Mathematics 350 Section 6.3 Introduction to Fractals
|
|
- Imogen Walsh
- 5 years ago
- Views:
Transcription
1 Mathematics 350 Section 6.3 Introduction to Fractals A fractal is generally "a rough or fragmented geometric shape that is self-similar, which means it can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." Mathematical fractals are usually based on one or more simple processes or equations that are applied recursively. This means that there is a starting figure or situation, the process is applied to it, then applied to the out put of the first application, then applied to the output of the second application, and so on infinitely many times. Classical fractals began to be studied by mathematicians over 100 years ago. But during the 1970s methods were developed to display them using a computer, and striking images appeared. Also, Mandelbrot began to champion them as a way of describing the geometry of many objects in the real world that were not smooth like the lines and circles of Euclidean geometry. First we will describe the process for creating a classical fractal called the Sierpinski Triangle. Start with the region enclosed by an equilateral triangle. This triangle is called the Stage 0 figure. Perform two steps: 1. Locate the midpoints of the sides of the Stage 0 triangle, and connect them with line segments. This divides the region inside the Stage 0 triangle into four congruent subregions bounded by equilateral triangles. 2. Remove the middle triangular region (this is usually done by coloring it dark), leaving three triangular regions touching only at vertices. You have now completed Stage 1 of the process and arrived at the Stage 1 figure, as shown on the next page.
2 Stage 1 Thereafter, at each stage steps 1 and 2 on the first page are applied to all the triangular regions remaining after the previous stage. Classwork 1 Create Stage 2 by applying the Process to the result of Stage 1 shown below. (The shaded triangle has been removed.)
3 Classwork 2 Create Stage 3 by applying the process to the result of Stage 2 shown below. (The shaded triangles have been removed.) The Sierpinski Triangle is what remains after completing an infinite number of stages. After a few stages the changes caused by the next stage cannot be seen by the naked eye, so an excellent approximation to the Sierpinski Triangle can be created by stopping after the first few (usually about 5-8) stages of the construction procedure. Because the instructions at each stage depend on the result of the previous stage, this is called a recursive procedure. (Compound interest is also a recursive procedure. Assuming you don t deposit or withdraw money from your account, your balance during the next period will be (1 + interest rate) x balance at end of current period. So the next balance is always computed from the current one.) It is important to note that the triangles that appear at a given stage are similar to, but at 1/2 scale of the triangles that appeared at the previous stage. Classwork 3 Complete the entries in the table on the next page. To make the patterns in the table more apparent, it is assumed that the sides of the Stage 0 triangle are each 1 unit long. Because the computation of the area of such a triangle in square units would involve 2 and obscure the patterns in the area columns, we have used the area of the Stage 1 triangle as a (nonstandard) unit of area and calculated the other entries as fractions of that area. The entries at a given stage may be calculated using the idea of scaling factor: a triangle that appears at a particular stage is the side length and 1 2 the area of a triangle that appeared at the previous stage.
4 Stage Number of Triangles Perimeter of a Triangle in units Sum of perimeters of all triangles (in units ) Area of each triangle as a fraction of Stage 0 area Total area as a fraction of Stage 0 area If you need it, here is the Stage 3 Triangle: Stage 3
5 Classwork 4 Answer the following questions about your table: 1. What pattern do you see in the numbers in the Number of triangles column? What should the entry be for Stage 4? For Stage n? 2. What pattern do you see in the numbers in the Perimeter of a triangle column? What should the entry be for Stage 4? For Stage n? 3. What pattern do you see in the numbers in the Sum of perimeters column? What should the entry be for Stage 4? For Stage n? 4. What pattern do you see in the numbers in the Area of each triangle column? What should the entry be for Stage 4? For Stage n? 5. What pattern do you see in the numbers in the Total area column? What should the entry be for Stage 4? For Stage n? 6. The picture at any Stage is just an approximation to the Sierpinski Triangle. The actual Sierpinski Triangle is what results if you do this process of middle triangle removing forever. More precisely, the Sierpinski Triangle is the limit of the process. What do you think should be the perimeter of the Sierpinski Triangle (i. e. of the limit of the perimeters of the Stages)? Why? 7. What should be the area of the Sierpinski Triangle (i.e. the limit of the areas of the Stages)?
6 The Sierpinski triangle is self-similar. That means that if we look at a small piece of the object under a microscope, it will look like the original object. For example, at left below is a Sierpinski triangle with 5 stages completed. At right is what we get by blowing up one of the stage 3 triangles in it. The two figures are similar. Many things in nature are structured like fractals, such as coastlines, clouds, plants, mountain ranges. Much of the animated versions of these that appear in the movies such as Star Wars have been generated from more advanced fractal techniques.
7 Fractal Dimension A point has no dimensions - no length, no width, no height. A That dot is obviously way too big to really represent a point. But we'll live with it, if we all just agree what a point really is. A line has one dimension - length. It has no width and no height, but infinite length. L Again, this model of a line is really not very good, but until we learn how to draw a line with 0 width and infinite length, it'll have to do. A plane has two dimensions - length and width, no depth. It's an absolutely flat tabletop extending out both ways to infinity. Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions. Obviously this isn't a good representation of 3-D. Besides its size, it's just a hexagon drawn to fool you into thinking it's a box. Fractals can have fractional (or fractal) dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate. Just as the images above weren't very good pictures of a point, line, plane, or space, the drawing meant to be the Sierpinski Triangle has limitations. Remember that fractals are really formed by infinitely many steps. So there are infinitely many smaller and smaller triangles inside the figure, and infinitely many holes (the black triangles that were removed). Let's look further at what we mean by dimension. Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment
8 Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do you get? Doubling the side gives eight copies. Let's organize our information into a table. Figure Line segment Square Cube Dimension Number of Copies When scale doubled 2 = = = 2 3 Do you see a pattern? It appears that the dimension is the exponent - and it is! So when we double the scale and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension. Let's add that as a row to the table.
9 Figure Line segment Square Cube Self-similar using n copies Dimension d Number of Copies When scale doubled 2 = = = 2 3 n = 2 d We can use this to figure out the dimension of the Sierpinski Triangle because when you double the length of the sides, you get another Sierpinski Triangle similar to the first. Start with a Sierpinski triangle of 1-inch sides. Double the length of the sides. Now how many copies of the original triangle do you have? (Remember that the black triangles are holes, so we can't count them.) Doubling the scale of the Sierpinski Triangle gives us 3 copies, so 3 = 2 d, where d = the dimension. But wait, 2 = 2 1, and 4 = 2 2, so what number could this be? It has to be somewhere between 1 and 2, right? Let's add this to our table. Figure Dimension Number of Copies When scale doubled Line segment Sierpinski Triangle Square Cube Self-similar using n copies So the dimension of Sierpinski's Triangle is between 1 and 2. 1? 2 3 d 2 = = 2? 4 = = 2 3 n = 2 d
10 Classwork 5 Use a calculator with an exponent key (usually ^) to find a value of d so that 2 d = 3. (For example, try 1.1. Type 2^1.1 and you get ) Determine d to 2 decimal places. Now solve 2 d = 3 algebraically by taking the logarithm of both sides and using a property of logarithms. Note: The base 2 in the number of copies column is of the table is the scale factor used to scale up to the larger similar copy. In other fractals the scale factor may be 3, or 4, or whatever. But in general, if scaling up by a scale factor s yields a similar copy made up of n copies of the original, then n = s d The Koch Fractal Use the triangular grids on the next page. In the first draw in pencil a horizontal line segment that is 9 units long. This is stage 0. Then: 1. Divide each straight line segment (at first there is only one) into three equal segments, and remove (erase) the middle part. 2. Replace each removed part with two segments each as long as the segment removed, making a v whose ends attach where the removed part was, so the v points outward. Here is a diagram of what happens at the first stage: each becomes The result of applying the steps to the original is called Stage 1. Create it in the top grid. It has 4 segments. To get Stage 2 apply the two steps above to each segment in Stage 1. You may do this in the first grid on top of Stage 1, or (if you don t like erasing) you may create it in the middle grid. To get stage 3 apply all the steps to the segments in Stage 2. You may create these in your Stage 2 or (if you don t like erasing) create it in the bottom grid. The Koch Fractal Curve is the result of iterating the process infinitely many times.
11 Once you have created the first three stages of the Koch Fractal, complete the chart below. Stage Number of Segments Length of each segment Total length at this stage Classwork 6 Answer the following questions about your table: 1. What pattern do you see in the numbers in the Number of segments column? What should the entry be for Stage 4? For Stage n? 2. What pattern do you see in the numbers in the Length of each segment column? What should the entry be for Stage 4? For Stage n? 3. What pattern do you see in the numbers in the Total length at this stage column? What should the entry be for Stage 4? For Stage n? 4. The picture at any Stage is just an approximation to the Koch Fractal. The actual Koch fractal is what results if you do this process forever. More precisely, the Koch fractal is the limit of the process. What do you think should be the total length of the Koch fractal (i. e. of the limit of the total lengths of the Stages)? Why? 5. What should be the fractal dimension of the Koch fractal? (Hint: How many copies do we need to put together to get a copy that is twice as big as the original?)
12 Homework for Section 6.3 Your homework is to repeat the work done in class, but with the Sierpinski Gasket. In this process we start with a square, think of it as 3 by 3 array of smaller squares whose sides are 1/3 as long as the original square, and then remove the middle square. Here is stage one, where the darkened square has been removed: 1. To create stage two, apply the process to each of the remaining 8 squares. (Use the figure above to show this.) 2. Suppose the outer square has sides of length 1 unit, and so has area 1 square unit. How much total area remains at stage 1? Justify your answer. 3. Continuing as in 2., how much total area remains at stage 2? Justify your answer.
13 4. Continuing as in 2. and 3., how much total area remains at stage 3? Justify your answer. 5. Continuing as in , how much total area remains at stage n? 6. The Sierpinski Gasket is the limit of this process of removing the inner square in each of the squares at the previous stage. Based on your answers to , what should be the area of the Sierpinski Gasket? Justify your answer. 7. What is the fractal dimension of the Sierpinski Gasket? Answer this exactly, and give a decimal approximation to 2 decimal places. Justify your answers. (Hint: The scale factor is not 2 here!)
14 References The beauty of fractal images is best appreciated on a computer. There are many web sites that provide java or other programs that are appropriate for teachers and students. Here are a few: 1. This site has a sequence of lesson on fractals that are appropriate for middle grades students. Some of the material in this handout was adapted from this web site This reference is to a NOVA program on fractals. Excellent images This site has a number of applets that generate fractals This lesson relates to construction of the Sierpinski Triangle. A good reference book is: Fractals for the Classroom, by Peitgen et. al., Springer Verlag, 1992
In this lesson, students build fractals and track the growth of fractal measurements using tables and equations. Enduring Understanding
LessonTitle: Fractal Functions Alg 5.8 Utah State Core Standard and Indicators Algebra Standards 2, 4 Process Standards 1-5 Summary In this lesson, students build fractals and track the growth of fractal
More informationSession 27: Fractals - Handout
Session 27: Fractals - Handout Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Benoit Mandelbrot (1924-2010)
More informationExploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss
Exploring Fractals through Geometry and Algebra Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Learning Objective and skills practiced Students will: Learn the three criteria
More informationFractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013
Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013 Background Fractal geometry is one of the most important developments in mathematics in the second half of the 20th century. Fractals
More informationApplications. 44 Stretching and Shrinking
Applications 1. Look for rep-tile patterns in the designs below. For each design, tell whether the small quadrilaterals are similar to the large quadrilateral. Explain. If the quadrilaterals are similar,
More informationFractals. Materials. Pencil Paper Grid made of triangles
Fractals Overview: Fractals are new on the mathematics scene, however they are in your life every day. Cell phones use fractal antennas, doctors study fractal-based blood flow diagrams to search for cancerous
More informationDiscovering. Algebra. An Investigative Approach. Condensed Lessons for Make-up Work
Discovering Algebra An Investigative Approach Condensed Lessons for Make-up Work CONDENSED L E S S O N 0. The Same yet Smaller Previous In this lesson you will apply a recursive rule to create a fractal
More informationFractal Geometry. LIACS Natural Computing Group Leiden University
Fractal Geometry Contents Introduction The Fractal Geometry of Nature Self-Similarity Some Pioneering Fractals Dimension and Fractal Dimension Cellular Automata Particle Systems Scope of Fractal Geometry
More informationFractal Coding. CS 6723 Image Processing Fall 2013
Fractal Coding CS 6723 Image Processing Fall 2013 Fractals and Image Processing The word Fractal less than 30 years by one of the history s most creative mathematician Benoit Mandelbrot Other contributors:
More informationChapel Hill Math Circle: Symmetry and Fractals
Chapel Hill Math Circle: Symmetry and Fractals 10/7/17 1 Introduction This worksheet will explore symmetry. To mathematicians, a symmetry of an object is, roughly speaking, a transformation that does not
More informationFun with Fractals and Functions. CHAMP at University of Houston March 2, 2015 Houston, Texas
Fun with Fractals and Functions CHAMP at University of Houston March 2, 2015 Houston, Texas Alice Fisher afisher@rice.edu Director of Technology Applications & Integration at Rice University School Mathematics
More informationFractal Geometry. Prof. Thomas Bäck Fractal Geometry 1. Natural Computing Group
Fractal Geometry Prof. Thomas Bäck Fractal Geometry 1 Contents Introduction The Fractal Geometry of Nature - Self-Similarity - Some Pioneering Fractals - Dimension and Fractal Dimension Scope of Fractal
More information5th Grade Mathematics Essential Standards
Standard 1 Number Sense (10-20% of ISTEP/Acuity) Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions, and percents. They understand the
More informationFractals and Self-Similarity
Fractals and Self-Similarity Vocabulary iteration fractal self-similar Recognize and describe characteristics of fractals. Identify nongeometric iteration. is mathematics found in nature? Patterns can
More informationFractals. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
Fractals Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Geometric Objects Man-made objects are geometrically simple (e.g., rectangles,
More informationScope and Sequence for the New Jersey Core Curriculum Content Standards
Scope and Sequence for the New Jersey Core Curriculum Content Standards The following chart provides an overview of where within Prentice Hall Course 3 Mathematics each of the Cumulative Progress Indicators
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,
More informationClouds, biological growth, and coastlines are
L A B 11 KOCH SNOWFLAKE Fractals Clouds, biological growth, and coastlines are examples of real-life phenomena that seem too complex to be described using typical mathematical functions or relationships.
More informationLecture 3: Some Strange Properties of Fractal Curves
Lecture 3: Some Strange Properties of Fractal Curves I have been a stranger in a strange land. Exodus 2:22 1. Fractal Strangeness Fractals have a look and feel that is very different from ordinary curves.
More informationNumber/Computation. addend Any number being added. digit Any one of the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9
14 Number/Computation addend Any number being added algorithm A step-by-step method for computing array A picture that shows a number of items arranged in rows and columns to form a rectangle associative
More informationGeneration of 3D Fractal Images for Mandelbrot and Julia Sets
178 Generation of 3D Fractal Images for Mandelbrot and Julia Sets Bulusu Rama #, Jibitesh Mishra * # Department of Computer Science and Engineering, MLR Institute of Technology Hyderabad, India 1 rama_bulusu@yahoo.com
More informationMathematics Background
Finding Area and Distance Students work in this Unit develops a fundamentally important relationship connecting geometry and algebra: the Pythagorean Theorem. The presentation of ideas in the Unit reflects
More information<The von Koch Snowflake Investigation> properties of fractals is self-similarity. It means that we can magnify them many times and after every
Jiwon MYP 5 Math Ewa Puzanowska 18th of Oct 2012 About Fractal... In geometry, a fractal is a shape made up of parts that are the same shape as itself and are of
More informationGrade 6 Math Circles. Shapeshifting
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Plotting Grade 6 Math Circles October 24/25, 2017 Shapeshifting Before we begin today, we are going to quickly go over how to plot points. Centre for Education
More informationUnit 1, Lesson 1: Moving in the Plane
Unit 1, Lesson 1: Moving in the Plane Let s describe ways figures can move in the plane. 1.1: Which One Doesn t Belong: Diagrams Which one doesn t belong? 1.2: Triangle Square Dance m.openup.org/1/8-1-1-2
More informationSimilarity - Using Mirrors to Find Heights
Similarity - Using Mirrors to Find Heights AUTHOR(S): DANA SHAMIR TEACH # 2 MENTOR: NANNETTE STRICKLAND DATE TO BE TAUGHT: 11/29/2007 LENGTH OF LESSON: 45 MINUTES GRADE LEVEL: 8 SOURCE OF THE LESSON: Connecting
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More informationTIMSS 2011 Fourth Grade Mathematics Item Descriptions developed during the TIMSS 2011 Benchmarking
TIMSS 2011 Fourth Grade Mathematics Item Descriptions developed during the TIMSS 2011 Benchmarking Items at Low International Benchmark (400) M01_05 M05_01 M07_04 M08_01 M09_01 M13_01 Solves a word problem
More informationGrade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:
More informationKoch Snowflake Go Figure The Koch Snowflake is a fractal based on a very simple rule.
Koch Snowflake The Koch Snowflake is a fractal based on a very simple rule. The Rule: Whenever you see a straight line, like the one on the left, divide it in thirds and build an equilateral triangle (one
More informationAnswer Key Lesson 5: Area Problems
Answer Key Lesson 5: Problems Student Guide Problems (SG pp. 186 187) Questions 1 3 1. Shapes will vary. Sample shape with an area of 12 sq cm: Problems Here are 12 square centimeters. A square centimeter
More information2 nd Grade Math Learning Targets. Algebra:
2 nd Grade Math Learning Targets Algebra: 2.A.2.1 Students are able to use concepts of equal to, greater than, and less than to compare numbers (0-100). - I can explain what equal to means. (2.A.2.1) I
More informationGrade 6 Math Circles February 19th/20th
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 19th/20th Tessellations Warm-Up What is the sum of all the angles inside
More informationThis lesson gives students practice in graphing
NATIONAL MATH + SCIENCE INITIATIVE 9 Mathematics Solving Systems of Linear Equations 7 5 3 1 1 3 5 7 LEVEL Grade, Algebra 1, or Math 1 in a unit on solving systems of equations MODULE/CONNECTION TO AP*
More informationDiscrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics
Discrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics Robert L. Devaney, Professor Department of Mathematics Boston University Boston, MA 02215 USA bob@bu.edu Abstract.
More informationGrade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More information17.2 Surface Area of Prisms
h a b c h a b c Locker LESSON 17. Surface Area of Prisms and Cylinders Texas Math Standards The student is expected to: G.11.C Apply the formulas for the total and lateral surface area of three-dimensional
More informationAREA Judo Math Inc.
AREA 2013 Judo Math Inc. 6 th grade Problem Solving Discipline: Black Belt Training Order of Mastery: Area 1. Area of triangles by composition 2. Area of quadrilaterals by decomposing 3. Draw polygons
More informationLesson 24: Surface Area
Student Outcomes Students determine the surface area of three-dimensional figures, those that are composite figures and those that have missing sections. Lesson Notes This lesson is a continuation of Lesson
More informationAbout Finish Line Mathematics 5
Table of COntents About Finish Line Mathematics 5 Unit 1: Big Ideas from Grade 1 7 Lesson 1 1.NBT.2.a c Understanding Tens and Ones [connects to 2.NBT.1.a, b] 8 Lesson 2 1.OA.6 Strategies to Add and Subtract
More informationA Review of Fractals Properties: Mathematical Approach
Science Journal of Applied Mathematics and Statistics 2017; 5(3): 98-105 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20170503.11 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)
More informationCCBC Math 081 Geometry Section 2.2
2.2 Geometry Geometry is the study of shapes and their mathematical properties. In this section, we will learn to calculate the perimeter, area, and volume of a few basic geometric shapes. Perimeter We
More informationFun with Fractals Saturday Morning Math Group
Fun with Fractals Saturday Morning Math Group Alistair Windsor Fractals Fractals are amazingly complicated patterns often produced by very simple processes. We will look at two different types of fractals
More informationWe can use square dot paper to draw each view (top, front, and sides) of the three dimensional objects:
Unit Eight Geometry Name: 8.1 Sketching Views of Objects When a photo of an object is not available, the object may be drawn on triangular dot paper. This is called isometric paper. Isometric means equal
More information2.) From the set {A, B, C, D, E, F, G, H}, produce all of the four character combinations. Be sure that they are in lexicographic order.
Discrete Mathematics 2 - Test File - Spring 2013 Exam #1 1.) RSA - Suppose we choose p = 5 and q = 11. You're going to be sending the coded message M = 23. a.) Choose a value for e, satisfying the requirements
More informationGrade 5 Mathematics MCA-III Item Sampler Teacher Guide
Grade 5 Mathematics MCA-III Item Sampler Teacher Guide Grade 5 Mathematics MCA Item Sampler Parent/Teacher Guide The purpose of the Item Samplers is to familiarize students with the online MCA test format.
More informationChapter 6 Rational Numbers and Proportional Reasoning
Chapter 6 Rational Numbers and Proportional Reasoning Students should build their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of
More informationThe Game of Criss-Cross
Chapter 5 The Game of Criss-Cross Euler Characteristic ( ) Overview. The regions on a map and the faces of a cube both illustrate a very natural sort of situation: they are each examples of regions that
More informationUnit 4 End-of-Unit Assessment Study Guide
Circles Unit 4 End-of-Unit Assessment Study Guide Definitions Radius (r) = distance from the center of a circle to the circle s edge Diameter (d) = distance across a circle, from edge to edge, through
More informationTable of Contents. Introduction to the Math Practice Series...1
Table of Contents Table of Contents Introduction to the Math Practice Series...1 Common Mathematics/Geometry Symbols and Terms...2 Chapter 1: Introduction To Geometry...13 Shapes, Congruence, Similarity,
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationGentle Introduction to Fractals
Gentle Introduction to Fractals www.nclab.com Contents 1 Fractals Basics 1 1.1 Concept................................................ 1 1.2 History................................................ 2 1.3
More informationNUMB3RS Activity: I It Itera Iteration. Episode: Rampage
Teacher Page 1 NUMB3RS Activity: I It Itera Iteration Topic: Iterative processes Grade Level: 9-12 Objective: Examine random iterative processes and their outcomes Time: about 45 minutes Materials: TI-83/84
More informationSection A Solids Grade E
Name: Teacher Assessment Section A Solids Grade E 1. Write down the name of each of these 3-D shapes, (i) (ii) (iii) Answer (i)... (ii)... (iii)... (Total 3 marks) 2. (a) On the isometric grid complete
More information1: #1 4, ACE 2: #4, 22. ACER 3: #4 6, 13, 19. ACE 4: #15, 25, 32. ACE 5: #5 7, 10. ACE
Homework Answers from ACE: Filling and Wrapping ACE Investigation 1: #1 4, 10 13. ACE Investigation : #4,. ACER Investigation 3: #4 6, 13, 19. ACE Investigation 4: #15, 5, 3. ACE Investigation 5: #5 7,
More informationBasic and Intermediate Math Vocabulary Spring 2017 Semester
Digit A symbol for a number (1-9) Whole Number A number without fractions or decimals. Place Value The value of a digit that depends on the position in the number. Even number A natural number that is
More informationFractals, Fibonacci numbers in Nature 17 mai 2015
1 Sommaire 1 Sommaire... 1 2 Presentation... 1 3 Fractals in nature... 3 3.1 The Von Koch curve... 3 3.2 The Sierpinski triangle... 3 3.3 The Sierpinski carpet... 3 3.4 Hilbert s fractal... 4 3.5 Cantor
More informationFRACTALS AND THE SIERPINSKI TRIANGLE
FRACTALS AND THE SIERPINSKI TRIANGLE Penelope Allen-Baltera Dave Marieni Richard Oliveira Louis Sievers Hingham High School Marlborough High School Ludlow High School Trinity High School The purpose of
More informationUsing the Best of Both!
Using the Best of Both! A Guide to Using Connected Mathematics 2 with Prentice Hall Mathematics Courses 1, 2, 3 2012, and Algebra Readiness MatBro111707BestOfBothPH10&CMP2.indd 1 6/7/11 11:59 AM Using
More informationAnswer 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
More informationSimplifying Expressions
Unit 1 Beaumont Middle School 8th Grade, 2017-2018 Math8; Intro to Algebra Name: Simplifying Expressions I can identify expressions and write variable expressions. I can solve problems using order of operations.
More informationFractals in Nature and Mathematics: From Simplicity to Complexity
Fractals in Nature and Mathematics: From Simplicity to Complexity Dr. R. L. Herman, UNCW Mathematics & Physics Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 1/41 Outline
More informationA C E. Applications. Applications Connections Extensions
A C E Applications Connections Extensions Applications 1. Suppose that the polygons below were drawn on centimeter grid paper. How many 1-centimeter cubes (some cut in pieces) would it take to cover each
More informationLesson 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.
More informationGrade 5 Mathematics MCA-III Item Sampler Teacher Guide
Grade 5 Mathematics MCA-III Item Sampler Teacher Guide Grade 5 Mathematics MCA Item Sampler Teacher Guide Overview of Item Samplers Item samplers are one type of student resource provided to help students
More informationFor many years, geometry in the elementary schools was confined to
SHOW 118 PROGRAM SYNOPSIS Segment 1 (1:39) MATHMAN: RECTANGLES Mathman is told to eat only the rectangles that appear on his video game board. He notes that rectangles must have four right angles (and
More informationPrime Time (Factors and Multiples)
CONFIDENCE LEVEL: Prime Time Knowledge Map for 6 th Grade Math Prime Time (Factors and Multiples). A factor is a whole numbers that is multiplied by another whole number to get a product. (Ex: x 5 = ;
More informationEngage NY Lesson 15: Representing Three-Dimensional Figures Using Nets
Name: Surface Area & Volume Packet Engage NY Lesson 15: Representing Three-Dimensional Figures Using Nets Classwork Cereal Box Similarities: Cereal Box Differences: Exercise 1 1. Some of the drawings below
More informationStandards Level by Objective Hits Goals Objs # of objs by % w/in std Title Level Mean S.D. Concurr.
Table 9. Categorical Concurrence Between Standards and Assessment as Rated by Six Reviewers Florida Grade 9 athematics Number of Assessment Items - Standards evel by Objective Hits Cat. Goals Objs # of
More informationGTPS Curriculum Mathematics Grade 8
4.2.8.B2 Use iterative procedures to generate geometric patterns: Fractals (e.g., the Koch Snowflake); Self-similarity; Construction of initial stages; Patterns in successive stages (e.g., number of triangles
More informationn! = 1 * 2 * 3 * 4 * * (n-1) * n
The Beauty and Joy of Computing 1 Lab Exercise 9: Problem self-similarity and recursion Objectives By completing this lab exercise, you should learn to Recognize simple self-similar problems which are
More informationCARDSTOCK MODELING Math Manipulative Kit. Student Activity Book
CARDSTOCK MODELING Math Manipulative Kit Student Activity Book TABLE OF CONTENTS Activity Sheet for L.E. #1 - Getting Started...3-4 Activity Sheet for L.E. #2 - Squares and Cubes (Hexahedrons)...5-8 Activity
More informationSome geometries to describe nature
Some geometries to describe nature Christiane Rousseau Since ancient times, the development of mathematics has been inspired, at least in part, by the need to provide models in other sciences, and that
More informationGeometry 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
More informationHillel Academy. Grade 9 Mathematics End of Year Study Guide September June 2013
Hillel Academy Grade 9 Mathematics End of Year Study Guide September 2012 - June 2013 Examination Duration Date The exam consists of 2 papers: Paper 1: Paper 2: Short Response No Calculators Allowed Structured
More informationGetting Ready to Teach Unit 6
Getting Ready to Teach Unit 6 Learning Path in the Common Core Standards In this unit, students study the attributes of triangles, quadrilaterals, and other polygons. They find perimeter and area of various
More informationCopyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND
Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,
More information6th 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
More informationGrade 6 Math Circles February 19th/20th. Tessellations
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 19th/20th Tessellations Introduction to Tessellations tessellation is a
More informationMathematics Numbers: Percentages. Science and Mathematics Education Research Group
F FA ACULTY C U L T Y OF O F EDUCATION E D U C A T I O N Department of Curriculum and Pedagogy Mathematics Numbers: Percentages Science and Mathematics Education Research Group Supported by UBC Teaching
More informationSimi imilar Shapes lar Shapes Nesting Squares Poly lyhedr hedra and E a and Euler ler s Form s Formula ula
TABLE OF CONTENTS Introduction......................................................... 5 Teacher s Notes....................................................... 6 NCTM Standards Alignment Chart......................................
More informationTTUSD Math Essential Standards Matrix 4/16/10 NUMBER SENSE
TTUSD Math Essential Standards Matrix 4/16/10 NUMBER SENSE 3 rd 4 th 5 th 6th 1.1 Read and write whole numbers in the millions 1.2 Order and compare whole numbers and decimals to two decimal places. 1.1
More informationWhat You ll Learn. Why It s Important
First Nations artists use their artwork to preserve their heritage. Haida artist Don Yeomans is one of the foremost Northwest Coast artists. Look at this print called The Benefit, created by Don Yeomans.
More informationLesson 21: Surface Area
Lesson 21: Surface Area Classwork Opening Exercise: Surface Area of a Right Rectangular Prism On the provided grid, draw a net representing the surfaces of the right rectangular prism (assume each grid
More informationLesson 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
More informationMiddle School Math Course 3
Middle School Math Course 3 Correlation of the ALEKS course Middle School Math Course 3 to the Texas Essential Knowledge and Skills (TEKS) for Mathematics Grade 8 (2012) (1) Mathematical process standards.
More informationAreas of Rectangles and Parallelograms
CONDENSED LESSON 8.1 Areas of Rectangles and Parallelograms In this lesson, you Review the formula for the area of a rectangle Use the area formula for rectangles to find areas of other shapes Discover
More informationName Period Date MATHLINKS GRADE 8 STUDENT PACKET 3 PATTERNS AND LINEAR FUNCTIONS 1
Name Period Date 8-3 STUDENT PACKET MATHLINKS GRADE 8 STUDENT PACKET 3 PATTERNS AND LINEAR FUNCTIONS 1 3.1 Geometric Patterns Describe sequences generated by geometric patterns using tables, graphs, and
More informationSomeone else might choose to describe the closet by determining how many square tiles it would take to cover the floor. 6 ft.
Areas Rectangles One way to describe the size of a room is by naming its dimensions. So a room that measures 12 ft. by 10 ft. could be described by saying its a 12 by 10 foot room. In fact, that is how
More informationGeometry Workbook WALCH PUBLISHING
Geometry Workbook WALCH PUBLISHING Table of Contents To the Student..............................vii Unit 1: Lines and Triangles Activity 1 Dimensions............................. 1 Activity 2 Parallel
More informationVolumes 1 and 2. Grade 5. Academic Standards in Mathematics. Minnesota. Grade 5. Number & Operation
Academic Standards in Mathematics Minnesota Volumes 1 and 2 2013 STANDARDS Number & Operation Divide multi-digit numbers; solve real-world and mathematical problems using arithmetic. 5.1.1.1 Divide multi-digit
More informationArea and Perimeter EXPERIMENT. How are the area and perimeter of a rectangle related? You probably know the formulas by heart:
Area and Perimeter How are the area and perimeter of a rectangle related? You probably know the formulas by heart: Area Length Width Perimeter (Length Width) But if you look at data for many different
More informationEuclid s Muse Directions
Euclid s Muse Directions First: Draw and label three columns on your chart paper as shown below. Name Picture Definition Tape your cards to the chart paper (3 per page) in the appropriate columns. Name
More informationTable of Contents. Student Practice Pages. Number Lines and Operations Numbers. Inverse Operations and Checking Answers... 40
Table of Contents Introduction... Division by Tens... 38 Common Core State Standards Correlation... Division of -Digit Numbers... 39 Student Practice Pages Number Lines and Operations Numbers Inverse Operations
More informationSection 7.5. Fractals
Section 7.5 Fractals To start out this section on fractals we will begin by answering several questions. The first question one might ask is what is a fractal? Usually a fractal is defined as a geometric
More information5 th Grade MCA3 Standards, Benchmarks, Examples, Test Specifications & Sampler Questions
5 th Grade 3 Standards, Benchmarks, Examples, Test Specifications & Sampler Questions Strand Standard No. Benchmark (5 th Grade) Sampler Item Number & Operation 18-22 11-14 Divide multidigit numbers; solve
More informationSOLIDWORKS: Lesson III Patterns & Mirrors. UCF Engineering
SOLIDWORKS: Lesson III Patterns & Mirrors UCF Engineering Solidworks Review Last lesson we discussed several more features that can be added to models in order to increase their complexity. We are now
More informationObjective: Use multiplication to calculate volume.
Lesson 4 Objective: Use multiplication to calculate volume. Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief Total Time (12 minutes) (5 minutes) (33 minutes)
More informationSECOND GRADE Mathematic Standards for the Archdiocese of Detroit
SECOND GRADE Mathematic Standards for the Archdiocese of Detroit Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. 2.OA. A. 1 Use addition and subtraction
More information