a. Plot the point (x, y, z) and understand it as a vertex of a rectangular prism. c. Recognize and understand equations of planes and spheres.
|
|
|
- Muriel Alexandra Hill
- 6 years ago
- Views:
Transcription
1 Standard: MM3G3 Students will investigate planes and spheres. a. Plot the point (x, y, z) and understand it as a vertex of a rectangular prism. b. Apply the distance formula in 3-space. c. Recognize and understand equations of planes and spheres. Essential Questions: How do I graph a point in 3-space? How do I determine the distance between two points in 3-space? What are the differences between the equations of a plane and sphere? Activating Strategies: Graph a circle with endpoints of diameter at (-2,-6) and (2, -6) by finding midpoint of circle and distance formula. Write the equation for the circle. Acceleration/Previewing: (Key Vocabulary) three-dimensional coordinate system or 3-space standard form for equation of sphere (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2. Teaching Strategies: Extension of knowledge of 2-space into 3-space. Discuss and demonstrate three-space being the intersection of an x-axis, a y-axis and a z-axis. The axes determine three coordinate planes: the xy-plane, the xz-plane, and the yz-plane. These planes divide the 3-sopace into eight octants. Each point in a 3-space is represented by an ordered triple (x, y, z). distance in 3-space d = (x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 +( z 2 - z 1 ) 2. Guided Practice - Model graphing a rectangular prism given the endpoints of the diagonal. a. (3,0,5) and (0,4,0) Find the length of the diagonal. (see attachment) Cooperative Pairs - Graph a rectangular prism given the endpoints of the diagonal. (0,-1,3) and (3,2,0). Find the length of the diagonal. Discuss and demonstrate the equation of a plane. A linear equation in three variables is a plane. Guided Practice Model graphing the equation of a plane. a. Graph the equation 2x 3y + 9z = -36. (see attachment) Cooperative Paris Graph the equation 2x + y 5z = 10. Discuss and demonstrate the equation of a sphere. Guided Practice Model writing the equation of a sphere in standard form given center and radius. (see attachment) a. Write the equation of a sphere in standard form with center ( 4,-2, 0) and radius r = 7. Cooperative Pairs Write the equation of a sphere in standard from with center ( 2, 0, 3) and r = 8. Figures in 3- space worksheet. (see attachment) Task: Distributed Guided Practice: see attachments Extending/Refining Strategies: Summarizing Strategies: Determine whether the equation represents a plane or a sphere. If it is a plane, graph the equation. a. x 2 + y 2 + z = 0 b. x 2y + z + 14 = 0
2 Guided Practice/Collaborative Pairs Lesson 2.5 (Using distance formula in 3-space) Draw a rectangular prism having a diagonal with endpoints Draw a rectangular prism having a diagonal with endpoints (3, 0, 5) and (0, 4, 0). Find the length of the diagonal. (0, -1, 3) and (3, 2, 0). Find the length of the diagonal. Step 1 Draw the rectangular prism. Plot and connect the points (3, 0, 5) and (0, 4, 0). Because the difference of the x-coordinates is, the prism has a length of unit(s). Because the difference of the y-coordinates is, the prism has a width of unit(s). Because the difference of the z- coordinates is, the prism has a height of unit(s). Draw the rectangular prism. Step 1 Draw the rectangular prism. Plot and connect the points (0, -1, 3) and (3, 2, 0). Because the difference of the x-coordinates is, the prism has a length of unit(s). Because the difference of the y-coordinates is, the prism has a width of unit(s). Because the difference of the z- coordinates is, the prism has a height of unit(s). Draw the rectangular prism. Step 2 Find the length of the diagonal. Let (x 1, y 1, z 1 ) = (3, 0, 5) and (x 2, y 2, z 2 ) = (0, 4, 0). d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 d = The length of the diagonal is units. Step 2 Find the length of the diagonal. Let (x 1, y 1, z 1 ) = (0, -1, 3) and (x 2, y 2, z 2 ) = (3, 2, 0). d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 d = The length of the diagonal is units.
3 Guided Practice/Collaborative Pairs Lesson 2.5 (Recognize and graph the equation of a plane) Graph the equation 2x 3y + 9z = -36 Graph the equation 2x + y - 5z = 10 The graph of a linear equation in three variables is a plane. In order to sketch a plane, complete the following steps. Step 1 Find three points at which the plane intersects the axes. Intersection of x-axis The graph of a linear equation in three variables is a plane. In order to sketch a plane, complete the following steps. Step 1 Find three points at which the plane intersects the axes. Intersection of x-axis The graph intersects the x-axis at (,, ). Intersection of y-axis The graph intersects the x-axis at (,, ). Intersection of y-axis The graph intersects the y-axis at (,, ). Intersection of z-axis The graph intersects the y-axis at (,, ). Intersection of z-axis The graph intersects the z-axis at (,, ). Step 2 Plot and connect the points. Then shade the resulting region. The graph intersects the z-axis at (,, ). Step 2 Plot and connect the points. Then shade the resulting region.
4 Guided Practice/Collaborative Pairs Lesson 2.5 (Write an equation for a sphere) Write an equation of a sphere in standard form with Write an equation of a sphere in standard form with center( (4, -2, 0) and a radius of 7 units. center( (2, 0, 3) and a radius of 8 units. Step 1 Write the standard form of an equation for a sphere. (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2 Step 1 Write the standard form of an equation for a sphere. (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2 Step 2 Identify x 0, y 0, and z 0. The center is at (4, -2, 0), so x o =, y 0 =, and z 0 =. Step 2 Identify x 0, y 0, and z 0. The center is at (2, 0, 3), so x o =, y 0 =, and z 0 =. Step 3 Find r 2. Because r =, r 2 =. Step 3 Find r 2. Because r =, r 2 =. The standard form of the equation is ( ) 2 + ( ) 2 + ( ) 2 = ( ) 2 + ( ) 2 + ( ) 2 =
5 Worksheet Figures in 3-Space Draw a rectangular prism having a diagonal with the given endpoints. Find the length of the diagonals. 1. (4, -5, 8) and (2, 2, 5) 2. (9, 5, 1) and (12, 0, -1) 3. (0, 0, -8) and (-2, 1, 0) Graph the equation of the plane. 4. 2x -4y 3z = x + 7y 3z + 42 = 0 Write the equation of the sphere in standard form with the given center and radius. 6. center (8, -2, 0) r = (10, -2, 3) r = (-2, 7, 3) r = 7
Three-Dimensional Coordinate Systems
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will
Rectangular Coordinates in Space
Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then
11.9 Three dimensional Coordinates
11.9 Three dimensional Coordinates Apr 1 10:06 AM 1 A Three Dimensional Coordinate System Dec 9 5:26 PM 2 With an ordered triple (x, y, z) Dec 9 5:26 PM 3 x axis y axis z axis Dec 9 5:26 PM 4 Dec 9 5:26
Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
The points (2, 2, 1) and (0, 1, 2) are graphed below in 3-space:
Three-Dimensional Coordinate Systems The plane is a two-dimensional coordinate system in the sense that any point in the plane can be uniquely described using two coordinates (usually x and y, but we have
The Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
Graphs of Equations. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Graphs of Equations
Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: sketch the graphs of equations, find the x- and y-intercepts
Vectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH6 2.1 Warm-Up: See Solved Homework questions 2.2 Cartesian coordinate system Coordinate axes: Two perpendicular lines that intersect at the origin O on each line.
7.3 3-D Notes Honors Precalculus Date: Adapted from 11.1 & 11.4
73 3-D Notes Honors Precalculus Date: Adapted from 111 & 114 The Three-Variable Coordinate System I Cartesian Plane The familiar xy-coordinate system is used to represent pairs of numbers (ordered pairs
11.2 RECTANGULAR COORDINATES IN THREE DIMENSIONS
11.2 Rectangular Coordinates in Three Dimensions Contemporary Calculus 1 11.2 RECTANGULAR COORDINATES IN THREE DIMENSIONS In this section we move into 3 dimensional space. First we examine the 3 dimensional
1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
Skills Practice Skills Practice for Lesson 2.1
Skills Practice Skills Practice for Lesson.1 Name Date Backyard Barbecue Introduction to Volume and Surface Area Vocabulary Write the term from the box that best completes each statement. surface area
Three-Dimensional Coordinates
CHAPTER Three-Dimensional Coordinates Three-dimensional movies superimpose two slightl different images, letting viewers with polaried eeglasses perceive depth (the third dimension) on a two-dimensional
WHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
Conic Sections and Locii
Lesson Summary: Students will investigate the ellipse and the hyperbola as a locus of points. Activity One addresses the ellipse and the hyperbola is covered in lesson two. Key Words: Locus, ellipse, hyperbola
Answer Key: Three-Dimensional Cross Sections
Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection
High School Geometry. Correlation of the ALEKS course High School Geometry to the ACT College Readiness Standards for Mathematics
High School Geometry Correlation of the ALEKS course High School Geometry to the ACT College Readiness Standards for Mathematics Standard 5 : Graphical Representations = ALEKS course topic that addresses
Unit 7: 3D Figures 10.1 & D formulas & Area of Regular Polygon
Unit 7: 3D Figures 10.1 & 10.2 2D formulas & Area of Regular Polygon NAME Name the polygon with the given number of sides: 3-sided: 4-sided: 5-sided: 6-sided: 7-sided: 8-sided: 9-sided: 10-sided: Find
Lesson 9. Three-Dimensional Geometry
Lesson 9 Three-Dimensional Geometry 1 Planes A plane is a flat surface (think tabletop) that extends forever in all directions. It is a two-dimensional figure. Three non-collinear points determine a plane.
Assignment Guide: Chapter 11 Geometry (L3)
Assignment Guide: Chapter 11 Geometry (L3) (136) 11.1 Space Figures and Cross Sections Page 692-693 #7-23 odd, 35 (137) 11.2/11.4 Surface Areas and Volumes of Prisms Page 703-705 #1, 2, 7-9, 11-13, 25,
Vocabulary. Term Page Definition Clarifying Example. cone. cube. cylinder. edge of a threedimensional. figure. face of a polyhedron.
CHAPTER 10 Vocabulary The table contains important vocabulary terms from Chapter 10. As you work through the chapter, fill in the page number, definition, and a clarifying example. cone Term Page Definition
Conic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
ALGEBRA II UNIT X: Conic Sections Unit Notes Packet
Name: Period: ALGEBRA II UNIT X: Conic Sections Unit Notes Packet Algebra II Unit 10 Plan: This plan is subject to change at the teacher s discretion. Section Topic Formative Work Due Date 10.3 Circles
Section 1.1 The Distance and Midpoint Formulas
Section 1.1 The Distance and Midpoint Formulas 1 y axis origin x axis 2 Plot the points: ( 3, 5), (0,7), ( 6,0), (6,4) 3 Distance Formula y x 4 Finding the Distance Between Two Points Find the distance
You may know these...
You may know these... Chapter 1: Multivariables Functions 1.1 Functions of Two Variables 1.1.1 Function representations 1.1. 3-D Coordinate System 1.1.3 Graph of two variable functions 1.1.4 Sketching
6th Grade Math. Parent Handbook
6th Grade Math Benchmark 3 Parent Handbook This handbook will help your child review material learned this quarter, and will help them prepare for their third Benchmark Test. Please allow your child to
The point (x, y) lies on the circle of radius r and center (h, k) iff. x h y k r
NOTES +: ANALYTIC GEOMETRY NAME LESSON. GRAPHS OF EQUATIONS IN TWO VARIABLES (CIRCLES). Standard form of a Circle The point (x, y) lies on the circle of radius r and center (h, k) iff x h y k r Center:
UNIT 5: GEOMETRIC AND ALGEBRAIC CONNECTIONS. Apply Geometric Concepts in Modeling Situations
UNIT 5: GEOMETRIC AND ALGEBRAIC CONNECTIONS This unit investigates coordinate geometry. Students look at equations for circles and use given information to derive equations for representations of these
Section 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
Math 8 Honors Coordinate Geometry part 3 Unit Updated July 29, 2016
Review how to find the distance between two points To find the distance between two points, use the Pythagorean theorem. The difference between is one leg and the difference between and is the other leg.
Name: Period 3/23/12 4/12/12 Pre-AP
Name: Period 3/23/12 4/12/12 Pre-AP UNIT 14: SOLIDS I can define, identify and illustrate the following terms: Face Edge Vertex Cross section Prism Height Surface area Lateral surface area Net Volume Scale
Exam 2 Review. 2. What the difference is between an equation and an expression?
Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? 2. What the difference is between an equation and an expression? 3. How to tell if an equation is linear? 4. How
Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V
Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find
Chapter 7. Description or Example. Found on Page. Vocabulary Term. Definition. base. center. circumference. chord. complex figure. cone.
C H A P T E R 7 This is an alphabetical list of new vocabulary terms you will learn in Chapter 7. As you complete the study notes for the chapter, you will see Build Your Vocabulary reminders to complete
SOL Chapter Due Date
Name: Block: Date: Geometry SOL Review SOL Chapter Due Date G.1 2.2-2.4 G.2 3.1-3.5 G.3 1.3, 4.8, 6.7, 9 G.4 N/A G.5 5.5 G.6 4.1-4.7 G.7 6.1-6.6 G.8 7.1-7.7 G.9 8.2-8.6 G.10 1.6, 8.1 G.11 10.1-10.6, 11.5,
Algebra II. Slide 1 / 181. Slide 2 / 181. Slide 3 / 181. Conic Sections Table of Contents
Slide 1 / 181 Algebra II Slide 2 / 181 Conic Sections 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 181 Review of Midpoint and Distance Formulas Introduction
Student Outcomes. Classwork. Opening Exercises 1 2 (5 minutes)
Student Outcomes Students use the Pythagorean Theorem to determine an unknown dimension of a cone or a sphere. Students know that a pyramid is a special type of cone with triangular faces and a rectangular
Ready To Go On? Skills Intervention 10-1 Solid Geometry
10A Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 10-1 Solid Geometry face edge vertex prism cylinder pyramid cone cube net cross
SOLIDS AND THEIR MEASUREMENTS
SOLIDS ND THEIR MESUREMENTS 9.. 9.. In this chapter, students examine three-dimensional shapes, known as solids. Students will work on visualizing these solids by building and then drawing them. Visualization
Math 165 Guided Activity to study ahead some concepts from sections 1.1 and 1.2 Name Section Distance and Midpoint Formula
Math 165 Guided Activity to study ahead some concepts from sections 1.1 and 1. Name Section 1.1 - Distance and Midpoint Formula Use the power point presentation for sections 1.1 and 1. to answer the following
The figures below are all prisms. The bases of these prisms are shaded, and the height (altitude) of each prism marked by a dashed line:
Prisms Most of the solids you ll see on the Math IIC test are prisms or variations on prisms. A prism is defined as a geometric solid with two congruent bases that lie in parallel planes. You can create
Vocabulary. Term Page Definition Clarifying Example. apothem. center of a circle. center of a regular polygon. central angle of a regular polygon
CHAPTER 9 Vocabulary The table contains important vocabulary terms from Chapter 9. As you work through the chapter, fill in the page number, definition, and a clarifying example. apothem Term Page Definition
Review Exercise. 1. Determine vector and parametric equations of the plane that contains the
Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, 42. 2. In question 1, there are a variety of different answers possible,
Name: Target 12.2: Find and apply surface of Spheres and Composites 12.2a: Surface Area of Spheres 12.2b: Surface Area of Composites Solids
Unit 12: Surface Area and Volume of Solids Target 12.0: Euler s Formula and Introduction to Solids Target 12.1: Find and apply surface area of solids 12.1a: Surface Area of Prisms and Cylinders 12.1b:
Warm-Up. Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)
Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) ) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,) 8.4 Graph and Write Equations of Ellipses What are the major parts of
UNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
Modeling and Prototypes
Modeling and Prototypes 4.4.1 Unit 4, Lesson 4 Explanation The Unit Big Idea The Engineering Design process is a systematic, iterative problem solving method which produces solutions to meet human wants
Lesson Polygons
Lesson 4.1 - Polygons Obj.: classify polygons by their sides. classify quadrilaterals by their attributes. find the sum of the angle measures in a polygon. Decagon - A polygon with ten sides. Dodecagon
Functions of Several Variables
. Functions of Two Variables Functions of Several Variables Rectangular Coordinate System in -Space The rectangular coordinate system in R is formed by mutually perpendicular axes. It is a right handed
Where we are. HOW do we apply the concepts of quadratics to better understand higher order polynomials?
A. Lesson Context BIG PICTURE of this UNIT: How & why do we build NEW knowledge in Mathematics? What NEW IDEAS & NEW CONCEPTS can we now explore with specific references to POLYNOMIAL FUNCTIONS AND RATIONAL
Lesson 1 4 Ordered Pairs and Relations.notebook. September 10, Lesson 1 4. Ordered Pairs and Relations
Lesson 1 4 Ordered Pairs and Relations 1 WHY? If "starting point" is (0,0), name the locations for Clue 1, Clue 2, and Clue 4. 2 Why Answer: Clue 1: (2, 1) Clue 2: (3, 2) Clue 4: (4, 5) 3 Quick Review
Geometry Foundations Planning Document
Geometry Foundations Planning Document Unit 1: Chromatic Numbers Unit Overview A variety of topics allows students to begin the year successfully, review basic fundamentals, develop cooperative learning
Section 7.2 Volume: The Disk Method
Section 7. Volume: The Disk Method White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3 White Board Challenge Calculate the volume V of the solid
Guided Problem Solving
-1 Guided Problem Solving GPS Student Page 57, Exercises 1 1: Match each rule with the correct translation. A. (x, y) (x, y 1 ) I. P(, 1) P (3, ) B. (x, y) (x 1 3, y) II. Q(3, 0) Q (3, ) C. (x, y) (x 1,
a b c d e f Guided practice worksheet Questions are targeted at the grades indicated Areas and perimeters
2 14.1 14.2 Areas and perimeters 1 Work out the area of the following shapes. 8 cm 7 cm 11 cm......... d e f 6 cm 6 cm 10 cm 9 cm......... 2 Work out the area of the following shapes. a b c 3 mm 3 mm 10
Geometry Foundations Pen Argyl Area High School 2018
Geometry emphasizes the development of logical thinking as it relates to geometric problems. Topics include using the correct language and notations of geometry, developing inductive and deductive reasoning,
Date Lesson Text TOPIC Homework. SA of Prisms & Pyramids Pg. 441 # 1, 3, 5a, 7b, 11bc, 16. Surface Area of Cylinders WS 6.6
UNIT 6 MEASUREMENT Date Lesson Text TOPIC Homework May 6.1 8.1 May 4 6. 8. The Pythagorean Theorem Pg. 4 # 1ac, ac, ab, 4ac, 5, 7, 8, 10 Perimeter and Area (NO CIRCLES) Pg. 4 # 1acde, abdf,, 4, 11, 14,
SPRINGBOARD UNIT 5 GEOMETRY
SPRINGBOARD UNIT 5 GEOMETRY 5.1 Area and Perimeter Perimeter the distance around an object. To find perimeter, add all sides. Area the amount of space inside a 2 dimensional object. Measurements for area
Chapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
Geometry CP Pen Argyl Area High School 2018
Geometry emphasizes the development of logical thinking as it relates to geometric problems. Topics include using the correct language and notations of geometry, developing inductive and deductive reasoning,
Write down a formula for the surface area of a Prism and a Cylinder
Write down a formula for the surface area of a Prism and a Cylinder Quiz Thursday Naming Figures Cross Sections Nets Lateral Area, Surface Area Prisms and cylinders have 2 congruent parallel bases. A lateral
The Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
11.4 Three-Dimensional Figures
11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called
12.6 Cylinders and Quadric Surfaces
12 Vectors and the Geometry of Space 12.6 and Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and We have already looked at two special types of surfaces:
Double Integrals over Polar Coordinate
1. 15.4 DOUBLE INTEGRALS OVER POLAR COORDINATE 1 15.4 Double Integrals over Polar Coordinate 1. Polar Coordinates. The polar coordinates (r, θ) of a point are related to the rectangular coordinates (x,y)
Grade 9 Math Terminology
Unit 1 Basic Skills Review BEDMAS a way of remembering order of operations: Brackets, Exponents, Division, Multiplication, Addition, Subtraction Collect like terms gather all like terms and simplify as
Geometry Practice. 1. Angles located next to one another sharing a common side are called angles.
Geometry Practice Name 1. Angles located next to one another sharing a common side are called angles. 2. Planes that meet to form right angles are called planes. 3. Lines that cross are called lines. 4.
Review for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
MATH 2023 Multivariable Calculus
MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set
EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS
EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS Steps to Find the Median of a Triangle: -Find the midpoint of a segment using the midpoint formula. -Use the vertex and midpoint to find the
Nets and Drawings for Visualizing Geometry. Unit 1 Lesson 1
Nets and Drawings for Visualizing Geometry Unit 1 Lesson 1 Students will be able to: Represent three-dimensional figures using nets. Make isometric and orthographic drawings. Key Vocabulary: Net Isometric
Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.
Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x
Math Geometry FAIM 2015 Form 1-A [ ]
![Math Geometry FAIM 2015 Form 1-A [ ]](/thumbs/80/81205175.jpg "Math Geometry FAIM 2015 Form 1-A [ ]")
Math Geometry FAIM 2015 Form 1-A [1530458] Student Class Date Instructions Use your Response Document to answer question 13. 1. Given: Trapezoid EFGH with vertices as shown in the diagram below. Trapezoid
6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
Integers & Absolute Value Properties of Addition Add Integers Subtract Integers. Add & Subtract Like Fractions Add & Subtract Unlike Fractions
Unit 1: Rational Numbers & Exponents M07.A-N & M08.A-N, M08.B-E Essential Questions Standards Content Skills Vocabulary What happens when you add, subtract, multiply and divide integers? What happens when
Work Toward Verification of a Geometric Construction The Regular Pentagon
Work Toward Verification of a Geometric Construction The Regular Pentagon Ed Ladenburger August 9, 007 Given: Steps for the geometric construction of a regular pentagon:. Construct a circle this circle,
13.1 2/20/2018. Conic Sections. Conic Sections: Parabolas and Circles
13 Conic Sections 13.1 Conic Sections: Parabolas and Circles 13.2 Conic Sections: Ellipses 13.3 Conic Sections: Hyperbolas 13.4 Nonlinear Systems of Equations 13.1 Conic Sections: Parabolas and Circles
0613ge. Geometry Regents Exam 0613
wwwjmaporg 0613ge 1 In trapezoid RSTV with bases and, diagonals and intersect at Q If trapezoid RSTV is not isosceles, which triangle is equal in area to? 2 In the diagram below, 3 In a park, two straight
EOC Review: Practice: 1. In the circle below, AB = 2BC. What is the probability of hitting the shaded region with a random dart?
EOC Review: Focus Areas: Trigonometric Ratios Area and Volume including Changes in Area/Volume Geometric Probability Proofs and Deductive Reasoning including Conditionals Properties of Polygons and Circles
FORMULAS to UNDERSTAND & MEMORIZE
1 of 6 FORMULAS to UNDERSTAND & MEMORIZE Now we come to the part where you need to just bear down and memorize. To make the process a bit simpler, I am providing all of the key info that they re going
Coordinate Geometry Lisa A. Over Robert Morris University Professor Trombetta EDUC 6400: Reading Across the Curriculum and ELL
Coordinate Geometry Lisa A. Over Robert Morris University Professor Trombetta EDUC 6400: Reading Across the Curriculum and ELL Overview According to the PA State Standards for mathematics, students must
Geometry Assessment. Eligible Texas Essential Knowledge and Skills
Geometry Assessment Eligible Texas Essential Knowledge and Skills STAAR Geometry Assessment Reporting Category 1: Geometric Structure The student will demonstrate an understanding of geometric structure.
At this station, you have a collection of prisms. Cling nets are included as well.
At this station, you have a collection of prisms. Cling nets are included as well. Sketch a picture of the net for a non- rectangular prism. Label the net to indicate the parts of the prism including edge,
Module 4A: Creating the 3D Model of Right and Oblique Pyramids
Inventor (5) Module 4A: 4A- 1 Module 4A: Creating the 3D Model of Right and Oblique Pyramids In Module 4A, we will learn how to create 3D solid models of right-axis and oblique-axis pyramid (regular or
Dolton School District 148 Curriculum Mapping Grade 7 Illinois Mathematics Assessment Frameworks
Dolton School District 148 Curriculum Mapping Grade 7 Mathematics works August/September Ch.1 Data Toolbox / Ch. 2 Number Theory & Algebraic Reasoning works Textbook Lessons Vocabulary/Posters Resources
Unit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all
6.7. POLAR COORDINATES
6.7. POLAR COORDINATES What You Should Learn Plot points on the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and
MPM 1D Learning Goals and Success Criteria ver1 Sept. 1, Learning Goal I will be able to: Success Criteria I can:
MPM 1D s and ver1 Sept. 1, 2015 Strand: Number Sense and Algebra (NA) By the end of this course, students will be able to: NA1 Demonstrate an understanding of the exponent rules of multiplication and division,
1.6 Modeling with Equations
1.6 Modeling with Equations Steps to Modeling Problems with Equations 1. Identify the variable you want to solve for. 2. Express all unknown quantities in terms of this variable. 3. Set up the model by
Anoka Hennepin K-12 Curriculum plan
Anoka Hennepin K-12 Curriculum plan Department: Elementary Math Unit Title: Packages and Polygons (Blue Book, Geo and Measurement) Triangles and Beyond (Blue Book, Geo and Measurement) Everyday Math: Volume
2 nd Semester Final Exam Review
2 nd Semester Final xam Review I. Vocabulary hapter 7 cross products proportion scale factor dilation ratio similar extremes scale similar polygons indirect measurements scale drawing similarity ratio
Beaumont Middle School Design Project April May 2014 Carl Lee and Craig Schroeder
Beaumont Middle School Design Project April May 2014 Carl Lee and Craig Schroeder 1 2 SketchUp 1. SketchUp is free, and you can download it from the website www.sketchup.com. For some K12 use, see www.sketchup.com/3dfor/k12-education.
In the first part of the lesson, students plot. Abridged
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
S3 (3.1) Volume.notebook March 02, 2016
Daily Practice 22.2.2016 Q1. Multiply out and simplify (3x - 2)(x 2-7x + 3) Q2. Factorise fully 3x 2-75 L.I: Today we will be revising how to find the volume of a prism. Q3. Calculate the value of a house
Lesson 10T ~ Three-Dimensional Figures
Lesson 10T ~ Three-Dimensional Figures Name Period Date Use the table of names at the right to name each solid. 1. 2. Names of Solids 3. 4. 4 cm 4 cm Cone Cylinder Hexagonal prism Pentagonal pyramid Rectangular
S3 (3.1) N5 Volume.notebook April 30, 2018
Daily Practice 16.3.2018 Q1. Multiply out and simplify (3x - 2)(x 2-7x + 3) Daily Practice 19.3.2018 Q1. Multiply out and simplify (2x + 3)(x 2 + 7x + 4) Q2. Factorise fully 3x 2-75 Q2. Simplify x 3 (x
Lines That Intersect Circles
LESSON 11-1 Lines That Intersect Circles Lesson Objectives (p. 746): Vocabulary 1. Interior of a circle (p. 746): 2. Exterior of a circle (p. 746): 3. Chord (p. 746): 4. Secant (p. 746): 5. Tangent of
Draw, construct, and describe geometrical figures and describe the relationships between them.
Focus Standards Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.A.2 7.G.A.3 Draw (freehand, with ruler and protractor, and with technology) geometric