Investigating Area Under a Curve


 Barnaby Carr
 2 years ago
 Views:
Transcription
1 Mathematics Investigating Area Under a Curve About this Lesson This lesson is an introduction to areas bounded by functions and the xaxis 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 nonlinear functions and the xaxis 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 nonlinear 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 GGPE.7 Use coordinates to compute perimeters of Analyze polygons and areas of triangles and rectangles, e.g., using the distance formula. FIF.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 FIF. 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
3 Teacher Overview Investigating Area Under a Curve LTF Content Progression Chart In the spirit of LTF s goal to connect mathematics across grade levels, the Content Progression Chart demonstrates how specific skills build and develop from sixth grade through precalculus. Each column, under a grade level or course heading, lists the concepts and skills that students in that grade or course should master. Each row illustrates how a specific skill is developed as students advance through their mathematics courses. th Grade Skills/Objectives Calculate and/or approximate area by counting squares or by using lefthand rectangles, righthand rectangles, or trapezoids in graphical contexts. (00_0.AC_E.0) Compare approximation methods and/or determine and justify whether an approximation is an overestimate or an underestimate. (00_0.AC_E.0) 7th Grade Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area by counting squares or by using lefthand rectangles, righthand rectangles, or trapezoids in graphical contexts. (00_07.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_07.AC_E.0) Algebra Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using lefthand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_A.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_A.AC_E.0) Geometry Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using lefthand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_GE.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_GE.AC_E.0) Algebra Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using lefthand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_A.AC_E.0) Compare approximation methods, determine and justify whether an approximation is PreCalculus Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using lefthand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_PC.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an an overestimate or an underestimate, underestimate, and/or interpret the and/or interpret the meaning of meaning of accumulated area. accumulated area. (00_A.AC_E.0) (00_PC.AC_E.0) Connection to AP* AP Calculus Topic: Accumulation *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Materials and Resources Student Activity pages Graphing calculators Assessments The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students apply knowledge to a new situation. The following additional assessments are located on the LTF website: Accumulation Algebra Free Response Questions Accumulation Algebra Multiple Choice Questions Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at iii
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 xaxis 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 xaxis 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 xaxis 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 (lefthand 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 lefthand rectangles or righthand 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 (PreCalculus) 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. Lefthand rectangles (0)(0) (0)(0) (0)(0) (0)() (0)() 700 square units ii. Righthand 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. Lefthand rectangles (0)(0) (0)(00) (0)(80) (0)(0) 8800 square units ii. Righthand 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. Lefthand rectangles ()(0) (0)() (0)() (0)(0) (0)() ()() 7 square units ii. Righthand 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 xaxis: 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 xaxis for the interval shown on the graph and in the table: i. by calculating the sum of lefthand rectangles with widths determined by the data in the table. ii. by calculating the sum of righthand rectangles with widths determined by the data in the table. iii. by averaging the sums of the lefthand and righthand 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 lefthand 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 righthand 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
More informationTransformation 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
More informationIn 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 multistep equations Mathematics Literal Equations Geometric Formulas
More informationIn 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
More informationSlope 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
More informationAlgebra 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.
More informationCreating, 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 informationThis 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 informationNorth 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.
More informationGeorgia Standards of Excellence 20152016 Mathematics
Georgia Standards of Excellence 20152016 Mathematics Standards GSE Coordinate Algebra K12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical
More informationHome 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
More informationBoxandWhisker Plots
Mathematics BoxandWhisker Plots About this Lesson This is a foundational lesson for boxandwhisker plots (boxplots), a graphical tool used throughout statistics for displaying data. During the lesson,
More informationLevel: High School: Geometry. Domain: Expressing Geometric Properties with Equations GGPE
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
More informationPolynomial 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 informationInfinite 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
More informationHigh 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 NQ.1. Use units as a way to understand problems
More informationAnalyzing 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
More informationPearson 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 informationPCHS 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
More informationWentzville 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
More informationMathematics 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 informationSolving 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
More informationGreater 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
More informationPowerTeaching 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
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More informationDistance, 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
More informationEstimating 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
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationFor 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
More informationCourse: 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 realworld situations, and the tables, graphs, and function rules that represent these
More informationREVIEW 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
More informationSection 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
More informationThe School District of Palm Beach County ALGEBRA 1 REGULAR Section 1: Expressions
MAFS.912.AAPR.1.1 MAFS.912.ASSE.1.1 MAFS.912.ASSE.1.2 MAFS.912.NRN.1.1 MAFS.912.NRN.1.2 MAFS.912.NRN.2.3 ematics Florida August 16  September 2 Understand that polynomials form a system analogous
More informationAlgebra 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 informationPHILOSOPHY 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
More informationMontana 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
More informationPARCC 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
More information4. 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
More informationGlencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 33, 58 84, 87 16, 49
Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 68 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationContent Emphases by ClusterKindergarten *
Content Emphases by ClusterKindergarten * Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers. Operations and Algebraic Thinking Understand
More informationProcedure 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
More informationREVISED 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
More informationPolygons 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
More informationMathematics Task Arcs
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
More informationMATH 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
More informationAlgebra 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
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 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
More informationDRAFT. 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 informationLesson 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
More informationPA Academic Standards. 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication
Curriculum Category Shift Standard Areas Grades K8 and High School PA Academic Standards 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication PA Common Core
More informationa. 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
More informationMathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions
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
More informationTennessee 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
More informationUnit 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
More informationCorrelation 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
More informationNEW 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
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationLaws 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
More informationAlgebra 1 Math Standards and I Can Statements
Algebra 1 Math Standards and I Can Statements Standard  CC.912.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic
More informationHigh 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 informationInteger 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 informationCalifornia 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
More informationCommon 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
More informationPrep 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
More informationMiddle Grades Mathematics 5 9
Middle Grades Mathematics 5 9 Section 25 1 Knowledge of mathematics through problem solving 1. Identify appropriate mathematical problems from realworld situations. 2. Apply problemsolving strategies
More informationNorth 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.
More informationGeorgia Department of Education. Calculus
K12 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
More informationMontana 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
More informationLAB 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.
More informationGrade: 8. Learning Goals:
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
More informationTopic: Solving Linear Equations
Unit 1 Topic: Solving Linear Equations NQ.1. Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and
More informationMichigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan PreAlgebra
Eighth Grade Mathematics Grade Level Content Expectations Michigan PreAlgebra 2010 Michigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan PreAlgebra Lessons
More informationAlgebra 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: AAPR.3: Identify zeros of polynomials
More information7.5 Approximating Definite Integrals
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationPrentice 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
More informationAP 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.
More informationPythagorean 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,
More informationThe 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,
More informationNumber and Numeracy SE/TE: 43, 49, 140145, 367369, 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,
More informationMEMORANDUM. 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
More informationOverview 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
More informationExam 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
More informationConstruction 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
More informationThis 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
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationOverview. 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
More informationComparing 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
More informationIn 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
More informationAP 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
More informationA 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
More informationPreAlgebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio PreAlgebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationFinding 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
More informationGeorgia 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
More informationQuadratic Functions: Complex Numbers
Algebra II, Quarter 1, Unit 1.3 Quadratic Functions: Complex Numbers Overview Number of instruction days: 1214 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Develop
More informationHIGH 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
More informationhttps://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 BellRinger Activity Students
More informationA Correlation of. to the. Common Core State Standards for Mathematics Grade 4
A Correlation of to the Introduction envisionmath2.0 is a comprehensive K6 mathematics curriculum that provides the focus, coherence, and rigor required by the CCSSM. envisionmath2.0 offers a balanced
More informationPA 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
More informationGeorgia Standards of Excellence Curriculum Map. Mathematics. GSE Algebra I
Georgia Standards of Excellence Curriculum Map Mathematics GSE Algebra I These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department
More information