Running head: A GEOMETRIC INTRODUCTION 1


 Rodney Stafford
 1 years ago
 Views:
Transcription
1 Running head: A GEOMETRIC INTRODUCTION A Geometric Introduction to Mathematical Induction Problems using the Sums of Consecutive Natural Numbers, the Sums of Squares, and the Sums of Cubes of Natural Numbers Nicholls State University Math 58 Fall 0 Galina Landa
2 A GEOMETRIC INTRODUCTION Abstract The Principle of Mathematical Induction (PMI) is a powerful method of proof. The illustrated examples often come from algebra rather than geometry. This paper explores mathematical induction in geometry. The traditional algebraic examples of the PMI involve proving formulas for the sums of consecutive natural numbers, the sums of squares of natural numbers, the sums of cubes, etc. However, it is not always clear how one would arrive at these formulas or in what circumstances they would be useful. This research paper presents several geometry problems to give context to such abstract formulas. It also demonstrates how one can use the PMI to prove important theorems in geometry. The problems considered in this research involve finding the number of lines through n points in a plane; finding the number of diagonals in convex polygons with n sides; and finding the number of squares of all sizes enclosed by a twodimensional checkerboard or the number of cubes of all sizes enclosed by a threedimensional checkerboard with each side length equal to n units.
3 A GEOMETRIC INTRODUCTION 3 Introduction A powerful method of proof, the Principle of Mathematical Induction (PMI) allows us to verify a particular property for all consecutive integers greater than some smallest one. It works like this: we start by checking if our conjecture is true for the first few initial values. Then, after assuming that the conjecture is true for the given element, we check whether we can show that it is also true for the next element. If we manage to complete both steps successfully, then the property is true for all elements. Mathematical induction is used to prove various sum formulas. However, it is not always clear how one would arrive at these formulas, why they would be legitimate candidates for verification, and in what circumstances they would be useful. In this paper, we will discover these formulas in the contexts of several geometry problems and then prove them using mathematical induction. Hopefully, by combining inductive investigation of geometry concepts with inductive proofs of the resulting sums, we will gain a new perspective on the meaning of these formulas and a better understanding of the PMI. The process of mathematical discovery is quite exciting, but a proper mathematical investigation can be time consuming. Thus, in this paper, we will limit our discussion to the following three sums: the sum of consecutive natural numbers the sum of squares of consecutive natural numbers the sum of cubes of consecutive natural numbers These three sums will give names to the three major sections of this paper, which constitute the body of related and original research. This body of research is called The Sums.
4 A GEOMETRIC INTRODUCTION The Sums The Sum of Consecutive Natural Numbers: 3... n We will consider this first sum in the context of the following geometry problem. Problem : Lines and Points Solution: Find the number of lines determined by a given number of points in a plane, no three of which are collinear (Wiscamb, p. 0). We are interested in the finite number of lines, so we will disregard a case of infinitely many lines through a single point. We will start with natural n>. Let us consider the first few cases for the number of points n=, n=3, n=, n=5, and n= (see Table ). Notice that while arbitrary points determine a single line, 3 noncollinear points determine 3 lines, points determine lines, 5 points determine 0 lines, and points determine 5 lines. Table. Lines and Points n, l n, ln 3... ( n ) ln pn pn points lines = ( ) 3 3 3=+ ( 3 3 ) (The table continues on the next page)
5 A GEOMETRIC INTRODUCTION 5 =++3 ( 3) 5 0 0=++3+ (5 ) 0 5 5= (5) 5 Note: Diagrams constructed with The Geometer's Sketchpad software. If we look for patterns in the table, we may notice two interesting details applicable to each case.. The number of lines determined by n points is equal to the sum of natural numbers from to n: ln 3... ( n ). The number of lines at each step is equal to half the product of the number of points at the current step and the number of points at the preceding step: ln ( pn pn )
6 A GEOMETRIC INTRODUCTION Based on these observations, we can make the following conjecture. Conjecture : For natural numbers greater than, 3... ( n ) n( n ) Proof: We can prove this conjecture by the PMI. Let P(n) be 3... ( n ) n( n ), n>. P() is true: ( ). Suppose P(n) is true for n>. 3. We will show that P(n) is true. (We want to show that, nn ( ) 3... n ) By the induction hypothesis, 3... ( n ) n( n ) Let us add n to both sides: 3... ( n ) n n( n ) n Pn ( ) Pn ( )
7 A GEOMETRIC INTRODUCTION 7 n( n ) n nn ( ) nn ( ) Thus, P(n) is true. Then by the PMI, P(n) is true for all natural numbers n>. Therefore, we have proved the formula: 3... ( n ) n( n ), for natural n> From now on, we can use it to find the number of lines determined by a given number of points in a plane, no three of which are collinear. For this particular task, it was convenient to use the formula for the sum of natural numbers from to n. We can also rewrite this formula for the sum of natural numbers from to n (like we did in our proof above for P(n)). Since we have already proven this result, we no longer call it a conjecture, but a theorem. Theorem (The Sum of Consecutive Natural Numbers): For natural numbers greater or equal to, nn ( ) 3... n,n We can now use this result to discover some other mathematical relationships. For example, Avital and Hansen (97) posed the following question.
8 A GEOMETRIC INTRODUCTION 8 Problem a: Diagonals of ngon What is the number of diagonals one can draw in a convex polygon of n vertices? (Avital & Hansen, p. 0) Solution: Clearly, a triangle has no diagonals; we will consider n > 3 (see Table a.) Table a. Diagonals of ngon Polygon n, vertices d, diagonals Triangle 3 0 Quadrilateral = Pentagon 5 5=+3 Hexagon 9=+3+ Heptagon 7 =+3++5 Note: Diagrams constructed with The Geometer's Sketchpad software.
9 A GEOMETRIC INTRODUCTION 9 After considering polygons with, 5,, and 7 sides, we find that the number of diagonals is, 5, 9, and, respectively. The last four numbers can be written as, +3, +3+, and Thus, the number of diagonals in each of these cases is the sum of natural numbers from to n, where n is the number of vertices. Let us denote the number of diagonals d; then for an ngon, d 3... ( n ) This resembles the sum from Theorem, minus, minus (n), and minus n. Then we can use the result from Theorem to find the expression for d: Thus, d 3... ( n ) 3... n ( n ) n Theorem Theorem nn ( ) n n n( n ) n nn ( ) nn ( 3) nn ( 3) 3... ( n ), n 3 From now on, we can use this result to find the number of diagonals in a convex ngon. We have demonstrated that this result follows from Theorem, so we will call it Corollary. Corollary : For natural numbers greater than 3, nn ( 3) 3... ( n ),n
10 A GEOMETRIC INTRODUCTION 0 This result can be verified by the PMI for n>3; however, we will omit the proof here because it is very similar to the proof of Theorem. The Sum of Squares: 3... n Wiscamb (970) suggested introducing the formula for the sum of squares in connection with the following problem. Problem : Checkerboard in D Solution: Given an nxn checkerboard with n small squares each measuring square unit, how many squares of all sizes does it contain? (Wiscomb, p. 03) Let us consider several cases for n=, n=, n=3, and n= (see Table ). Table. Checkerboard in D Squares: x x 3x3 x Total Number of Squares n= n= 5 n=3 9 3 n= Note: Square images above were formatted with MS Word Picture Tools. We can see that that for n=, there is only one square. For n=, there are four x squares and one x square. For n=3, there are nine x squares, four x squares, and one 3x3 square. Finally, for
11 A GEOMETRIC INTRODUCTION n=, there are sixteen x squares, nine x squares, four 3x3 squares, and one x square. The total number of squares in each case is the sum of squares of natural numbers from to n (see Table ). Continuing in this manner, we can conjecture that an nxn checkerboard will contain 3... n squares of all sizes, but what is this sum equal to? How can we guess the closedform expression for the sum of squares of natural numbers? To answer this question, we will try to use an analogy with the formula for the sum of natural numbers from Theorem : nn ( ) 3... n This formula has a polynomial of the second degree on the right side. Thus, we will assume that the sum of squares of natural numbers should add up to a polynomial of the third degree, something like 3 an bn cn d, for real coefficients a, b, c, d. Our guess is that Where, n an bn cn d, (Eq.) a, b, c, d However, what are these real coefficients a, b, c, d? If we rewrite Equation four times for n=, n=, n=3, and n=, respectively, we will get a system of four linear equations in four variables:
12 A GEOMETRIC INTRODUCTION n : n : n 3: n : a b c d 3 a b c d a 3 b 3c d 3 a b c d 30 a b c d 8a b c d 5 7a 9b 3c d a b c d 30 We can solve this system by hand (using a method like Cramer s rule, Gaussian elimination, or several others), or we can use a calculator to save time and paper space. The solutions are: Then, Then we can make the following conjecture. a, b, c, d 0 3 n n n 3 3 n 3n n n(n 3n ) n( n )(n ). 3 an bn cn d 3 Conjecture : For natural numbers greater or equal to, Proof: We can prove this conjecture by the PMI. Let P(n) be, n( n )(n ) 3... n,n n( n )(n ) 3... n,n
13 A GEOMETRIC INTRODUCTION 3. P() is true:. Suppose P(n) is true. ( )( ) 3. We will show that P(n+) is true. (We want to show that, By the induction hypothesis, Let us add ( n ) ( n) (( n) ) 3... n ( n ) ) ( n ) to both sides. Then, n( n )(n ) 3... n 3... n ( n ) n( n )(n ) ( n ) Pn ( ) Pn ( ) n( n )(n ) ( n ) ( n ) n(n ) ( n ) ( n ) n n n ( n ) n 7n ( n ) n (n3) ( n ) ( n) (( n) ) Thus, P(n+) is true. Then by the PMI, P(n) is true for all natural numbers.
14 A GEOMETRIC INTRODUCTION Therefore, we have proved the formula for the sum of squares of natural numbers, n( n )(n ) 3... n,n From now on, we can use it to find the number of squares of all sizes contained in a twodimensional checkerboard. Since we have already proved this result, we will no longer call it a conjecture, but a theorem. Theorem (The Sum of Squares of Consecutive Natural Numbers): For natural numbers greater or equal to, n( n )(n ) 3... n,n We will now consider a similar problem for a threedimensional checkerboard. We will use this problem to introduce the formula for the sum of cubes of natural numbers. The Sum of Cubes: n Problem 3: Checkerboard in 3D Solution: How many cubes of all sizes are there in a nxnxn threedimensional checkerboard? (Wiscomb, p. 0) Let us consider several cases for n=, n=, and n=3 (see Table 3.).
15 A GEOMETRIC INTRODUCTION 5 Table 3. Checkerboard in 3D Cubes: xx xx 3x3x3 Total Number of Cubes n= n= n= Note: Cube images above were formatted with MS Word Picture Tools. We can see that that for n=, there is only one cube. For n=, there are eight x cubes and one x cube. For n=3, there are twentyseven x cubes, eight x cubes, and one 3x3 cube. The total number of cubes in each case is the sum of cubes of natural numbers from to n. Continuing in this manner, we can conjecture that an nxnxn checkerboard will contain n cubes of all sizes, but what is this sum equal to? How can we guess the closedform expression for the sum of cubes of natural numbers? Let us consider the sum of cubes when n=, n=, n=3, and n=. The numbers that we get in each case are perfect squares:, 9, 3, and 00, respectively (see Table 3., left column). They can be written as,3,,0. Consider the sequence, 3,, 0. Notice that we can rewrite the terms of this sequence as, +, ++3, ++3+ (see Table 3., right column).
16 A GEOMETRIC INTRODUCTION Table 3. Checkerboard in 3D Total Number of Cubes k i n= 3 n= ( ) n= n= ( 3) ( 3 ) Note: Total Number of Cubes expressed in two different ways. Notice that in each case, the sum of cubes can be written as the square of the sum of consecutive natural numbers from to n: 3... n ( 3... n) The right side of this expression contains the sum 3... n, for which we already know the formula (from Theorem ): 3... n nn ( ) Then, 3... n n ( n) and therefore, n 3... n ( n) Then we can make the following conjecture. Conjecture 3: For natural numbers greater or equal to, Proof: We can prove this conjecture by the PMI n ( n) 3... n,n
17 A GEOMETRIC INTRODUCTION 7 Let P(n) be. P() is true: n 3... n ( n) 3 ( ). Suppose P(n) is true. 3. We are going to show that P(n+) is true. (We want to show that By the induction hypothesis, n ( n) (( n) ) 3 ( n ) ) n 3... n ( n) Let us add 3 ( n ) to both sides. Then, n ( n) n ( n ) n Pn ( ) Pn ( ) 3 n ( n ) ( n ) ( n ) ( n n ) ( n) ( n) ( n)( n) ( n)(( n) ) ( n) (( n) )
18 A GEOMETRIC INTRODUCTION 8 Thus, P(n+) is true. Then by the PMI, P(n) is true for all natural numbers. Therefore, we have proved the formula for the sum of cubes of natural numbers: n ( n) 3... n,n From now on, we can use it to find the number of cubes of all sizes contained in a threedimensional checkerboard. Since we have already proved this result, we will no longer call it a conjecture, but a theorem. Theorem 3 (The Sum of Cubes of Consecutive Natural Numbers): For natural numbers greater or equal to, n ( n) 3... n,n Recommendations for Further Research There are additional geometry problems that could provide interesting contexts to various formulas used to illustrate the Principle of Mathematical Induction. There are also cases when the PMI proves indispensable to verifying important results in geometry. This mutually beneficial relationship between geometry and the Principle of Mathematical Induction deserves further consideration. Unfortunately, the limited scope of this paper does not allow us to explore these topics in more detail; however, below are some sample questions that could be investigated in future research.
19 A GEOMETRIC INTRODUCTION 9 Questions about n lines in the plane: Given n concurrent lines, how many pairs of supplementary angles are formed? How many pairs of vertical angles are formed? What is the maximum number of regions in the plane determined by n lines? (Wiscamb, 970) Questions about polygons with n sides: What are the sums of interior angles of various convex polygons? Into how many triangles can a convex polygon be divided by nonintersecting diagonals? How many such nonintersecting diagonals are needed to divide a convex polygon into these triangles? What are the answers to these questions for a concave polygon? It takes some time to completely work through questions like these in order to actively explore geometry concepts, to find patterns, to formulate conjectures in terms of natural numbers, and to write induction proofs. However, by not skipping any steps of the process, we can gain not only a better understanding of mathematical induction, but also a feeling for the manner in which mathematics is being created (Avital & Hansen, p. ).
20 A GEOMETRIC INTRODUCTION 0 References Avital, S., & Hansen, R. T. (97). Mathematical induction in the classroom. Educational Studies in Mathematics, 7(), 399. Wiscamb, M. (970). A geometric introduction to mathematical induction. The Mathematics Teacher, 3(5), 00.
A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.
hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.
More informationLesson 6: Polygons and Angles
Lesson 6: Polygons and Angles Selected Content Standards Benchmark Assessed: G.4 Using inductive reasoning to predict, discover, and apply geometric properties and relationships (e.g., patty paper constructions,
More informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationPolygons are figures created from segments that do not intersect at any points other than their endpoints.
Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures created from segments that do not intersect at any points other than their endpoints. A polygon is convex if all of the interior angles
More informationUnit 3: Triangle Bisectors and Quadrilaterals
Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More informationPerformance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will
Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles
More informationLESSON PLAN #1: Discover a Relationship
LESSON PLAN #1: Discover a Relationship Name Alessandro Sarra Date 4/14/03 Content Area Math A Unit Topic Coordinate Geometry Today s Lesson Sum of the Interior Angles of a Polygon Grade Level 9 NYS Mathematics,
More informationUnit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period
Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period 1 2 3 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,
More informationSection 62 Mathematical Induction
6 Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More information1.2. Successive Differences
1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers
More information22.1 Interior and Exterior Angles
Name Class Date 22.1 Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Resource Locker Explore 1 Exploring Interior
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationExterior Angles of Polygons
easures, hape & pace EXEMPLAR 14: Exterior Angles of Polygons Objective: To explore the angle sum of the exterior angles of polygons Key Stage: 3 Learning Unit: Angle related with Lines and Rectilinear
More informationInvestigating Relationships of Area and Perimeter in Similar Polygons
Investigating Relationships of Area and Perimeter in Similar Polygons Lesson Summary: This lesson investigates the relationships between the area and perimeter of similar polygons using geometry software.
More informationQuadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid
Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationAppendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
More informationPreAlgebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio PreAlgebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More information11.3 Curves, Polygons and Symmetry
11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon
More informationAcquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours
Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours Essential Question: LESSON 4 FINITE ARITHMETIC SERIES AND RELATIONSHIP TO QUADRATIC
More informationVISUALIZING THE HANDSHAKE PROBLEM
Grade: 5/Math VISUALIZING THE HANDSHAKE PROBLEM Brief Description of the Lesson: The students will investigate various forms of the handshake problem by using visualization problem solving like drawing,
More informationWhich shapes make floor tilings?
Which shapes make floor tilings? Suppose you are trying to tile your bathroom floor. You are allowed to pick only one shape and size of tile. The tile has to be a regular polygon (meaning all the same
More informationThe Geometer s Sketchpad: NonEuclidean Geometry & The Poincaré Disk
The Geometer s Sketchpad: NonEuclidean Geometry & The Poincaré Disk Nicholas Jackiw njackiw@kcptech.com KCP Technologies, Inc. ICTMT11 2013 Bari Overview. The study of hyperbolic geometry and noneuclidean
More informationTopological Treatment of Platonic, Archimedean, and Related Polyhedra
Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 15341178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic
More informationPolygon Properties and Tiling
! Polygon Properties and Tiling You learned about angles and angle measure in Investigations and 2. What you learned can help you figure out some useful properties of the angles of a polygon. Let s start
More informationGeometry Chapter 1 Vocabulary. coordinate  The real number that corresponds to a point on a line.
Chapter 1 Vocabulary coordinate  The real number that corresponds to a point on a line. point  Has no dimension. It is usually represented by a small dot. bisect  To divide into two congruent parts.
More informationUNIT H1 Angles and Symmetry Activities
UNIT H1 Angles and Symmetry Activities Activities H1.1 Lines of Symmetry H1.2 Rotational and Line Symmetry H1.3 Symmetry of Regular Polygons H1.4 Interior Angles in Polygons Notes and Solutions (1 page)
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationArea, Perimeter, Surface Area and Change Overview
Area, Perimeter, Surface Area and Change Overview Enduring Understanding: (5)ME01: Demonstrate understanding of the concept of area (5)ME02: Demonstrate understanding of the differences between length
More informationWarm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.
Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationVocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.
CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion
More informationMaximizing Angle Counts for n Points in a Plane
Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationGeometry 81 Angles of Polygons
. Sum of Measures of Interior ngles Geometry 81 ngles of Polygons 1. Interior angles  The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.
More informationIntermediate Math Circles October 10, 2012 Geometry I: Angles
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationExploring Geometric Figures Using Cabri Geometry II
Exploring Geometric Figures Using Cabri Geometry II Regular Polygons Developed by: Charles Bannister. Chambly County High School Linda Carre.. Chambly County High School Manon Charlebois Vaudreuil Catholic
More informationG C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.
More informationUnit 6 Grade 7 Geometry
Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson
More information1.1 Identify Points, Lines, and Planes
1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms  These words do not have formal definitions, but there is agreement aboutwhat they mean.
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationWORK SCHEDULE: MATHEMATICS 2007
, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check
More informationAssessment Anchors and Eligible Content
M07.AN The Number System M07.AN.1 M07.AN.1.1 DESCRIPTOR Assessment Anchors and Eligible Content Aligned to the Grade 7 Pennsylvania Core Standards Reporting Category Apply and extend previous understandings
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More informationProblem of the Month: Cutting a Cube
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationSituation: Proving Quadrilaterals in the Coordinate Plane
Situation: Proving Quadrilaterals in the Coordinate Plane 1 Prepared at the University of Georgia EMAT 6500 Date Last Revised: 07/31/013 Michael Ferra Prompt A teacher in a high school Coordinate Algebra
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationGlencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 33, 58 84, 87 16, 49
Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 68 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationRectangles with the Same Numerical Area and Perimeter
About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection
More informationStudents will understand 1. use numerical bases and the laws of exponents
Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More informationPRINCIPLES OF PROBLEM SOLVING
PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problemsolving process and to
More informationSOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
More informationRadius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge
TMME,Vol.1, no.1,p.9 Radius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge Introduction I truly believe learning mathematics can be a fun experience for children of all ages.
More informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationE XPLORING QUADRILATERALS
E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationActually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.
1: 1. Compute a random 4dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3dimensional polytopes do you see? How many
More informationStandards for Mathematical Practice: Commentary and Elaborations for 6 8
Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:
More informationMcDougal Littell California:
McDougal Littell California: PreAlgebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California PreAlgebra Components: Pupil Edition (PE), Teacher s Edition (TE),
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationArchimedes Calculation of Pi Barbara J. Meidinger Helen Douglas. Part I: Lesson Plans
Archimedes Calculation of Pi Barbara J. Meidinger Helen Douglas Part I: Lesson Plans Day One: Introduction Target Learners: Second semester geometry students Student Materials: Scientific or graphing calculators
More informationGeometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning:
Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Conjecture: Advantages: can draw conclusions from limited information helps us to organize
More informationPick s Theorem. Tom Davis Oct 27, 2003
Part I Examples Pick s Theorem Tom Davis tomrdavis@earthlink.net Oct 27, 2003 Pick s Theorem provides a method to calculate the area of simple polygons whose vertices lie on lattice points points with
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationMTH History of Math Reading Assignment #5  Answers Fall 2012
MTH 6610  History of Math Reading Assignment #5  Answers Fall 2012 Pat Rossi Name Instructions. Read pages 141181 to find the answers to these questions in your reading. 1. What factors made Aleandria
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationInduction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
More informationPrentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)
New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct
More informationEstimating Angle Measures
1 Estimating Angle Measures Compare and estimate angle measures. You will need a protractor. 1. Estimate the size of each angle. a) c) You can estimate the size of an angle by comparing it to an angle
More information55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationGeometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment
Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points
More informationKEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 447) Surveys and
More informationMaths class 11 Chapter 7. Permutations and Combinations
1 P a g e Maths class 11 Chapter 7. Permutations and Combinations Fundamental Principles of Counting 1. Multiplication Principle If first operation can be performed in m ways and then a second operation
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationGeometry, Technology, and the Reasoning and Proof Standard inthemiddlegradeswiththegeometer ssketchpad R
Geometry, Technology, and the Reasoning and Proof Standard inthemiddlegradeswiththegeometer ssketchpad R Óscar Chávez University of Missouri oc918@mizzou.edu Geometry Standard Instructional programs from
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twentyfold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or pingpong balls)
More informationA Correlation of Pearson Texas Geometry Digital, 2015
A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations
More informationG333 Building Pyramids
G333 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids. Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons Vocabulary:
More informationTheory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras
Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information