Investigating Area Under a Curve

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Mathematics Investigating Area Under a Curve About this Lesson This lesson is an introduction to areas bounded by functions and the x-axis on a given interval. Since the functions in the beginning of the lesson are linear, or piecewise linear, the enclosed regions form rectangles, triangles, or trapezoids. Within the lesson, the concept of accumulation is extended to approximating areas enclosed by non-linear functions and the x-axis on a given interval using rectangles and trapezoids. Prior to the lesson, students should be familiar with evaluating linear equations, as well as computing areas of plane figures. This lesson is included in Module Accumulation. Objectives Students will determine the area of a bounded region formed by linear functions. use rectangles to approximate the area of a bounded region formed by non-linear functions. Level Algebra Common Core State Standards for Mathematical Content This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Explicitly addressed in this lesson Code Standard Level of Thinking G-GPE.7 Use coordinates to compute perimeters of Analyze polygons and areas of triangles and rectangles, e.g., using the distance formula. F-IF.7a Graph functions expressed symbolically and Analyze show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph linear and quadratic functions and show intercepts, maxima, and minima. Depth of Knowledge III III Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at i

2 Teacher Overview Investigating Area Under a Curve Code Standard Level of Thinking F-IF. Relate the domain of a function to its graph Apply and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Depth of Knowledge II Common Core State Standards for Mathematical Practice These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. LTF incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. Implicitly addressed in this lesson Code Standard Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at ii

4 Teacher Overview Investigating Area Under a Curve Teaching Suggestions In question, all of the functions are linear, so the students should calculate the exact area of the bounded region. Demonstrate how to drop the vertical boundaries from the function to the x-axis at the endpoints of the interval, and then discuss using only vertical lines to divide the figure into triangles, rectangles, and/or trapezoids. Calculate the areas of the geometric figures then sum (accumulate) the values to determine the total area of the region. In questions and, students are asked to approximate the area of the region bounded by a curve and the x-axis on a given interval. Since the functions are curved, the exact area cannot be determined using only algebraic techniques. Explain that the vertical lines drawn from known points of the function to the x-axis will be used to create the sides of rectangles, These vertical line segments are not the same length; therefore, the students must determine where to locate the tops of the rectangles. One of two rules is generally applied in this lesson: () draw the top of the rectangle from the top of the vertical segment on the left (left-hand rectangles), or () draw the top of the rectangle from the top of the vertical segment on the right side of the rectangle. The directions in the questions tell the students to use left-hand rectangles or right-hand rectangles. Work question a with the students to demonstrate this concept. Students who are successful with questions and will also enjoy the following lessons: Applying Accumulation (Geometry) Adaptation of AP Calculus 00 AB (Geometry) Accumulation with a Quadratic Function (Algebra ) Accumulation (Pre-Calculus) Modality LTF emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using these representations to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at iv

5 Answers. a. trapezoid: b b h A ()( ) square units b. trapezoid: b b h Teacher Overview Investigating Area Under a Curve A ()( ) 8 square units 9 c. triangle height = () units 9 7 A () square units 8 d. trapezoid: b ( ) b () h A () 9 square units e. rectangle: b h trapezoid: b b h A ()() ()( ) 7 square units f. rectangle: b h trapezoid: b b h A ()() ()( ) square units g. trapezoid(): b 0 b h rectangle: b h trapezoid(): b b 8 h A ()( ) ()() ()( ) 0 square units h. trapezoid: b 0 b h rectangle: b h triangle: b h A ()( ) ()() ()() square units Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at v

6 Teacher Overview Investigating Area Under a Curve. a. i. Left-hand rectangles (0)(0) (0)(0) (0)(0) (0)() (0)() 700 square units ii. Right-hand rectangles (0)(0) (0)(0) (0)() (0)() (0)(70) 0 square units iii. Average square units The average is a better approximation because the average gives the sum of areas of trapezoids whose upper edges are closer to the graph of f than the upper edges of the rectangles. b. i. Left-hand rectangles (0)(0) (0)(00) (0)(80) (0)(0) 8800 square units ii. Right-hand rectangles (0)(00) (0)(80) (0)(0) (0)(0) 00 square units iii. Average square units The average is a better approximation because the average gives the sum of areas of trapezoids whose upper edges are closer to the graph of f than the upper edges of the rectangles. c. i. Left-hand rectangles ()(0) (0)() (0)() (0)(0) (0)() ()() 7 square units ii. Right-hand rectangles ()() (0)() (0)(0) (0)() (0)() ()(0) 7 square units iii. Average square units Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at vi

7 Teacher Overview Investigating Area Under a Curve. a b. x f(x) c. ()() ()(9) ()() 9 square units d. ()(9) ()() ()(0) 90 square units e square units. Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at vii

8 Mathematics Investigating Area Under a Curve. For the given interval, determine the area between the graph of the given function and the x-axis: a. Interval: x b. Interval: x Function: y x Function: y x c. Interval: x d. Interval: x Function: y x Function: y x Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at

9 Student Activity Investigating Area Under a Curve e. Interval: 0 x f. Interval: x Function:, x f( x) x, x Function:, x f( x) x, x g. Interval: 0 x h. Interval: 0 x 8 Function: x, x f( x), x 8 x, x x, x Function: f( x), x 8 x, x Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at

10 Student Activity Investigating Area Under a Curve. Approximate the area between the function and the x-axis for the interval shown on the graph and in the table: i. by calculating the sum of left-hand rectangles with widths determined by the data in the table. ii. by calculating the sum of right-hand rectangles with widths determined by the data in the table. iii. by averaging the sums of the left-hand and right-hand rectangles and explaining why this may be a better approximation for the actual area under the curve. a x f(x) b x f(x) Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at

11 Student Activity Investigating Area Under a Curve c x f(x) a. Use a graphing calculator to sketch the graph of the function 0 x. y x on the interval b. Complete the table of values for the function. x f(x) 0 Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at

12 Student Activity Investigating Area Under a Curve c. Use the data in the table to approximate the area under the graph of the function by calculating the sum of left-hand rectangles with widths indicated by the data table. d. Use the data in the table to approximate the area under the graph of the function by calculating the sum of the right-hand rectangles with widths indicated by the data table. e. Calculate the average of your answers to parts (c) and (d). Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at

Volumes of Revolution

Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students

Transformation of Rational Functions

Mathematics Transformation of Rational Functions About this Lesson In this lesson, students will apply transformations to the graphs of rational functions, describe the transformations, and graph the transformed

In this lesson, students practice the essential skill

LEVEL NATIONAL MATH + SCIENCE INITIATIVE a b c a b b c b a c b a c b a c b Since a 0, a c b Algebra 1 or Math 1 in a unit on solving multi-step equations Mathematics Literal Equations Geometric Formulas

In the first part of the lesson, students plot

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Using Linear Equations to Define Geometric Solids Level Geometry within a unit on volume applications Module/Connection to AP* Area and Volume *Advanced Placement

Slope Investigation. Grade 8, Algebra 1, or Math 1 in a unit with rate of change and slope

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Slope Investigation LEVEL Grade 8, Algebra 1, or Math 1 in a unit with rate of change and slope MODULE/CONNECTION TO AP* Rate of Change *Advanced Placement

Algebra I Common Core Standards

Algebra I Common Core Standards Algebra 1 Overview Number and Quantity The Real Number System Extend the properties of exponents to rational exponents. Use properties of rational and irrational numbers.

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

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

North Carolina Math 1

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.

Georgia Standards of Excellence 2015-2016 Mathematics

Georgia Standards of Excellence 2015-2016 Mathematics Standards GSE Coordinate Algebra K-12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical

Home Runs, Statistics, and Probability

NATIONAL MATH + SCIENCE INITIATIVE Mathematics American League AL Central AL West AL East National League NL West NL East Level 7 th grade in a unit on graphical displays Connection to AP* Graphical Display

Box-and-Whisker Plots

Mathematics Box-and-Whisker Plots About this Lesson This is a foundational lesson for box-and-whisker plots (boxplots), a graphical tool used throughout statistics for displaying data. During the lesson,

Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Translate between the geometric

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.

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

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

Analyzing Piecewise Functions

Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including

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).

PCHS ALGEBRA PLACEMENT TEST

MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If

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

Solving Equations with One Variable

Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Solve linear equations in one variable

Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

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

Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

Distance, Midpoint, and Pythagorean Theorem

Geometry, Quarter 1, Unit 1.1 Distance, Midpoint, and Pythagorean Theorem Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Find distance and midpoint. (2 days) Identify

Estimating the Average Value of a Function

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

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

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

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

REVIEW SHEETS COLLEGE ALGEBRA MATH 111

REVIEW SHEETS COLLEGE ALGEBRA MATH 111 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts that are taught in the specified math course. The sheets present

Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

The School District of Palm Beach County ALGEBRA 1 REGULAR Section 1: Expressions

MAFS.912.A-APR.1.1 MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.1.2 MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 MAFS.912.N-RN.2.3 ematics Florida August 16 - September 2 Understand that polynomials form a system analogous

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.

PHILOSOPHY OF THE MATHEMATICS DEPARTMENT

PHILOSOPHY OF THE MATHEMATICS DEPARTMENT The Lemont High School Mathematics Department believes that students should develop the following characteristics: Understanding of concepts and procedures Building

Montana Common Core Standard

Algebra I Grade Level: 9, 10, 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: A must pass course Course Description This course covers the real number system, solving

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

4. How far have they traveled at 3 hours? Write the answer using function notation.

6. Bike Lovers A Solidify Understanding Task 013 http://flic.kr/p/896re8 Michelle and Rashid love going on long bike rides. Every Saturday, they have a particular route they bike together that takes four

Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

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

Content Emphases by Cluster--Kindergarten *

Content Emphases by Cluster--Kindergarten * Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers. Operations and Algebraic Thinking Understand

Procedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:

Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series

REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011

REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working

Polygons and Area. Overview. Grade 6 Mathematics, Quarter 4, Unit 4.1. Number of instructional days: 12 (1 day = minutes)

Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area Overview Number of instructional days: 12 (1 day = 45 60 minutes) Content to be learned Calculate the area of polygons by composing into rectangles

Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

MATH 132: CALCULUS II SYLLABUS

MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

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

Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

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

Lesson 3. Numerical Integration

Lesson 3 Numerical Integration Last Week Defined the definite integral as limit of Riemann sums. The definite integral of f(t) from t = a to t = b. LHS: RHS: Last Time Estimate using left and right hand

PA Academic Standards. 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication

Curriculum Category Shift Standard Areas Grades K-8 and High School PA Academic Standards 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication PA Common Core

a. Assuming your pattern continues, explain how you would build the 10 th figure.

Tile Patterns 1. The first 3 s in a pattern are shown below. Draw the 4 th and 5 th s. 1 st Figure 2 nd Figure 3 rd Figure 4 th Figure 5 th Figure a. Assuming your pattern continues, explain how you would

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

Tennessee Department of Education. Task: Sally s Car Loan

Tennessee Department of Education Task: Sally s Car Loan Sally bought a new car. Her total cost including all fees and taxes was \$15,. She made a down payment of \$43. She financed the remaining amount

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

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

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

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

Laws of Sines and Cosines and Area Formula

Precalculus, Quarter 3, Unit 3.1 Laws of Sines and Cosines and Area Formula Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Derive and use the formula y = 1 2 absin

Algebra 1 Math Standards and I Can Statements

Algebra 1 Math Standards and I Can Statements Standard - CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic

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

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.

California Common Core State Standards Comparison- FOURTH GRADE

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics. Standards

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

Prep for College Algebra

Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

Middle Grades Mathematics 5 9 Section 25 1 Knowledge of mathematics through problem solving 1. Identify appropriate mathematical problems from real-world situations. 2. Apply problem-solving strategies

North Carolina Math 2

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.

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

Montana Common Core Standard

Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in

LAB 4: APPROXIMATING REAL ZEROS OF POLYNOMIAL FUNCTIONS

LAB 4: APPROXIMATING REAL ZEROS OF POLYNOMIAL FUNCTIONS Objectives: 1. Find real zeros of polynomial functions. 2. Solve nonlinear inequalities by graphing. 3. Find the maximum value of a function by graphing.

Title: Building with Tiles CCGPS Standards Addressed: Grade: 8 Learning Goals: BIG Idea: Linear Patterns & Algebraic Thinking MCC8.F.2 Compare properties of two functions each represented in a different

Topic: Solving Linear Equations

Unit 1 Topic: Solving Linear Equations N-Q.1. Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and

Michigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan Pre-Algebra

Eighth Grade Mathematics Grade Level Content Expectations Michigan Pre-Algebra 2010 Michigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan Pre-Algebra Lessons

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

7.5 Approximating Definite Integrals

WileyPLUS: Home Help Contact us Logout Hughes-Hallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration

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

AP Calculus AB Syllabus

Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.

Pythagorean Theorem. Overview. Grade 8 Mathematics, Quarter 3, Unit 3.1. Number of instructional days: 15 (1 day = minutes) Essential questions

Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean Theorem Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Prove the Pythagorean Theorem. Given three side lengths,

The standards in this course continue the work of modeling situations and solving equations.

Algebra II Overview View unit yearlong overview here Building on the understanding of linear, quadratic and exponential functions from Algebra I, this course will extend function concepts to include polynomial,

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

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

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number

This lesson introduces students to decimals.

NATIONAL MATH + SCIENCE INITIATIVE Elementary Math Introduction to Decimals LEVEL Grade Five OBJECTIVES Students will compare fractions to decimals. explore and build decimal models. MATERIALS AND RESOURCES

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

Overview. Essential Questions. Precalculus, Quarter 1, Unit 1.4 Analyzing Exponential and Logarithmic Functions

Analyzing Exponential and Logarithmic Functions Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Rewrite radical expressions using the properties of exponents. Rewrite

Comparing Fractions and Decimals

Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals Overview Number of Instructional Days: 10 (1 day = 45 60 minutes) Content to be Learned Explore and reason about how a number representing

In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

A Correlation of Pearson Algebra 1, Geometry, Algebra 2 Common Core 2015

A Correlation of Pearson,, Common Core 2015 To the North Carolina High School Mathematics Alignment to Traditional Text - MATH II A Correlation of Pearson,, Common Core, 2015 Introduction This document

Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

Finding Parallelogram Vertices

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade

Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade

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

HIGH SCHOOL: GEOMETRY (Page 1 of 4)

HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course

https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...

of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students

A Correlation of. to the. Common Core State Standards for Mathematics Grade 4

A Correlation of to the Introduction envisionmath2.0 is a comprehensive K-6 mathematics curriculum that provides the focus, coherence, and rigor required by the CCSSM. envisionmath2.0 offers a balanced