The Divine Proportion. What is the Divine Proportion? What is the Divine Proportion? 10/15/2011. MA 341 Topics in Geometry Lecture 20
|
|
- Randell Greene
- 6 years ago
- Views:
Transcription
1 The Divine Proportion MA 341 Topics in Geometry Lecture 20 What is the Divine Proportion? In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. 14-Oct-2011 MA What is the Divine Proportion? Other names frequently used for the golden ratio are the golden section golden cut golden proportion mean ratio divine proportion mean of Phidias golden mean golden number extreme ratio medial section divine section Denoted by phi = φ 14-Oct-2011 MA
2 Value? Does this ratio have a number associated with it, like π = Oct-2011 MA If a/b = φ, then Value? φ 2 = φ + 1 φ 2 - φ 1 = 0 14-Oct-2011 MA Value? Which is it? Is it + or -? What do we know about φ? a > b so φ > 1 and 14-Oct-2011 MA
3 Phi Oct-2011 MA Properties of φ From earlier we have that Therefore, 14-Oct-2011 MA Properties of φ 14-Oct-2011 MA
4 Φ and φ Sometimes authors use: Φ = and φ = = 1/Φ 14-Oct-2011 MA History Euclid's Elements provides first known written definition of golden mean: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. Euclid gives a construction for cutting a line "in extreme and mean ratio", i.e. the golden ratio. Several propositions and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number. 14-Oct-2011 MA Golden Rectangle A rectangle is called a golden rectangle if its sides are in the ratio of the golden mean: a b 14-Oct-2011 MA
5 Golden Rectangle Remove a square from the golden rectangle: a b b b a-b 14-Oct-2011 MA Golden Rectangle The remaining rectangle is a golden rectangle! Do it again! a b b b a-b 14-Oct-2011 MA Golden Rectangle This actually gives us a construction for a golden rectangle using a compass and straightedge thus GeoGebra or Sketchpad Start with a square a 14-Oct-2011 MA
6 Golden Rectangle Find midpoint of the base and split square in two. a 14-Oct-2011 MA Golden Rectangle Construct diagonal MB. MC = a/2 and BC = a MB =? A B a D M C 14-Oct-2011 MA Golden Rectangle Construct circle with radius MB centered at M. A B a D M C 14-Oct-2011 MA
7 Golden Rectangle Mark point of intersection E. A B a C D M E 14-Oct-2011 MA Golden Rectangle Construct perpendicular at E. A B a C D M E 14-Oct-2011 MA Golden Rectangle Extend AB to meet this perpendicular. A B F a D C E 14-Oct-2011 MA
8 Golden Rectangle AFED is a golden rectangle. A F a D E 14-Oct-2011 MA Golden Spiral Construct a golden rectangle ABCD. A F D E 14-Oct-2011 MA Golden Spiral Construct a square inside XBCY. A X B D Y C 14-Oct-2011 MA
9 Golden Spiral Construct another square inside the smaller golden rectangle.. 14-Oct-2011 MA Again Golden Spiral 14-Oct-2011 MA Again Golden Spiral 14-Oct-2011 MA
10 Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA
11 Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA Golden Spiral In each square construct a quarter circle: 14-Oct-2011 MA
12 Golden Spiral This does give a logarithmic spiral: Θ = a ln(b r), in polar coordinates 14-Oct-2011 MA Golden Spirals? 14-Oct-2011 MA Chambered nautilus shell Golden Spirals? Spiral galaxies 14-Oct-2011 MA
13 Golden Spirals? Cyclones 14-Oct-2011 MA Golden Spirals? Mandelbrot Set 14-Oct-2011 MA Golden Spirals? Flower heads 14-Oct-2011 MA
14 Golden Spirals? Phyllotaxis ( 14-Oct-2011 MA How? Consider the following list of numbers: 1,1,2,3,5,8,13,21,34,55,89,144, This is the Fibonacci sequence {F n }. We are interested in the quotients F n+1 /F n 14-Oct-2011 MA F 0 =1 F 1 =1 F 2 =2 F 3 =3 F 4 =5 F 5 =8 F 6 =13 F 7 =21 F 8 =34 Fibonacci Connection F 9 =55 F 5 /F 4 = 1.6 F 10 =89 F 6 /F 5 = F 11 =144 F 7 /F 6 = F 12 =233 F 8 /F 7 = F 13 =377 F 9 /F 8 = F 1 /F 0 = 1 F 10 /F 9 = F 2 /F 1 = 2 F 11 /F 10 = F 3 /F 2 =1.5 F 12 /F 11 = F 4 /F 3 = F 13 /F 12 = Oct-2011 MA
15 Fibonacci Connection Is it true that: Note: Since φ 2 = φ + 1, multiplying by φ n-1 gives a Fibonacci type relationship: φ n+1 = φ n + φ n-1 (F n+1 = F n + F n-1 ) And it so happens that 14-Oct-2011 MA Other representations? Consider the following sequence: What is lim a n? First, we need to know that {a n } has a limit. This can be shown with calculus. Let L = lim a n Then, 14-Oct-2011 MA Other representations? Hey!!! L = φ, so 14-Oct-2011 MA
16 Another representations Consider the following sequence: 14-Oct-2011 MA Other representations? What is lim b n? First, we need to know that {b n } has a limit. This can be shown with calculus. Let L = lim b n Then, Again, L = φ. 14-Oct-2011 MA Golden Triangle A golden triangle is an isosceles triangle where the ratio of the longer side to the base is φ. 1 φ φ 14-Oct-2011 MA
17 Golden Triangle What are the angles? cos(α)= ½/φ α = 72º, making summit angle 36º 1 α φ φ 14-Oct-2011 MA Golden Triangle 1 + φ = φ 2, so the larger triangle is similar to the smaller with similarity φ. φ φ φ Oct-2011 MA Golden Triangle Where do we find a golden triangle? 14-Oct-2011 MA
18 In fact: Red/green = green/blue = blue/pink = φ Golden Triangle 14-Oct-2011 MA Golden Angle If ratio of arcs a/b = φ, then angle subtended by smaller arc is called golden angle. It measures approximately , or about radians. 14-Oct-2011 MA It is exactly Golden Angle 14-Oct-2011 MA
19 Golden Ratio and Art Proportion.html 14-Oct-2011 MA The Parthenon 14-Oct-2011 MA The Parthenon 14-Oct-2011 MA
20 The Acropolis, Porch of the Maidens 14-Oct-2011 MA Chartes Cathedral & UN Building 14-Oct-2011 MA Taj Mahal 14-Oct-2011 MA
21 Fra Luca Pacioli 14-Oct-2011 MA Pacioli s De divina proportione Written in Milan in , Published in Venice in 1509 The subject - mathematical and artistic proportion, especially mathematics of golden ratio and application in architecture. Leonardo da Vinci drew illustrations of regular solids in De divina proportione while living with and taking mathematics lessons from Pacioli. Discusses use of perspective by painters such as Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano 14-Oct-2011 MA Leonardo da Vinci 14-Oct-2011 MA
22 Leonardo da Vinci 14-Oct-2011 MA Facial Study 14-Oct-2011 MA Mona Lisa 14-Oct-2011 MA
23 Mona Lisa 14-Oct-2011 MA The Last Supper 14-Oct-2011 MA The Annunciation 14-Oct-2011 MA
24 Madonna and Child with St. Anne and St. John 14-Oct-2011 MA Michaelangelo & Raphael 14-Oct-2011 MA The Holy Family - Michelangelo 14-Oct-2011 MA
25 The Crucifixion - Raphael 14-Oct-2011 MA Self Portrait - Rembrandt Red line divides base into golden mean. 14-Oct-2011 MA The Bathers - Seurat 14-Oct-2011 MA
26 The Perfect Face Dr. Stephen Marquardt ( tm) Claims that this gives the most beautiful shape of human face Used decagons and pentagons and embodies φ in all their dimensions. 14-Oct-2011 MA The Perfect Face This mask of the human face is based on the Golden Ratio. The proportions of the length of the nose, the position of the eyes and the length of the chin, all conform to some aspect of the Golden Ratio. 14-Oct-2011 MA The Perfect Face? 14-Oct-2011 MA
27 The Perfect Face? 14-Oct-2011 MA The Perfect Smile Front two teeth form a golden rectangle Also a golden ratio in height to width of center two teeth. Ratio of the width of the 2 center teeth to those next to them is φ. Ratio of width of smile to 3 rd tooth from center is φ 14-Oct-2011 MA See Donald Duck in Mathemagic Land by Disney Studios gah9zc 14-Oct-2011 MA
28 Without mathematics, there is no art. - Fra. Luca Pacioli 14-Oct-2011 MA
Algebra 2. TMT 3 Algebra 2: Student Lesson 2 140
A.1(B) collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.
More informationObjective: Use a compass and straight edge to construct congruent segments and angles.
CONSTRUCTIONS Objective: Use a compass and straight edge to construct congruent segments and angles. Oct 1 8:33 AM Oct 2 7:42 AM 1 Introduction to Constructions Constructions: The drawing of various shapes
More informationDa Vinci and the Divine Proportion in Art Composition
Da Vinci and the Divine Proportion in Art Composition July 7, 2014 by Gary Meisner 10 Comments Leonardo Da Vinci has long been associated with the golden ratio. This association was reinforced in popular
More information1. Find the length of c of the shaded rectangle so that it is a gnomon to the white rectangle with sides 3 and 9.
This problem set will not be collected, so that you can use it in your studying. Nonetheless, you should have it completely done by Monday or at the latest, by Tuesday. 1. Find the length of c of the shaded
More informationObjective: Use a compass and straight edge to construct congruent segments and angles.
CONSTRUCTIONS Objective: Use a compass and straight edge to construct congruent segments and angles. Introduction to Constructions Constructions: The drawing of various shapes using only a pair of compasses
More informationWorksheet 10 Memorandum: Construction of Geometric Figures. Grade 9 Mathematics
Worksheet 10 Memorandum: Construction of Geometric Figures Grade 9 Mathematics For each of the answers below, we give the steps to complete the task given. We ve used the following resources if you would
More informationConstructions. Unit 9 Lesson 7
Constructions Unit 9 Lesson 7 CONSTRUCTIONS Students will be able to: Understand the meanings of Constructions Key Vocabulary: Constructions Tools of Constructions Basic geometric constructions CONSTRUCTIONS
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 3: Constructing Polygons Instruction
rerequisite Skills This lesson requires the use of the following skills: using a compass copying and bisecting line segments constructing perpendicular lines constructing circles of a given radius Introduction
More informationDownloaded from
1 IX Mathematics Chapter 8: Quadrilaterals Chapter Notes Top Definitions 1. A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order. 2. A diagonal
More informationActivity: Fold Four Boxes
ctivity: Fold Four Boxes 1. Cut out your copy of the crease pattern for the square-base twist box but only cut along the solid lines. 2. Look at this key: mountain crease valley crease When folded, a mountain
More informationStep 2: Extend the compass from the chosen endpoint so that the width of the compass is more than half the distance between the two points.
Student Name: Teacher: Date: District: Miami-Dade County Public Schools Test: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the given
More informationSymmetry: A Visual Presentation
Symmetry: A Visual Presentation Line Symmetry Shape has line symmetry when one half of it is the mirror image of the other half. Symmetry exists all around us and many people see it as being a thing of
More informationDesign Fundamentals I: AAID-101 Spring 2012: PROPORTION AND ORDERING SYSTEMS
Design Fundamentals I: AAID-101 Spring 2012: PROPORTION AND ORDERING SYSTEMS From the patterning of the seed in the sunflower To the edges of the Universe A spiral, created by drawing arcs connecting the
More information4 b. Measure the dimensions of given items using standard (English and metric) measurements. Institute Content Based on MS Framework
Algebra/Geometry Institute Summer 2006 Faculty Name: Rene Branson School: Grenada Middle School Grenada, MS Grade Level: 7 th 1 Teaching objective(s) 2007 Mississippi Curriculum Framework 4 b. Measure
More informationHow to Construct a Logarithmic Rosette (Without Even Knowing it) Paul A. Calter
Nexus00/01_017-102 31-05-2001 17:27 Pagina 25 Paul A. Calter How to Construct a Logarithmic Rosette (Without Even Knowing it) Paul Calter explains what a logarithmic rosette is and gives some examples
More informationElementary Geometric Drawings Angles. Angle Bisector. Perpendicular Bisector
Lessons and Activities GEOMETRY Elementary Geometric Drawings Angles Angle Bisector Perpendicular Bisector 1 Lessons and Activities POLYGONS are PLANE SHAPES (figures) with at least 3 STRAIGHT sides and
More informationProject Maths Geometry Notes
The areas that you need to study are: Project Maths Geometry Notes (i) Geometry Terms: (ii) Theorems: (iii) Constructions: (iv) Enlargements: Axiom, theorem, proof, corollary, converse, implies The exam
More informationThe Magic Circle Basic Lesson. Developed by The Alexandria Seaport Foundation
The Magic Circle Basic Lesson Developed by The Alexandria Seaport Foundation The Tools Needed Compass Straightedge Pencil Paper (not graph paper, 8.5 x 11 is fine) Your Brain (the most important tool!)
More informationMeet #3 January Intermediate Mathematics League of Eastern Massachusetts
Meet #3 January 2009 Intermediate Mathematics League of Eastern Massachusetts Meet #3 January 2009 Category 1 Mystery 1. How many two-digit multiples of four are there such that the number is still a
More informationTrigonometry. David R. Wilkins
Trigonometry David R. Wilkins 1. Trigonometry 1. Trigonometry 1.1. Trigonometric Functions There are six standard trigonometric functions. They are the sine function (sin), the cosine function (cos), the
More informationConstructing Perpendiculars to a Line. Finding the Right Line. Draw a line and a point labeled P not on the line, as shown above.
Page 1 of 5 3.3 Intelligence plus character that is the goal of true education. MARTIN LUTHER KING, JR. Constructing Perpendiculars to a Line If you are in a room, look over at one of the walls. What is
More information3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm.
1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify
More informationFibonacci Numbers ANSWERS Lesson 1 of 10, work individually or in pairs
Lesson 1 of 10, work individually or in pairs In 1202, the mathematician Leonardo Pisano Fibonacci (pronounced fi-buh-nah-chee) published a book with the famous Fibonacci sequence in it. (A sequence is
More informationThursday 2 November 2017 Morning Time allowed: 1 hour 30 minutes
Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature GCSE MATHEMATICS Foundation Tier Paper 1 Non-Calculator F Thursday 2 November 2017 Morning
More informationSURDS, SPIRALS, DYNAMIC GEOMETRY AND CAS
SURDS, SPIRALS, DYNAMIC GEOMETRY AND CAS Kaye Stacey and Elizabeth Price University of Melbourne and Canterbury Girls Secondary College This paper aims to enrich the teaching of surds. We use explorations
More information0810ge. Geometry Regents Exam y # (x $ 3) 2 % 4 y # 2x $ 5 1) (0,%4) 2) (%4,0) 3) (%4,%3) and (0,5) 4) (%3,%4) and (5,0)
0810ge 1 In the diagram below, ABC! XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements
More informationTable of Contents. Constructions Day 1... Pages 1-5 HW: Page 6. Constructions Day 2... Pages 7-14 HW: Page 15
CONSTRUCTIONS Table of Contents Constructions Day 1...... Pages 1-5 HW: Page 6 Constructions Day 2.... Pages 7-14 HW: Page 15 Constructions Day 3.... Pages 16-21 HW: Pages 22-24 Constructions Day 4....
More informationJune 2016 Regents GEOMETRY COMMON CORE
1 A student has a rectangular postcard that he folds in half lengthwise. Next, he rotates it continuously about the folded edge. Which three-dimensional object below is generated by this rotation? 4) 2
More information2. Here are some triangles. (a) Write down the letter of the triangle that is. right-angled, ... (ii) isosceles. ... (2)
Topic 8 Shapes 2. Here are some triangles. A B C D F E G (a) Write down the letter of the triangle that is (i) right-angled,... (ii) isosceles.... (2) Two of the triangles are congruent. (b) Write down
More informationJMG. Review Module 1 Lessons 1-20 for Mid-Module. Prepare for Endof-Unit Assessment. Assessment. Module 1. End-of-Unit Assessment.
Lesson Plans Lesson Plan WEEK 161 December 5- December 9 Subject to change 2016-2017 Mrs. Whitman 1 st 2 nd Period 3 rd Period 4 th Period 5 th Period 6 th Period H S Mathematics Period Prep Geometry Math
More informationUnit 6 Lesson 1 Circle Geometry Properties Project
Unit 6 Lesson 1 Circle Geometry Properties Project Name Part A Look up and define the following vocabulary words. Use an illustration where appropriate. Some of this vocabulary can be found in the glossary
More informationCircles Assignment Answer the following questions.
Answer the following questions. 1. Define constructions. 2. What are the basic tools that are used to draw geometric constructions? 3. What is the use of constructions? 4. What is Compass? 5. What is Straight
More informationmatics A presentation by Fernando Corbalán
y matics A presentation by Fernando Corbalán JORNADAS SOBRE EL APRENDIZAJE Y LA ENSEÑANZA DE LAS MATEMÁTICAS Centro de Arte y Creación Industrial 1. 3. 4. 5. In Search for Beauty: The Common Territory
More informationContents. Introduction 4. Leonardo da Vinci 7. Christopher Wren 21. Antoni Gaudí 33. Pablo Picasso 47. Frida Kahlo 59. Glossary 71
Contents Introduction 4 Leonardo da Vinci 7 Christopher Wren 21 Antoni Gaudí 33 Pablo Picasso 47 Frida Kahlo 59 Glossary 71 Leonardo da Vinci 1452 1519 the man who painted the Mona Lisa I had many careers
More information3. Given the similarity transformation shown below; identify the composition:
Midterm Multiple Choice Practice 1. Based on the construction below, which statement must be true? 1 1) m ABD m CBD 2 2) m ABD m CBD 3) m ABD m ABC 1 4) m CBD m ABD 2 2. Line segment AB is shown in the
More informationThe Human Body: Phi & Proportion
The Human Body: Phi & Proportion What is the Golden Ratio? Greek letter "phi" shown right Special number denoting beauty & order Appears many times in geometry, art, architecture and nature including the
More informationWinter Quarter Competition
Winter Quarter Competition LA Math Circle (Advanced) March 13, 2016 Problem 1 Jeff rotates spinners P, Q, and R and adds the resulting numbers. What is the probability that his sum is an odd number? Problem
More informationObjectives. Materials
. Objectives Activity 8 To plot a mathematical relationship that defines a spiral To use technology to create a spiral similar to that found in a snail To use technology to plot a set of ordered pairs
More informationSec Geometry - Constructions
Sec 2.2 - Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have
More informationGeometry SOL G.4 Constructions Name Date Block. Constructions
Geometry SOL G.4 Constructions Mrs. Grieser Name Date Block Constructions Grab your compass and straight edge - it s time to learn about constructions!! On the following pages you will find instructions
More informationh r c On the ACT, remember that diagrams are usually drawn to scale, so you can always eyeball to determine measurements if you get stuck.
ACT Plane Geometry Review Let s first take a look at the common formulas you need for the ACT. Then we ll review the rules for the tested shapes. There are also some practice problems at the end of this
More informationUNIT 1 GEOMETRY. (revision from 1 st ESO) Unit 8 in our books
UNIT 1 GEOMETRY (revision from 1 st ESO) Unit 8 in our books WHAT'S GEOMETRY? Geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. In geometry, one explores
More information0809ge. Geometry Regents Exam Based on the diagram below, which statement is true?
0809ge 1 Based on the diagram below, which statement is true? 3 In the diagram of ABC below, AB # AC. The measure of!b is 40. 1) a! b 2) a! c 3) b! c 4) d! e What is the measure of!a? 1) 40 2) 50 3) 70
More informationSELECTED GEOMETRICAL CONSTRUCTIONS
FACULTY OF NATURAL SCIENCES CONSTANTINE THE PHILOSOPHER UNIVERSITY IN NITRA ACTA MATHEMATICA 17 SELECTED GEOMETRICAL CONSTRUCTIONS ABSTRACT. This article deals with selected classical geometric constructions
More informationMethods in Mathematics (Linked Pair Pilot)
Centre Number Surname Candidate Number For Examiner s Use Other Names Candidate Signature Examiner s Initials Methods in Mathematics (Linked Pair Pilot) Unit 2 Geometry and Algebra Monday 11 November 2013
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 2: Constructing Lines, Segments, and Angles Instruction
Prerequisite Skills This lesson requires the use of the following skills: using a compass understanding the geometry terms line, segment, ray, and angle Introduction Two basic instruments used in geometry
More informationClass VI Mathematics. Time: 2 hour Total Marks: 50
Class VI Mathematics Time: 2 hour Total Marks: 50 1. Correct answer: A 1.35 = Solution Section A 2. Correct answer: A Data collected from a group of 40 students is an example of primary data. 3. Correct
More informationSolutions to Exercise problems
Brief Overview on Projections of Planes: Solutions to Exercise problems By now, all of us must be aware that a plane is any D figure having an enclosed surface area. In our subject point of view, any closed
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission. Junior Certificate Examination Mathematics. Paper 2 Higher Level
2016. S35 Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2016 Mathematics Paper 2 Higher Level Monday 13 June Morning 9:30 to 12:00 300 marks Examination number
More informationLesson 9.1 Assignment
Lesson 9.1 Assignment Name Date Earth Measure Introduction to Geometry and Geometric Constructions Use a compass and a straightedge to complete Questions 1 and 2. 1. Construct a flower with 12 petals by
More informationThe Texas Education Agency and the Texas Higher Education Coordinating Board Geometry Module Pre-/Post-Test. U x T'
Pre-/Post-Test The Texas Education Agency and the Texas Higher Education Coordinating Board Geometry Module Pre-/Post-Test 1. Triangle STU is rotated 180 clockwise to form image STU ' ' '. Determine the
More informationUNIT 3 CIRCLES AND VOLUME Lesson 3: Constructing Tangent Lines Instruction
Prerequisite Skills This lesson requires the use of the following skills: understanding the relationship between perpendicular lines using a compass and a straightedge constructing a perpendicular bisector
More informationSpirals and the Golden Section
John Sharp Spirals and the Golden Section The author examines different types of spirals and their relationships to the Golden Section in order to provide the necessary background so that logic rather
More informationConstructions. Learning Intention: By If you use 1 litre of orange, you will use 4 litres of water (1:4).
Constructions Scales Scales are important in everyday life. We use scales to draw maps, to construct building plans, in housing, street construction... It is impossible to draw building plans with the
More informationOne of the classes that I have taught over the past few years is a technology course for
Trigonometric Functions through Right Triangle Similarities Todd O. Moyer, Towson University Abstract: This article presents an introduction to the trigonometric functions tangent, cosecant, secant, and
More informationLesson 27: Sine and Cosine of Complementary and Special Angles
Lesson 7 M Classwork Example 1 If α and β are the measurements of complementary angles, then we are going to show that sin α = cos β. In right triangle ABC, the measurement of acute angle A is denoted
More informationName. Ms. Nong. Due on: Per: Geometry 2 nd semester Math packet # 2 Standards: 8.0 and 16.0
Name FRIDAY, FEBRUARY 24 Due on: Per: TH Geometry 2 nd semester Math packet # 2 Standards: 8.0 and 16.0 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume
More informationGeometry by Jurgensen, Brown and Jurgensen Postulates and Theorems from Chapter 1
Postulates and Theorems from Chapter 1 Postulate 1: The Ruler Postulate 1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1. 2. Once
More information3 Kevin s work for deriving the equation of a circle is shown below.
June 2016 1. A student has a rectangular postcard that he folds in half lengthwise. Next, he rotates it continuously about the folded edge. Which three-dimensional object below is generated by this rotation?
More informationKangourou Mathematics 2008 Levels 7-8
3 points 1) How many pieces of string are there in the picture? A) 3 B) 4 C) 5 D) 6 E) 7 2) In a class there are 9 boys and 13 girls. Half of the children in this class have got a cold. How many girls
More informationis formed where the diameters intersect? Label the center.
E 26 Get Into Shape Hints or notes: A circle will be folded into a variety of geometric shapes. This activity provides the opportunity to assess the concepts, vocabulary and knowledge of relationships
More information1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw.
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 4 Name Date 1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw.
More informationCopying a Line Segment
Copying a Line Segment Steps 1 4 below show you how to copy a line segment. Step 1 You are given line segment AB to copy. A B Step 2 Draw a line segment that is longer than line segment AB. Label one of
More informationPENNSYLVANIA. List properties, classify, draw, and identify geometric figures in two dimensions.
Know: Understand: Do: CC.2.3.4.A.1 -- Draw lines and angles and identify these in two-dimensional figures. CC.2.3.4.A.2 -- Classify twodimensional figures by properties of their lines and angles. CC.2.3.4.A.3
More informationHow to Do Trigonometry Without Memorizing (Almost) Anything
How to Do Trigonometry Without Memorizing (Almost) Anything Moti en-ari Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ c 07 by Moti en-ari. This work is licensed under the reative
More information1 st Subject: 2D Geometric Shape Construction and Division
Joint Beginning and Intermediate Engineering Graphics 2 nd Week 1st Meeting Lecture Notes Instructor: Edward N. Locke Topic: Geometric Construction 1 st Subject: 2D Geometric Shape Construction and Division
More informationDIFFERENT SEQUENCES. Learning Outcomes and Assessment Standards T 2 T 3
Lesson 21 DIFFERENT SEQUENCES Learning Outcomes and Assessment Standards Learning Outcome 1: Number and number relationships Assessment Standard Investigate number patterns including but not limited to
More informationItalian Renaissance Art
OLLI at Duke Winter 2017 Kris Door, lecturer kristinedoor.com North Carolina Museum of Art Lectures: Wednesdays, 11:00-12:30 Italian Renaissance Art February 15 Italian Renaissance and Mannerism Stylize
More informationHANOI STAR - APMOPS 2016 Training - PreTest1 First Round
Asia Pacific Mathematical Olympiad for Primary Schools 2016 HANOI STAR - APMOPS 2016 Training - PreTest1 First Round 2 hours (150 marks) 24 Jan. 2016 Instructions to Participants Attempt as many questions
More informationLesson 1. Unit 4. Golden Ratio. Ratio
Lesson 1 Ratio Golden Ratio The golden ratio is a special ratio that is found in nature. In a nautilus shell it is found in the spiral. The spiral forms squares as shown. The rectangle formed reflects
More informationArt of the Renaissance
Art of the Renaissance Changes in Art & Learning The rise of Humanism can be seen in paintings created by Renaissance artists. During the Medieval period, art and learning were centered on the church and
More informationConstruction Junction, What s your Function?
Construction Junction, What s your Function? Brian Shay Teacher and Department Chair Canyon Crest Academy Brian.Shay@sduhsd.net @MrBrianShay Session Goals Familiarize ourselves with CCSS and the GSE Geometry
More information1. Convert 60 mi per hour into km per sec. 2. Convert 3000 square inches into square yards.
ACT Practice Name Geo Unit 3 Review Hour Date Topics: Unit Conversions Length and Area Compound shapes Removing Area Area and Perimeter with radicals Isosceles and Equilateral triangles Pythagorean Theorem
More informationGCSE APPLICATIONS OF MATHEMATICS (LINKED PAIR)
Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature GCSE APPLICATIONS OF MATHEMATICS (LINKED PAIR) Foundation Tier Unit 2 Geometry and Measures
More information(Geometry) Academic Standard: TLW use appropriate tools to perform basic geometric constructions.
Seventh Grade Mathematics Assessments page 1 (Geometry) Academic Standard: TLW use appropriate tools to perform basic geometric constructions. A. TLW use tools to draw squares, rectangles, triangles and
More informationGROWING UP IN MORECAMBE 2008 GROWING UP IN MORECAMBE The Mathematics of Shell Construction. and other patterns
GROWING UP IN MORECAMBE 2008 GROWING UP IN MORECAMBE 2008 The Mathematics of Shell Construction and other patterns The Mathematics of Shell Construction and other patterns Contents 3 Fibonnaci Numbers
More informationGeometry Unit 3 Note Sheets Date Name of Lesson. Slopes of Lines. Partitioning a Segment. Equations of Lines. Quiz
Date Name of Lesson Slopes of Lines Partitioning a Segment Equations of Lines Quiz Introduction to Parallel and Perpendicular Lines Slopes and Parallel Lines Slopes and Perpendicular Lines Perpendicular
More information7th Grade Drawing Geometric Figures
Slide 1 / 53 Slide 2 / 53 7th Grade Drawing Geometric Figures 2015-11-23 www.njctl.org Slide 3 / 53 Topics Table of Contents Determining if a Triangle is Possible Click on a topic to go to that section
More informationChapter 11: Constructions and Loci
Chapter 11: Section 11.1a Constructing a Triangle given 3 sides (sss) Leave enough room above the line to complete the shape. Do not rub out your construction lines. They show your method. 1 Section 11.1b
More informationStudent Name: Teacher: Date: District: Rowan. Assessment: 9_12 T and I IC61 - Drafting I Test 1. Form: 501
Student Name: Teacher: Date: District: Rowan Assessment: 9_12 T and I IC61 - Drafting I Test 1 Description: Test 4 A (Diagrams) Form: 501 Please use the following figure for this question. 1. In the GEOMETRIC
More informationDaniel Plotnick. November 5 th, 2017 Mock (Practice) AMC 8 Welcome!
November 5 th, 2017 Mock (Practice) AMC 8 Welcome! 2011 = prime number 2012 = 2 2 503 2013 = 3 11 61 2014 = 2 19 53 2015 = 5 13 31 2016 = 2 5 3 2 7 1 2017 = prime number 2018 = 2 1009 2019 = 3 673 2020
More informationConstructing Perpendicular and Parallel Lines. Adapted from Walch Education
Constructing Perpendicular and Adapted from Walch Education Perpendicular Lines and Bisectors Perpendicular lines are two lines that intersect at a right angle (90 ). A perpendicular line can be constructed
More informationIndicate whether the statement is true or false.
MATH 121 SPRING 2017 - PRACTICE FINAL EXAM Indicate whether the statement is true or false. 1. Given that point P is the midpoint of both and, it follows that. 2. If, then. 3. In a circle (or congruent
More informationGeometer s Skethchpad 8th Grade Guide to Learning Geometry
Geometer s Skethchpad 8th Grade Guide to Learning Geometry This Guide Belongs to: Date: Table of Contents Using Sketchpad - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 17, :30 to 3:30 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 17, 2017 12:30 to 3:30 p.m., only Student Name: School Name: The possession or use of any communications
More informationGOAL Practise techniques for creating various types of geometric lines by constructing and reproducing figures. sheet of letter-sized white paper
TECHNIQUE STUDENT BOOK Chapter 11, page 340 TOOLBOX Pages 62 67 GOAL Practise techniques for creating various types of geometric lines by constructing and reproducing figures. MATERIALS drawing board T-square
More informationDivine Composition With Fibonacci s Ratio (The Rule of Thirds on Steroids)- AC Core
Are you a stickler for little details? Well, if you re a photographer, you had better be. Discovering the rule of thirds is a big milestone for any photographer. Suddenly, you realize that all you ever
More information6th FGCU Invitationdl Math Competition
6th FGCU nvitationdl Math Competition Geometry ndividual Test Option (E) for all questions is "None of the above." 1. MC = 12, NC = 6, ABCD is a square. 'h What is the shaded area? Ans ~ (A) 8 (C) 25 2.
More informationMATHEMATICS UNIT 2: CALCULATOR-ALLOWED FOUNDATION TIER
Surname Centre Number Candidate Number Other Names 0 GCSE NEW 3300U20-1 S17-3300U20-1 MATHEMATICS UNIT 2: CALCULATOR-ALLOWED FOUNDATION TIER TUESDAY, 20 JUNE 2017 AFTERNOON 1 hour 30 minutes For s use
More informationCONSTRUCTION #1: Segment Copy
CONSTRUCTION #1: Segment Copy Objective: Given a line segment, construct a line segment congruent to the given one. Procedure: After doing this Your work should look like this Start with a line segment
More informationPerry High School. Geometry: Week 3
Geometry: Week 3 Monday: Labor Day! Tuesday: 1.5 Segments and Angle Bisectors Wednesday: 1.5 - Work Thursday: 1.6 Angle Pair Relationships Friday: 1.6-Work Next Week 1.7, Review, Exam 1 on FRIDAY 1 Tuesday:
More informationNumber Fun December 3,
Number Fun December 3, 2008 John L. Lehet jlehet@mathmaverick.com www.mathmaverick.com Numbers Fibonacci Numbers Digital Roots Vedic Math Original Puzzles MathMagic Tricks Predict the Sum? (PredictTheSum.xls)
More informationParallel and Perpendicular Lines on the Coordinate Plane
Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane 1.5 Learning Goals Key Term In this lesson, you will: Determine whether lines are parallel. Identify and write the
More informationSlopes of Lines Notes What is slope?
Slopes of Lines Notes What is slope? Find the slope of each line. 1 Find the slope of each line. Find the slope of the line containing the given points. 6, 2!!"#! 3, 5 4, 2!!"#! 4, 3 Find the slope of
More information5.3. Area of Polygons and Circles Play Area. My Notes ACTIVITY
Area of Polygons and Circles SUGGESTED LEARNING STRATEGIES: Think/Pair/Share ACTIVITY 5.3 Pictured below is an aerial view of a playground. An aerial view is the view from above something. Decide what
More informationMaterials: Computer lab or set of calculators equipped with Cabri Geometry II and lab worksheet.
Constructing Perpendiculars Lesson Summary: Students will complete the basic compass and straight edge constructions commonly taught in first year high school Geometry. Key Words: perpendicular, compass,
More informationName Date Class Practice A. 5. Look around your classroom. Describe a geometric pattern you see.
Practice A Geometric Patterns Identify a possible pattern. Use the pattern to draw the next figure. 5. Look around your classroom. Describe a geometric pattern you see. 6. Use squares to create a geometric
More informationMathematics Foundation Tier, June /1F (Paper 1, non-calculator)
Link to past paper on AQA website: www.aqa.org.uk The associated question paper is available to download freely from the AQA website. To navigate around the website, choose QUALIFICATIONS, GCSE, MATHS,
More informationSenior Math Circles: Geometry III
University of Waterloo Faculty of Mathematics entre for Education in Mathematics and omputing Senior Math ircles: Geometry III eview of Important Facts bout Trigonometry Most famous trig identity: sin
More informationBasic Mathematics Review 5232
Basic Mathematics Review 5232 Symmetry A geometric figure has a line of symmetry if you can draw a line so that if you fold your paper along the line the two sides of the figure coincide. In other words,
More information