Instructing Group Theory Concepts from Pre-Kindergarten to College Through Movement Activities

Instructing Group Theory Concepts from Pre-Kindergarten to College through Movement Activities

AThesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Mathematical Science in the Graduate School of The Ohio State University

Jessica Wheeler, B.A.

Graduate Program in Mathematics

The Ohio State University

2016

Master’s Examination Committee:

Dr. Bart Snapp, Advisor Dr. Rodica Costin, Advisor Dr. Herb Clemens c Copyright by Jessica Wheeler

2016 Abstract

Group theory is a mathematical topic that is usually reserved for upper-level un- dergraduate courses. In this thesis, we will explicitly show the connection between group theory and fundamental mathematics taught in grade school, and present activities designed for pre-k through 3rd,4th through 9th,and7th through 12th grade levels with culminating activities for college level students. Our annotated activities are explicitly connected to standards from the common core, they aim to provide hands- on learning environments for students, and demonstrate that topics from higher level courses have deep connections to fundamental mathematics.

ii This thesis is dedicated to my parents for their support of my education and their

unwavering encouragement.

iii Vita

2010 ...... Karns High School

2014 ...... B.A. Mathematics, Radford University

2014 ...... B.A. Dance, Radford University

2014-present ...... Graduate Teaching Associate, The Ohio State University

Fields of Study

Major Field: Mathematics

iv Table of Contents

Page

Abstract...... ii

Dedication ...... iii

Vita ...... iv

Chapters:

1. Introduction ...... 1

2. Functions,Groups,andGroupActions ...... 3

2.1 Functions ...... 3 2.2 Groups ...... 6 2.2.1 Free Group ...... 7 2.2.2 GroupAction...... 18

3. Pre-Kindergarten to Third Grade ...... 20

3.1 Rational Tangles ...... 20 3.1.1 AllTangledUp! ...... 20 3.1.2 AddingupTangles ...... 25 3.2 Contra Dancing ...... 28 3.2.1 DancingAroundaSquare ...... 28

4. FourthGradetoNinthGrade ...... 34

4.1 Rational Tangles ...... 34 4.1.1 Tangles & Fractions ...... 34 4.2 Contra Dancing ...... 40 4.2.1 Contra Dance Functions ...... 40

v 5. Seventh to Twelfth Grade ...... 46

5.1 Rational Tangles ...... 46 5.1.1 Tangles & Functions ...... 46 5.2 Contra Dancing ...... 53 5.2.1 Math-y Dance ...... 53 5.2.2 Dancing Functions ...... 58

6. College ...... 66

6.1 Rational Tangles ...... 66 6.1.1 Tangles ...... 66 6.2 Contra Dancing ...... 74 6.2.1 PropertiesofContraDanceFigures...... 74 6.2.2 Generators and Relations ...... 81 6.2.3 Free Group ...... 84

7. Irrational Tangles ...... 88

7.1 Rational and Irrational Tangles ...... 88 7.1.1 Connections to the Tangle Group ...... 94

Appendices:

A. Pre-KindergartentoThirdGradeActivities ...... 96

A.1 AllTangledUp! ...... 96 A.2 AddingupTangles ...... 98 A.3 DancingAroundaSquare ...... 100

B. FourthGradetoNinthGradeActivities ...... 105

B.1 Tangles & Fractions ...... 105 B.2 Contra Dance Functions ...... 109

C. SeventhGradetoTwelfthGradeActivities...... 113

C.1 Tangles & Functions ...... 113 C.2 Math-yDance ...... 118 C.3 Dancing Functions ...... 122

vi D. College Activities ...... 128

D.1 Tangles ...... 128 D.2 PropertiesofContraDanceFigures...... 134 D.3 Generators and Relations ...... 140 D.4 Free Group ...... 142

Bibliography ...... 145

vii Chapter 1: Introduction

Group theory is a fundamental topic in mathematics that is typically introduced at the college level. In particular group theory is notably absent from the kindergarten through twelfth grade curriculum. In this thesis we will explicitly show the connection between group theory and fundamental mathematics, and give activities designed for pre-kindergarten through third grade, fourth grade through ninth grade, and seventh grade through twelfth grade with a culmination chapter on activities for college level students.

The subsets of grade levels chosen for these activities are motivated by the grade level cuto↵s for teacher licensures in Ohio. Three main categories for educators in the state of Ohio to be licenced are Early Childhood Licensure, Middle Childhood

Licensure, and Adolescence to Young Adult Licensure. These are separated by the grade levels pre-k – 3, 4 – 9, and 7 – 12, respectively [8].

In this thesis we provide movement activities to help instruct concepts of group theory in a friendly and engaging manner. The activities are based on the ideas of rational tangles and square dancing [7, 14]. Rational tangles is an activity ﬁrst devised by John Conway and has various mathematical applications [5]. Contra dancing is a folk dance with many dance ﬁgures. The activities in the following chapters simplify

1 this folk dance and use only a few basic moves. This allows the dance to be accessible to more students and illustrates the symmetries of a square discussed in Chapter 2.

These activities illustrate ways for students at any level to learn about groups and functions. Each activity corresponds to standards from the Common Core State

Standards Initiative [12]. It is our hope that this may facilitate the use of activities such as these in an actual classroom.

The chapters in this thesis provide annotated activities, and blank activities can be found in the appendix. We also include a ﬁnal chapter that makes connections between the real numbers and the tangle group discussed in Chapter 2.

2 Chapter 2: Functions, Groups, and Group Actions

2.1 Functions

The concept of a function and functional notation are taught in grade school.

As we will see, functions are also fundamental to group theory. Functions describe relationships between sets.

Deﬁnition 1. A set is an aggregation of elements such that one always knows if one given element is in the set or not.

We notate x X to indicate that an object x is an element of the set X.Sets 2 can contain any type of object; they are not restricted to numbers.

Example 2.1.1. S = cat, dog, mouse is a set with 3 elements such that dog S, { } 2 but frog / S. 2 Deﬁnition 2. The cartesian product of two sets X and Y is the set X Y with ⇥ elements (x, y)suchthatx X and y Y . 2 2

Two elements (x, y) X Y and (x0,y0) X Y are equal if and only if x = x0 2 ⇥ 2 ⇥ and y = y0.

Example 2.1.2. Let S = cat, dog, mouse and T = spots, furry .Then { } { } (dog, spots) S T while (dog, bark) / S T . 2 ⇥ 2 ⇥ 3 Deﬁnition 3. A function with domain X and range Y is a subset f X Y that ✓ ⇥ satisﬁes the following three properties.

For all x X,thereexistsy Y such that (x, y) f.Thismeansthata • 2 2 2 function must map every element in the domain, or else that element is not in

the domain.

For all y Y ,thereexistsx X such that (x, y) f.Thismeansthata • 2 2 2 function must map to every element in the range or else that element is not in

the range.

If (x, a) f and (x, b) f,thena = b.Thismeansthatforeveryelementin • 2 2 the domain, there is exactly 1 element in the range.

A function is the correspondence between X and Y given as an ordered pair.

Given a function f = (x, y) X Y :(x, y) f ,wewritef : X Y and notate { 2 ⇥ 2 } ! f(x)=y for the mapping of a given element x X to y Y . 2 2

Example 2.1.3. The function f(x)=x is an identity function because it returns the same element as the input.

Example 2.1.4. f(x, y, z)=xyz deﬁnes a function with domain X = R3 and range

Y = R.

1 1 The inverse of a function f is denoted by f and we write f = (y, x) Y X : { 2 ⇥ (x, y) X Y . 2 ⇥ }

1 Deﬁnition 4. Afunctionf : X Y is invertible if for each y Y , f (y)has ! 2 exactly one element.

4 An invertible function satisﬁes the properties of a function as well as the property that for every element in the range, there is exactly 1 element in the domain.

Not every function is invertible. Consider the function f(x, y, z)=xyz, f : R3 ! R from Example 2.1.4. 60 is an element in the range that has at least two elements in the domain, (1, 2, 30) and (10, 3, 2), such that f(1, 2, 30) = 60 and f(10, 3, 2) = 60.

Functions can be combined by using the operation of composition.

Deﬁnition 5. The composition of two functions f : X Y and g : Y Z is the ! ! function g f : X Z such that g f = (x, z) X Z :(f(x),z) g . ! { 2 ⇥ 2 }

Taking the composition of two functions, g f,isreadfromrighttoleftsuchthat f is mapped to some element in its range, which g then maps to some element of its range.

Theorem 2.1.1 (Functional composition is always associative). Let f, g, and h be functions such that f : X Y , g : Y Z, and h : Z W . Then h (g f)= ! ! ! (h g) f.

Proof. We want to show that h (g f)=(h g) f. By deﬁnition of composition of functions,

g f = (x, z) X Z :(f(x),z) g . { 2 ⇥ 2 }

Thus,

h (g f)(x)= (x, w) X W :(g f(x),w) h . { 2 ⇥ 2 }

Now, by deﬁnition of composition of functions,

h g = (y, w) Y W :(g(y),w) h . { 2 ⇥ 2 }

5 Thus,

(h g) f = (x, w) X W :(f(x),w) h g . { 2 ⇥ 2 }

If (f(x),w) h g,then(f(x),w) Y W such that (g(f(x)),w) h.Thus, 2 2 ⇥ 2

(h g) f = (x, w) X W :(g f(x),w) h . { 2 ⇥ 2 }

Therefore, h (g f)=(h g) f. 2.2 Groups

As previously mentioned, functions are fundamental to group theory. We will see their connection to groups by ﬁrst introducing the deﬁnition.

Deﬁnition 6. A group is a nonempty set G equipped with an operation, ,such that

1. G is closed under the operation ,thatis,iff,g G then f g G. 2 2

2. The operation is associative, meaning that for any elements f,g,h G, f 2 (g h)=(f g) h.

3. G contains an element e,calledtheidentity,suchthatg e = e g = g for all g G. 2

1 1 1 4. Every element g G has an inverse, g ,suchthatg g = e and g g = e. 2

We can in fact think of the deﬁnition of a group as an algebraic description of what it means to be a set of invertible functions that are closed under composition.

Asetofinvertiblefunctionsthatareclosedundercompositionimmediatelysatisﬁes

1 properties 1 and 4. Property 3 is satisﬁed since f f = e for any function in our set. 6 Functional composition is always associative by Theorem 2.1.1, satisfying property 2.

Thus, our claim above is true.

2.2.1 Free Group

While there are many excellent choices of groups to start with: the integers, cyclic group, dihedral group, symmetric group, etc., we choose to start with free groups.

The deﬁnition of a free group has many nice connections to grade school curriculum.

It also allows us to discuss groups in terms of generators and relations. We will use this characterization of a group to discuss two main examples that will be referenced in the following chapters.

Consider the set A = a . Using A as an alphabet,wecancreatewords through { } the operation of concatenation a

aaa

aaaa

··· It is convenient to notate these words as a1, a2, a3, a4,etc.,respectively.Words can also be created by concatenating two words together. If aj and ak are words, then ajak = aj+k is also a word. Also, words are equal if and only if their string lengths are equal. For example, am and an are equal if and only if m = n.Wewilldenote the set of words on A by F [A]. Thus, concatenation is a binary operation mapping

F [A] F [A] F [A]. ⇥ ! The deﬁnition of concatenation is identical to the exponential rules found in high school. When multiplying two exponents of the same base, the rule is to add the

7 exponents. For example, x4x7 = x4+7 = x11.Thisisidenticaltoconcatenatingtwo words together.

The concatenation operation is also associative. Given a word ajakal we see that

(ajak)al = aj+kal

= aj+k+l

= ajak+l

= aj(akal)

Thus, F [A]isasemigroup.

Deﬁnition 7. A semigroup is a nonempty set G equipped with an operation, , such that

1. G is closed under the operation .Iff,g G then f g G. 2 2

2. The operation is associative. For any elements f,g,h G, f (g h)=(f g) h. 2

In concatenation we simply omit the symbol ,thatis,a a is denoted by aa. The word containing no letters is denoted by e and is called the empty word,and it is conveniently denoted by a0.Ifweconcatenateanywordwiththeemptyword, we are left with the original word. Consider the word ak.Then

ake = aka0

= ak+0

= ak and eak = a0ak

= a0+k

= ak

8 We again see the connection to the high school exponential rules since we choose to denote the empty word by a0:anyvalueraisedtothepowerof0is1.Thus, x0xk =1xk = xk. So, concatenating words with the empty word directly relates to the property of multiplying values by 1.

The empty word is an identity element for our set A = a under the operation { } of concatenation. Now including the identity element, our set of words formed by the alphabet A, F [A], is a monoid.

Deﬁnition 8. A monoid is a nonempty set G equipped with an operation, ,such that

1. G is closed under the operation .Iff,g G then f g G. 2 2

2. The operation is associative. For any elements f,g,h G, f (g h)=(f g) h. 2

3. G contains an element e,calledtheidentity,suchthatg e = e g = g for all g G. 2

1 Let a denote a new letter, called the inverse element of a,whichweassumeto

1 1 obey the rule aa = e = a a.Weextendthedeﬁnitionofawordformedbythe

1 alphabet A by allowing strings of both a and a in any order, where repetition is

1 1 allowed. A word is then reduced if all instances of aa and a a are replaced by e.LetF [A]denotethesetofallreducedwordsformedbythealphabetA with the operation of concatenation. F [A]isnowagroup.

In general, if A = a is any set, we can further extend the deﬁnition of a word { i} on A and reduced words.

Deﬁnition 9. A word on A is a sequence of the elements a A and their inverses i 2 1 ai joined by concatenation where repetition is allowed.

9 ✏1 ✏2 ✏3 Deﬁnition 10. [11] A word a1 a2 a3 ... is a reduced word if it satisﬁes the following.

1 1. a and a are not adjacent for all a A. i i 2

2. If a = e then a = e for all j k. k j

Example 2.2.1. Consider a set of red and black chips. Let each red chip represent

1andeachblackchiprepresent+1.Thus,everycombinationof1redchipand1 1 black chip returns an overall value of 0. We can think that a =ablackchipanda = a red chip. Addition of positive and negative values can be represented by using this chip model. For example, 4 7=4+( 7) can be represented by combining 4 black chips with 7 red chips. We can match up one red chip and one black chip at a time until there are no more matches. Since 4 red chips can match up with 4 black chips, we are left with 3 single red chips. The pairs all have a value of 0 and the three red chips have a value of 3. Thus, the value of 4 7is0+0+0+0 3= 3.

Example 2.2.1 illustrates the process of reducing a word. Part 1 of Definition 10 corresponds to the process of matching pairs of red and black chips. Inverse elements cancel each other out just as a red chip cancels out a black chip. Part 2 is illustrated in the last step of Example 2.2.1. In our chip model, we replace 0 + 0 + 0 + 0 3with 3. Once part 1 is satisfied, we remove e unless the word overall is e. Again, F [A]asdefinedasthesetofallreducedwordsonA is a group. As we have defined F [A], A is a generating set of F [A].

Deﬁnition 11. Let S be a subset of a group G. S is said to generate G if it has the property such that all the elements of G can be written as products of elements of S and their inverses.

10 Example 2.2.2. Consider a wagon wheel with 8 spokes:

As the wagon moves to the right, the wheel must rotate. We can see the rotational symmetry in this wheel as it rotates in increments of 45. Notate a 45 rotation

2 3 clockwise by ⇢.Thus,⇢ is a 90 rotation, ⇢ is a 135 rotation, and so on. If the wagon were to move left, the wheel would rotate counterclockwise. Since this movement is the opposite of rotating clockwise, we can denote a 45 rotation counterclockwise by

1 ⇢ .

Example 2.2.2 illustrates how ⇢ is a function that generates all possible positions that map the wheel onto itself. A given position of the wheel can be described as some string of ⇢ and its inverse.

Given a group G and a set S that generates G, G deﬁned to be a free group when it is free of all relations.Moreexplicitly,theelementsinthesetA are free of relations.

Deﬁnition 12. Equations in a group G that are satisﬁed by the generators are called relations.

In Example 2.2.2 ⇢8 = e.Thisrelationiscalledanequationsatisﬁedbythe generator, ⇢.

Deﬁnition 13. If A is a set such that the elements satisfy no relations, and F [A]is the set of all reduced words on A,thenF [A]isthefree group generated by A.

The universal property of a free group uniquely deﬁnes the relationship between asetA,agroupG,andafreegroupF [A].

11 Deﬁnition 14. Let G and H be groups under the operations and ,respectively. • Afunctionf : G H is a homomorphism if !

f(x y)=f(x) f(y) • for all x, y G. 2

AhomomorphismrelatesthestructuresofthetwogroupsG and H.

Example 2.2.3. Consider the following two groups: the integers, Z, under the operation of addition and the free group on one generator, F [A], where A = a .There { } is a homomorphism where : F [A] Z. Using our convenient notation of words on ! A as ak, we define the homomorphism as (ak)=k.Wecancheckthatthisfunction satisfies the conditions from Definition 14. For all k, l Z, 2

(akal)=(ak+l)=k + l and

(ak)+(al)=k + l

Thus, (akal)=(ak)+(al). Note that we saw this example of a homomorphism in the chip activity from Example 2.2.1.

Theorem 2.2.1 (Universal Property of Free Groups). (see [11]) Let F [A] be the free group on a set A and i : A F [A] the inclusion function, that is, the canonical ! embedding of A in F [A].IfG is a group and f : A G a function, then there exists ! a unique homomorphism of groups f¯ : F [A] G such that f¯ i = f. !

Thus, every group is the homomorphic image of a free group. The following diagram illustrates the universal property of free groups.

12 i A F [A]

f¯ f G

Example 2.2.4. The free group on one generator is isomorphic to the integers, Z. We can see this by ﬁrst demonstrating how the universal property of free groups applies to the group Z under the operation of addition. Note that the generator of Z is 1 because all elements of Z can be written as a list of repeated addition of 1 and its inverse. Let A = a .Theni(a)=a and F [A]isthefreegroupononegenerator { } with elements ak where k Z.Wecandeﬁnef(a)=1andf¯(ak)=k. 2

Example 2.2.5. We can also see how the universal property of free groups applies to the group Z3 under the operation of addition. Note that the generator of Z3 is 1 because all elements of Z3 can be written as a list of repeated addition of 1 and its inverse. Let A = a .Theni(a)=a and F [A]isthefreegroupononegenerator { } with elements ak where k Z.Wecandeﬁnef(a)=1andf¯(ak)=k (mod 3). 2

Example 2.2.6. Symmetries of a Square

Deﬁnition 15. The nth dihedral group, denoted Dn,isthegroupofsymmetries of the regular n-gon.

In this thesis we are speciﬁcally interested in the dihedral group of degree 4, D4, that describes the symmetries of the square. Let 1, 2, 3, and 4 denote the four vertices of a square.

13 4 1

3 2

Let r denote a 90 rotation clockwise about the center of the square and let s denote the reﬂection about the line of symmetry from vertex 1 to 3 (dashed line). We can deﬁne D4 based on its generators, r and s.

Let A = r, s .TheelementsofA satisfy the following relations: { }

1. r4 = e

4 1 3 4 2 3 1 2 4 1 r r r r

3 2 2 1 1 4 4 3 3 2

2. s2 = e

4 1 2 1 4 1 s s

3 2 3 4 3 2

1 3. s r = r s

14 s r

4 1 3 4 1 4 r s

3 2 2 1 2 3

1 r s

4 1 2 1 1 4 1 s r

3 2 3 4 2 3

The set of generators A = r, s under the operation of composition with the { } three relations listed above form a group. Any combination of r and s produces a rigid motion of the square and is thus a binary operation. Associativity is satisﬁed by Theorem 2.1.1. The identities are listed under the relations: r4, s2. The inverse

3 of r is r and the inverse of s is s.Thus,D4 as characterized by its generators and relations is a group.

We can see how the universal property holds for D4 in the following way. Let

A = ⇢, .Theni(⇢)=⇢ and i()=. F [A]isthefreegrouponthissetA.The { } function f maps f(⇢)=r and f()=s,andthefunctionf¯ maps f¯(⇢)=r and

4 2 1 f¯()=s where r = e, s = e,ands r = r s. i A F [A]

f¯ f

15 We also note that D is a non-commutative group, meaning that the operation 4 on our group is not commutative. For example, r s = s r,asshownbelow. 6 r s

4 1 2 1 3 2 s r

3 2 3 4 4 1

s r

4 1 3 4 1 4 r s

3 2 2 1 2 3

Example 2.2.7. Tangle Group

The rational tangle activities that follow this chapter are based on a free group with two generators that satisﬁes two relations. We will refer to this group as the tangle group, and denote it by T .

Let the two lines (as shown below) represent two ropes.

Deﬁne the function T as a twist of the two ropes as shown below.

Deﬁne the function R as a clockwise rotation of the two ropes by 90.

16 R

We can deﬁne T based on its generators, T and R.LetA = T,R .Theelements { } of A satisfy the following relations:

1. R R = e R R

R R

1 1 2. R T R T = T R R T R T

T R T R

17 1 1 T R

1 1 R T

The set of generators A = T,R under the operation of composition with the { } two relations listed above form a group. Any combination of T and R produces a tangle of the two ropes and is thus a binary operation. Associativity is satisﬁed by

Theorem 2.1.1 because our set is formed by composing the functions T and R. T includes the identity elements R R and R T R T R T ,asshownabove.The inverse of T is R T R T R and the inverse of R is R.Thus,weseethatthe set T satisﬁes all the conditions for the deﬁnition of a group as characterized by its generators and relations.

We can see how the universal property holds for T in the following way. Let

A = ⇢, .Theni(⇢)=⇢ and i()=. F [A]isthenthefreegrouponthissetA. { } The function f is deﬁned as f(⇢)=R and f()=T ,andthefunctionf¯ is deﬁned as f¯(⇢)=R and f¯()=T where R R = e and R T R T R T = e. i A F [A]

f¯ f T

2.2.2 Group Action

Agroupactiondescribeshowagroupactsuponagivenset.Thenotionofan action is a useful tool for understanding and studying the structure of a given group.

18 Deﬁnition 16 (See [9]). A group action of a group G on a set A is a function from G A to A (written as g a,forallg G and a A)satisfyingthefollowing ⇥ · 2 2 properties:

1. g (g a)=(g g ) a,forallg ,g G, a A,and 1 · 2 · 1 · 2 · 1 2 2 2

2. e a = a,foralla A. · 2

The activities presented in this thesis provide the opportunity to learn about group structures of D4 and T through group actions. The contra dancing activities allow us to “see” D4 acting upon the set of the conﬁguration of dancers found in contra dance. The rational tangle activities allow us to “see” T acting upon the set of the conﬁgurations of two ropes and their tangles.

19 Chapter 3: Pre-Kindergarten to Third Grade

This chapter presents annotated activities based on rational tangles and square dancing for pre-kindergarten through third grade.

3.1 Rational Tangles

The following rational tangle activities provide an environment for students to practice counting, addition, and properties of addition. The twisting action in the rope allows students to visualize how their counting number directly correlates to the amount of tangle in the rope. The students also work through the process of combining two sets of tangles in order to model addition and gain an intuitive understanding of the commutative property of addition. The untwist move introduced in this activity is the inverse of a twist. Understanding that addition and subtraction are inverse operations is a foundational step towards understanding inverse functions.

3.1.1 All Tangled Up!

Set up

20 like the picture below with students A, B, C, and D. The shapes represent placeholders for the four positions. The ﬁfth student is the caller and stands at the front of the group.

A D

B C

Caller

Twist

The ﬁrst movement we will learn is a twist.Studentsinthestarandcirclepositions switch places with the student in the circle position lifting their rope over the other student.

A C

B D

Caller

Untwist

Our second move is called the untwist.Startintheoriginalplacement:

A D

B C

Caller

21 Students in the diamond and rectangle positions switch places with the student in the diamond position lifting their rope over the other student.

B D

A C

Caller

Counting Game

Roll two dice. Count up to the total number that you rolled by performing twists.

The total number is the sum of the values from the two dice. As you count you must say the numbers out loud (as a group) and do a twist for each number.

1) How many total twists have we created?

Answer. The total amount of twists is the total number from the two dice. Students should perform these twists as a one-to-one correspondence with the dots that show up on the dice. They should see that the total number of dots on the two dice is the same as the total number of twists.

Once you have counted up to the total number on the dice, you must untangle the rope! We do this by counting up to the same number and doing an untwist for each number we count. Be sure to count out loud as a group!

2) What happens to the ropes at the end of the game?

Answer. The ropes should be completely untangled. Students may answer by de- scribing what the ropes look like with words similar to ”undone”, ”separate”, ”not

22 connected”, or by drawing a picture of the two ropes with making sure they are not touching.

Continue playing by rolling the dice again and switching the students’ positions around on the ropes.

CCSS.MATH.CONTENT.K.CC.A.1

Count to 100 by ones and by tens.

In this activity, students have the opportunity to practice counting up to 12 by using two dice in the Counting Game. More advanced students can use more than two dice for more of a challenge.

CCSS.MATH.CONTENT.K.CC.B.4.B

Understand that the last number name said tells the number of objects counted.

The number of objects is the same regardless of their arrangement or the order

in which they were counted.

Students will see that the last number name said tells the number of twists that are in the ropes. This is conﬁrmed when they must do that same number of untwists to undo the ropes.

CCSS.MATH.CONTENT.K.CC.B.4.C

Understand that each successive number name refers to a quantity that is one

larger.

Each successive number name will be represented as another twist in the ropes.

This visualization can help students see that more twists in the rope represents a larger amount. Advanced students can recognize that a larger amount of untwists

23 corresponds to less twists in the rope since we are taking away (or subtracting) that amount of twists.

CCSS.MATH.CONTENT.K.OA.A.1

Represent addition and subtraction with objects, ﬁngers, mental images, draw-

ings, sounds (e.g., claps), acting out situations, verbal explanations, expressions,

or equations.

In this activity addition is represented by the twisting action while subtraction is represented by untwist.

24 3.1.2 Adding up Tangles

Addition Game 1

Partner up with another group of 5 students and stand side by side in the original placements.

Group 1 Group 2

A D A D

B C B C

Caller Caller

Each group rolls two dice separately. Count up to the number (out loud) and do atwistforeachnumberthatyoucount.

Now we are going to combine the two groups’ ropes. The student in the star position from group 1 ties their rope together with the student in the rectangle position in group 2. Similarly, the student in the circle position from group 1 ties their rope together with the student in the diamond position in group 2. Now we have one long tangle:

Group 1 Group 2

A knot D

......

B knot C

Caller Caller

25 1) How many total twists are in the combined ropes?

Answer. The total amount of twists is the amount from group 1’s tangle plus the amount from group 2.

2) How many untwists can we do to fully untangle the ropes?

Answer. We need to do the same amount of untwists as there are twists. So, the answer is the same value from number 1. Students should check their answer here by doing that amount of untwists to see if the combined ropes become fully untangled.

Addition Game 2

Reset all the ropes and tangles.

Group 1 twist up to 5 total twists. Group 2 twist up to 8 total twists.

3) How many total twists will we have if we combine the tangles with group 1 on the left and group 2 on the right?

Answer. There will be 5 + 8 = 13 total twists.

4) How many total twists will we have if we combine the tangles with group 2 on the left and group 1 on the right?

Answer. There will be 8 + 5 = 13 total twists.

5) Are there any similarities to the answers for questions 3 and 4?

Answer. Yes; the answers are the same. Students should check their answers to these 3 questions by setting group 1 and group 2 side by side (group 1 on the left, then group 1 on the right). Ask the students if their answer to this question would change if the numbers were di↵erent.

26 CCSS.MATH.CONTENT.1.OA.B.3

Apply properties of operations as strategies to add and subtract. Examples: If

8+3=11 is known, then 3+8=11 is also known. (Commutative property of

addition.) To add 2+6+4, the second two numbers can be added to make a ten,

so 2+6+4=2+10=12. (Associative property of addition.)

Students use the commutative property of addition when deciding the total number of twists when combining two groups in di↵erent orders.

CCSS.MATH.CONTENT.2.OA.B.2

Fluently add and subtract within 20 using mental strategies. By end of Grade

2, know from memory all sums of two one-digit numbers.

Advanced students should be more ﬂuent when counting up the total number of twists.

27 3.2 Contra Dancing

The following activity teaches students a few basic contra dancing moves that provide an environment for them to model the shape of a square and its symmetries.

The questions based on the dance moves are building students’ intuition of how afunction(dancemove)mapsaset(4dancers,initialposition)toanotherset(4 dancers, ending position).

3.2.1 Dancing Around a Square

Set up

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Assign four group members a letter: A, B, C, and D. The ﬁfth memeber is the caller.

Stand in a square formation like the picture below. Everyone should face the middle. The shapes represent the placeholders of the four positions. The caller stands at the front of the group.

A D middle B C

Caller

We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

28 A D B A

B C C D

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

A D C B

B C D A

Caller Caller

3. Chain

The student in the star position switches places with the student in the diamond

position.

A D A B

B C D C

Caller Caller

Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following two dances without stopping between the moves.

29 1. Quarter Circle left, Chain, Half Circle Left

2. Chain, Quarter Circle Left, Half Circle Left

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two dances (use three moves for each):

1. Dance 1:

2. Dance 2:

Answer. Any three moves are valid.

Let’s practice writing down our positions.

Start in our original position:

A D

B C

Caller

2) Dance a chain and then a half circle left.Pausewhenyouﬁnishandﬁllinthe blank circles to indicate your ending position.

30 Answer. C D

B A Students may use the letters A, B, C, and D, or they may ﬁll in the circles with the students’ names.

Go back to your original position.

A D

B C

Caller

3) Dance a half circle left and then a half circle left.Pausewhenyouﬁnishand

ﬁll in the blank circles to indicate your ending position.

Answer. A D

B C

What do you notice about the ending position compared to the beginning position?

We end up in the same place! Can you do it again with another dance move?

4) List a sequence of two dance moves (other than half circle left) that will dance you back to the original position.

31 Answer. Chain, Chain

Solve these problems!

For each of the following, always begin the question by starting in our original position:

A D

B C

Caller

5) What single dance move should we do to end up in the following position?

C B

D A

Answer. Half Circle Left

6) What two dance moves should we do to end up in the following position?

D A

C B

Answer. Chain, Quarter Circle Left

32 7) What position do we end up in if we do 9 chains in a row? Dance this out and then ﬁll in the circles.

Answer. A B

D C

CCSS.MATH.CONTENT.K.G.B.5

Model shapes in the world by building shapes from components (e.g., sticks and

clay balls) and drawing shapes.

Students model a square with 4 students throughout the whole activity.

33 Chapter 4: Fourth Grade to Ninth Grade

This chapter presents annotated activities based on rational tangles and contra dancing for the fourth through ninth grade levels.

4.1 Rational Tangles

The following rational tangle activity provides an opportunity for students to practice with fractions. Students must work with adding 1 to fractions and taking reciprocals in order to solve the questions. Part 2 of the activity applies the students’ fraction skills to help solve problems in a hands-on activity with untangling ropes.

4.1.1 Tangles & Fractions

Tangles & Fractions, Part 1

Introduction

We will give you a number. Use the rules listed below to bring the number down to zero!

Rules

m n 1. If the number is positive, take the negative reciprocal. ( ) n ! m

2. If the number is negative, add 1.

34 3. Continue Rules 1 and 2 until you have reached 0.

Example

2 Given number: 3

2 3 Use Rule 1: 3 ! 2 3 1 Use Rule 2: 2 ! 2 1 1 Use Rule 2: 2 ! 2 1 2 Use Rule 1: 2 ! 1 Use Rule 2: 2 1 ! Use Rule 2: 1 0 ! DONE!

Use the rules to bring the following numbers to zero.

1 1) 2

1 Answer. Given number: 2

1 1 Use Rule 2: 2 ! 2 1 2 Use Rule 1: 2 ! 1 Use Rule 2: 2 1 ! Use Rule 2: 1 0 !

4 2) 3

4 Answer. Given number: 3

4 3 Use Rule 1: 3 ! 4 3 1 Use Rule 2: 4 ! 4 1 4 Use Rule 1: 4 ! 1 Use Rule 2: 4 3 ! 35 Use Rule 2: 3 2 ! Use Rule 2: 2 1 ! Use Rule 2: 1 0 !

3 3) 4

3 Answer. Given number: 4

3 4 Use Rule 1: 4 ! 3 4 1 Use Rule 2: 3 ! 3 1 2 Use Rule 2: 3 ! 3 2 3 Use Rule 1: 3 ! 2 3 1 Use Rule 2: 2 ! 2 1 1 Use Rule 2: 2 ! 2 1 2 Use Rule 1: 2 ! 1 Use Rule 2: 2 1 ! Use Rule 2: 1 0 !

Tangles & Fractions, Part 2

We will apply our rules from part 1 to tangle and untangle ropes! We need four volunteers to hold up our two ropes in the front of the class. The rest of the class will help us decide how to tangle and untangle the ropes.

Tangle Rules

1. Original Position: Each student is represented by the letters A, B, C, and D in

the picture below. The lines represent the ropes and the shapes represent the

placeholders for the positions. The original position of the ropes is zero.

36 A D

B C

Class

We have two movements:

2. Twist:Thestudentinthestarpositionliftstheirropeupandoverthestudent

in the circle position as they trade places. Every twist adds one to our current

number.

A D A C Twist

B C B D

Class

3. Rotate:Everyonemovesoveroneplaceclockwise.Everytimewerotate,we

take the opposite reciprocal of our current number.

A D B A Rotate

B C C D

Class

Practice these moves without keeping track of numbers for now.

37 After the students are comfortable with the moves, take the ropes back to the original position

4) What number does this tangle represent?

Answer. This tangle is 0.

Use the ropes to create the following tangle:

T wist, T wist, Rotate, T wist

5) What number does this tangle represent?

1 Answer. This tangle is 2 .Itwillhelpiftheclasskeepscountofthetangleaftereach move.

We want to use the twist and rotate moves to get back to our original position.

How can we do this?

Remember our rules from part 1 about getting fractions to 0:

Rules

m n 1. If the number is positive, take the opposite reciprocal. ( ) n ! m

2. If the number is negative, add 1.

3. Continue Rules 1 and 2 until you have reached 0.

6) What sequence of twists and rotates should we use?

Answer. Rotate ( 2), Twist ( 1), Twist (0)

38 7) Write out a new list of of 5 twists or rotates.

Using our tangle rules, what number does this represent? •

What list of twists and rotates should we use to untangle the ropes? •

Answer. The list can be any combination of twists and rotates. The answers here will depend on the chosen list.

CCSS.MATH.CONTENT.5.NF.A.1

Add and subtract fractions with unlike denominators (including mixed numbers)

by replacing given fractions with equivalent fractions in such a way as to produce

an equivalent sum or di↵erence of fractions with like denominators. For example,

2/3+5/4=8/12 + 15/12 = 23/12. (In general, a/b + c/d =(ad + bc)/bd.)

Students use fraction addition throughout the activity in order to help untangle the ropes.

39 4.2 Contra Dancing

The following activity teaches students how to contra dance with the goal of students gaining an intuitive understanding of functions. The dance moves allow students to visualize the symmetries of a square and how they a↵ect the positions of the dancers when performed multiple times. The activity can also lead towards discussion of inverse and identity functions.

4.2.1 Contra Dance Functions

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

Stand in a square formation like the picture below. Everyone should face the middle. The shapes represent the placeholders for the position of the dancers and the caller stands at the front of the group.

L1 F2 middle

F1 L2

Caller

40 We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

3. Chain

The student in the star position switches places with the student in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

41 Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following without stopping between the moves.

1. Quarter Circle left, Chain, Half Circle Left, Chain

2. Chain, Quarter Circle Left, Half Circle Left, Chain

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two dances (using at least 3 moves for each):

1. Dance 1:

2. Dance 2:

Answer. Any list of moves is valid.

2) Start in our original placement...

L1 F2

F1 L2

Caller

...and dance the following list of moves.

half circle left, chain, quarter circle left, chain, chain

Pause after you have ﬁnished the whole dance and ﬁll in the blank circles to demonstrate your ending position.

42 Caller

Answer. F1 L2

L1 F2

Caller Functions

A function is a rule that assigns to each input exactly one output. We can think of our contra dance moves as functions. Notice that if we start in our original position and then dance a quarter circle left, we end up in a new formation. Our original formation is the input and the new formation is the output.

Start in the original formation.

3) What is the output if we perform a chain?

Answer. The output is the ending formation:

L1 F1

F2 L2

Caller

Number of moves

Let’s explore some cool properties of our functions (dance moves!).

43 Start in the original position. Notice that if we dance chain two times in a row, we get back to our original position. So, 2 is the least number of times that we must dance chain in order to get back to the beginning.

How many times must we perform the following functions to get back to the original position (when starting in the original position)? Fill in the table with your answers. Dance these functions with your group to ﬁnd the answers!

Dance Move Least number of times to get back to beginning Quarter Circle Left Half Circle Left Chain 2

Dance Move Least number of times to get back to beginning Quarter Circle Left 4 Answer. Half Circle Left 2 Chain 2 These answers can lead into a discussion of what other number of times we could dance each move to get back to the beginning. For example, the chain move will take the formation back to the original position whenever it is danced 2n times, where n is any positive integer.

Reset to our original position. Dancing any sequence (of any length) of our moves, how many di↵erent ending positions can you get to?

Answer. There are 8 possible distinct positions.

CCSS.MATH.CONTENT.8.F.A.1

Understand that a function is a rule that assigns to each input exactly one output.

44 The graph of a function is the set of ordered pairs consisting of an input and the

corresponding output.

Students work with contra dance moves as functions and can see that the input is a single formation and the output is also a formation.

CCSS.MATH.CONTENT.8.G.A.1

Verify experimentally the properties of rotations, reﬂections, and translations:

Students experiment with rotations and reﬂections in relation to the square while performing the dance moves.

CCSS.MATH.CONTENT.8.G.A.1.A

Lines are taken to lines, and line segments to line segments of the same length.

CCSS.MATH.CONTENT.8.G.A.1.B

Angles are taken to angles of the same measure.

CCSS.MATH.CONTENT.8.G.A.1.C

Parallel lines are taken to parallel lines.

The above properties are used when students must keep their square formation as they move in and out of dance moves.

45 Chapter 5: Seventh to Twelfth Grade

This chapter presents annotated activities based on rational tangles and contra dancing for the seventh through twelfth grade levels.

5.1 Rational Tangles

The following rational tangle activity motivates students to work with functions and functional notation with a hands-on activity. Students are challenged in this activity to create functions that correctly describe the actions happening with the ropes. They also work with inverses and identities for which they must determine how to check if their answers are correct. Perseverance will be key as students learn the value of trial and error to solve problems.

5.1.1 Tangles & Functions

Setup

The class needs four volunteers to hold up the two ropes in front of the class. The two ropes should be held up parallel to the classroom as in the picture below with students A, B, C, and D. The shapes represent placeholders for the position of the students.

46 A D

B C

Class

We will refer to this position as the 0 position. Note that the 0 position is deﬁned by the position of the ropes rather than the students. Thus, any position in which the ropes are parallel to the class and untangled represents a 0 position.

Twist

The ﬁrst move we will learn is a twist.Fromthe0position,thestudentinthe circle position lifts their end of the rope up and over the student in the star position as they switch places. For example:

A D A C Twist

B C B D

Class

Two twists in a row:

A D A D Twist, Twist

B C B C

Class

47 Rotate

The next move we will learn is a rotate.Thefourstudentsholdingtheropeswill rotate 90 clockwise.

Starting in the 0 position,

A D

B C

Class

one rotate will move the students and ropes around as shown in the below picture.

B A

C D

Class

One more rotate would bring the formation back to the 0 position.

C B

D A

Class

We tangle up the two ropes by using the two moves, twist and rotate, in any order and with repetition. Practice these two moves by having the class shout out commands of twist or rotate.

48 Inverse

Answer the following questions by experimenting with the ropes.

1) What is the inverse of twist?

1 Answer. t = r t r t r,orrotate,twist,rotate,twist,rotate.Studentsmight answer rotate, twist. The following directions should guide them away from this answer.

Denote the string of moves that you answered as the inverse of twist as ?.Check your answer by starting in the 0 position and performing ?, ?,twist,twist.

2) If ? is truly the inverse of twist, what position should the ropes end in?

Answer. The 0 position.

3) What is the inverse of rotate?

1 Answer. r = r,orrotate.

Identity

Use your answers from the above inverse questions to answer the following.

4) List two identities using twists and rotates:

49 Answer. 1. r t r t r t 2. r r

Functions

Every twist adds 1 to our current tangle number. For example, if we start in the

0 position

A D

B C

Class

and twist two times,

A D

B C

Class

our tangle number is 2.

Denote the twist function as t(x).

5) How should we deﬁne t(x)? t(x)=

Answer. t(x)=x +1

Denote the rotate function as r(x). Deﬁning r(x)isnotstraightforward.Experi- ment with the twist and rotate moves together in order to ﬁgure out the e↵ect rotate has on the tangle number.

50 6) How should we deﬁne r(x)? r(x)=

1 Answer. r(x)= x

Untangle

Read o↵the following list of twists and rotates for the four volunteers to do with the ropes.

Twist, Twist, Twist, Rotate, Twist, Rotate, Twist, Twist, Twist, Twist, Twist,

Rotate, Twist

7) What tangle number does this represent?

5 Answer. 7

8) As a class determine what moves we should do in order to undo the tangle (to end in the 0 position).

Answer. rotate, twist, twist, rotate, twist, twist, rotate, twist, twist, twist. The class should start to realize that there is a pattern to these choices: rotate when the number is positive and twist when it is negative.

CCSS.MATH.CONTENT.HSA.CED.A.2

Create equations in two or more variables to represent relationships between

quantities; graph equations on coordinate axes with labels and scales.

Students create the twist and rotate equations.

51 CCSS.MATH.CONTENT.HSF.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and inter-

pret statements that use function notation in terms of a context.

Students use function notation for the twists and rotates.

CCSS.MATH.CONTENT.HSF.BF.A.1

Write a function that describes a relationship between two quantities.

The twist and rotate functions describe the relationship between the current tangle and ending tangle.

CCSS.MATH.CONTENT.HSF.BF.B.4

Find inverse functions.

The inverse of twist and rotate are deﬁned.

CCSS.MATH.CONTENT.HSF.BF.B.4.B

Verify by composition that one function is the inverse of another.

Students verify their inverse functions in the activity.

52 5.2 Contra Dancing

The following activities allow students to explore functions through contra dance activities. The goal of the first activity is to introduce contra dancing in order to explore the symmetries of the square. Activity 2 introduces formal functional notation of the dance figures and provides practice for functional composition. Students then work with finding inverse functions and can see how these a↵ect their dancing positions.

5.2.1 Math-y Dance

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

Stand in a square formation like the picture below. Everyone should face the middle, and the caller stands at the front of the group. The shapes represent placeholders of the positions of the dancers.

53 L1 F2

F1 L2

Caller

We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

3. Chain

54 The student in the star position switches places with the student in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following without stopping in between the moves.

1. Quarter Circle left, Chain, Half Circle Left, Chain

2. Chain, Quarter Circle Left, Half Circle Left, Chain

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two:

Dance 1: •

Dance 2: •

Answer. Any list of moves is valid.

Practice our dance moves in your group to help answer the following questions.

Questions

2) What happens when you perform multiple half circles in a row? multiple chains?

55 Answer. Two half circles in a row brings the formation back to the original placement. The same is true for dancing two chains in a row.

3) How many times do you have to do a certain move to get back to your original placement (when starting in your original placement)? Fill in the table below. Dance Move Number of times to get back to original placement Quarter Circle Left Half Circle Left Chain

Answer. Dance Move Number of times to get back to original placement Quarter Circle Left 4 Half Circle Left 2 Chain 2 These answers can lead into a discussion of what other number of times we could dance each move to get back to the beginning. For example, the chain move will take the formation back to the original position whenever it is danced 2n times, where n is any positive integer.

4) Stand in your original placement. Dance a Quarter Circle Left. Can you dance any move or string of moves to undo the quarter circle left?

Answer. We can dance a half circle left and quarter circle left.

5) If we consider the “undo the quarter circle left” as a single move, what might you call the move?

Answer. Three quarter circle left or quarter circle right

56 6) Begin in your original placement. Can you make a string of 10 moves that brings you back to the original placement? Dance this sequence without any pauses between moves and write the sequence below.

Answer. There are many correct answers. For example: quarter circle left, chain, half circle left, chain, half circle left, quarter circle left, chain, chain, quarter circle left, quarter circle left

CCSS.MATH.CONTENT.HSF.IF.A.1

Understand that a function from one set (called the domain) to another set

(called the range) assigns to each element of the domain exactly one element of

the range. If f is a function and x is an element of its domain, then f(x) denotes

the output of f corresponding to the input x. The graph of f is the graph of the

equation y = f(x).

Students work with the inputs and outputs of the dancing functions.

57 5.2.2 Dancing Functions

Review

Separate into groups of ﬁve with new classmates. Let’s review our contra dance moves. We start with the original set up:

L1 F2

F1 L2

Caller

We have a Quarter Circle left:

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

1) Do you remember Half Circle Left? Fill in the circles below:

L1 F2

F1 L2

Caller Caller

Answer. L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

58 2) Do you remember Chain? Fill in the circles below:

L1 F2

F1 L2

Caller Caller

Answer. L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller 3) We also created a new move called three quarters circle left. Fill in the circles below:

L1 F2

F1 L2

Caller Caller

Answer. L1 F2 F2 L2

F1 L2 L1 F1

Caller Caller 4) What would a whole circle left move be deﬁned as? Fill in the circles below:

L1 F2

F1 L2

Caller Caller

59 Answer. L1 F2 L1 F2

F1 L2 F1 L2

Caller Caller Let’s introduce some notation so that we can talk about these moves more clearly.

Notation:

1. R1:QuarterCircleLeft

2. R2: Half Circle Left

3. R3:ThreeQuartersCircleLeft

4. R0: Whole Circle Left

5. Ch:Chain

The letter R represents rotate.So,R1 means to rotate once to the left.

5) What does R3 mean?

Answer. R3 means to rotate three times to the left.

We will now consider our dance moves as functions. For example:

L1 F2 F1 L1 R = 1 F L L F 1 2 2 2

Caller Caller

6) Now try the function for Chain:

60 L1 F2 =

Ch F1 L2

Caller Caller

L1 F2 L1 F1 Answer. =

Ch F1 L2 F2 L2

Caller Caller 7) ...and for Three Quarters Circle Left:

L1 F2 R = 3 F L 1 2

Caller Caller

L1 F2 F2 L2 Answer. R = 3 F L L F 1 2 1 1

Caller Caller 8) What if we started in a di↵erent position and then danced a Half Circle Left?

F2 F1 R = 2 L L 2 1

Caller Caller

61 F2 F1 L1 L2 Answer. R = 2 L L F F 2 1 1 2

Caller Caller Recall that in part 1 we practiced the dance:

Quarter Circle left, Chain, Half Circle Left, Chain

How can we write this whole dance down as a function? We use functional composition!

For our example above, we can write:

Ch(R2(Ch(R1(original position))))

9) Why does it look like the list is backwards?

Answer. With composition of functions we start with the inner most function.

10) Write out the composition of functions for the following dances.

Half Circle Left, Quarter Circle Left, Chain, Whole Circle Left •

Chain, Chain, Chain, Quarter Circle left •

Answer. R (Ch(R (R (original position)))) • 0 1 2

R (Ch(Ch(Ch(original position)))) • 1

62 In your groups, practice the above two dances.

11) Fill in the circles for the following composition of functions.

F2 F1 R = 2 L L Ch 2 1

Caller Caller

F2 F1 L1 F1 Answer. R = Ch 2 L2 L1 L2 F2 Caller Caller

F2 F1 R R = 2 2 L L 2 1

Caller Caller

F2 F1 F2 F1 Answer. R R = 2 2 L2 L1 L2 L1 Caller Caller

12) What happened when we took the composition of R2 and R2?

Answer. We end up back in the starting position.

We call these functions identity functions! They are our “do nothing” functions.

63 13) Are there any other “do nothing” functions?

Answer. Afewexamples:R , Ch Ch, R R 0 1 3

Recall the Quarter Circle Left function, R1.

L1 F2 F1 L1 R = 1 F L L F 1 2 2 2

Caller Caller

14) What function (a single function) would we have to apply to the output in order to return to the input (the original position)?

Answer. R3

We call this type of function an inverse function! For example, we found that the inverse of R1 is R3.Wenotatethisas:

1 R1 = R3

15) What is the inverse of R0?

1 Answer. R0 = R0

16) What is the inverse of Ch?

1 Answer. Ch = Ch

17) What is the inverse of R3?

64 1 Answer. R3 = R1

CCSS.MATH.CONTENT.HSF.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and inter-

pret statements that use function notation in terms of a context.

Functional notation is used throughout the activity for all of the contra dance moves.

CCSS.MATH.CONTENT.HSF.BF.A.1

Write a function that describes a relationship between two quantities.

Students deﬁne the function R3.

CCSS.MATH.CONTENT.HSF.BF.A.1.C

Compose functions. For example, if T (y)isthetemperatureintheatmosphere

as a function of height, and h(t)istheheightofaweatherballoonasafunction

of time, then T (h(t)) is the temperature at the location of the weather balloon

as a function of time.

Composition of functions is used to notate multiple dance moves in a row.

CCSS.MATH.CONTENT.HSF.BF.B.4

Find inverse functions.

The inverse of all dance functions are deﬁned in the activity.

65 Chapter 6: College

This chapter presents annotated activities based on rational tangles and contra dancing for the college level.

6.1 Rational Tangles

The following rational tangle activity will allow students to work with ropes in order to visualize a free group on two generators and two relations. The activity starts by working students through the process of creating the twist and rotate functions and then gives an opportunity for them to prove that their identity and inverse functions are correct. This activity can lead into a more detailed discussion about the group structure.

6.1.1 Tangles

Setup

The class needs four volunteers to hold up the two ropes in front of the class. The two ropes should be help up parallel to the classroom as in the picture below with students A, B, C, and D.

66 A D

B C

Class

Twist

The ﬁrst move we will learn is a twist.Fromthe0position,thestudentinthe circle position lifts their end of the rope up and over the student in the star position as they switch places. For example:

A D A C Twist

B C B D

Class

Two twists in a row:

A D A D Twist, Twist

B C B C

Class

67 Rotate

The next move we will learn is a rotate.Thefourstudentsholdingtheropeswill rotate 90 clockwise.

Starting in the 0 position,

A D

B C

Class

one rotate will move the students and ropes around as shown in the below picture.

B A

C D

Class

One more rotate would bring the formation back to the 0 position.

C B

D A

Class

We tangle up the two ropes by using the two moves, twist and rotate, in any order and with repetition. Practice these two moves by having the class shout out commands of twist or rotate.

68 Inverse

Answer the following questions by experimenting with the ropes.

1) What is the inverse of twist?

1 Answer. t = r t r t r,orrotate,twist,rotate,twist,rotate.Studentsmight answer rotate, twist. The following directions should guide them away from this answer.

Denote the string of moves that you answered as the inverse of twist as ?.Check your answer by starting in the 0 position and performing ?, ?,twist,twist.

2) If ? is truly the inverse of twist, what position should the ropes end in?

Answer. The 0 position.

3) What is the inverse of rotate?

1 Answer. r = r,orrotate.

Identity

Use your answers from the above inverse questions to answer the following.

4) List two identities using twists and rotates:

69 Answer. 1. r t r t r t 2. r r

Functions

Every twist adds 1 to our current tangle number. For example, if we start in the

0 position

A D

B C

Class

and twist two times,

A D

B C

Class

our tangle number is 2.

Denote the twist function as t(x).

5) How should we deﬁne t(x)? t(x)=

Answer. t(x)=x +1

Denote the rotate function as r(x). Deﬁning r(x)isnotstraightforward.Experi- ment with the twist and rotate moves together in order to ﬁgure out the e↵ect rotate has on the tangle number.

70 6) How should we deﬁne r(x)? r(x)=

1 Answer. r(x)= x

Untangle

Read o↵the following list of twists and rotates for the four volunteers to do with the ropes.

Twist, Twist, Twist, Rotate, Twist, Rotate, Twist, Twist, Twist, Twist, Twist,

Rotate, Twist

7) What tangle number does this represent?

5 Answer. 7

8) As a class determine what moves we should do in order to undo the tangle (to end in the 0 position).

9) Consider all the combinations of r and t as a set. What mathematical object can describe this set?

Answer. This set can be described as a group. Speciﬁcally, this describes the free group on two generators with two relations. The class should give reasoning as to why this is a group. This discussion should include all four axioms of a group.

71 Tangles: Extension

Use the functions deﬁned in questions 5 and 6 to answer the following.

10) Prove that r t r t r t and r r are identities.

Answer. r t r t r t(x)=r t r t r(x +1) 1 = r t r t( ) x +1 1 = r t r( +1) x +1 x = r t r( ) x +1 x 1 = r t( ) x x 1 = r( +1) x 1 = r( ) x x = 1 = x 1 r r(x)=r( ) x x = 1 = x

1 1 11) Prove that t (x) = t r. Why might it seem like t (x)=t r when using the 6 ropes?

Answer. t r t(x)=t r(x +1) 1 = t( ) x +1 1 = +1 x +1 x = x +1 72 1 It might seems like t (x)=t r when using the ropes from the 0 position because t r t(0) = 0 = 0 =0. 0+1 1

CCSS.MATH.PRACTICE.MP1

Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2

Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3

Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4

Model with mathematics.

CCSS.MATH.PRACTICE.MP6

Attend to precision.

CCSS.MATH.PRACTICE.MP7

Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8

Look for and express regularity in repeated reasoning.

Students use the above mathematical practices as they work though trial and error to answer the questions in this activity.

73 6.2 Contra Dancing

The following activities introduce students to contra dancing for hands-on learning of functions and their properties. Students will explore ideas such as identities and inverse functions that lead into a discussion about groups. Throughout activities

1and2,studentsareworkingwiththestructureofD4 and work with the idea of generators and relations for this group. The last activity leads students towards the connection of a free group to the group described by the dancing functions.

6.2.1 Properties of Contra Dance Figures

Introduction

Today we will be learning how to contra dance!

Activity

Separate into groups of ﬁve. Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

Stand in a square formation like the picture below. The shapes represent the four corners of a square and are placeholders for the dancers’ positions. Everyone should face the middle. The caller stands in the same position throughout the dance.

74 L1 F2

middle

F1 L2

Caller

We will learn a few contra dance ﬁgures. Practice each move to get the hang of it as the caller calls out the ﬁgure’s name.

1. Quarter Circle Left

The square formation rotates 90 clockwise.

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Quarter Circle Right

The square formation rotates 90 counterclockwise.

L1 F2 F2 L2

F1 L2 L1 F1

Caller Caller

3. Half Circle Left

The square formation rotates 180 clockwise.

75 L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

4. Half Circle Right

The square formation rotates 180 counterclockwise.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

5. Chain

The dancer in the star position switches places with the dancer in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

6. Swing on Side

The dancers in the square and star positions switch places, and the dancers in

the diamond and circle positions switch places.

76 L1 F2 F2 L1

F1 L2 L2 F1

Caller Caller

7. California Twirl

The dancers in the square and diamond positions switch places, and the dancers

in the star and circle positions switch places.

L1 F2 F1 L2

F1 L2 L1 F2

Caller Caller

Now that you’ve practiced each of these dance ﬁgures separately, let’s string together the moves to create a dance. Have the caller call out the following lists and try to dance the ﬁgures without stopping in between each move.

1. Quarter Circle left, Chain, Half Circle Left, Swing on Side, Half Circle Right

2. California Twirl, Quarter Circle right, Half Circle Left, Chain, Swing on Side

Now you’re contra dancing!

In your group, practice creating new strings of moves. Write down your favorite two:

1) Dance 1:

77 2) Dance 2:

Properties of the Dance Figures

We can think of our dance ﬁgures as functions that describe the relationship between the initial and ending dance formation. Answer the following questions about these functions by practicing our dance ﬁgures in your group.

3) What functional operation can be used to notate the action of dancing multiple

ﬁgures in a row?

Answer. Functional composition.

4) What is the inverse of each dance ﬁgure? Fill in the table below. Note that there may be more than one correct answer. Dance Figure Inverse Quarter Circle Left Quarter Circle Right Half Circle Left Half Circle Right Chain Swing on Side California Twirl

78 Answer. Dance Figure Inverse Quarter Circle Left Quarter Circle Right Quarter Circle Right Quarter Circle Left Half Circle Left Half Circle Left Half Circle Right Half Circle Right Chain Chain Swing on Side Swing on Side California Twirl California Twirl

5) Can you describe any identity functions from our set of dance ﬁgures?

Answer. There are many correct answers. A few examples are:

(Quarter Circle Left) (Quarter Circle Right) •

(Chain) (Chain) •

(Half Circle Left) (Half Circle Right) •

6) Are there any combinations of dance ﬁgures that are commutative?

Answer. Yes. There are many examples, such as:

(Quarter Circle Left) (Half Circle Left) •

(Chain) (Chain) (Chain) •

(Half Circle Left) (Half Circle Right) •

7) Are there any combinations of dance ﬁgures that are not commutative?

79 Answer. Yes. There are many examples, such as:

(Chain) (Quarter Circle Left) •

(Quarter Circle Right) (California Twirl) •

8) Are there any combinations of dance ﬁgures that are associative?

Answer. Yes. In fact functional composition is always associative.

9) Are there any combinations of dance ﬁgures that are not associative?

Answer. No. Students may argue from the fact that functional composition is always associative.

10) What mathematical object describes our set of dance ﬁgures (functions)?

Answer. Our set of dance moves under the operation of composition form a group.

80 6.2.2 Generators and Relations

In the ﬁrst activity we discovered that our set of dance moves form a non- commutative group. Our goal in this activity is to describe this group in terms of generators and relations.

Generators

For the following questions you may notate quarter circle left as r and chain as c.

1) Consider the e↵ect each dance ﬁgure has on the initial position. Using only Quarter

Circle Left and Chain, can you redeﬁne the rest of the moves (using composition) so that the e↵ect on the formation is the same? Fill in the table below. Dance Figure New Description Quarter Circle Right Half Circle Left Half Circle Right Swing on Side California Twirl

Answer. Dance Figure New Description Quarter Circle Right r r r Half Circle Left r r Half Circle Right r r Swing on Side r c California Twirl r r r c Thus, quarter circle left (r)andchain(c) are generators of this group.

Relations

Equations in a group that are satisﬁed by the generators are called relations.

2) List three expressions that equal the identity, e,byusingonlyr and c.

81 1. = e

2. = e

3. = e

Answer. 1. c c = e

2. r r r r = e

3. c r c r = e

3) How many possible ending dance formations are there?

Answer. There are 8 possible positions.

4) List all the elements of this group by only using the functions r and c.

Answer.

r, r2,r3,r4 = e, c, r c, r2 c, r3 c

5) Fill out the multiplication table below.

e r r2 r3 c rc r2c r3c e r r2 r3 c rc r2c r3c

82 e r r2 r3 c rc r2c r3c e e r r2 r3 c rc r2c r3c r r r2 r3 e rc r2c r3c c r2 r2 r3 e r r2c r3c c rc 3 3 2 3 2 Answer. r r e r r r c c rc r c c c r3c r2c rc e r3 r2 r rc rc c r3c r2c r e r3 r2 r2c r2c rc c r3c r2 r e r3 r3c r3c r2c rc c r3 r2 r e

83 6.2.3 Free Group

A free group is a group that is free of relations. We discovered in activity 2 that our group has 3 relations, and thus is not a free group. At the end of this activity we will see the connections between our group described in activities 1 and 2, and a free group.

Elements of the Free Group on 2 Generators

Let A = a, b .Wecancreatewords on the set A by concatenating elements of { } 1 1 A and their inverses, denoted by a and b ,inanyorder.Forexample,

1 1 1 1 aaabbb b b ba is a word.

1) Create ﬁve new words on A.

1 1 Answer. Answers can be any combination of a, b, a ,andb .

A reduced word is one in which we simplify the word. Any instances of an element and its inverse concatenated together must be removed. For example, 1 1 1 1 1 1 aaabbb b b ba a = aaabb a

1 = aaaa

= aa

84 is a reduced word.

2) Reduce your words from Exercise 1.

Answer. The answers will vary based on the words created in Exercise 1.

The free group on the set A is the set of all possible reduced words on A.

Connections to Contra Dance

The universal property of free groups gives a nice connection between groups and free groups:

Let F [A]bethefreegrouponasetA and i : A F [A]theinclusionfunction such that i(a)=a.IfG is a group and f : A !G amapofsets,thenthere exists a unique homomorphism of groups f¯ : F [!A] G such that f¯ i = f. !

To apply this property to our group from activity 2, we deﬁne the functions i, f, and f¯ in the following manner:

85 i(a)=a

i(b)=b

f(a)=r

f(b)=c

f¯(a)=r

f¯(b)=c where r4 = s2 =(rs)2 = e.

The homomorphism of groups relates the structure of the group and free group.

3) For each of the elements in the free group below, write the corresponding element in the group from activity 2, as described by the function f¯ from above. Free group element Dance group element 1 ababa b bbbbbb 1 1 a a b 1 1 bab a abbbba

Answer. Free group element Dance group element 1 1 1 3 ababa b rcrcr c = r c = r c bbbbbb c6 = e 1 1 1 1 3 3 6 2 a a b r r c = r r c = r c = r c 1 1 1 1 3 3 2 aba b rcr c = rcr c = rcr rcr = rccr = rr = r abbbbba rcccccr = rcr = c

CCSS.MATH.PRACTICE.MP1

Make sense of problems and persevere in solving them.

86 CCSS.MATH.PRACTICE.MP2

Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3

Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4

Model with mathematics.

CCSS.MATH.PRACTICE.MP5

Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP6

Attend to precision.

CCSS.MATH.PRACTICE.MP7

Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8

Look for and express regularity in repeated reasoning.

Students us the above mathematical practices throughout the contra dance activities to reason and solve problems.

87 Chapter 7: Irrational Tangles

The rational tangle activities presented in this paper are based on the ideas of

John Conway’s demonstration of rational tangles [5]. As an extended topic to connect dance, art, and mathematics, we present the following chapter on irrational tangles.

7.1 Rational and Irrational Tangles

Deﬁnition 17. (see [1]) A tangle in a knot or link projection is a region in the projection plane surrounded by a circle such that the knot or link crosses the circle exactly four times.

A tangle is a portion of a knot or link that has four ends. Our activities illustrate these projections with four students holding the ends of two ropes.

Deﬁnition 18. (see [1]) A rational tangle is a tangle that can be constructed through the use of twists and rotates from the 0 tangle:

As demonstrated in our previous activities, rational tangles can be constructed from sequences of twists and rotates. These sequences also correlate to a given rational

1 number where T (x)=x +1andR(x)= x and x is the current tangle number.

88 Given a speciﬁc rational number, we can determine the sequence needed to tangle up to that value by using continued fractions.

Deﬁnition 19. (see [13]) A continued fraction is an expression of the form

b1 a1 + b2 a2 + b3 a + 3 a + 4 ··· where ai and bi may be any real or complex values and the expression may be either

ﬁnite or inﬁnite.

Every rational number can be written as a ﬁnite continued fraction [13]. If we write rational numbers in the form

1 a0 1 a1 1 a2 ···ak where ai are integers, the sequence of twists and rotates can be easily read o↵: ak twists, rotate, ...,rotate,a2 twists, rotate, a1 twists, rotate, a0 twists. This goal can

p be accomplished for a number q by using Euclid’s algorithm for ﬁnding the GCD of p and q.

4 Example 7.1.1. We can ﬁnd determine the continued fraction for 11 by ﬁrst determining gcd(4, 11): 4=1(11) 7 11 = 2(7) 3 7=3(3) 2 3=2(2) 1 2=2(1) 0 89 We can take each of the above equations and write equivalent statements: 4 7 =1 11 11 11 3 =2 7 7 7 2 =3 3 3 3 1 =2 2 2 2=2

Then starting at the ﬁrst equation we can make substitutions. 4 1 =1 11 11 /7 1 =1 2 3 7 1 =1 1 2 7 /3 1 =1 1 2 3 2 3 1 =1 1 2 1 3 3 /2 1 =1 1 2 3 1 2 1 2 So, we have 4 1 =1 11 1 2 3 1 2 1 2 4 Thus, the sequence to tangle up to the value of 11 is 2 twists, rotate, 2 twists, rotate, 3 twists, rotate, 2 twists, rotate, 1 twist. In function notation we see that

T R T 2 R T 3 R T 2 R T 2(0) = 4 ,andapictureofthetwistedropesis 11 displayed below.

90 If tangles can be rational, what does an irrational tangle look like? Since every irrational number can be written as an inﬁnite continued fraction, we can use that characterization to deﬁne an irrational tangle.

Example 7.1.2. The golden ratio is an irrational number with the following continued fraction: 1+p5 1 =1+ 2 1 1+ 1 1+ 1+ ··· Since the continued fraction is inﬁnite, we cannot read o↵a list of twists and rotates directly from the continued fraction to represent the golden ratio. Instead, we can use this information to create a list of approximations.

The following ﬁgures illustrate the process of determining the irrational tangle of

1+p5 the golden ratio, 2 . We can start with the following ﬁrst step:

1 5 1 1+ = =2 1 3 3 1+ 1+1 Thus, our ﬁrst step is the sequence: 3 twists, rotate, 2 twists.

91 Next, we use the following fraction

1 13 1 1+ = =2 1 8 1 1+ 3 1 3 1+ 1 1+ 1+1 which is represented by the sequence: 3 twists, rotate, 3 twists, rotate, 2 twists.

Next, we use the following fraction

1 34 1 1+ = =2 1 21 1 1+ 3 1 1 1+ 3 1 3 1+ 1 1+ 1 1+ 1+1 which is represented by the sequence: 3 twists, rotate, 3 twists, rotate, 3 twists, rotate, 2 twists.

92 This pattern continues in both the pictures and sequence. For example, see the next two ﬁgures if we were to continue our approximations.

Thus, we can deﬁne the sequence of twists and rotates that represent the golden ratio, 1+p5 ,asT 3RT 3R T 3RT 2 (if read from left to right). 2 ···

93 7.1.1 Connections to the Tangle Group

1 Recall that we deﬁned T (x)=x +1 and R(x)= x .ThefunctionsT and R generate our tangle group, T ,andwecancheckagainthatthefollowingtwo relations are satisﬁed.

1. R R = e 1 R R(x)=R( ) x 1 = 1 x = x 1. R T R T R T = e R T R T R T (x)=R T R T R(x +1) 1 = R T R T ( ) x +1 1 = R T R( +1) x +1 x = R T R( ) x +1 x 1 = R T ( ) x x 1 = R( +1) x 1 = R( ) x = x This group acts on the real numbers in a “dance” of sorts. Applying the functions

T and R to any rational number will result in another rational number, and the same is true for irrationals. However, the rational numbers’ “dance” is completely separate from the irrationals’ “dance”, meaning that any sequence of T and R applied to a rational number can never result in an irrational number, and vice versa.

94 Considering the structure of our tangle group, T ,wenotethatT is isomorphic to the group SL2(Z). The group SL2(Z)isdeﬁnedas

ab : a, b, c, d, Z, ad bc =1 cd 2 ⇢ where the following two matrices generate the group [4].

0 1 S = 10  11 T = 01 

95 Appendix A: Pre-Kindergarten to Third Grade Activities

A.1 All Tangled Up!

Set up

Split up into groups of 5 students. Each group needs 2 ropes. Line the ropes up side by side to that they are straight and do not cross. Four group members should hold on to the ends of the ropes and lift them o↵the ground. The group should look like the picture below with students A, B, C, and D. The shapes represent placeholders for the four positions. The ﬁfth student is the caller and stands at the front of the group.

A D

B C

Caller

Twist

The ﬁrst movement we will learn is a twist.Studentsinthestarandcirclepositions switch places with the student in the circle position lifting their rope over the other student.

96 A C

B D

Caller

Untwist

Our second move is called the untwist.Startintheoriginalplacement:

A D

B C

Caller

Students in the diamond and rectangle positions switch places with the student in the diamond position lifting their rope over the other student.

B D

A C

Caller

Counting Game

Roll two dice. Count up to the total number that you rolled by performing twists.

The total number is the sum of the values from the two dice. As you count you must say the numbers out loud (as a group) and do a twist for each number.

97 1) How many total twists have we created?

2) What happens to the ropes at the end of the game?

Continue playing by rolling the dice again and switching the students’ positions around on the ropes.

A.2 Adding up Tangles

Addition Game 1

Partner up with another group of 5 students and stand side by side in the original placements.

Group 1 Group 2

A D A D

B C B C

Caller Caller

Each group rolls two dice separately. Count up to the number (out loud) and do atwistforeachnumberthatyoucount.

Now we are going to combine the two groups’ ropes. The student in the star position from group 1 ties their rope together with the student in the rectangle position

98 in group 2. Similarly, the student in the circle position from group 1 ties their rope together with the student in the diamond position in group 2. Now we have one long tangle:

Group 1 Group 2

A knot D

......

B knot C

Caller Caller

1) How many total twists are in the combined ropes?

2) How many untwists can we do to fully untangle the ropes?

Addition Game 2

Reset all the ropes and tangles.

Group 1 twist up to 5 total twists. Group 2 twist up to 8 total twists.

3) How many total twists will we have if we combine the tangles with group 1 on the left and group 2 on the right?

4) How many total twists will we have if we combine the tangles with group 2 on the left and group 1 on the right?

5) Are there any similarities to the answers for questions 3 and 4?

99 A.3 Dancing Around a Square

Set up

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Assign four group members a letter: A, B, C, and D. The ﬁfth memeber is the caller.

Stand in a square formation like the picture below. Everyone should face the middle. The shapes represent the placeholders of the four positions. The caller stands at the front of the group.

A D middle B C

Caller

We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

A D B A

B C C D

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

100 A D C B

B C D A

Caller Caller

3. Chain

The student in the star position switches places with the student in the diamond

position.

A D A B

B C D C

Caller Caller

Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following two dances without stopping between the moves.

1. Quarter Circle left, Chain, Half Circle Left

2. Chain, Quarter Circle Left, Half Circle Left

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two dances (use three moves for each):

1. Dance 1:

101 2. Dance 2:

Let’s practice writing down our positions.

Start in our original position:

A D

B C

Caller

2) Dance a chain and then a half circle left.Pausewhenyouﬁnishandﬁllinthe blank circles to indicate your ending position.

Go back to your original position.

A D

B C

Caller

3) Dance a half circle left and then a half circle left.Pausewhenyouﬁnishand

ﬁll in the blank circles to indicate your ending position.

102 What do you notice about the ending position compared to the beginning position?

We end up in the same place! Can you do it again with another dance move?

4) List a sequence of two dance moves (other than half circle left) that will dance you back to the original position.

Solve these problems!

For each of the following, always begin the question by starting in our original position:

A D

B C

Caller

5) What single dance move should we do to end up in the following position?

C B

D A

6) What two dance moves should we do to end up in the following position?

D A

C B

103 7) What position do we end up in if we do 9 chains in a row? Dance this out and then ﬁll in the circles.

104 Appendix B: Fourth Grade to Ninth Grade Activities

B.1 Tangles & Fractions

Tangles & Fractions, Part 1

Introduction

We will give you a number. Use the rules listed below to bring the number down to zero!

Rules

m n 1. If the number is positive, take the negative reciprocal. ( ) n ! m

2. If the number is negative, add 1.

3. Continue Rules 1 and 2 until you have reached 0.

Example

2 Given number: 3

2 3 Use Rule 1: 3 ! 2 3 1 Use Rule 2: 2 ! 2 1 1 Use Rule 2: 2 ! 2 1 2 Use Rule 1: 2 ! 1 105 Use Rule 2: 2 1 ! Use Rule 2: 1 0 ! DONE!

Use the rules to bring the following numbers to zero.

1 1) 2

4 2) 3

3 3) 4

Tangles & Fractions, Part 2

Tangle Rules

1. Original Position: Each student is represented by the letters A, B, C, and D in

the picture below. The lines represent the ropes and the shapes represent the

placeholders for the positions. The original position of the ropes is zero.

A D

B C

Class

We have two movements:

106 2. Twist:Thestudentinthestarpositionliftstheirropeupandoverthestudent

in the circle position as they trade places. Every twist adds one to our current

number.

A D A C Twist

B C B D

Class

3. Rotate:Everyonemovesoveroneplaceclockwise.Everytimewerotate,we

take the opposite reciprocal of our current number.

A D B A Rotate

B C C D

Class

Practice these moves without keeping track of numbers for now.

After the students are comfortable with the moves, take the ropes back to the original position

4) What number does this tangle represent?

Use the ropes to create the following tangle:

T wist, T wist, Rotate, T wist

5) What number does this tangle represent?

107 We want to use the twist and rotate moves to get back to our original position.

How can we do this?

Remember our rules from part 1 about getting fractions to 0:

Rules

m n 1. If the number is positive, take the opposite reciprocal. ( ) n ! m

2. If the number is negative, add 1.

3. Continue Rules 1 and 2 until you have reached 0.

6) What sequence of twists and rotates should we use?

7) Write out a new list of of 5 twists or rotates.

Using our tangle rules, what number does this represent? •

What list of twists and rotates should we use to untangle the ropes? •

108 B.2 Contra Dance Functions

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

L1 F2 middle

F1 L2

Caller

We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

109 L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

3. Chain

The student in the star position switches places with the student in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following without stopping between the moves.

110 1. Quarter Circle left, Chain, Half Circle Left, Chain

2. Chain, Quarter Circle Left, Half Circle Left, Chain

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two dances (using at least 3 moves for each):

1. Dance 1:

2. Dance 2:

2) Start in our original placement...

L1 F2

F1 L2

Caller

...and dance the following list of moves.

half circle left, chain, quarter circle left, chain, chain

Pause after you have ﬁnished the whole dance and ﬁll in the blank circles to demonstrate your ending position.

Caller

111 Functions

Start in the original formation.

3) What is the output if we perform a chain?

Number of moves

Let’s explore some cool properties of our functions (dance moves!).

Start in the original position. Notice that if we dance chain two times in a row, we get back to our original position. So, 2 is the least number of times that we must dance chain in order to get back to the beginning.

Dance Move Least number of times to get back to beginning Quarter Circle Left Half Circle Left Chain 2

Reset to our original position. Dancing any sequence (of any length) of our moves, how many di↵erent ending positions can you get to?

112 Appendix C: Seventh Grade to Twelfth Grade Activities

C.1 Tangles & Functions

Setup

A D

B C

Class

Twist

113 The ﬁrst move we will learn is a twist.Fromthe0position,thestudentinthe circle position lifts their end of the rope up and over the student in the star position as they switch places. For example:

A D A C Twist

B C B D

Class

Two twists in a row:

A D A D Twist, Twist

B C B C

Class

Rotate

The next move we will learn is a rotate.Thefourstudentsholdingtheropeswill rotate 90 clockwise.

Starting in the 0 position,

A D

B C

Class

one rotate will move the students and ropes around as shown in the below picture.

114 B A

C D

Class

One more rotate would bring the formation back to the 0 position.

C B

D A

Class

We tangle up the two ropes by using the two moves, twist and rotate, in any order and with repetition. Practice these two moves by having the class shout out commands of twist or rotate.

Inverse

Answer the following questions by experimenting with the ropes.

1) What is the inverse of twist?

Denote the string of moves that you answered as the inverse of twist as ?.Check your answer by starting in the 0 position and performing ?, ?,twist,twist.

2) If ? is truly the inverse of twist, what position should the ropes end in?

115 3) What is the inverse of rotate?

Identity

Use your answers from the above inverse questions to answer the following.

4) List two identities using twists and rotates:

Functions

Every twist adds 1 to our current tangle number. For example, if we start in the

0 position

A D

B C

Class

and twist two times,

A D

B C

Class

our tangle number is 2.

Denote the twist function as t(x).

116 5) How should we deﬁne t(x)? t(x)=

Denote the rotate function as r(x). Deﬁning r(x)isnotstraightforward.Experi- ment with the twist and rotate moves together in order to ﬁgure out the e↵ect rotate has on the tangle number.

6) How should we deﬁne r(x)? r(x)=

Untangle

Read o↵the following list of twists and rotates for the four volunteers to do with the ropes.

Twist, Twist, Twist, Rotate, Twist, Rotate, Twist, Twist, Twist, Twist, Twist,

Rotate, Twist

7) What tangle number does this represent?

8) As a class determine what moves we should do in order to undo the tangle (to end in the 0 position).

117 C.2 Math-y Dance

Today we will be learning how to contra dance! Separate into groups of ﬁve.

Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

L1 F2

F1 L2

Caller

We will learn a few contra dance moves. Practice each move to get the hang of it.

1. Quarter Circle Left

Turn towards the student to your left and walk around the square to stand in

the next placeholder.

118 L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Half Circle Left

Turn towards the student to your left and walk around the square to the second

placeholder away.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

3. Chain

The student in the star position switches places with the student in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

Now that you’ve practiced each of these moves separately, let’s string together the moves to create a dance. Try the following without stopping in between the moves.

119 1. Quarter Circle left, Chain, Half Circle Left, Chain

2. Chain, Quarter Circle Left, Half Circle Left, Chain

Now you’re contra dancing!

1) In your group, practice creating new strings of moves. Write down your favorite two:

Dance 1: •

Dance 2: • Practice our dance moves in your group to help answer the following questions.

Questions

2) What happens when you perform multiple half circles in a row? multiple chains?

4) Stand in your original placement. Dance a Quarter Circle Left. Can you dance any move or string of moves to undo the quarter circle left?

5) If we consider the “undo the quarter circle left” as a single move, what might you call the move?

120 6) Begin in your original placement. Can you make a string of 10 moves that brings you back to the original placement? Dance this sequence without any pauses between moves and write the sequence below.

121 C.3 Dancing Functions

Review

Separate into groups of ﬁve with new classmates. Let’s review our contra dance moves. We start with the original set up:

L1 F2

F1 L2

Caller

We have a Quarter Circle left:

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

1) Do you remember Half Circle Left? Fill in the circles below:

L1 F2

F1 L2

Caller Caller

2) Do you remember Chain? Fill in the circles below:

122 L1 F2

F1 L2

Caller Caller

3) We also created a new move called three quarters circle left. Fill in the circles below:

L1 F2

F1 L2

Caller Caller

4) What would a whole circle left move be deﬁned as? Fill in the circles below:

L1 F2

F1 L2

Caller Caller

Let’s introduce some notation so that we can talk about these moves more clearly.

Notation:

1. R1:QuarterCircleLeft

2. R2: Half Circle Left

3. R3:ThreeQuartersCircleLeft

4. R0: Whole Circle Left

123 5. Ch:Chain

The letter R represents rotate.So,R1 means to rotate once to the left.

5) What does R3 mean?

We will now consider our dance moves as functions. For example:

L1 F2 F1 L1 R = 1 F L L F 1 2 2 2

Caller Caller

6) Now try the function for Chain:

L1 F2 =

Ch F1 L2

Caller Caller

7) ...and for Three Quarters Circle Left:

L1 F2 R = 3 F L 1 2

Caller Caller

8) What if we started in a di↵erent position and then danced a Half Circle Left?

124 F2 F1 R = 2 L L 2 1

Caller Caller

Recall that in part 1 we practiced the dance:

Quarter Circle left, Chain, Half Circle Left, Chain

How can we write this whole dance down as a function? We use functional composition!

For our example above, we can write:

Ch(R2(Ch(R1(original position))))

9) Why does it look like the list is backwards?

10) Write out the composition of functions for the following dances.

Half Circle Left, Quarter Circle Left, Chain, Whole Circle Left •

Chain, Chain, Chain, Quarter Circle left •

In your groups, practice the above two dances.

11) Fill in the circles for the following composition of functions.

125 F2 F1 R = 2 L L Ch 2 1

Caller Caller

F2 F1 R R = 2 2 L L 2 1

Caller Caller

12) What happened when we took the composition of R2 and R2?

We call these functions identity functions! They are our “do nothing” functions.

13) Are there any other “do nothing” functions?

Recall the Quarter Circle Left function, R1.

L1 F2 F1 L1 R = 1 F L L F 1 2 2 2

Caller Caller

14) What function (a single function) would we have to apply to the output in order to return to the input (the original position)?

126 We call this type of function an inverse function! For example, we found that the inverse of R1 is R3.Wenotatethisas:

1 R1 = R3

15) What is the inverse of R0?

16) What is the inverse of Ch?

17) What is the inverse of R3?

127 Appendix D: College Activities

D.1 Tangles

Setup

The class needs four volunteers to hold up the two ropes in front of the class. The two ropes should be help up parallel to the classroom as in the picture below with students A, B, C, and D.

A D

B C

Class

Twist

The ﬁrst move we will learn is a twist.Fromthe0position,thestudentinthe circle position lifts their end of the rope up and over the student in the star position as they switch places. For example:

128 A D A C Twist

B C B D

Class

Two twists in a row:

A D A D Twist, Twist

B C B C

Class

Rotate

The next move we will learn is a rotate.Thefourstudentsholdingtheropeswill rotate 90 clockwise.

Starting in the 0 position,

A D

B C

Class

one rotate will move the students and ropes around as shown in the below picture.

129 B A

C D

Class

One more rotate would bring the formation back to the 0 position.

C B

D A

Class

We tangle up the two ropes by using the two moves, twist and rotate, in any order and with repetition. Practice these two moves by having the class shout out commands of twist or rotate.

Inverse

Answer the following questions by experimenting with the ropes.

1) What is the inverse of twist?

Denote the string of moves that you answered as the inverse of twist as ?.Check your answer by starting in the 0 position and performing ?, ?,twist,twist.

2) If ? is truly the inverse of twist, what position should the ropes end in?

130 3) What is the inverse of rotate?

Identity

Use your answers from the above inverse questions to answer the following.

4) List two identities using twists and rotates:

Functions

Every twist adds 1 to our current tangle number. For example, if we start in the

0 position

A D

B C

Class

and twist two times,

A D

B C

Class

our tangle number is 2.

Denote the twist function as t(x).

131 5) How should we deﬁne t(x)? t(x)=

Denote the rotate function as r(x). Deﬁning r(x)isnotstraightforward.Experi- ment with the twist and rotate moves together in order to ﬁgure out the e↵ect rotate has on the tangle number.

6) How should we deﬁne r(x)? r(x)=

Untangle

Read o↵the following list of twists and rotates for the four volunteers to do with the ropes.

Twist, Twist, Twist, Rotate, Twist, Rotate, Twist, Twist, Twist, Twist, Twist,

Rotate, Twist

7) What tangle number does this represent?

8) As a class determine what moves we should do in order to undo the tangle (to end in the 0 position).

9) Consider all the combinations of r and t as a set. What mathematical object can describe this set?

Tangles: Extension

Use the functions deﬁned in questions 5 and 6 to answer the following.

132 10) Prove that r t r t r t and r r are identities.

1 1 11) Prove that t (x) = t r. Why might it seem like t (x)=t r when using the 6 ropes?

133 D.2 Properties of Contra Dance Figures

Introduction

Today we will be learning how to contra dance!

Activity

Separate into groups of ﬁve. Denote each group member as one of the following:

1. Leader 1 (L1)

2. Follower 1 (F1)

3. Leader 2 (L2)

4. Follower 2 (F2)

5. Caller

L1 F2

middle

F1 L2

Caller

We will learn a few contra dance ﬁgures. Practice each move to get the hang of it as the caller calls out the ﬁgure’s name.

134 1. Quarter Circle Left

The square formation rotates 90 clockwise.

L1 F2 F1 L1

F1 L2 L2 F2

Caller Caller

2. Quarter Circle Right

The square formation rotates 90 counterclockwise.

L1 F2 F2 L2

F1 L2 L1 F1

Caller Caller

3. Half Circle Left

The square formation rotates 180 clockwise.

L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

4. Half Circle Right

The square formation rotates 180 counterclockwise.

135 L1 F2 L2 F1

F1 L2 F2 L1

Caller Caller

5. Chain

The dancer in the star position switches places with the dancer in the diamond

position.

L1 F2 L1 F1

F1 L2 F2 L2

Caller Caller

6. Swing on Side

The dancers in the square and star positions switch places, and the dancers in

the diamond and circle positions switch places.

L1 F2 F2 L1

F1 L2 L2 F1

Caller Caller

7. California Twirl

136 The dancers in the square and diamond positions switch places, and the dancers

in the star and circle positions switch places.

L1 F2 F1 L2

F1 L2 L1 F2

Caller Caller

1. Quarter Circle left, Chain, Half Circle Left, Swing on Side, Half Circle Right

2. California Twirl, Quarter Circle right, Half Circle Left, Chain, Swing on Side

Now you’re contra dancing!

In your group, practice creating new strings of moves. Write down your favorite two:

1) Dance 1:

2) Dance 2:

Properties of the Dance Figures

137 3) What functional operation can be used to notate the action of dancing multiple

ﬁgures in a row?

5) Can you describe any identity functions from our set of dance ﬁgures?

6) Are there any combinations of dance ﬁgures that are commutative?

7) Are there any combinations of dance ﬁgures that are not commutative?

8) Are there any combinations of dance ﬁgures that are associative?

9) Are there any combinations of dance ﬁgures that are not associative?

138 10) What mathematical object describes our set of dance ﬁgures (functions)?

139 D.3 Generators and Relations

In the ﬁrst activity we discovered that our set of dance moves form a non- commutative group. Our goal in this activity is to describe this group in terms of generators and relations.

Generators

For the following questions you may notate quarter circle left as r and chain as c.

1) Consider the e↵ect each dance ﬁgure has on the initial position. Using only Quarter

Thus, quarter circle left (r)andchain(c) are generators of this group.

Relations

Equations in a group that are satisﬁed by the generators are called relations.

2) List three expressions that equal the identity, e,byusingonlyr and c.

1. = e

2. = e

3. = e

3) How many possible ending dance formations are there?

140 4) List all the elements of this group by only using the functions r and c.

5) Fill out the multiplication table below.

e r r2 r3 c rc r2c r3c e r r2 r3 c rc r2c r3c

141 D.4 Free Group

Elements of the Free Group on 2 Generators

Let A = a, b .Wecancreatewords on the set A by concatenating elements of { } 1 1 A and their inverses, denoted by a and b ,inanyorder.Forexample,

1 1 1 1 aaabbb b b ba is a word.

1) Create ﬁve new words on A.

A reduced word is one in which we simplify the word. Any instances of an element and its inverse concatenated together must be removed. For example,

1 1 1 1 1 1 aaabbb b b ba a = aaabb a

1 = aaaa

= aa

142 is a reduced word.

2) Reduce your words from Exercise 1.

The free group on the set A is the set of all possible reduced words on A.

Connections to Contra Dance

The universal property of free groups gives a nice connection between groups and free groups:

To apply this property to our group from activity 2, we deﬁne the functions i, f, and f¯ in the following manner:

i(a)=a

i(b)=b

f(a)=r

f(b)=c

f¯(a)=r

f¯(b)=c

143 where r4 = s2 =(rs)2 = e.

The homomorphism of groups relates the structure of the group and free group.

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