# The mathematical art of juggling: using mathematics to predict, describe and create

Save this PDF as:

Size: px
Start display at page:

Download "The mathematical art of juggling: using mathematics to predict, describe and create"

## Transcription

1 Bridges 2011: Mathematics, Music, Art, Architecture, Culture The mathematical art of juggling: using mathematics to predict, describe and create Mike Naylor Norwegian center for mathematics education (NSMO) Norwegian Technology and Science University (NTNU) 7491 Trondheim Norway Abstract Mathematics has the power to describe, predict and create patterns, a power that is very well demonstrated in the patternrich world of juggling. In this paper we examine a simple method of describing juggling patterns using mathematical notation, and then use this notation to predict new juggling patterns. We conclude with a demonstration of how mathematics has been used to create beautiful patterns that did not exist before these mathematical methods had been used, and how mathematical names are now used by jugglers worldwide a powerful demonstration of mathematics advising the arts. Introduction Mathematics is the science of patterns, and nowhere can we see that better than in the field of mathematical art. By recognizing, extending and creating patterns we can create artistic representations of stunning beauty, either visually as in prints and sculpture, acoustically as in music or soundscapes, or dynamically as in dance or performance pieces. One of the most powerful examples of this interplay between mathematics and art can be demonstrated in the pattern-rich world of juggling. The power of mathematics lies in that mathematical tools and thinking enable us to observe phenomena and use these observations to describe, predict and create. This power is well demonstrated in the world of juggling, where mathematics can be and is used to describe juggling patterns, predict which sequences of notation describe viable patterns, and then use notation to create new and beautiful patterns that have never existed before. Perhaps most surprising is that the mathematic notation and names of some of these patterns have entered the common language of juggling and are used by non-mathematician jugglers worldwide. Figure 1 lends a fanciful introduction into our mathematical exploration of juggling. Titled Nothing but Jugglers, it displays the inevitable result of creating a scene with nothing but jugglers. In order to properly represent jugglers, the people in the scene must be juggling something, but what can those objects be if the only objects allowed are Figure 1: Nothing but Jugglers; a jugglers? Mathematicians are quick to conclude the scene must be juggler juggling jugglers juggling recursive with the number of jugglers growing exponentially to infinity as jugglers... (image by author) their size decreases to nothing. (An animated version is available at Mathematicians often find juggling appealing, perhaps because juggling is rich in patterns and combinations. In fact it was computer programmers who first discovered some of the mathematics of juggling in their efforts to create juggling pattern simulators. The first notation was created by Paul Klimek in Santa Cruz in 1981 and further developed by others in the following years [1]. Many juggling notation systems have been created since then, including systems involving multiple jugglers, but none match the elegance and simplicity of the original system known as Vanilla Siteswap Notation, or more simply as siteswap. 33

2 Naylor Vanilla siteswap describes juggling patterns using only a sequence of integers that signifies the length of time each object is in the air. In performance, a juggler may throw objects behind the back, under the leg, or from the back of the hands or elbows. Items may be bounced off the forehead or knees, thrown with palms upwards or downwards, bounced off the floor, or rolled across the chest. Multiple objects may be caught and thrown with one hand, and hands may throw asynchronously or simultaneously. Siteswap notation ignores these tricks, making the assumption that each hand throws just one object at a time, alternately in a simple arc. Despite this severe limitation, this system describes a surprisingly large number of juggling patterns and has been used to create new juggling moves that are now enjoyed by jugglers worldwide. Encoding throws and basic patterns A juggler engaged in a repeating pattern moves to the steady beat of the throws and the steady rhythm of the balls slapping into the juggler s hands like a metronome. In siteswap notation the unit of measurement is this beat ; each beat corresponding to a point in time in which a throw can be made. Left and right hands alternate, so that on the first beat, the left hand throws, on the second beat, the right hand throws, and so on. Each throw in a juggling pattern is labeled by the number of beats which transpire until the thrown object is thrown again. In practice there are about 3 or 4 beats per second. Figure 2: Three-ball cascade with path visualizations The basic three-ball juggling pattern is known as the cascade because it evokes images of water tumbling over itself (many juggling patterns are given names inspired by water-images). In the three-ball cascade, the balls travels in arcs of identical height, from right hand to left and left to right. Since the juggling takes place in a plane parallel to the front of the juggler, we could represent the entire pattern, including the element of time, by stacking snapshots of the location of the balls to create a three-dimensional (four-dimensional?) woven structure as shown in Figure 2. (An interactive animation of this structure can be found at A simplified version of a type which we will use to analyze juggling pattern is shown in Figure 3. Each string of arcs represents the path of one ball through time as it travels from hand to hand. Every throw lands three places forward of where it originates in this diagram; each of these places corresponds to a beat. In this case each throw has a duration of three beats. If we were to count 1, 2, 3, 1, 2, 3, 1, 2, 3... in time to the beats as this pattern were juggled, the same ball would be thrown on the same numbered count every time. Figure 3: Simplified three-ball cascade space-time diagram We can encode this pattern by writing down the number of beats of each throw: By adopting a rule similar to repeating decimals and designating the pattern solely in terms of its repeating digits, the above pattern can be simplified to a single digit: 3. 34

3 The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create Figure 4 shows the pattern and the space-time diagram for the basic four ball juggling pattern. Every throw is a 4, and the pattern name in siteswap notation is therefore also written as 4. Although the paths appear to be crossing in this diagram, the balls do not cross paths at all every ball thrown from the left hand returns to the left hand and likewise for the right hand. Because of the parity of this system, even numbered throws return to the same hand while odd numbered throws change hands. This pattern is known to jugglers as the four-ball fountain. (The basic patterns with an even number of objects are called fountains because the balls seem to spout like water from the center of the pattern and fall outwards.) Figure 4: Four-ball fountain and its space-time diagram In general, a one-digit pattern notation denotes the basic juggling pattern for the same number of objects. A 5 is the 5-ball cascade, a 6 is the 6-ball fountain, etc. (See Table 1). Consider yourself lucky if you get the chance to witness a 7, 8, or 9. Very few jugglers have mastered the 7-ball cascade, while perhaps less than a dozen can juggle nine or more. # as a throw as a pattern 0 empty Hand both hands empty 1 fast hand-off one ball shuttled back and forth between 2 hold one in each hand, no motion 3 chin-level arc from one hand to other three-ball cascade 4 head level throw returning to same hand 5 above head-level throw from one hand to the other 6 high throw returning to same hand four-ball fountain five-ball cascade six-ball fountain 7... etc... Table 1: Digits, throws and patterns 35

4 Naylor Throws corresponding to digits less than 3 warrant some explanation. A 0 is used to denote an empty hand for one beat. A 1 is a quick hand-off to the other hand. A 2 would, in theory, leave the hand with a very short toss, and be caught immediately with that hand taking no action between throw and catch. In practice, however, this is almost never done. Instead, a 2 designates a pause; the hand holds the ball for that beat without throwing it. The pattern designated by the single digit 2, then, would be right-hand pause, left-hand pause, right-hand pause, left-hand pause... in other words, this pattern is simply holding one ball in each hand and not doing anything a juggling pattern most everyone can do! Two-Digit Patterns Showers. Most non-jugglers believe that the proper way to juggle three balls is in a circle, one hand throwing the balls in a high arc, the other hand catching and passing the balls quickly to the throwing hand. This pattern is called the shower (another water name!) and it is actually quite a bit more difficult to perform than the cascade. The space-time diagram for the three ball shower is shown in Figure 5. Figure 5: Space-time diagram for the three-ball shower In this case, the left hand is doing the high throws, the right hand is quickly shuttling each ball to the left hand where it is immediately thrown again. The balls that leave the left hand land five beats later, and the balls that leave the right hand land one beat later. This patterns requires two digits to encode as there are two different types of throws being made. It is encoded as the repeating unit of digits, that is, 5-1. By convention, we usually name the pattern with the greatest digit first, although this is not necessary. One could easily say that a 1-5 is the same pattern as a 5-1 but in the opposite direction. Owing to the ambidextrous nature of juggling and the arbitrary choice of which hand to start the pattern with, we will not make a distinction and instead refer to patterns and their reflections by the same name. By increasing the height of the high throw, we make room for additional balls. A four-ball shower would be a 7-1, a five-ball shower is a 9-1, and so on. You may wish to verify these with space-time diagrams. The record for objects juggled in a shower pattern is eight an incredible 15-1 pattern performed by a woman on the island of Tonga in the Pacific [3]. In Tonga, juggling is an activity for girls, many of whom juggle limes or tui-tui nuts. When Western jugglers first came to Tonga, islanders thought it was very funny to see men juggling! Half-Showers. Another class of juggling patterns are the half-showers. They are similar to showers in that one hand throws the balls over the top of the pattern, but instead of the other hand making a quick hand-off, the ball is sent across in short arc. Half-showers are somewhat slower and easier to perform than showers. The diagram for the four-ball half-shower is shown in Figure 6. Here, the left hand always makes high a throw to the right hand, which returns the ball to the left in a shorter arc. This pattern is a 5-3. Figure 6: Four-ball half-shower 36

5 The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create One-Hand Patterns. Two balls in two hands is just not challenging enough for most jugglers; when we speak of a two-ball juggle, we generally mean two balls in one hand as shown in Figure 7. Figure 7: Two balls, one hand The zero denotes the empty hand; this pattern is encoded as 4-0. Notice how similar the path diagram is to the four-ball fountain. In effect, the paths of the two balls in the right hand have been removed, and a zero entered as a place-holder. Similarly, three balls in one hand (an impressive feat!) is encoded 6-0, and four balls in one hand is an 8-0. Suppose we were juggling two balls in one hand as above (the 4-0 ), and picked up a ball in the other hand and simply held it. This three-ball pattern would be no more difficult than the 4-0, but it would have a different siteswap name: a 4-2 (see Figure 8). Likewise, juggling three in one hand and holding a fourth in the other hand would be designated Figure 8: Two in one with the other hand just holding a ball Patterns of Patterns. The two-digit patterns have been organized in Table 2, and the patterns of patterns within the table have been extended both downwards and to the right to include such patterns as the 6-4, a five-ball pattern in which one hand juggles three by itself, the other hand juggles two. 1 hand shower 1 hand + hold 1/2 shower 1 hand + 2 in 1 high shower 1 hand + 3 in 1 super shower 1 hand + 4 in 1 1 ball: ball: ball: ball: ball: ball: ball: ball: ball: Table 2: Two-Digit Patterns This listing quickly extends beyond the capabilities of human jugglers, but it accommodates all possible twodigit siteswap patterns. Notice that the sums of the digits of each of the two-digit patterns is twice the number of objects being juggled a property of these patterns we will generalize shortly. 37

6 Naylor Three Digit Patterns A quick and dirty way to generate some three-digit patterns is to remove one or two paths from the three-ball cascade. Removing one ball from the pattern gives us the two-ball pattern (see Figure 9) Figure 9: Two balls in three-ball cascade pattern Removing one more path reduces it to a one-ball pattern: (see Figure 10) Figure 10: A very simple 3-digit pattern Notice that the is a two-ball pattern with a digit sum of 6, and the is a one-ball pattern with a digit sum of 3 both sums are three times the number of objects being juggled. Incidentally, the and the are excellent patterns to practice when one is learning the 3-ball cascade. Let s replace the zero in the with a different digit. This digit must represent a throw that will land in an empty hand, so throws of 1, 2, 4 or 5 will not work; these throws would result in two balls landing in a hand. A six, however will work - six spaces to the right of a zero in the space-time diagram is another zero which represents an empty hand and thus a potential landing spot for this additional object. Changing the 0 to a 6 actually creates two new paths in the pattern, since each new set of arcs connects every other empty space in the pattern (see Figure 11). The or equivalently 6-3-3, is a four-ball pattern wherein two balls cascade back and forth between hands while the other two balls stay one in each hand in alternating high throws. This pattern has a digit sum of 12, again, 3 times the number of objects. If instead of changing the zero to a six, we change it to a nine, there will be room for an additional object in the pattern. A is a five-ball pattern, and the digit sum is exactly three times the number of objects. Figure 11: The four-ball pattern Properties of Legal Patterns As we have seen, one digit patterns are equal to the number of objects juggled, in two digit patterns the digit sum is twice the number of objects in the pattern, and in three-digit patterns the digit sum is thrice the number of objects being juggled. In fact, the mean of the digits in a siteswap pattern is exactly the number of balls in the pattern. If the mean of the digits is not an integer, the pattern will not work. A 2-1, for instance, tells the juggler to hold a ball in the left hand, make a quick pass from the right, hold the left, quick pass from the right... the right hand never receives any balls, and the left hand never gets rid of any balls, an impossible situation. The following proof that the mean of the digits equals the number of objects being juggled is based on a clever argument by Warren Nichols of Florida State University [4]: 38

7 The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create Assume we have a legal pattern of n digits and a digit sum of m. In the example below we consider the legal pattern ; this pattern has 5 digits and a digit sum of 15. Arrange these digits in a circle and construct a space-time diagram which, instead of being linear, wraps around the circle to join itself in a continuous manner (Figure 12). Since this is a legal pattern, the paths are uniformly distributed around the circle, that is, there could not be a place in the space-time diagram where there were, say 3 paths, and another place where there were 4 paths, as this would indicate that the number of objects being juggled changes mid-pattern. The circumference of this circle is n beats since there are n digits, and the total length of the paths is the sum of the digits or m beats, so the paths wrap around the circle m/n times. The paths are uniformly distributed, so at every point in time during this pattern there are m/n paths, which means there are m/n objects being juggled. The wrapped space-time diagram for appears below. The circumference is 5 beats, the total length of the paths is 15, indicating that the paths must wrap around the circle 15/5 = 3 times is thus a three-ball pattern. Conversely, if m/n is not an integer, then the paths cannot wrap around the circle an integral number of times. The number of objects being juggled must change during the pattern, and therefore the pattern is not legal. Figure 12: The total length of the paths is 15 and the circumference is 5; the paths wrap around the circle 15/5 = 3 times. The fact that the mean of the digits must be an integer is a necessary but not sufficient condition for the digits to represent a legal pattern. One can create strings of digits which average to an integer, but which are impossible to juggle , for example, has an integral average of 4. The space-time diagram in Figure 13 shows what happens when this pattern is attempted. L R L R L R Figure 13: Collision! All three balls land at the same time in the same hand! While in real-life, this would be an impressive feat, in siteswap notation this situation is referred to a collision, and it is not allowed. In order for a pattern to be legal, only one object may land in a hand at one time. The problem with this pattern is not the choice of digits, but rather the order of the digits. If we permute the digits as 5-3-4, the pattern works just fine as shown in Figure 14. Figure 14: Collision fixed 39

8 Naylor Construction of a space-time diagram is not necessary to determine if a pattern is legal. Other mathematical tools can be found in a paper included in the references [4]. The Lost 3-Ball Pattern and Other Patterns We now have all the tools necessary to create and test legal patterns. All that is left it to use these tools to creatively explore the space of possible juggling patterns. One of the truly great powers of mathematics is exemplified in the story of siteswap notation. As jugglers studied pattern notation, they discovered patterns that were beautiful and unknown. One of the most striking is the pattern (Figure 15). This pattern consists of two staggered throws straight up, then a quick hand-off which is immediately sent up as the first of a different pair of staggered fours (Figure 18). This pattern is beautiful to see and satisfying to perform, and was not discovered until mathematics showed the way! Figure 15: The ( The lost 3-ball pattern ) is referred to by jugglers worldwide as the While many juggling patterns have fanciful names such as the cascade, the shower, and the fountain, in juggling circles the pattern is called by its siteswap name. The mathematical terms have entered into juggling culture worldwide a powerful testament to the interplay between mathematics and the arts. Other siteswap pattern names broadly used by jugglers include the and the Juggling is one of the few arts in which practitioners regularly use names derived from mathematical notation! These pattern names exemplify the true power of mathematics. We can observe a phenomenon and create mathematical notation to describe it, look for patterns and generate rules within our notation, and then use these rules to create new symbolic patterns which correspond to new discoveries within the natural world. Not only does mathematics have the power to describe mathematics has the power to create! Finale Several siteswap juggling pattern simulators are available for free on the internet, but siteswap notation is more than just a powerful tool for creating electronic jugglers. Siteswap can aid jugglers while practicing complicated patterns by focusing attention on individual throws, and the notation has been used to discover new patterns which were previously unknown anywhere. Incidentally, there are 11 legal three-digit three-ball patterns finding them all and visualizing what they look like is a nice challenge. Regardless of whether or not a juggler understands or uses siteswap, juggling is still a sport of kinesthetic mathematics. One cannot help but be mystified and drawn into the intricate web of translations, inversions, dilations, and permutations as a skilled juggler weaves patterns upon patterns of order balancing on the edge of chaos. References [1] Polster, B., The Mathematics of Juggling, Springer, [2] Knutson, A., Site-Swap FAQ, [Online] Available: [January 12, 2012]. [3] Truzzi, M., On Keeping Things Up in the Air, Natural History, 88, 10, December [4] Naylor, M., Patterns of patterns: the mathematics of juggling, The Journal of Recreational Mathematics, 32, 1,

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 1 Introduction to Counting

1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### 1 Number System. Introduction. Face Value and Place Value of a Digit. Categories of Numbers

RMAT Success Master 3 1 Number System Introduction In the decimal number system, numbers are expressed by means of symbols 0, 1,, 3, 4, 5, 6, 7, 8, 9, called digits. Here, 0 is called an insignificant

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### NS5-38 Remainders and NS5-39 Dividing with Remainders

:1 PAGE 89-90 NS5-38 Remainders and NS5-39 Dividing with Remainders GOALS Students will divide with remainders using pictures, number lines and skip counting. Draw: 6 3 = 2 7 3 = 2 Remainder 1 8 3 = 2

### Sudoku puzzles and how to solve them

Sudoku puzzles and how to solve them Andries E. Brouwer 2006-05-31 1 Sudoku Figure 1: Two puzzles the second one is difficult A Sudoku puzzle (of classical type ) consists of a 9-by-9 matrix partitioned

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### The Physics and Math of Ping-pong and How It Affects Game Play. By: Connor Thompson & Andrew Johnson

The Physics and Math of Ping-pong and How It Affects Game Play 1 The Physics and Math of Ping-pong and How It Affects Game Play By: Connor Thompson & Andrew Johnson The Practical Applications of Advanced

### VOLLEYBALL. Rules, Regulations and Scoring Team Composition 1. A team is comprised of six players:

VOLLEYBALL Volleyball is a game played with two teams of six players each. The object of the game is to use skills such as the forearm pass, set, spike and block to gain an advantage on the opponent by

### III. Applications of Force and Motion Concepts. Concept Review. Conflicting Contentions. 1. Airplane Drop 2. Moving Ball Toss 3. Galileo s Argument

III. Applications of Force and Motion Concepts Concept Review Conflicting Contentions 1. Airplane Drop 2. Moving Ball Toss 3. Galileo s Argument Qualitative Reasoning 1. Dropping Balls 2. Spinning Bug

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence

### Kenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract

Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 7, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills.

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### Youth Volleyball Coaches Drill Book

Youth Volleyball Coaches Drill Book Volleyball - 4 X 2 Pepper Drill Practice ball control with a wide variety of types of contacts. Put players in pairs. Player one hits the ball down to player two. Player

### Number 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

### Combinations and Permutations

Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

### Grade 7/8 Math Circles November 3/4, 2015. M.C. Escher and Tessellations

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Tiling the Plane Grade 7/8 Math Circles November 3/4, 2015 M.C. Escher and Tessellations Do the following

### Math 181 Handout 16. Rich Schwartz. March 9, 2010

Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =

### Balls, Hoops and Odds & Ends

KIWIDEX RUNNING / WALKING 181 Balls, Hoops and Odds & Ends On 1 Feb 2012, SPARC changed its name to Sport NZ. www.sportnz.org.nz 182 Section Contents Suggestions 183 Cats and Pigeons 184 Geared Up Relays

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### or (ⴚ), to enter negative numbers

. Using negative numbers Add, subtract, multiply and divide positive and negative integers Use the sign change key to input negative numbers into a calculator Why learn this? Manipulating negativ e numbers

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### 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

### Tossing a Biased Coin

Tossing a Biased Coin Michael Mitzenmacher When we talk about a coin toss, we think of it as unbiased: with probability one-half it comes up heads, and with probability one-half it comes up tails. An ideal

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Utah Core Curriculum for Mathematics

Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions

### We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

### Review Chapters 2, 3, 4, 5

Review Chapters 2, 3, 4, 5 4) The gain in speed each second for a freely-falling object is about A) 0. B) 5 m/s. C) 10 m/s. D) 20 m/s. E) depends on the initial speed 9) Whirl a rock at the end of a string

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton

LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton 1. Take a two-digit number and write it down three times to form a six-digit number. For example,

### Optical Illusions Essay Angela Wall EMAT 6690

Optical Illusions Essay Angela Wall EMAT 6690! Optical illusions are images that are visually perceived differently than how they actually appear in reality. These images can be very entertaining, but

### Topic: 1 - Understanding Addition and Subtraction

8 days / September Topic: 1 - Understanding Addition and Subtraction Represent and solve problems involving addition and subtraction. 2.OA.1. Use addition and subtraction within 100 to solve one- and two-step

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### Fifth Grade Physical Education Activities

Fifth Grade Physical Education Activities 89 Inclement Weather PASS AND COUNT RESOURCE Indoor Action Games for Elementary Children, pg. 129 DESCRIPTION In this game, students will be ordering whole numbers.

### Why Vedic Mathematics?

Why Vedic Mathematics? Many Indian Secondary School students consider Mathematics a very difficult subject. Some students encounter difficulty with basic arithmetical operations. Some students feel it

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Notre Dame Academy Soccer Program

Notre Dame Academy Soccer Program 12 Week Training Program you get out of it, what you put in it Notre Dame Academy Soccer Program 12 Week Training Program "Besides pride, loyalty, discipline, heart, and

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS

### EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

### Pythagorean 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

### WPA World Artistic Pool Championship. Official Shot / Challenge Program. November 8, 2011 1

WPA World Artistic Pool Championship 2012 Official Shot / Challenge Program November 8, 2011 1 Revision History November 30, 2010: Initial version of shot program. January 10, 2011: February 14, 2011:

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold 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 ping-pong balls)

### Swavesey Primary School Calculation Policy. Addition and Subtraction

Addition and Subtraction Key Objectives KS1 Foundation Stage Say and use number names in order in familiar contexts Know that a number identifies how many objects in a set Count reliably up to 10 everyday

### Regions in a circle. 7 points 57 regions

Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an

### 1.2. 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

### Freehand Sketching. Sections

3 Freehand Sketching Sections 3.1 Why Freehand Sketches? 3.2 Freehand Sketching Fundamentals 3.3 Basic Freehand Sketching 3.4 Advanced Freehand Sketching Key Terms Objectives Explain why freehand sketching

### ICPC Challenge. Game Description. March 26, 2009

ICPC Challenge Game Description March 26, 2009 The ICPC Challenge game, Capture, is played on a field that looks something like the following figure. A red player and a blue player compete on the 800 800

### Knowing and Using Number Facts

Knowing and Using Number Facts Use knowledge of place value and Use knowledge of place value and addition and subtraction of two-digit multiplication facts to 10 10 to numbers to derive sums and derive

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Chapter 13: Fibonacci Numbers and the Golden Ratio

Chapter 13: Fibonacci Numbers and the Golden Ratio 13.1 Fibonacci Numbers THE FIBONACCI SEQUENCE 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, The sequence of numbers shown above is called the Fibonacci

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

Graduate Management Admission Test (GMAT) Quantitative Section In the math section, you will have 75 minutes to answer 37 questions: A of these question are experimental and would not be counted toward

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### GMAT Math: Exponents and Roots (Excerpt)

GMAT Math: Exponents and Roots (Excerpt) Jeff Sackmann / GMAT HACKS January 201 Contents 1 Introduction 2 2 Difficulty Levels Problem Solving 4 4 Data Sufficiency 5 5 Answer Key 6 6 Explanations 7 1 1

### This is a game for two players and one referee. You will need a deck of cards (Ace 10).

PLAYER SALUTE This is a game for two players and one referee. You will need a deck of cards (Ace 10). Each player draws a card and, without looking, holds the card on his/her forehead facing outward. The

### NEW 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

### Mental Math Addition and Subtraction

Mental Math Addition and Subtraction If any of your students don t know their addition and subtraction facts, teach them to add and subtract using their fingers by the methods taught below. You should

### Prime Factorization 0.1. Overcoming Math Anxiety

0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF

### Figure 1. A typical Laboratory Thermometer graduated in C.

SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

### Style Guide For Writing Mathematical Proofs

Style Guide For Writing Mathematical Proofs Adapted by Lindsey Shorser from materials by Adrian Butscher and Charles Shepherd A solution to a math problem is an argument. Therefore, it should be phrased

### Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### Math Games For Skills and Concepts

Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,

### 23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

### Mathematical 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

### 2 Step 2 Dance Studio Cha-Cha Basics. Cha-Cha Basics. Pre-requisite None (These are basic patterns)

Pre-requisite Introduction The Music None (These are basic patterns) Cha-Cha Basics Although Cha-cha is one of the Cuban family of dances that is danced throughout the Latin American countries, it is one

### Grade 6 Grade-Level Goals. Equivalent names for fractions, decimals, and percents. Comparing and ordering numbers

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### Common Core State Standards for Mathematics Accelerated 7th Grade

A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting

### Beating Roulette? An analysis with probability and statistics.

The Mathematician s Wastebasket Volume 1, Issue 4 Stephen Devereaux April 28, 2013 Beating Roulette? An analysis with probability and statistics. Every time I watch the film 21, I feel like I ve made the

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Primary Curriculum 2014

Primary Curriculum 2014 Suggested Key Objectives for Mathematics at Key Stages 1 and 2 Year 1 Maths Key Objectives Taken from the National Curriculum 1 Count to and across 100, forwards and backwards,