# Polynomials and Polynomial Functions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions Overview Number of instruction days: (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Prove polynomial identities and use them in describing relationships between numbers. Use the characteristics and structure of an expression to recognize ways to rewrite it. Know and apply the Remainder Theorem. Identify zeros of polynomials when suitable factorizations are available. Calculate and interpret the average rate of change of a polynomial function over a specified interval. 4 Model with mathematics. Model real-world situations using technology to represent regressions 5 Use appropriate tools strategically. Use graph paper and technology (TI-Nspire) to graph and analyze polynomial functions. 7 Look for and make use of structure. Rewrite polynomial expressions in equivalent forms. Interpret key features of graphs and tables. Create and graph polynomial equations. Graph polynomial functions, identify zeros when factorizations are available, and show and describe end behavior. Recognize structural similarities between integers and polynomials. Discern patterns in factoring polynomials. Explain why the x-coordinates of the points of intersection are the solutions of the equation f(x) = g(x). Providence Public Schools D-28

2 Polynomials and Polynomial Functions (13-15 days) Algebra II, Quarter 1, Unit 1.4 Essential Questions How do you simplify polynomial expressions? Why is it important to rewrite polynomial expressions in different forms? What can you determine from different features of the graph of a polynomial function? What is the relationship between zeros and factors of polynomials? How is division of polynomials connected to the Remainder Theorem? How can key features of a polynomial function be identified given different representations? What real-world situations can be modeled with polynomial functions? Standards Common Core State Standards for Mathematical Content Algebra Arithmetic with Polynomials and Rational Expressions A-APR Understand the relationship between zeros and factors of polynomials A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Use polynomial identities to solve problems A-APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Providence Public Schools D-29

3 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (13-15 days) Seeing Structure in Expressions A-SSE Interpret the structure of expressions [Polynomial and rational] 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 ). [A-SSE.2 is embedded throughout most Algebra II units.] Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Represent and solve equations and inequalities graphically A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models] F-IF.4 F-IF.6 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. D-30 Providence Public Schools

4 Polynomials and Polynomial Functions (13-15 days) Algebra II, Quarter 1, Unit 1.4 c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Providence Public Schools D-31

7 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (13-15 days) Analyze the characteristics of graphs and tables of polynomial functions. Solve polynomial equations with and without technology. Determine the number and types of roots of polynomials. Calculate the average rate of change of a function. Use multiple representations to determine the characteristics of polynomial graphs. Build polynomial functions to relate problem situations. Graph polynomial function derived from given roots or solutions. Identify the roots, extreme values and symmetry of a quadratic equation and interpret these in a real-world situation using factoring. Find the number and types of zeros of polynomial functions. Instruction Learning Objectives Students will be able to Apply the rules of polynomial identities when simplifying expressions. Generate equivalent polynomials expressions. Use long division to divide and simplify polynomial expressions. Use synthetic division to divide, simplify polynomial expressions. Apply the Remainder Theorem using synthetic division to evaluate a function. Determine whether a binomial is a factor of polynomial by using synthetic substitution. Determine the number and or types of roots for a polynomial functional. Read, understand, and interpret characteristics of graphs and tables of polynomial functions. Sketch a graph of a polynomial function given the zeros. Use graphing technology to find the solution(s) of polynomial equations. D-34 Providence Public Schools

8 Polynomials and Polynomial Functions (13-15 days) Algebra II, Quarter 1, Unit 1.4 Calculate and determine the average rate of change of a function over a specified interval. Use the degree of the polynomial to identify the key characteristics. Use a variety of techniques (including factoring) to solve polynomials. Build polynomial equations to relate and solve problem situations. Demonstrate understanding of concepts and skills learned in this unit. Resources Algebra 2, Glencoe McGraw-Hill, 2010: Student/Teacher Editions Sections (pp ) Extend 6-3 Algebra Lab, Polynomial Functions and Rate of Change (pp. 356) Chapter 6 Resources Masters Glencoe McGraw-Hill Online Interactive Classroom CD (PowerPoint Presentations) Exam View Assessment Suite Representing Polynomials: See the Supplementary Materials section of this binder for notes for this activity. Graphic Organizer: Synthetic Division. See the Supplementary Materials section of this binder for notes for this activity. Graphic Organizer: Finding the zeros for a Polynomial Function. See the Supplementary Materials section of this binder for notes for this activity. TI-Nspire Teacher Software Education.TI.com: End Behavior of Polynomial Functions. See the Supplementary Materials Section of this Binder for the Student and Teacher materials. Education.TI.com: Polynomials Factors, Roots, and Zeros. See the Supplementary Materials Section of this Binder for the Student and Teacher materials. Providence Public Schools D-35

9 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (13-15 days) Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TI-Nspire Graphing Calculator, gridded chart paper, markers, glue, construction paper and sentence strips Instructional Considerations Key Vocabulary depressed polynomial synthetic division synthetic substitution end behavior Remainder Theorem Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: binomial, coefficient, degree, monomial, terms, degree of a polynomial, trinomial, long division. roots, zeros, polynomial function, binomial, coefficient, intercepts, intervals of increasing, decreasing, positive, negative, relative maximums and minimums; symmetries, zero product property, degree of a polynomial, completing the square, and solutions. The Mathematics Assessment Project, also known as Math Shell, provides a variety of projects for formative and summative assessment that may help make knowledge and reasoning visible to students. These CCSSM aligned resources include a variety of lessons focused on concept development and problem solving. It also includes numerous tasks designed for expert, apprentice and novice learners. The Representing Polynomials lesson is intended to help teachers assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in: Recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials. Recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x)... D-36 Providence Public Schools

10 Polynomials and Polynomial Functions (13-15 days) Algebra II, Quarter 1, Unit 1.4 Use the link below to access detailed information for the Representing Polynomials lesson: The lesson resources are also provided in the supplementary materials section of this curriculum frameworks binder. Post essential questions for this unit around the classroom at the beginning of the unit. Use posted Essential Questions upon completion of an objective to review and close the lesson. Essential Questions can also be as an Exit Ticket to formatively assess student understanding. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples and/or diagrams to accompany the definitions. Additionally, on page 388 at the beginning of Chapter 6, the foldable organizer has students use nonlinguistic representations. Using physical models while summarizing and taking notes helps students make sense of the attributes of polynomials and polynomial functions. The Glencoe resources listed above should provide ample choices to engage students in operations with polynomials. The Math in Motion and Personal Tutors animations on Glencoe.com also offer additional visual support for students struggling with these concepts. Have the class discuss the following: Given 4x 2 x, identify the leading coefficient, degree, and base. Explain what the degree tells you about possible solutions. Then break students into groups of two or three. Give each group of students two different polynomials (i.e., 3x 4 + 4x 2 + x 3 and 3x 2 3) written on large strips of paper. Individually, students should compare the two polynomials to help them decide what similarities and differences are present. Students then write the similarities and differences on individual sentence strips. Then have the group members discuss their strips together. When they agree on what they consider to be the accurate and complete list of similarities and differences, have students place their strips into a Venn diagram. Groups will share their finished task. Each group will evaluate the other groups work. Students should use their prior knowledge of previously studied functions to extend their conceptual understanding of polynomial functions. Students should be encouraged to use technology and connect the study of polynomial functions to earlier work with quadratic functions. Provide opportunities for students to display their results on gridded chart paper as they analyze functions and their graphs using multiple representations. The more representations that students see of applications of polynomial functions, the better connected their understanding of the concept. Students should be able to discern from a graph the solutions of a polynomial and write that polynomial and give all its characteristics. A group of students could also look at a graph and be given the task of writing their own polynomials given one or two roots. Have students then share out their polynomials with the whole group, giving their justification for the polynomial they have constructed. Algebra II students should learn how to find zeros from factored form and should also learn why this works (the zero product property). Technology can also be used to support this learning, including Providence Public Schools D-37

11 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (13-15 days) mathematical notations indicating exact or approximate solutions. The End Behavior of Polynomial Functions activity on education.ti.com provides the opportunity for students to use technology to find the similarities and differences among functions. The Polynomials Factors, Roots, and Zeros activity uses technology to analyze multiple representation s of polynomial functions. Additional TI-Nspire resources can be found using the TI-Nspire Teacher Software on your work computer. The content tab of the TI-Nspire desktop software contains links to the Algebra 2, Glencoe McGraw-Hill, 2010 textbook. These resources are accessible by chapter and section. Students could use the song The Twelve Days of Christmas to use a real-world example of a cubic polynomial (constant third difference). The table below shows the cubic relationship. Number of Days Total Gifts Received Problems 1. Doctors examine cardiac output in potential heart attack patients by checking the concentration of dye after a known amount is injected in a vein near the heart. In a healthy heart, the amount of dye in the bloodstream after t seconds can be expressed by the function f(t) = 0.006t t t t. Explain how the roots of this equation can be used in cardiology. Include an explanation of what the roots of this equation reveal about the concentration of dye in a healthy heart. 2. The average amount P (in dollars), donated by a company annually to any charitable organization, can be modeled by the equation P = 20(t 4 2.4t 3 28t t +756), where t = 0 represents the year What is the trend for the donations of this company? 3. Jim is saving a percentage of his income each month for a vacation. His savings for the last 6 months can be modeled by the equation S = 0.333m m m m 26.66, where S represents amount saved, given as a percentage, and m represents the number of months. Explain Jim s savings plan. 4. An estimated calorie requirement (in kilocalories) of physically active males and females can be modeled by the equations C m = 0.002x x x x and C f = x x x x , where C m represents calorie requirements of males, C f represents calorie requirements of females, and x represents age. What are the limitations of this model? D-38 Providence Public Schools

12 Polynomials and Polynomial Functions (13-15 days) Algebra II, Quarter 1, Unit The minimum and maximum mean temperature (in degrees Fahrenheit) of a particular city for the months January to December can be modeled by the equation T min = 0.04x x x x and T max = 0.042x x x x , respectively. What are the limitations of this model? 6. A garden bed is filled with potting soil in the shape of a rectangular prism. The length is depth squared and the width is 3 times the depth. The gardener plans to change the dimensions of the bed. The new volume of potting soil can be modeled by the equation V(h) = 2h 4 + 9h h h + 16, where h is the depth of the potting soil. What are the limitations of this equation? 7. The weight w, in pounds, of a patient during a 7-week illness is modeled by the cubic equation w(t) = 0.1n 3 0.6n , where n is the number of weeks since the patient became ill. What trends in the patient s weight does the graph suggest? Is it reasonable to assume the trend will continue indefinitely? The following rubric could be used to evaluate students work on the problems above. Points Graph 5 Graph contains an appropriate title, scales are uniform, and axes are labeled. 5 Roots of the function are accurate and labeled. 10 Intervals of the function are accurate and indicated. 10 x- and y-intercepts are accurate and labeled. Explanation 20 Explain the similarities of zeros, roots, solutions, and x-intercepts. 10 Explain rates of change between the intervals. 10 Explain all local maximum and minimum points. 10 Explain all the points of inflection and end behavior of the function. Justify 10 Appropriate domain and range of the function. 10 An accurate solution to its given problem. 100 TOTAL Providence Public Schools D-39

13 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (13-15 days) In this unit students connect division of polynomials with longg division of integers. They will learn how to find the quotient of two polynomials through synthetic division and long division. Students should be able to explain how synthetic division can be used to write the factorization of a polynomial. Be sure to connect division of polynomials by monomials to operations with rational numbers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. This unit develops the structural similarities between the system of polynomials andd the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Consider using a synthetic division matching activity as a Doo Now or an exit activity. In this activity, students match the polynomials on the left with their roots on the right. Sample tables are providedd below. Notes D-40 Providence Public Schools

Algebra II, Quarter 1, Unit 1.3 Quadratic Functions: Complex Numbers Overview Number of instruction days: 12-14 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Develop

Mathematical Models with Applications, Quarter 3, Unit 3.1 Quadratic and Linear Systems Overview Number of instruction days: 5-7 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be

### 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.

### 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

### Unit 2 Quadratic Equations and Polynomial Functions Algebra 2

Number of Days: 29 10/10/16 11/18/16 Unit Goals Stage 1 Unit Description: Students will build on their prior knowledge of solving quadratic equations. In Unit 2, solutions are no longer limited to real

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Modeling in Geometry

Modeling in Geometry Overview Number of instruction days: 8-10 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Use geometric shapes and their components to represent

### Overview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices

Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with

### 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

### 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

### Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres

Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders,

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### Algebra Nation MAFS Videos and Standards Alignment Algebra 2

Section 1, Video 1: Linear Equations in One Variable - Part 1 Section 1, Video 2: Linear Equations in One Variable - Part 2 Section 1, Video 3: Linear Equations and Inequalities in Two Variables Section

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.4 Interpret, Solve, and Graph Inverse Trigonometric Functions

Trigonometric Functions Overview Number of instruction days: 3 5 (1 day = 53 minutes) Content to Be Learned Use restricted domains in order to construct inverse Use inverse trigonometric functions to solve

### 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.

### 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

### Overview. Essential Questions. Grade 7 Mathematics, Quarter 3, Unit 3.3 Area and Circumference of Circles. Number of instruction days: 3 5

Area and Circumference of Circles Number of instruction days: 3 5 Overview Content to Be Learned Develop an understanding of the formulas for the area and circumference of a circle. Explore the relationship

### 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).

### PARCC MODEL CONTENT FRAMEWORK FOR MATHEMATICS. Algebra I Overview FOR ALGEBRA I

PARCC MODEL CONTENT FRAMEWORK FOR MATHEMATICS FOR ALGEBRA I Algebra I Overview Numerals in parentheses designate individual content standards that are eligible for assessment in whole or in part. Underlined

### Grade 4 Mathematics, Quarter 4, Unit 4.3 Using Place Value to Add and Subtract Whole Numbers to the Millions. Overview

Whole Numbers to the Millions Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Round multi-digit whole numbers using understanding of place value. Recognize that the

### 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

### Describing and Solving for Area and Perimeter

Grade 3 Mathematics, Quarter 2,Unit 2.2 Describing and Solving for Area and Perimeter Overview Number of instruction days: 8-10 (1 day = 90 minutes) Content to Be Learned Distinguish between linear and

### Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities. Number of instruction days: 5 7 (1 day = 53 minutes)

Precalculus, Quarter, Unit.5 Proving Trigonometric Identities Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Verify proofs of Pythagorean identities. Apply Pythagorean,

### Course Title: Honors Algebra Course Level: Honors Textbook: Algebra 1 Publisher: McDougall Littell

Course Title: Honors Algebra Course Level: Honors Textbook: Algebra Publisher: McDougall Littell The following is a list of key topics studied in Honors Algebra. Identify and use the properties of operations

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### 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

### Wentzville School District Algebra 1: Unit 9 Stage 1 Desired Results

Wentzville School District Algebra 1: Unit 9 Stage 1 Desired Results Unit 9 - Quadratic Functions Unit Title: Quadratics Functions Course: Algebra I Brief Summary of Unit: At the end of this unit, students

### Georgia Department of Education. Calculus

K-12 Mathematics Introduction Calculus The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety

### PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

### Understanding Place Value of Whole Numbers and Decimals Including Rounding

Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned

### West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12

West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12 Unit 1: Polynomials and Factoring Course & Grade Level: Algebra I Part 2, 9 12 This unit involves knowledge and skills relative

### Common Core State Standards. Standards for Mathematical Practices Progression through Grade Levels

Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for

### 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

### Overview. Essential Questions. Grade 4 Mathematics, Quarter 1, Unit 1.1 Applying Place Value Up to the 100,000s Place

to the 100,000s Place Overview Number of instruction days: 8 10 (1 day = 90 minutes) Content to Be Learned Compare whole numbers within 1,000,000 using >,

### 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

### Georgia Standards of Excellence Mathematics

Georgia Standards of Excellence Mathematics Standards GSE Algebra II/Advanced Algebra K-12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical

### 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

### ALG 1A Algebra I, First Semester PR-10254, BK (v.3.0) To the Student:

ALG 1A Algebra I, First Semester PR-10254, BK-10255 (v.3.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG 1A. WHAT

### Problem of the Month: Perfect Pair

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student

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

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

### Overview of Math Standards

Grade 8A Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### 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:

### 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,

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### Overview. Essential Questions. Grade 4 Mathematics, Quarter 4, Unit 4.1 Dividing Whole Numbers With Remainders

Dividing Whole Numbers With Remainders Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Solve for whole-number quotients with remainders of up to four-digit dividends

### Standards for Mathematical Practice: Commentary and Elaborations for 6 8

Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:

### ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### Algebra I Support Lab CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT

TRENTON PUBLIC SCHOOLS Department of Curriculum and Instruction 108 NORTH CLINTON AVENUE TRENTON, NEW JERSEY 08609 Secondary Schools Algebra I Support Lab CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT The

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

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

### Correlation to the Common Core State Standards for Mathematics Algebra 1. Houghton Mifflin Harcourt Algerbra

Correlation to the Common Core State Standards for Mathematics Algebra 1 Houghton Mifflin Harcourt Algerbra 1 2015 Houghton Mifflin Harcourt Algebra I 2015 correlated to the Common Core State Standards

### 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.

### 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

### 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

### 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,

### Algebra. Indiana Standards 1 ST 6 WEEKS

Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Larson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN

ALG B Algebra II, Second Semester #PR-0, BK-04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO

### CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology

### 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

### High School Mathematics Algebra

High School Mathematics Algebra This course is designed to give students the foundation of understanding algebra at a moderate pace. Essential material will be covered to prepare the students for Geometry.

### Algebra I Texas Mathematics: Unpacked Content

Algebra I Texas Mathematics: Unpacked Content What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards expect a student

### DRAFT. Algebra 2 EOC Item Specifications

DRAFT Algebra 2 EOC Item Specifications The release of the updated FSA Test Item Specifications is intended to provide greater specificity for item writers in developing items to be field tested in 2016.

### Common Core State Standards - Mathematics Content Emphases by Cluster Grade K

Grade K Not all of the content in a given grade is emphasized equally in the standards. Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to

### Passaic County Technical Institute ALGEBRA 2 HONORS

Passaic County Technical Institute ALGEBRA 2 HONORS September 2012 Course Description Algebra 2 Honors The Algebra honors curriculum is design for academically motivated students who are proficient in

### 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

### Unit 1: Place value and operations with whole numbers and decimals

Unit 1: Place value and operations with whole numbers and decimals Content Area: Mathematics Course(s): Generic Course Time Period: 1st Marking Period Length: 10 Weeks Status: Published Unit Overview Students

### CAMI Education linked to CAPS: Mathematics

- 1 - TOPIC 1.1 Whole numbers _CAPS Curriculum TERM 1 CONTENT Properties of numbers Describe the real number system by recognizing, defining and distinguishing properties of: Natural numbers Whole numbers

### Algebra 1 Course Objectives

Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in

### 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

### Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate

### Georgia Standards of Excellence Mathematics

Georgia Standards of Excellence Mathematics Standards GSE Algebra I K-12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical understanding

### 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

### 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

### Examples of Tasks from CCSS Edition Course 3, Unit 5

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

### Division with Whole Numbers and Decimals

Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers

### 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

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### 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

### CCSS: Mathematics. Operations & Algebraic Thinking. CCSS: Grade 5. 5.OA.A. Write and interpret numerical expressions.

CCSS: Mathematics Operations & Algebraic Thinking CCSS: Grade 5 5.OA.A. Write and interpret numerical expressions. 5.OA.A.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate

### 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

### Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### Derive and Apply Geometric Formulas

Derive and Apply Geometric Formulas Overview Number of instruction days: 8 10 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Write an explanation of the formula for

### Algebra 1-2. A. Identify and translate variables and expressions.

St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### Course: Algebra 1 Unit #5: Quadratic Functions

Course: Algebra 1 Unit #5: Overarching Question: What patterns of change are modeled by quadratic functions as seen in real-world situations, and the tables, graphs, and function rules that represent these

### Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

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

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

### 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

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties