# Volumes of Revolution

Size: px
Start display at page:

Transcription

1 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 will determine the area of twodimensional figures created on a coordinate plane. In addition, students will determine the volume of three-dimensional figures created by revolving the region on the coordinate plane about a horizontal or vertical line. Prior to the lesson, students should be able to plot points on a coordinate plane, calculate areas of plane figures, and calculate volumes of solids. This lesson is included in Module : Areas and Volumes. Objective Students will plot the points for a plane figure on a coordinate plane. determine the area of that plane figure. draw and describe the solid formed by revolving the plane figure about a vertical or horizontal line. calculate the volume of the solid. compare the volumes determined when revolving about different axes. Level Geometry Common Core State Standards for Mathematical Content This lesson addresses the following Common Core 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. A-REI. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Analyze Depth of Knowledge III III Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at i

2 Teacher Overview Volumes of Revolution Code Standard Level of Thinking G-GMD. Identify the shapes of two-dimensional crosssections Analyze of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. G-GMD. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Analyze Depth of Knowledge III III 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 connection across grade levels. This lesson allows teachers to address the following Common Core 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. 6 Attend to precision. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at ii

3 Teacher Overview Volumes of Revolution 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 pre-calculus. 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. 6th Grade Skills/Objectives 7th Grade Skills/Objectives Algebra Skills/Objectives Geometry Skills/Objectives Algebra Skills/Objectives Pre-Calculus Skills/Objectives Given or coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. (00_06.AV_B0) Given or coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. (00_06.AV_B0) Given or coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. (00_07.AV_B0) Given or coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. (00_07.AV_B0) Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/or determine the equations of the lines that bound the figure. (00_A.AV_B0) Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the region formed by revolving the region about the line that is horizontal or vertical. (00_A.AV_B0) Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. (00_GE.AV_B0) Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations conic equations and/or determine the and/or determine the equations of the lines equations of the lines and circles that and circles that bound the figure. bound the figure. (00_A.AV_B0) (00_PC.AV_B0) Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed by revolving the region about a by revolving the horizontal or vertical region about a line. (00_GE.AV_B0) horizontal or vertical line. (00_A.AV_B0) Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed by revolving the region about a horizontal or vertical line. (00_PC.AV_B0) Connections to AP* AP Calculus Topics: Areas and Volumes *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 Student Activity pages Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at iii

4 Teacher Overview Volumes of Revolution Assessment The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students construct and use manipulatives. Students apply knowledge to a new situation. The following additional assessments are located on the LTF website: Areas and Volumes Geometry Free Response Questions Areas and Volumes Geometry Multiple Choice Questions Teaching Suggestions The concept of revolving a region about an axis is fundamental to integral calculus. This lesson includes calculating perimeter and area of a planar region and then the volume generated by rotating the region around the x-axis or the y-axis. To help students visualize the solid generated by revolving the figure about an axis, have them glue or tape a triangle onto a stick or dowel and then rotate the triangle horizontally and vertically between their hands so the students can see the cone that will be generated. An extension of this lesson is to bring D objects to class and having the students draw crosssections. Being able to visualize solids is a valuable tool for calculus. When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region about the axis of rotation. Connect significant points and their reflections with ellipses. 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. P Physical V Verbal A Analytical N Numerical G Graphical Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at iv

5 Answers Volumes of Revolution Answers. a) b) units c) 0 sq. units d) e) a cylinder with a cylinder removed f) V R h r h h R r () 0 cu. units Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at v

6 Answers Volumes of Revolution. a) b) units c) 6 sq. units d) e) cone f) cu. units g) sq. units (remember the base) h) The volume is 6 cu. units, so the y-axis rotation has greater volume. The surface area, 6 sq. units, is greater in the y-axis rotation as well. i) Region bounded by y x, y 0 and x 0. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at vi

7 Answers Volumes of Revolution. a) b) ( +) units c) sq. units d) e) sphere g) cu. units h) 6 sq. units i) The geometric solid would be the top half of the sphere, called a hemisphere. The volume would be half the volume of the sphere, or 6 cu. units. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at vii

9 Mathematics Volumes of Revolution. Sketch the region bounded by the lines y =, y =, x =, x = 6. c) d) Determine the perimeter of the region. e) Determine the area of the region. f) Draw a picture of the region being revolved about the x-axis g) Describe the geometric solid formed by revolving the region about the x-axis. h) Determine the volume of the geometric solid. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at

10 Student Activity Volumes of Revolution. Sketch the region bounded by the lines y = x, y = 0, x = 0. a) b) Determine the perimeter of the region. c) Determine the area of the region. d) Draw a picture of the region being revolved about the x-axis e) What geometric figure is formed by revolving the region about the x-axis? f) Determine the volume of the geometric solid. g) Determine the surface area of the geometric solid. h) If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Compare the surface areas. i) Name another region that could be revolved about the x-axis to create exactly the same geometric solid. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at

11 Student Activity Volumes of Revolution. Sketch the region bounded by the curve y = x and the line y = 0. a) b) Determine the perimeter of the region. c) Determine the area of the region. d) Draw a picture of the region being revolve about the x-axis e) What geometric figure is formed by revolving the region about the x-axis? f) Determine the volume of the geometric solid. g) Determine the surface area of the geometric solid. h) If the region were rotated about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at

12 Student Activity Volumes of Revolution. A region is bounded by the graphs y ( x ) and y x. a) Draw a picture of the region. b) Draw a picture of the region rotated around the x-axis. y x y x c) Draw a picture of the region rotated around the y-axis. y x When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region across the axis of revolution. Connect significant points and their reflections with ellipses. Copyright 0 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at

### Investigating Area Under a Curve

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

### 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,

### 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,

### 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

### Activity Set 4. Trainer Guide

Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### How does one make and support a reasonable conclusion regarding a problem? How does what I measure influence how I measure?

Middletown Public Schools Mathematics Unit Planning Organizer Subject Mathematics Grade/Course Grade 7 Unit 3 Two and Three Dimensional Geometry Duration 23 instructional days (+4 days reteaching/enrichment)

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### 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,

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### Basic Understandings

Activity: TEKS: Exploring Transformations Basic understandings. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential to understanding underlying

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### Further Steps: Geometry Beyond High School. Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu

Further Steps: Geometry Beyond High School Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu Geometry the study of shapes, their properties, and the spaces containing

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Grade 6 Mathematics Performance Level Descriptors

Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this

### 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

### 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

### Prentice Hall Connected Mathematics 2, 7th Grade Units 2009

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems

### 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

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

### Open-Ended Problem-Solving Projections

MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### 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

### 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

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Problem of the Month: Cutting a Cube

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:

### Geometry Notes VOLUME AND SURFACE AREA

Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

### 4.4 Transforming Circles

Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different

### 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

### 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

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes

Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical

### 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

### Everyday Mathematics GOALS

Copyright Wright Group/McGraw-Hill GOALS The following tables list the Grade-Level Goals organized by Content Strand and Program Goal. Content Strand: NUMBER AND NUMERATION Program Goal: Understand the

### GRADE 5 SKILL VOCABULARY MATHEMATICAL PRACTICES Evaluate numerical expressions with parentheses, brackets, and/or braces.

Common Core Math Curriculum Grade 5 ESSENTIAL DOMAINS AND QUESTIONS CLUSTERS Operations and Algebraic Thinking 5.0A What can affect the relationship between numbers? round decimals? compare decimals? What

### N Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### 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

### Everyday Mathematics CCSS EDITION CCSS EDITION. Content Strand: Number and Numeration

CCSS EDITION Overview of -6 Grade-Level Goals CCSS EDITION Content Strand: Number and Numeration Program Goal: Understand the Meanings, Uses, and Representations of Numbers Content Thread: Rote Counting

### Grade 5 Math Content 1

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

### 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

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter A. Elementary

Elementary 111.A. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter A. Elementary Statutory Authority: The provisions of this Subchapter A issued under the Texas Education Code,

### Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:

Solid Geometry Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. Three Dimensions It is called three-dimensional or 3D because there are three dimensions: width,

### McDougal Littell California:

McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE),

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

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### 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

### Voyager Sopris Learning Vmath, Levels C-I, correlated to the South Carolina College- and Career-Ready Standards for Mathematics, Grades 2-8

Page 1 of 35 VMath, Level C Grade 2 Mathematical Process Standards 1. Make sense of problems and persevere in solving them. Module 3: Lesson 4: 156-159 Module 4: Lesson 7: 220-223 2. Reason both contextually

### Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions

Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function

### NCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5

NCTM Curriculum Focal Points and, Grade 5 NCTM Curriculum Focal Points for Grade 5 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers Students

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Problem of the Month: William s Polygons

Problem of the Month: William s Polygons 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

### Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### OPEN LESSON SAMPLE LESSONS FOR THE CLASSROOM FROM LAYING THE FOUNDATION

OPEN LESSON SAMPLE LESSONS FOR THE CLASSROOM FROM LAYING THE FOUNDATION Middle Grades Science Running the Stairs Measuring Work, Energy, and Power About this Lesson This activity can be used to introduce

Academic Standards for Grades Pre K High School Pennsylvania Department of Education INTRODUCTION The Pennsylvania Core Standards in in grades PreK 5 lay a solid foundation in whole numbers, addition,

### Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

### WORK SCHEDULE: MATHEMATICS 2007

, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check

### Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

### Mathematics. Mathematical Practices

Mathematical Practices 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

### Discovering Math: Exploring Geometry Teacher s Guide

Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional

### 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

### Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.

3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

### Functional Math II. Information CourseTitle. Types of Instruction

Functional Math II Course Outcome Summary Riverdale School District Information CourseTitle Functional Math II Credits 0 Contact Hours 135 Instructional Area Middle School Instructional Level 8th Grade

### Grades K-6. Correlated to the Common Core State Standards

Grades K-6 Correlated to the Common Core State Standards Kindergarten Standards for Mathematical Practice Common Core State Standards Standards for Mathematical Practice Kindergarten The Standards for

### A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Unit 5 Area. What Is Area?

Trainer/Instructor Notes: Area What Is Area? Unit 5 Area What Is Area? Overview: Objective: Participants determine the area of a rectangle by counting the number of square units needed to cover the region.

### 1. Mathematics Content/Alignment with the Standards Correlation to California Algebra Readiness Standards

PROGRAM DESCRIPTION The goal of Prentice Hall Connecting to Algebra is to fully prepare students for success in Algebra 1 by thoroughly covering the Algebra Readiness standards outlined by the California

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Warning! Construction Zone: Building Solids from Nets

Brief Overview: Warning! Construction Zone: Building Solids from Nets In this unit the students will be examining and defining attributes of solids and their nets. The students will be expected to have

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### 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

### Mathematics Scope and Sequence, K-8

Standard 1: Number and Operation Goal 1.1: Understands and uses numbers (number sense) Mathematics Scope and Sequence, K-8 Grade Counting Read, Write, Order, Compare Place Value Money Number Theory K Count

### Ministry of Education. The Ontario Curriculum. Mathematics. Mathematics Transfer Course, Grade 9, Applied to Academic

Ministry of Education The Ontario Curriculum Mathematics Mathematics Transfer Course, Grade 9, Applied to Academic 2 0 0 6 Contents Introduction....................................................... 2

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### with functions, expressions and equations which follow in units 3 and 4.

Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

### G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

### G3-33 Building Pyramids

G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids. Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons Vocabulary:

### Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

### We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

### Common Core State Standards for Mathematics Accelerated 7th Grade

A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting

### 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

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

### GRADES 7, 8, AND 9 BIG IDEAS

Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year