3. How many winning lines are there in 5x5 Tic-Tac-Toe? 4. How many winning lines are there in n x n Tic-Tac-Toe?

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Download "3. How many winning lines are there in 5x5 Tic-Tac-Toe? 4. How many winning lines are there in n x n Tic-Tac-Toe?"

Transcription

1 Winning Lines in Tic-Tac-Toe 1. The standard Tic-Tac-Toe is played on a 3 x 3 board, where there are vertical winning lines, horizontal winning lines, diagonal winning lines. This is a grand total of winning lines in the standard 3x3 Tic-Tac-Toe. 2. In 4 x 4 Tic-Tac-Toe, the winning lines are again 4-in-a-row horizontally, vertically or diagonally. How many winning lines are there in 4 x 4 Tic-Tac-Toe? 3. How many winning lines are there in 5x5 Tic-Tac-Toe? 4. How many winning lines are there in n x n Tic-Tac-Toe? Pairing Strategies in n x n Tic-Tac-Toe: Below is what is called a Pairing Strategy Draw for Player 2 in 6 x 6 Tic-Tac-Toe: \ / / \ Can you create a Pairing Strategy Draw for Player 2 in 5 x 5 Tic-Tac-Toe?

2 Can you create a Pairing Strategy Draw for Player 2 in 4 x 4 Tic-Tac-Toe? Can you create a Pairing Strategy Draw for Player 2 in 3 x 3 Tic-Tac-Toe? Q What do you need to happen to HPE that a Pairing Strategy Draw exists for Player 2? Notation: (To hopefully make our life easier.) We will consider Tic-Tac-Toe to be played by two players who select points from the plane. In n x n Tic-Tac-Toe, the points the players select are in the set {(1,1), (1,2),.,(1,n), (2,1), (2,2),.., (2,n) (n,n)} This type of notation may help us when we play Tic-Tac-Toe in higher dimensions. Note: When we think of a horizontal winning line in Tic-Tac-Toe, we can think of this as a series of points, where the y-coordinate is fixed and the x-coordinate varies. Similarly, a vertical winning line has the x-coordinate fixed and the y-coordinate varying. A diagonal winning line has both the x-coordinate and the y-coordinate varying.

3 You may know about three-dimensional Tic-Tac-Toe, where the winning lines either have one, two or three coordinates varying. Notation: In the interest of my sanity, we will say n 3 TTT is n x n x n Tic-Tac-Toe. This game will be played on a cube where each side is of length n. 5. How many winning lines are there in 2 3 TTT? 6. How many winning lines are there in 3 3 TTT? 7. How many winning lines are there in 4 3 TTT? 8. Find a formula for the number of winning lines in n 3 TTT. Tricks Here are two ways for determining the number of winning lines in n d TTT (d-dimensional Tic-Tac-Toe where each winning line is of length n). For this game, n and d are any positive integers and the game is played on an n d hypercube. a. Consider each winning line as having d-coordinates (since this is d- dimensional Tic-Tac-Toe) where each coordinate is element of the set {1, 2,, n,, }. means the coordinate is varying up. means the coordinate is varying down. How many winning lines are there if at least one coordinate is varying?

4 b. In 3x3 Tic-Tac-Toe, draw a one square border around the board. Draw in your complete winning lines for the 3x3 board and extend them into your 5 x 5 rectangle. Notice that each winning line hits exactly two points in the shaded border. How many squares are there in the shaded region? How many winning lines were there in 3x3 Tic-Tac-Toe? If you draw in a one square border around the 7x7 Tic-Tac-Toe board and how many squares would there be in the border? How many winning lines are there in 7x7 Tic-Tac-Toe? If you draw in a If you draw in a one square border around the n x n Tic-Tac-Toe board and how many squares would there be in the border? How many winning lines are there in n x n Tic-Tac-Toe? Trying the same trick for n d board. How many winning lines are there in the n 3 board.. using Trick a? using Trick b? The number of directions in n d TTT: In n x n Tic-Tac-Toe, there are four different directions: \, /. How many directions are there in n 3 TTT? How many directions are there in n 4 TTT? How many directions are there in n d TTT?

5 Tic-Tac-Toe on the Torus Pretend that the Tic-Tac-Toe is wrapped around a sphere, so that there is no top/bottom of the board or left/right of the board. All squares are effectively equivalent. Winning Lines in 3 x 3 Tic-Tac-Toe on a torus: Besides the 8 usual lines, there are the following additional lines. Let s play the game: We know that Player 2 can force a draw in standard 3x3 Tic-Tac-Toe. Q: Can Player 2 force a draw in 3 x 3 Tic-Tac-Toe on a torus? Play a few games with your classmates. See if you can figure it out. Then analyze the game move by move. Winning Lines in Tic-Tac-Toe on a torus How many winning lines are there in 4 x 4 Tic-Tac-Toe on a torus? How many winning lines are there in 5 x 5 Tic-Tac-Toe on a torus? How many winning lines are there in n x n Tic-Tac-Toe on a torus? How many winning lines are there in n 3 Tic-Tac-Toe on a torus?

Algorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure

Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm

Math. Understanding Exponents. Answers 1) 9 4 2) 4 2 3) 4 3 4) 6 3 5) 8 3 6) ) 9 9 8) 4 4 9) )

1) 9 4 2) 4 2 3) 4 3 4) 6 3 5) 8 3 6) 4 4 4 4 4 7) 9 9 8) 4 4 9) 3 3 3 3 10) 5 5 5 5 5 11) 2 2 12) 8 3 13) 9 3 14) 6 2 15) 6 3 16) What is 8 to the power of two? 17) What is 5 squared? 18) What is 2 to

Math Circles: Invariants Part 1

Math Circles: Invariants Part 1 March 23, 2011 What is an Invariant? In the simplest terms, an invariant is something that doesn t change. As easy as that sounds, invariance is a very powerful property

TEACHER NOTES MATH NSPIRED

Math Objectives Students will predict and identify the shapes of two-dimensional cross-sections are formed when three-dimensional figures are cut by horizontal and vertical planes. Students will draw,

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1

. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n

Geometric Series. A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio.

Introduction Geometric Series A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio. A simple example wound be: where the first

MD5-26 Stacking Blocks Pages 115 116

MD5-26 Stacking Blocks Pages 115 116 STANDARDS 5.MD.C.4 Goals Students will find the number of cubes in a rectangular stack and develop the formula length width height for the number of cubes in a stack.

Roots of Real Numbers

Roots of Real Numbers Math 97 Supplement LEARNING OBJECTIVES. Calculate the exact and approximate value of the square root of a real number.. Calculate the exact and approximate value of the cube root

Game 6M2: Area. Copyright Ann Moran

Game 6M2: Area Strand: Measures Strand Unit: Area Curriculum Objectives Covered: Recognise that the length of the perimeter of a rectangular shape does not determine the area of the shape Calculate the

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

Open GeoGebra & Format Worksheet

Open GeoGebra & Format Worksheet 1. Close the Algebra view tab by clicking on the in the top right corner. 2. Show the grid by clicking on the show grid icon, located under the toolbar. 3. Select the Move

Prime Time Notes. Problem 1.1

Problem 1.1 factor - one of two or more whole numbers that when multiplied together give a product proper factor - all the factors of that number, except the number itself abundant numbers - numbers whose

Strong Induction. Catalan Numbers.

.. The game starts with a stack of n coins. In each move, you divide one stack into two nonempty stacks. + + + + + + + + + + If the new stacks have height a and b, then you score ab points for the move.

Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

GLASS CHESS SET GENERAL RULES AND INFORMATION

GLASS CHESS SET 47884 GENERAL RULES AND INFORMATION THANK YOU for choosing a HARBOR FREIGHT TOOLS product. For future reference, please complete the owner s record below: Model Serial No. Purchase Date

1.5. Transformations of graphs. Translating graphs vertically

Let c be a positive number. 1.5. Transformations of graphs Translating graphs vertically Translation upward. The vertical translation c units upward shifts given point onto the point which is located in

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

acute angle acute triangle Cartesian coordinate system concave polygon congruent figures

acute angle acute triangle Cartesian coordinate system concave polygon congruent figures convex polygon coordinate grid coordinates dilatation equilateral triangle horizontal axis intersecting lines isosceles

Introduction to Matrices

Introduction to Matrices Tom Davis tomrdavis@earthlinknet 1 Definitions A matrix (plural: matrices) is simply a rectangular array of things For now, we ll assume the things are numbers, but as you go on

MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane.

MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane. Symmetry When you can fold a figure in half, with both sides congruent, the fold line

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

Mathematical Content: Numbers and Operations

Mathematical Content: Numbers and Operations What are numbers? How can I count forward from any number other than 1? How can I write numbers up to 20 and show numbers of objects from 0-20? What is the

SMMG February 25 th, 2006 featuring Dr. Alistair Windsor Fun with Fractals

SMMG February 25 th, 2006 featuring Dr. Alistair Windsor Fun with Fractals 1. If f(z) is a function, various behaviors can arise when f is iterated. Gaston Julia studied the iteration of polynomials and

Lines That Pass Through Regions

: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

Galena Park ISD. Friday, September 9, 2016 (Due Date) Summer Assignment. Pre-AP Geometry. (Name)

Summer Assignment (Name) Friday, September 9, 2016 (Due Date) Pre-AP Geometry Summer Assignment: Pre-AP Geometry This assignment should serve as a review of the skills necessary for success in Pre-AP Geometry.

Date: Period: Symmetry

Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into

Most classrooms are built in the shape of a rectangular prism. You will probably find yourself inside a polyhedron at school!

3 D OBJECTS Properties of 3 D Objects A 3 Dimensional object (3 D) is a solid object that has 3 dimensions, i.e. length, width and height. They take up space. For example, a box has three dimensions, i.e.

What You ll Learn. Why It s Important

These students are setting up a tent. How do the students know how to set up the tent? How is the shape of the tent created? How could students find the amount of material needed to make the tent? Why

Students 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?

Objectives. score only one 2-point field goal and point field goals even though the ordered

Objectives Graph systems of linear inequalities Investigate the concepts of constraints and feasible polygons Activity 3 Introduction The graph of a system of linear inequalities can create a region defined

Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) August-September (12 days) Unit #1 : Geometry

Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Common Core and Research from the CCSS Progression Documents Geometry Students learn to analyze and relate categories

Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7

Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You

x button, the number 13, close

Calculator Skills: Know the difference between the negative button (by the decimal point) and the subtraction button (above the plus sign). Know that your calculator automatically does order of operations,

Year 9 mathematics test

Ma KEY STAGE 3 Year 9 mathematics test Tier 6 8 Paper 1 Calculator not allowed First name Last name Class Date Please read this page, but do not open your booklet until your teacher tells you to start.

RIT scores between 191 and 200

Measures of Academic Progress for Mathematics RIT scores between 191 and 200 Number Sense and Operations Whole Numbers Solve simple addition word problems Find and extend patterns Demonstrate the associative,

( ) = sin x so that it is a one-to-one function

(Section 4.7: Inverse Trig Functions) 4.72 SECTION 4.7: INVERSE TRIG FUNCTIONS You may want to review Section 1.8 on inverse functions. PART A : GRAPH OF sin -1 x (or arcsin x) Warning: Remember that f

Vocabulary Cards and Word Walls Revised: June 29, 2011

Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,

Common Core Standards for Mathematics Kindergarten. Counting & Cardinality Date Taught

Counting & Cardinality Know number names and the count sequence. K.CC.1. Count to 100 by ones and by tens. K.CC.2. Count forward beginning from a given number within the known sequence (instead of having

MTH2132 The Nature and Beauty of Mathematics Burkard Polster

MTH2132 The Nature and Beauty of Mathematics Burkard Polster Playing Billiards With Jugs In the movie Die Hard: With a Vengeance John (Bruce Willis) and Zeus (Samuel Jackson) are having a little problem

Standard: 8.EE.1: Know and apply properties of integer exponents to generate equivalent numerical expressions.

Standard: 8.EE.1: Know and apply properties of integer exponents to generate equivalent numerical expressions. Learning Goal: Students will be able to develop and use the rules for multiplying expressions

A BRIEF GUIDE TO ABSOLUTE VALUE. For High-School Students

1 A BRIEF GUIDE TO ABSOLUTE VALUE For High-School Students This material is based on: THINKING MATHEMATICS! Volume 4: Functions and Their Graphs Chapter 1 and Chapter 3 CONTENTS Absolute Value as Distance

THE DISTANCE FORMULA

THE DISTANCE FORMULA In this activity, you will develop a formula for calculating the distance between any two points in a coordinate plane. Part 1: Distance Along a Horizontal or Vertical Line To find

Inequalities (Year 9)

Inequalities (Year 9) Contents What is an inequality? Displaying inequalities on a number line Solving inequalities 4 Showing Inequalities on a graph 4 What is an inequality? An inequality is a mathematical

Kindergarten Mathematics CCSS I Can Statements!

Kindergarten Mathematics I Can Statements I Can Domain: Counting and Cardinality Cluster: Know number names and the count sequence. K.CC.1 Count to 100 by ones and by tens. I can count to 100 by ones.

Archdiocese 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,

Transformations Worksheet. How many units and in which direction were the x-coordinates of parallelogram ABCD moved? C. D.

Name: ate: 1. Parallelogram ABC was translated to parallelogram A B C. 2. A shape is shown below. Which shows this shape transformed by a flip? A. B. How many units and in which direction were the x-coordinates

*1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles.

Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review

(Knight) 3 : A Graphical Perspective of the Knight's Tour on a Multi-Layered Chess Board

Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-2-2016 (Knight) 3 : A Graphical Perspective of the Knight's

Grade 8 Unit 2 Pythagorean Theorem Applications. Assessment Plan

Grade 8 Unit 2 Pythagorean Theorem Applications Work with radicals and integer exponents Assessment Plan 8. EE.2 Use square root and cube root symbols to represent solutions to equations. Evaluate square

Class 9 Coordinate Geometry

ID : in-9-coordinate-geometry [1] Class 9 Coordinate Geometry For more such worksheets visit www.edugain.com Answer t he quest ions (1) Find the coordinates of the point shown in the picture. (2) Find

Elements of a graph. Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT

UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

Geometry Notes VOLUME AND SURFACE AREA

Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

Tic-Tac-Toe. For more crafts, literacy, and math units, please visit my TPT store! Borders in this item are from My TPT Store

Tic-Tac-Toe Thank you for downloading my Sight Words & Spelling Tic-Tac-Toe Activity. I hope you and your kiddies enjoy it! For more games, units, and freebies, please visit my blog All Students Can SHINE

What Does Your Quadratic Look Like? EXAMPLES 1. An equation such as y = x 2 4x + 1 descries a type of function known as a quadratic function. Review with students that a function is a relation in which

ascending order decimal denominator descending order Numbers listed from largest to smallest equivalent fraction greater than or equal to SOL 7.

SOL 7.1 ascending order Numbers listed in order from smallest to largest decimal The numbers in the base 10 number system, having one or more places to the right of a decimal point denominator The bottom

FOUR DIMENSIONAL CHESS

FOUR DIMENSIONAL CHESS BEN Abstract. This is a quick introduction to four dimensional chess, with the mathematics in mind. 1. FIDE Chess We shall assume a basic knowledge of chess, but desire to reintroduce

The equation of the axis of symmetry is. Therefore, the x-coordinate of the vertex is 2.

1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. Here, a = 2, b = 8, and c

Your child may do whatever portion of this math packet you would like to make sure that he/she is prepared for Course 1.

Monday, May 23 rd, 2016 Dear parents of incoming Math Course 1 students, The math packet for incoming students for Math Course 1 is optional, and I would like to give you a bit more information about this

Board Games. With Monkey Friends. 15 Multiplication Board Games to practice multiplication up to 10 x 10

Board Games With Monkey Friends 15 Multiplication Board Games to practice multiplication up to 10 x 10 By TeachersPayTeachers.com Teresa Evans 2012 Contents Monkey Multiplication Maker - Create Multiplication

Math Games with a Deck of Cards

Math Games with a Deck of Cards Multiplication/Division Chart Games to play at home to practice math skills Tips for playing math card games: You can play with a regular deck of cards. Some games may need

Integers (pages 294 298)

A Integers (pages 294 298) An integer is any number from this set of the whole numbers and their opposites: { 3, 2,, 0,, 2, 3, }. Integers that are greater than zero are positive integers. You can write

Engineering Drawing. Anup Ghosh. September 12, Department of Aerospace Engineering Indian Institute of Technology Kharagpur

Department of Aerospace Engineering Indian Institute of Technology Kharagpur September 12, 2011 Example -1 1 A vertical square prism (50mm side) 2 A horizontal square prism (35mm side) with axis to VP

Rectangles and Factors

Rectangles and Factors Background information: Unlike addition, in multiplication, each factor has a different purpose; one identifies the number in each group and the other, the number of groups. Therefore

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

Frogs, Fleas and Painted Cubes: Homework Examples from ACE

Frogs, Fleas and Painted Cubes: Homework Examples from ACE Investigation 1: Introduction to Quadratic Functions, ACE #7, 14 Investigation 2: Quadratic Expressions, ACE #17, 24, 26, 36, 51 Investigation

Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

Category 3 Number Theory Meet #1, October, 2000

Category 3 Meet #1, October, 2000 1. For how many positive integral values of n will 168 n be a whole number? 2. What is the greatest integer that will always divide the product of four consecutive integers?

Fraction and Decimal

Fraction and Decimal Helping your students convert fractions and decimals, one bingo ball at a time! Created By: Stephanie Moorman Board 1 Board 2 1/2 1/4 3/4 1/3 1/5 1/4 3/5 0.40 2/5 1.00 0.5 0.25 0.75

Fractions of a Whole. Unit. Practice. Write fraction words to describe the shaded part. A fraction that names the shaded part is.

Fractions of a Whole Home Link 5-1 Family Note Today your child worked with fraction circle pieces to explore fractions as equal parts of wholes. Children covered fraction circle pieces with equal parts

Exercise 1: Challenges using cubes

Curious Cubes Exercise 1: Challenges using cubes We are learning that shapes may be different but can still have the same properties. Equipment: Multi-link cubes AC EA AA AM AP 1) Here is one way of drawing

Cambridge University Press A Primer of Analytic Number Theory: From Pythagoras to Riemann Jeffrey Stopple Excerpt More information

Chapter 1 Sums and Differences Imet a traveller from an antique land Who said: Two vast and trunkless legs of stone Stand in the desert Near them, on the sand, Half sunk, a shattered visage lies Percy

Cornerstones: Click, Clack, Moo Flash Card Games: Instructions

Flash Card Games: Instructions The Secret Word Distribute a card to each student, and tell them to keep their word secret. One at a time, each student signs or says a series of clues a word that begins

Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

angle Definition and illustration (if applicable): a figure formed by two rays called sides having a common endpoint called the vertex

angle a figure formed by two rays called sides having a common endpoint called the vertex area the number of square units needed to cover a surface array a set of objects or numbers arranged in rows and

100 Math Facts 6 th Grade

100 Math Facts 6 th Grade Name 1. SUM: What is the answer to an addition problem called? (N. 2.1) 2. DIFFERENCE: What is the answer to a subtraction problem called? (N. 2.1) 3. PRODUCT: What is the answer

Lesson 11. Playing Board Games. 3 D Objects

Math 5 Lesson 11 3 D Objects Playing Board Games Zach has a new board game that he wants to play with his friends. He notices that the box the game is stored in is a lot like the prism he learned about

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

Florida Early Learning and Developmental Standards for Four-Year-Olds Aligned to the Kindergarten Common Core State Standards for Math

Kindergarten - Counting and Cardinality Know number names and the count sequence 1 CCM.K.CC.1 Count to 100 by ones and by tens. Benchmark b: Child counts up through 31 by understanding the pattern of adding

Closest Pair Problem

Closest Pair Problem Given n points in d-dimensions, find two whose mutual distance is smallest. Fundamental problem in many applications as well as a key step in many algorithms. p q A naive algorithm

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

Paper 1. Mathematics test. Calculator not allowed. First name. Last name. School KEY STAGE TIER

Ma KEY STAGE 3 TIER 4 6 2005 Mathematics test Paper 1 Calculator not allowed Please read this page, but do not open your booklet until your teacher tells you to start. Write your name and the name of your

New Zealand Mathematical Olympiad Committee. Induction

New Zealand Mathematical Olympiad Committee Induction Chris Tuffley 1 Proof by dominoes Mathematical induction is a useful tool for proving statements like P is true for all natural numbers n, for example

Chapter 1: Coordinates and Design

Chapter 1: Coordinates and Design TRUE/FALSE 1. A change in a figure that results in a different position or orientation is known as a transformation. T DIF: Easy OBJ: Section 1.3 NAT: SS5 KEY: transformation

ABOUT IZZI 2 ABOUT IZZI. Using Puzzles to Teach Problem Solving TEACHER S GUIDE TO IZZI AND IZZI 2

Using Puzzles to Teach Problem Solving TEACHER S GUIDE TO IZZI AND IZZI 2 BENEFITS IZZI and IZZI 2 are a pattern matching games that are an exciting way to teach elements of problem solving pattern recognition

Computational Geometry

and range trees Lecture 6: and range trees Lecture 7: and range trees Database queries 1D range trees Databases Databases store records or objects Personnel database: Each employee has a name, id code,

Warm-Up. 1) Create a model to solve this problem *Use decomposition(break-apart) to solve. A 577. *Solve two ways.

Warm-Up Review: 3NS2.4 1) Create a model to solve this problem. 7 8 CST: 3NS2.4 *Use decomposition(break-apart) to solve. CST: 3NS2.4 CST: 4NS3.2 A 577 B 25, 872 C 26, 400 D 26, 872 528 49 = *Which answer

Lesson 18: Determining Surface Area of Three Dimensional Figures

Lesson 18: Determining the Surface Area of Three Dimensional Figures Student Outcomes Students determine that a right rectangular prism has six faces: top and bottom, front and back, and two sides. They

Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

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

Running head: A GEOMETRIC INTRODUCTION 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