Hard Math Middle School
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
CC Geometry H
CC Geometry H Aim #8: How do we find perimeter and area of polygons in the Cartesian Plane? Do Now: Find the area of the triangle below using decomposition and then using the shoelace formula (also known as Green's theorem). 1) Given rectangle ABCD, a. Identify the vertices. b. Find the exact perimeter. c. Find the area using the area formula. d. List the vertices starting with A moving counterclockwise and apply the shoelace formula. Does the method work for quadrilaterals? 2) Calculate the area using the shoelace formula and then find the perimeter (nearest hundredth). 3) Find the perimeter (to the nearest hundredth) and the area of the quadrilateral with vertices A(-3,4), B(4,6), C(2,-3), and D(-4,-4). 4) A textbook has a picture of a triangle with vertices (3,6) and (5,2). Something happened in printing the book and the coordinates of the third vertex are listed as (-1, ). The answers in the back of the book give the area of the triangle as 6 square units. What is the y-coordinate of this missing vertex? 5) Find the area of the pentagon with vertices A(5,8), B(4,-3), C(-1,-2), D(-2,4), and E(2,6). 6) Show that the shoelace formula used on the trapezoid shown confirms the traditional formula for the area of a trapezoid: 1 (b1 + b2)h 2 D (x3,y) C (x2,y) A B (x ,0) (0,0) 1 7) Find the area and perimeter (exact and nearest hundredth) of the hexagon shown. -
Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities Student Outcomes . Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities. Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem. Lesson Notes In previous lessons, students found the area of polygons in the plane using the “shoelace” method. In this lesson, we give a name to this method—Green’s theorem. Students will draw polygons described by a system of inequalities, find the perimeter of the polygon, and use Green’s theorem to find the area. Classwork Opening Exercises (5 minutes) The opening exercises are designed to review key concepts of graphing inequalities. The teacher should assign them independently and circulate to assess understanding. Opening Exercises Graph the following: a. b. Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 135 Date: 8/28/14 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY d. c. Example 1 (10 minutes) Example 1 A parallelogram with base of length and height can be situated in the coordinate plane as shown. Verify that the shoelace formula gives the area of the parallelogram as . What is the area of a parallelogram? Base height . The distance from the -axis to the top left vertex is some number . -
Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M4 GEOMETRY Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane Student Outcomes . Students find the perimeter of a quadrilateral in the coordinate plane given its vertices and edges. Students find the area of a quadrilateral in the coordinate plane given its vertices and edges by employing Green’s theorem. Classwork Opening Exercise (5 minutes) Scaffolding: . Give students time to The Opening Exercise allows students to practice the shoelace method of finding the area struggle with these of a triangle in preparation for today’s lesson. Have students complete the exercise questions. Add more individually, and then compare their work with a neighbor’s. Pull the class back together questions as necessary to for a final check and discussion. scaffold for struggling If students are struggling with the shoelace method, they can use decomposition. students. Consider providing students with pre-graphed Opening Exercise quadrilaterals with axes on Find the area of the triangle given. Compare your answer and method to your neighbor’s, and grid lines. discuss differences. Go from concrete to abstract by starting with finding the area by decomposition, then translating one vertex to the origin, and then using the shoelace formula. Post the shoelace diagram and formula from Lesson 9. Shoelace Formula Decomposition Coordinates: 푨(−ퟑ, ퟐ), 푩(ퟐ, −ퟏ), 푪(ퟑ, ퟏ) Area of Rectangle: ퟔ units ⋅ ퟑ units = ퟏퟖ square Area Calculation: units ퟏ Area of Left Rectangle: ퟕ. ퟓ square units ((−ퟑ) ∙ (−ퟏ) + ퟐ ∙ ퟏ + ퟑ ∙ ퟐ − ퟐ ∙ ퟐ − (−ퟏ) ∙ ퟑ − ퟏ ∙ (−ퟑ)) ퟐ Area of Bottom Right Triangle: ퟏ square unit Area: ퟔ. -
1 Introduction 2 Problems Without the Formula
DG Kim Spokane Math Circle The Shoelace Theorem March 3rd 1 Introduction The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices. In this lecture, we'll explore the Shoelace Theorem and its applications. 2 Problems without the Formula Find the areas of the following shapes: 1 DG Kim Spokane Math Circle The Shoelace Theorem March 3rd 3 Answers to Exercises 1. 4 × 4 = 16 2. 1=2 × 6 × 2 = 6 3. 6 × 4 − 1=2 × 4 × 4 = 24 − 8 = 16 4. 4 × 6 − 1=2 × 4 × 1 − 1=2 × 4 × 2 = 24 − 2 − 4 = 18 5. 8 × 3 − 1=2 × 8 × 1 = 24 − 4 = 20 4 The Cartesian Plane One quick note that you should know is about the Cartesian Plane. Cartesian Planes are in an (x; y) format. The first number in a Cartesian point is the number of spaces it goes horizontally, and the second number is the number of spaces it goes vertically. You should be able to find the Cartesian coordinates in all the diagrams that were provided earlier. 5 The Shoelace Formula! n−1 n−1 1 X X A = x y + x y − x y − x y 2 i i+1 n 1 i+1 i 1 n i=1 i=1 Okay, so this looks complicated, but now we'll look at why the shoelace formula got its name. 2 DG Kim Spokane Math Circle The Shoelace Theorem March 3rd The reason this formula is called the shoelace formula is because of the method used to find it. -
Areas and Shapes of Planar Irregular Polygons
Forum Geometricorum b Volume 18 (2018) 17–36. b b FORUM GEOM ISSN 1534-1178 Areas and Shapes of Planar Irregular Polygons C. E. Garza-Hume, Maricarmen C. Jorge, and Arturo Olvera Abstract. We study analytically and numerically the possible shapes and areas of planar irregular polygons with prescribed side-lengths. We give an algorithm and a computer program to construct the cyclic configuration with its circum- circle and hence the maximum possible area. We study quadrilaterals with a self-intersection and prove that not all area minimizers are cyclic. We classify quadrilaterals into four classes related to the possibility of reversing orientation by deforming continuously. We study the possible shapes of polygons with pre- scribed side-lengths and prescribed area. In this paper we carry out an analytical and numerical study of the possible shapes and areas of general planar irregular polygons with prescribed side-lengths. We explain a way to construct the shape with maximum area, which is known to be the cyclic configuration. We write a transcendental equation whose root is the radius of the circumcircle and give an algorithm to compute the root. We provide an algorithm and a corresponding computer program that actually computes the circumcircle and draws the shape with maximum area, which can then be deformed as needed. We study quadrilaterals with a self-intersection, which are the ones that achieve minimum area and we prove that area minimizers are not necessarily cyclic, as mentioned in the literature ([4]). We also study the possible shapes of polygons with prescribed side-lengths and prescribed area. -
Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M4 GEOMETRY Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane Student Outcomes . Students find the perimeter of a quadrilateral in the coordinate plane given its vertices and edges. Students find the area of a quadrilateral in the coordinate plane given its vertices and edges by employing Green’s theorem. Classwork Scaffolding: Opening Exercise (5 minutes) . Give students time to The Opening Exercise allows students to practice the shoelace method of finding the area struggle with these of a triangle in preparation for today’s lesson. Have students complete the exercise questions. Add more individually, and then compare their work with a neighbor’s. Pull the class back together questions as necessary to for a final check and discussion. scaffold for struggling If students are struggling with the shoelace method, they can use decomposition. students. Consider providing students with pre-graphed Opening Exercise quadrilaterals with axes on Find the area of the triangle given. Compare your answer and method to your neighbor’s and grid-lines. discuss differences. Go from concrete to abstract by starting with finding the area by decomposition, then translating one vertex to the origin, then using the shoelace formula. Post shoelace diagram and formula from Lesson 9. Shoelace Formula Decomposition Coordinates: Area of Rectangle: square units Area Calculation: Area of Left Rectangle: square units Area of Bottom Right Triangle: square unit ( ) Area of Top Right Triangle: square units Area: square units Area of Shaded Triangle: square units Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane Date: 124 This work is licensed under a © 2014 Common Core, Inc. -
Campus Close-Up
USA FREE PRESS MARCH 31 - APRIL 6 02KSCOLBY.DAT TUESDAY0OCALTV0ISTINGSMARCH 31 6 PM 6:30 7 PM 7:30sponsored8 PM 8:30 9 PM by9:30 10thePM 10:30 11 PM 11:30 Campus KLBY/ABC News (N) Enter- Accord- Accord- Dancing With the Cupid Pilot (Series News (N) Nightline Jimmy Kimmel Live USAH hFREE(CC) PRESStainmentMARCHing-Jim 31ing-Jim - APRILStars 6 (Live) 02KSCOLBY.DAT (CC) Premiere) (N) (CC) (CC) (N) (CC) (N) (CC) KSNK/NBC News (N) Wheel of The Biggest Loser Recent winners return. Law & Order: SVU News (N) The Tonight Show Late L j Fortune (N) (CC) With Jay Leno (CC) Night KBSL/CBS News (N) Inside NCIS Dead The Mentalist Without a Trace News (N) Late Show With Late 1< NX (CC) Edition Reckoning (N) (CC) Russet Potatoes (N) Daylight (N) (CC) (CC) David Letterman Late WEEKDAYS MARCH 31 - APRIL 6 K15CG The NewsHour Doctors Second- Frontline Sick Nova Last Extinction World Ro- Charlie Rose (N) USAd FREEWith PRESS Jim Lehrer (N)MARCHon Call 31Health - APRILAround 6 America 02KSCOLBY.DAT (N) (N) (CC) (DVS) News mance (CC) ESPN Women’s College Basketball: NCAA Women’s College Basketball: NCAA SportsCenter (Live) NFL Live College Close-Up O_ Tournament6 AM Regional6:30 Final7 A --M Teams7:30 TBA. Tournament8 AM Regional8:30 Final9 A --M Teams9:30 TBA. 10(CC)AM 10:30 11(N) (CC)AM Gmday11:30 KLBY/ABCUSA GoodNCIS SplitMorning Decision GoodHouse Morning Euphoria AmericaHouse Euphoria TheHouse Martha No Reason Stewart TheLaw View & Order: SVU Million-Law & Order:News SVU Colby Free Press WEEKDAYSHP^h Kansas(CC) (Part 1 of 2) (CC) (Part 2 of 2) (CC) Show(CC) -
Flames Heating up for Veterans Degrees at the Same Time
FRIDAY 161st YEAR • NO. 264 MARCH 4, 2016 CLEVELAND, TN 22 PAGES • 50¢ CAP, Brinkley using ‘Hats’ to comfort cancer patients CADETS with the local Civil Air By CHRISTY ARMSTRONG Patrol and their Banner Staff Writer commander, Maj. The Cleveland Composite Squadron of the Civil Air Patrol recently col- Linda Quiett, join lected more than 100 T-shirts which will be turned into hats to help Becky Brinkley cancer patients. for a photo as Cleveland resident Becky Brinkley has for two years been using her they present her sewing talents to comfort cancer patients through a program she calls with T-shirts for Comfort Hats. Having learned about her cause, CAP Squadron Commander Maj. her Comfort Hats Linda Quiett decided to have her cadets collect T-shirts as a service proj- program. Banner photo, ect. CHRISTY ARMSTRONG Comfort Hats started simply, with Brinkley setting out to meet a need described by people she knew who battled the disease. “It just seemed like everybody you talked to had cancer,” Brinkley said. “One of the things the ones who lost their hair would mention is how hard it was for them to find a good, affordable hat.” See CAP, Page 10 Inside Today CSCC finalizing details on ‘Early College’ program Initiative slated in fall semester By CHRISTY ARMSTRONG Banner Staff Writer Cleveland State Community Banner photo, LARRY C. BOWERS College has been in the process of Mustangs ride away LEE UNIVERSITY BASEBALL coach Mark Brew, left, and local veteran officials met at Cleveland’s Fort Hill Cemetery recently. finalizing plans for its new early Proceeds from Lee’s Military Appreciation Day April 16 will be used for upgrades to the veterans section of the cemetery, especially the college program for high school with the region title pavilion and dais area. -
And Generalization of Shoelace Algorithm Usage in Convex Polygon
A new approach (extra vertex) and generalization of Shoelace Algorithm usage in convex polygon (Point-in-Polygon) Rakhmanov Ochilbek Department of Computer Science Nile University of Nigeria Abuja, Nigeria [email protected] Abstract- In this paper we aim to bring new approach into Of course, the shape of polygon is also important in usage of Shoelace Algorithm for area calculation in convex application of those methods. Some will show high polygons on Cartesian coordinate system, with concentration efficiency only for convex polygons, but may not do same on point in polygon concept. Generalization of usage of the for concave polygons. While others can deal with concave concept will be proposed for line segment and polygons. polygons, but time complexity may increase. Testing of new method will be done using Python language. Results of tests show that the new approach is more effective This paper aims to bring a new approach to “Sum of than the current one. area” method in a way of calculation and coding. Generalization of usage of this method will be proposed as Keywords— Shoelace Algorithm, Point in polygon, Area of well. Python will be used as a medium for tests. polygon, Python The well-known shoelace algorithm (shoelace formula) is used to calculate the area related problems in polygons. The I. INTRODUCTION algorithm is called so, since it looks like a shoelace during Point-in-polygon (PiP) is one of the fundamental cross product calculation of matrix. It was proposed by operations of Geographic Information Systems. Yet, Gauss, in 1795 [2]. The basic idea be-hind the algorithm is to nowadays this concept is also getting big attention in divide the polygon into triangles and calculate sum of area of graphical programming, mobile game programming, and all formed triangles. -
FAITH in POP CULTURE Religious Representation on Television
Powell 1 1Nss1Do7n7Ei/R 0/ 8 M&91O S2DO4E3C2LI0EI3NT2GY5 9 W/1 F3Oe3bRrKuary 2004 FAITH IN POP CULTURE Religious Representation on Television JARED POWELL University of Kansas Abstract This study is concerned with the presence of religion and religious characters in contemporary popular culture, specifically which groups receive media attention and in what ways they are portrayed. After a careful review of studies in media, religion, and culture, American television is analyzed with a focus on the Fox medical drama House. Using both quantitative and qualitative data found in 155 episodes, the author argues that the world‐renowned television show implicitly holds an agnostic view on religious matters, leaving interpretation up to the audience. Despite many characters that are religious, atheist, or ambiguous about their beliefs, a sense of uncertainty and a lack of specificity appear to promote this idea. Caricaturizing specific beliefs and using them as plot devices also helps to hide religious representation in plain sight. Findings suggest that American popular culture has a tendency to target the broadest possible audience with regard to religiosity by presenting the most basic representation of various views without promoting any single standpoint. More research must be done to further the understanding of how religion is represented via worldwide media and how it is received by consumers. Powell 2 In an age of diverse religious plurality and global media popularity, a few questions must be asked: how do the realms of faith and popular culture intersect, intertwine, interact? How are those in power using our most complex and subtle cultural tools to construct images of differing worldviews? Religious institutions often serve as sources of social pedagogy, but the media does this as well ‐ with the additional power of teaching us how to think about other pedagogical forces (such as religion, philosophy, and academic institutions). -
Kinematic and Experimental Evaluation of Commercial Wheelchair-Mounted
Kinematic and Experimental Evaluation of Commercial Wheelchair-Mounted Robotic Arms by John William Capille, Jr. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Rajiv Dubey, Ph.D. Redwan Alqasemi, Ph.D. Stephanie Carey, Ph.D. Nathan Crane, Ph.D. Date of Approval: October 15th 2010 Keywords: assistive, design, manipulator, rehabilitation, comparison Copyright © 2010, John William Capille, Jr. Dedication I dedicate this work to my colleagues, friends, and family who constantly inspire me and give me the courage and drive to succeed. Acknowledgments I would like to acknowledge my committee members for their contributions to this work. To Dr. Rajiv Dubey, for allowing me to pursue this project and his financial support. To Dr. Redwan Alqasemi, for his insight and guidance throughout this project. To Dr. Stephanie Carey, for her contribution to my understanding of working with participants and certification processes. To Dr. Nathan Crane, for his interest in this work. I would like to acknowledge my colleagues, Derek Lura and Karan Khokar. Their assistance in programming, robotic theory, and experience in high academics was critical to the success of this project. And to Shawn Gicka, for being willing and eager to assist in project protocols and performing his role as Key Safety Personnel. Table of Contents List of Tables ....................................................................................................... -
Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities Student Outcomes . Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities. Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem. Lesson Notes In previous lessons, students found the area of polygons in the plane using the “shoelace” method. In this lesson, the method is given a name—Green’s theorem. Students draw polygons described by a system of inequalities, find the perimeter of the polygon, and use Green’s theorem to find the area. Classwork Opening Exercise (5 minutes) The opening exercises are designed to review key concepts of graphing inequalities. The teacher should assign them independently and circulate to assess understanding. Opening Exercise Graph the following: a. 풚 ≤ ퟕ b. 풙 > −ퟑ Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 135 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M4-TE-1.3.0 -09.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY ퟏ ퟐ 풚 < 풙 − ퟒ d. 풚 ≥ − 풙 + ퟓ c. ퟐ ퟑ Example 1 (10 minutes) Example 1 A parallelogram with base of length 풃 and height 풉 can be situated in the coordinate plane, as shown.