficients. For example, 2x2y and -8x2y are like terms because they have the same variables raised to the same exponents. 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 polynomial a monomial or sum of monomials, like 4x2 + 3x 10 special product term a product resulting from binomial multiplication that has certain characteristics. For example x2 25 is called a special product because both its terms are perfect squares and it can be factored into (x + 5)(x 5). a value in a sequence--the first value in a sequence is the 1st term, the second value is the 2nd term, and so on; a term is also any of the monomials that make up a polynomial "#$)

17 Unit 8 Common Core NROC Algebra 1--An Open Course Unit 8 Mapped to Common Core State Standards, Mathematics Algebra 1 Polynomials Operations on Monomials Multiplying and Dividing Monomials Grade: Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). Algebra 1 Polynomials Operations on Polynomials Polynomials Grade: Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-CED. Creating Equations Create equations that describe numbers or relationships. EXPECTATION A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling "#$*

18 context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.F. Functions CATEGORY / CLUSTER F-BF. Building Functions Build a function that models a relationship between two quantities. EXPECTATION F-BF.1. Write a function that describes a relationship between two quantities. GRADE EXPECTATION F-BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Algebra 1 Polynomials Operations on Polynomials Adding and Subtracting Polynomials Grade: Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, "#$"

19 subtraction, and multiplication; add, subtract, and multiply polynomials. Algebra 1 Polynomials Operations on Polynomials Multiplying Polynomials Grade: 7 - Adopted 2010 STRAND / DOMAIN CC.7.EE. Expressions and Equations CATEGORY / CLUSTER Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Grade: Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. "#$+

20 Algebra 1 Polynomials Operations on Polynomials Special Products of Polynomials Grade: 7 - Adopted 2010 STRAND / DOMAIN CC.7.EE. Expressions and Equations CATEGORY / CLUSTER Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Grade: Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials EdGate Correlation Services, LLC. All Rights reserved EdGate Correlation Services, LLC. All Rights reserved. Contact Us - Privacy - Service Agreement "#%,

Learning Objectives 9.2. Media Run Times 9.3

Learning Objectives 9.2. Media Run Times 9.3 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

More information

Unit 5: Analyze, Solve, and Graph Linear Inequalities

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,

More information

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

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

More information

Unit 12: Introduction to Factoring. Learning Objectives 12.2

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

More information

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

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

More information

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

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

More information

Polynomial Operations and Factoring

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.

More information

Curriculum Alignment Project

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

More information

Unit 10: Solving Equations and Inequalities. Learning Objectives 10.2

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

More information

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

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

More information

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

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:

More information

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 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

More information

Polynomials and Quadratics

Polynomials and Quadratics 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

More information

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

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

More information

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

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

More information

Mathematics Curriculum

Mathematics Curriculum Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,

More information

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

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

More information

CAHSEE on Target UC Davis, School and University Partnerships

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,

More information

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

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,

More information

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 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...

More information

POLYNOMIAL FUNCTIONS

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

More information

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

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

More information

Pearson Algebra 1 Common Core 2015

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).

More information

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 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

More information

Factors and Products

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

More information

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

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

More information

Tennessee Department of Education

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

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

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

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

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,

More information

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

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

More information

Factoring Trinomials: The ac Method

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

More information

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).

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,

More information

Multiplying and Factoring Notes

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

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

Factoring Quadratic Trinomials

Factoring Quadratic Trinomials 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

More information

How To Factor By Gcf In Algebra 1.5

How To Factor By Gcf In Algebra 1.5 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

More information

Veterans Upward Bound Algebra I Concepts - Honors

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

More information

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

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

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

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:

More information

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

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?

More information

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

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

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

A Quick Algebra Review

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

More information

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

More information

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 % Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

More information

Polynomials and Polynomial Functions

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

More information

1.3 Algebraic Expressions

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,

More information

1.3 Polynomials and Factoring

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.

More information

CPM Educational Program

CPM Educational Program CPM Educational Program A California, Non-Profit Corporation Chris Mikles, National Director (888) 808-4276 e-mail: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few

More information

Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Algebra I Teacher Notes Expressions, Equations, and Formulas Review Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

Factoring Polynomials

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

More information

2.3. Finding polynomial functions. An Introduction:

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

More information

ALGEBRA I (Created 2014) Amherst County Public Schools

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

More information

A Systematic Approach to Factoring

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

More information

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.

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

More information

DRAFT. Algebra 1 EOC Item Specifications

DRAFT. Algebra 1 EOC Item Specifications DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations. Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable

More information

Factoring Quadratic Expressions

Factoring Quadratic Expressions Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

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

More information

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

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

More information

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

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

More information

Algebra II Unit Number 4

Algebra II Unit Number 4 Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

Chapter R.4 Factoring 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

More information

Factoring and Applications

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

More information

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

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

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

Prentice Hall. California Edition of Algebra 1 - Classics Edition (Smith/Charles) 2008. Grade 8

Prentice Hall. California Edition of Algebra 1 - Classics Edition (Smith/Charles) 2008. Grade 8 Prentice Hall Grade 8 California Edition of Algebra 1 - Classics Edition (Smith/Charles) 2008 C O R R E L A T E D T O California s Map for a Basic Grade Level Program Grade 8 PROGRAM DESCRIPTION Prentice

More information

Factor Polynomials Completely

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

More information

Answer Key for California State Standards: Algebra I

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.

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

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.

More information

a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one.

a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one. FACTORING TRINOMIALS USING THE AC METHOD. Factoring trinomial epressions in one unknown is an important skill necessary to eventually solve quadratic equations. Trinomial epressions are of the form a 2

More information

Prerequisite: MATH 0302, or meet TSI standard for MATH 0305; or equivalent.

Prerequisite: MATH 0302, or meet TSI standard for MATH 0305; or equivalent. 18966.201610 COLLIN COLLEGE COURSE SYLLABUS Course Number: MATH 0305.XS1 Course Title: Beginning Algebra Course Description: With an emphasis on developing critical thinking skills, a study of algebraic

More information

Algebra 1 Course Information

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

More information

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

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,...}

More information

Manhattan Center for Science and Math High School Mathematics Department Curriculum

Manhattan Center for Science and Math High School Mathematics Department Curriculum Content/Discipline Algebra 1 Semester 2: Marking Period 1 - Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types

More information

Higher Education Math Placement

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

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

15.1 Factoring Polynomials

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

More information

Polynomials and Factoring. Unit Lesson Plan

Polynomials and Factoring. Unit Lesson Plan Polynomials and Factoring Unit Lesson Plan By: David Harris University of North Carolina Chapel Hill Math 410 Dr. Thomas, M D. 2 Abstract This paper will discuss, and give, lesson plans for all the topics

More information

6.1 Add & Subtract Polynomial Expression & Functions

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

More information

MATH 90 CHAPTER 6 Name:.

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

More information

Algebra 1. Curriculum Map

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

More information

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

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

More information

Mathematics Georgia Performance Standards

Mathematics Georgia Performance Standards Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by

More information

Florida Math for College Readiness

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

More information

MAT 096, ELEMENTARY ALGEBRA 6 PERIODS, 5 LECTURES, 1 LAB, 0 CREDITS

MAT 096, ELEMENTARY ALGEBRA 6 PERIODS, 5 LECTURES, 1 LAB, 0 CREDITS 1 LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK MATHEMATICS, ENGINEERING and COMPUTER SCIENCE DEPARTMENT FALL 2015 MAT 096, ELEMENTARY ALGEBRA 6 PERIODS, 5 LECTURES, 1 LAB, 0 CREDITS Catalog

More information

SPECIAL PRODUCTS AND FACTORS

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

More information

Grade 5 Math Content 1

Grade 5 Math Content 1 Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Mathematics Common Core Sample Questions

Mathematics Common Core Sample Questions New York State Testing Program Mathematics Common Core Sample Questions Grade The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

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

More information

Prentice Hall Mathematics, Algebra 1 2009

Prentice Hall Mathematics, Algebra 1 2009 Prentice Hall Mathematics, Algebra 1 2009 Grades 9-12 C O R R E L A T E D T O Grades 9-12 Prentice Hall Mathematics, Algebra 1 Program Organization Prentice Hall Mathematics supports student comprehension

More information

POLYNOMIALS and FACTORING

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

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information