Instructing Group Theory Concepts from Pre-Kindergarten to
College through Movement Activities
A Thesis
Presented in Partial Fulfillment of the Requirements for the Degree
Master of Mathematical Science in the Graduate School of The Ohio
State University
By
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 undergraduate 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 , and 7th 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 handson 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, and Group Actions . . . . . . . . . . . . . . . . . . .
3
2.1
2.2
3.
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3
6
7
18
Pre-Kindergarten to Third Grade . . . . . . . . . . . . . . . . . . . . . .
20
3.1
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28
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Fourth Grade to Ninth Grade . . . . . . . . . . . . . . . . . . . . . . . .
34
4.1
34
34
40
40
3.2
4.
4.2
Functions . . . . . .
Groups . . . . . . .
2.2.1 Free Group .
2.2.2 Group Action
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Rational Tangles . . . . . . . . .
3.1.1 All Tangled Up! . . . . .
3.1.2 Adding up Tangles . . . .
Contra Dancing . . . . . . . . . .
3.2.1 Dancing Around a Square
Rational Tangles . . . . . . . .
4.1.1 Tangles & Fractions . .
Contra Dancing . . . . . . . . .
4.2.1 Contra Dance Functions
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5.
Seventh to Twelfth Grade . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.1
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College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.1
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Irrational Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1
88
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5.2
6.
6.2
7.
Rational Tangles . . . . . .
5.1.1 Tangles & Functions
Contra Dancing . . . . . . .
5.2.1 Math-y Dance . . .
5.2.2 Dancing Functions .
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Rational Tangles . . . . . . . . . . . . . .
6.1.1 Tangles . . . . . . . . . . . . . . .
Contra Dancing . . . . . . . . . . . . . . .
6.2.1 Properties of Contra Dance Figures
6.2.2 Generators and Relations . . . . .
6.2.3 Free Group . . . . . . . . . . . . .
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Rational and Irrational Tangles . . . . . . . . . . . . . . . . . . . .
7.1.1 Connections to the Tangle Group . . . . . . . . . . . . . . .
Appendices:
A.
Pre-Kindergarten to Third Grade Activities . . . . . . . . . . . . . . . .
96
A.1 All Tangled Up! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.2 Adding up Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.3 Dancing Around a Square . . . . . . . . . . . . . . . . . . . . . . . 100
B.
Fourth Grade to Ninth Grade Activities . . . . . . . . . . . . . . . . . . 105
B.1 Tangles & Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Contra Dance Functions . . . . . . . . . . . . . . . . . . . . . . . . 109
C.
Seventh Grade to Twelfth Grade Activities . . . . . . . . . . . . . . . . . 113
C.1 Tangles & Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.2 Math-y Dance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.3 Dancing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
vi
D.
College Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D.1
D.2
D.3
D.4
Tangles . . . . . . . . . . . . . . .
Properties of Contra Dance Figures
Generators and Relations . . . . .
Free Group . . . . . . . . . . . . .
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128
134
140
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 2 X to indicate that an object x is an element of the set X. Sets
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 2 S,
but f rog 2
/ S.
Definition 2. The cartesian product of two sets X and Y is the set X ⇥ Y with
elements (x, y) such that x 2 X and y 2 Y .
Two elements (x, y) 2 X ⇥ Y and (x0 , y 0 ) 2 X ⇥ Y are equal if and only if x = x0
and y = y 0 .
Example 2.1.2. Let S = {cat, dog, mouse} and T = {spots, f urry}. Then
(dog, spots) 2 S ⇥ T while (dog, bark) 2
/ S ⇥ T.
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 2 X, there exists y 2 Y such that (x, y) 2 f . This means that a
function must map every element in the domain, or else that element is not in
the domain.
• For all y 2 Y , there exists x 2 X such that (x, y) 2 f . This means that a
function must map to every element in the range or else that element is not in
the range.
• If (x, a) 2 f and (x, b) 2 f , then a = b. This means that for every element in
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) 2 X ⇥ Y : (x, y) 2 f }, we write f : X ! Y and notate
f (x) = y for the mapping of a given element x 2 X to y 2 Y .
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.
The inverse of a function f is denoted by f
1
and we write f
1
= {(y, x) 2 Y ⇥X :
(x, y) 2 X ⇥ Y }.
Definition 4. A function f : X ! Y is invertible if for each y 2 Y , f
exactly one element.
4
1
(y) has
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) 2 X ⇥ Z : (f (x), z) 2 g}.
Taking the composition of two functions, g f , is read from right to left such that
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
(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) 2 X ⇥ Z : (f (x), z) 2 g}.
Thus,
h (g f )(x) = {(x, w) 2 X ⇥ W : (g f (x), w) 2 h}.
Now, by definition of composition of functions,
h g = {(y, w) 2 Y ⇥ W : (g(y), w) 2 h}.
5
(g
f) =
Thus,
(h g) f = {(x, w) 2 X ⇥ W : (f (x), w) 2 h g}.
If (f (x), w) 2 h g, then (f (x), w) 2 Y ⇥ W such that (g(f (x)), w) 2 h. Thus,
(h g) f = {(x, w) 2 X ⇥ W : (g f (x), w) 2 h}.
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 , that is, if f, g 2 G then f
2. The operation
(g h) = (f
g 2 G.
is associative, meaning that for any elements f, g, h 2 G, f
g) h.
3. G contains an element e, called the identity, such that g
e = e g = g for all
g 2 G.
4. Every element g 2 G has an inverse, g 1 , such that g g
1
= e and g
1
g = e.
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.
A set of invertible functions that are closed under composition immediately satisfies
properties 1 and 4. Property 3 is satisfied since f f
6
1
= e for any function in our set.
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, we can create words through
the operation of concatenation
a
aa
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
aj ak = 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. We will denote
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, x4 x7 = x4+7 = x11 . This is identical to concatenating two
words together.
The concatenation operation is also associative. Given a word aj ak al we see that
(aj ak )al = aj+k al
= aj+k+l
= aj ak+l
= aj (ak al )
Thus, F [A] is a semigroup.
Definition 7. A semigroup is a nonempty set G equipped with an operation, ,
such that
1. G is closed under the operation . If f, g 2 G then f
g 2 G.
2. The operation is associative. For any elements f, g, h 2 G, f (g h) = (f g) h.
In concatenation we simply omit the symbol , that is, 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 . If we concatenate any word with the empty word,
we are left with the original word. Consider the word ak . Then
ak e = ak a0
= ak+0
= ak
and
eak = a0 ak
= 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 : any value raised to the power of 0 is 1. Thus,
x0 xk = 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 . If f, g 2 G then f
g 2 G.
2. The operation is associative. For any elements f, g, h 2 G, f (g h) = (f g) h.
3. G contains an element e, called the identity, such that g
e = e g = g for all
g 2 G.
Let a
1
denote a new letter, called the inverse element of a, which we assume to
obey the rule aa
1
= e = a 1 a. We extend the definition of a word formed by the
alphabet A by allowing strings of both a and a
1
in any order, where repetition is
allowed. A word is then reduced if all instances of aa
1
and a 1 a are replaced by
e. Let F [A] denote the set of all reduced words formed by the alphabet A with the
operation of concatenation. F [A] is now a group.
In general, if A = {ai } is any set, we can further extend the definition of a word
on A and reduced words.
Definition 9. A word on A is a sequence of the elements ai 2 A and their inverses
ai
1
joined by concatenation where repetition is allowed.
9
Definition 10. [11] A word a✏11 a✏22 a✏33 . . . is a reduced word if it satisfies the following.
1. ai and ai
1
are not adjacent for all a 2 A.
2. If ak = e then aj = e for all j
k.
Example 2.2.1. Consider a set of red and black chips. Let each red chip represent
1 and each black chip represent +1. Thus, every combination of 1 red chip and 1
black chip returns an overall value of 0. We can think that a = a black chip and a
1
=
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
7 is 0 + 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
3 with
3. Once part 1 is satisfied, we remove e unless the word overall is e.
Again, F [A] as defined as the set of all reduced words on A 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
clockwise by ⇢. Thus, ⇢2 is a 90 rotation, ⇢3 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. More explicitly, the elements in the set A 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. This relation is called an equation satisfied by the
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, then F [A] is the free group generated by A.
The universal property of a free group uniquely defines the relationship between
a set A, a group G, and a free group F [A].
11
Definition 14. Let G and H be groups under the operations
and •, respectively.
A function f : G ! H is a homomorphism if
f (x y) = f (x) • f (y)
for all x, y 2 G.
A homomorphism relates the structures of the two groups G 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. We can check that this function
satisfies the conditions from Definition 14. For all k, l 2 Z,
(ak al ) = (ak+l ) = k + l
and
(ak ) + (al ) = k + l
Thus, (ak al ) = (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]. If G 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}. Then i(a) = a and F [A] is the free group on one generator
with elements ak where k 2 Z. We can define f (a) = 1 and f¯(ak ) = k.
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}. Then i(a) = a and F [A] is the free group on one generator
with elements ak where k 2 Z. We can define f (a) = 1 and f¯(ak ) = k (mod 3).
Example 2.2.6. Symmetries of a Square
Definition 15. The nth dihedral group, denoted Dn , is the group of symmetries
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}. The elements of A satisfy the following relations:
1. r4 = e
r4
4
1
3
4
r
3
2
2
3
r
2
1
1
2
r
1
4
4
3
s2
1
2
1
s
3
3. s r = r
1
2
4
1
3
2
s
3
4
s
14
1
3
2
r
2. s2 = e
4
4
s r
4
1
3
4
1
4
1
2
3
1
1
4
2
3
r
3
2
s
2
r
4
1
1
s
2
s
3
2
r
3
4
1
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
of r is r3 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 = {⇢, }. Then i(⇢) = ⇢ and i( ) = . F [A] is the free group on this set A. The
function f maps f (⇢) = r and f ( ) = s, and the function f¯ maps f¯(⇢) = r and
f¯( ) = s where r4 = e, s2 = e, and s r = r
i
A
1
s.
F [A]
f¯
f
D4
15
We also note that D4 is a non-commutative group, meaning that the operation
on our group is not commutative. For example, r s 6= s r, as shown below.
r s
4
1
2
1
3
2
4
4
1
4
1
4
2
3
s
3
2
r
3
s r
4
1
3
r
3
2
s
2
1
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.
T
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. Let A = {T, R}. The elements
of A satisfy the following relations:
1. R R = e
R R
R
2. R T
R T =T
1
R
R
1
R T
T
R
17
R T
T
R
T
R
1
R
1
1
T
1
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
inverse of T is R
T
R
T
R T
R T , as shown above. The
R and the inverse of R is R. Thus, we see that the
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 = {⇢, }. Then i(⇢) = ⇢ and i( ) = . F [A] is then the free group on this set A.
The function f is defined as f (⇢) = R and f ( ) = T , and the function f¯ is defined
as f¯(⇢) = R and f¯( ) = T where R R = e and R T
i
A
R T
R T = e.
F [A]
f¯
f
T
2.2.2
Group Action
A group action describes how a group acts upon a given set. The notion of an
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, for all g 2 G and a 2 A) satisfying the following
properties:
1. g1 · (g2 · a) = (g1 · g2 ) · a, for all g1 , g2 2 G, a 2 A, and
2. e · a = a, for all a 2 A.
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
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
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. Students in the star and circle positions
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. Start in the original placement:
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 describing 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, drawings, 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
a twist for each number that you count.
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
A
Group 2
D
knot
...
B
...
C
knot
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
a function (dance move) maps a set (4 dancers, initial position) to another set (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. Pause when you finish and fill in the
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. Pause when you finish and
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
1. If the number is positive, take the negative reciprocal. ( m
!
n
2. If the number is negative, add 1.
34
n
)
m
3. Continue Rules 1 and 2 until you have reached 0.
Example
2
3
Given number:
Use Rule 1:
2
3
!
Use Rule 2:
3
2
!
Use Rule 2:
1
2
!
Use Rule 1:
1
2
!
3
2
1
2
1
2
2
1
Use Rule 2:
2!
Use Rule 2:
1!0
1
DONE!
Use the rules to bring the following numbers to zero.
1)
1
2
Answer. Given number:
Use Rule 2:
Use Rule 1:
1
2
1
2
1
2
!
!
2
1
Use Rule 2:
2!
Use Rule 2:
1!0
2)
1
2
1
4
3
Answer. Given number:
Use Rule 1:
Use Rule 2:
Use Rule 1:
Use Rule 2:
4
3
!
3
4
1
4
3
4
1
4
!
!
4
3
4
1
4!
3
35
Use Rule 2:
3!
2
Use Rule 2:
2!
1
Use Rule 2:
1!0
3)
3
4
Answer. Given number:
Use Rule 1:
3
4
!
Use Rule 2:
4
3
!
Use Rule 2:
1
3
!
Use Rule 1:
2
3
!
Use Rule 2:
3
2
!
Use Rule 2:
1
2
!
Use Rule 1:
1
2
!
3
4
4
3
1
3
2
3
3
2
1
2
1
2
2
1
Use Rule 2:
2!
Use Rule 2:
1!0
1
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: The student in the star position lifts their rope up and over the student
in the circle position as they trade places. Every twist adds one to our current
number.
A
D
A
C
B
D
Twist
B
C
Class
3. Rotate: Everyone moves over one place clockwise. Every time we rotate, we
take the opposite reciprocal of our current number.
A
D
B
A
C
D
Rotate
B
C
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?
Answer. This tangle is 12 . It will help if the class keeps count of the tangle after each
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
1. If the number is positive, take the opposite reciprocal. ( m
!
n
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
n
)
m
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
Answer.
2
Dance Move
Least number of times to get back to beginning
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.
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. From the 0 position, the student in the
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
B
D
Twist
B
C
Class
Two twists in a row:
A
D
A
D
B
C
Twist, Twist
B
C
Class
47
Rotate
The next move we will learn is a rotate. The four students holding the ropes will
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?
Answer. t
1
= r t r t r, or rotate, twist, rotate, twist, rotate. Students might
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?
Answer. r
1
= r, or rotate.
Identity
Use your answers from the above inverse questions to answer the following.
4) List two identities using twists and rotates:
1.
2.
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) is not straight forward. Experiment 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) =
Answer. r(x) =
1
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?
Answer.
5
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 interpret 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
Answer.
Caller
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
Answer.
Caller
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
Answer.
Caller
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 : Quarter Circle Left
2. R2 : Half Circle Left
3. R3 : Three Quarters Circle Left
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:
R1
L1
F2
F1
L1
L2
F2
=
F1
L2
Caller
6) Now try the function for Chain:
60
Caller
Ch
L1
F2
=
F1
L2
Caller
Answer.
Ch
L1
Caller
F2
L1
F1
F2
L2
=
F1
L2
Caller
Caller
7) ...and for Three Quarters Circle Left:
R3
L1
F2
=
F1
L2
Caller
Answer.
R3
L1
Caller
F2
F2
L2
L1
F1
=
F1
L2
Caller
Caller
8) What if we started in a di↵erent position and then danced a Half Circle Left?
R2
F2
F1
=
L2
L1
Caller
61
Caller
Answer.
R2
F2
F1
L1
L2
F1
F2
=
L2
L1
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.
• R0 (Ch(R1 (R2 (original position))))
• R1 (Ch(Ch(Ch(original position))))
62
In your groups, practice the above two dances.
11) Fill in the circles for the following composition of functions.
Ch R2
F2
F1
=
L2
L1
Caller
Answer.
Ch R2
Caller
F2
F1
L2
L1
=
L1
F1
L2
F2
Caller
R2 R2
F2
Caller
F1
=
L2
L1
Caller
Answer.
R2 R2
Caller
F2
F1
L2
L1
=
Caller
F2
F1
L2
L1
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. A few examples: R0 , Ch Ch, R1 R3
Recall the Quarter Circle Left function, R1 .
R1
L1
F2
F1
L1
L2
F2
=
F1
L2
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 . We notate this as:
R1 1 = R3
15) What is the inverse of R0 ?
Answer. R0 1 = R0
16) What is the inverse of Ch?
Answer. Ch
1
= Ch
17) What is the inverse of R3 ?
64
Answer. R3 1 = R1
CCSS.MATH.CONTENT.HSF.IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret 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) is the temperature in the atmosphere
as a function of height, and h(t) is the height of a weather balloon as a function
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
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. From the 0 position, the student in the
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
B
D
Twist
B
C
Class
Two twists in a row:
A
D
A
D
B
C
Twist, Twist
B
C
Class
67
Rotate
The next move we will learn is a rotate. The four students holding the ropes will
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?
Answer. t
1
= r t r t r, or rotate, twist, rotate, twist, rotate. Students might
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?
Answer. r
1
= r, or rotate.
Identity
Use your answers from the above inverse questions to answer the following.
4) List two identities using twists and rotates:
1.
2.
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) is not straight forward. Experiment 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) =
Answer. r(x) =
1
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?
Answer.
5
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.
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
)
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
= r t r t(
=x
r r(x) = r(
=
1
)
x
x
1
=x
11) Prove that t 1 (x) 6= t r. Why might it seem like t 1 (x) = t r when using the
ropes?
Answer.
t r t(x) = t r(x + 1)
1
)
x+1
1
=
+1
x+1
x
=
x+1
= t(
72
It might seems like t 1 (x) = t r when using the ropes from the 0 position because
t r t(0) =
0
0+1
=
0
1
= 0.
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
1 and 2, students are working with the structure of D4 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)
• (Chain)
(Quarter Circle Right)
(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)
• (Chain)
(Chain)
• (Half Circle Left)
(Half Circle Left)
(Chain)
(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 noncommutative 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) and chain (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, by using only r 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
e
r
r2
r3
c
rc
r2 c
r3 c
82
c
rc
r2 c
r3 c
Answer.
e
r
r2
r3
c
rc
r2 c
r3 c
e
e
r
r2
r3
c
rc
r2 c
r3 c
r
r
r2
r3
e
rc
r2 c
r3 c
c
r2
r2
r3
e
r
r2 c
r3 c
c
rc
r3
r3
e
r
r2
r3 c
c
rc
r2 c
c
c
r3 c
r2 c
rc
e
r3
r2
r
rc
rc
c
r3 c
r2 c
r
e
r3
r2
r2 c
r2 c
rc
c
r3 c
r2
r
e
r3
r3 c
r3 c
r2 c
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}. We can create words on the set A by concatenating elements of
A and their inverses, denoted by a
1
and b 1 , in any order. For example,
aaabbb 1 b 1 b 1 ba
1
is a word.
1) Create five new words on A.
1.
2.
3.
4.
5.
Answer. Answers can be any combination of a, b, a 1 , and b 1 .
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,
aaabbb 1 b 1 b 1 ba 1 a = aaabb 1 a
= aaaa
= aa
84
1
1
is a reduced word.
2) Reduce your words from Exercise 1.
1.
2.
3.
4.
5.
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] be the free group on a set A and i : A ! F [A] the inclusion function
such that i(a) = a. If G is a group and f : A ! G a map of sets, then there
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
a 1a 1b
bab 1 a
1
abbbba
Answer.
Free group element
Dance group element
1
ababa b
rcrcr 1 c = r 1 c = r3 c
bbbbbb
c6 = e
a 1a 1b
r 1 r 1 c = r 3 r 3 c = r 6 c = r2 c
aba 1 b
1
rcr 1 c
abbbbba
1
= rcr3 c = rcr3 rcr = rccr = rr = r2
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
number where T (x) = x + 1 and R(x) =
1
x
88
and x is the current tangle number.
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 +
a3 +
b3
a4 + · · ·
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
a2
···
1
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
be accomplished for a number
p
q
by using Euclid’s algorithm for finding the GCD of
p and q.
Example 7.1.1. We can find determine the continued fraction for
mining gcd(4, 11):
4 = 1(11)
7
11 = 2(7)
3
7 = 3(3)
2
3 = 2(2)
1
2 = 2(1)
0
89
4
11
by first deter-
We can take each of the above equations and write equivalent statements:
4
11
11
7
7
3
3
2
7
11
3
7
2
3
1
2
=1
=2
=3
=2
2=2
Then starting at the first equation we can make substitutions.
4
=1
11
=1
=1
=1
=1
=1
1
11/
7
1
3
7
2
1
2
1
7/
3
1
2
1
3
2
3
1
2
1
1
3/
2
3
1
2
1
3
1
2
1
2
So, we have
4
=1
11
1
2
1
3
1
2
Thus, the sequence to tangle up to the value of
1
2
4
11
is 2 twists, rotate, 2 twists,
rotate, 3 twists, rotate, 2 twists, rotate, 1 twist. In function notation we see that
T
R
T2
R
T3
R
T2
R
T 2 (0) =
displayed below.
90
4
,
11
and a picture of the twisted ropes is
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:
p
1+ 5
=1+
2
1
1
1+
1
1 + ···
Since the continued fraction is infinite, we cannot read o↵ a list of twists and
1+
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
the golden ratio,
p
1+ 5
.
2
We can start with the following first step:
1
1+
=
5
=2
3
1
3
1
1+1
Thus, our first step is the sequence: 3 twists, rotate, 2 twists.
1+
91
Next, we use the following fraction
1
1+
=
1
1+
13
=2
8
1
1+
1+
1
1
3
3
1
1+1
which is represented by the sequence: 3 twists, rotate, 3 twists, rotate, 2 twists.
Next, we use the following fraction
1
1+
=
1
1+
1
1+
1
1
3
3
1
1+
34
=2
21
1
3
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,
p
1+ 5
,
2
as T 3 RT 3 R · · · T 3 RT 2 (if read from left to right).
93
7.1.1
Connections to the Tangle Group
Recall that we defined T (x) = x + 1 and R(x) =
1
.
x
The functions T and
R generate our tangle group, T , and we can check again that the following two
relations are satisfied.
1. R R = e
1
)
x
R R(x) = R(
=
1
1
x
=x
1. R T
R T
R T =e
R T
R T
R T (x) = R T
=R
=R
= R(
= R(
R(x + 1)
1
)
x+1
1
T R(
+ 1)
x+1
x
T R(
)
x+1
x 1
T(
)
x
x 1
+ 1)
x
1
)
x
=R T
=R
R T
R T(
=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 , we note that T is isomorphic
to the group SL2 (Z). The group SL2 (Z) is defined as
⇢
a b
: a, b, c, d, 2 Z, ad
c d
bc = 1
where the following two matrices generate the group [4].

0
1
1 0

1 1
T =
0 1
S=
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. Students in the star and circle positions
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. Start in the original placement:
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?
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?
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
a twist for each number that you count.
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
A
Group 2
D
knot
...
B
...
C
knot
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. Pause when you finish and fill in the
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. Pause when you finish and
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
1. If the number is positive, take the negative reciprocal. ( m
!
n
2. If the number is negative, add 1.
3. Continue Rules 1 and 2 until you have reached 0.
Example
Given number:
Use Rule 1:
2
3
!
Use Rule 2:
3
2
!
Use Rule 2:
1
2
!
Use Rule 1:
1
2
!
2
3
3
2
1
2
1
2
2
1
105
n
)
m
Use Rule 2:
2!
Use Rule 2:
1!0
1
DONE!
Use the rules to bring the following numbers to zero.
1)
1
2
2)
4
3
3)
3
4
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.
A
D
B
C
Class
We have two movements:
106
2. Twist: The student in the star position lifts their rope up and over the student
in the circle position as they trade places. Every twist adds one to our current
number.
A
D
A
C
B
D
Twist
B
C
Class
3. Rotate: Everyone moves over one place clockwise. Every time we rotate, we
take the opposite reciprocal of our current number.
A
D
B
A
C
D
Rotate
B
C
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
1. If the number is positive, take the opposite reciprocal. ( m
!
n
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
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
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
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?
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.
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
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
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.
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
113
The first move we will learn is a twist. From the 0 position, the student in the
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
B
D
Twist
B
C
Class
Two twists in a row:
A
D
A
D
B
C
Twist, Twist
B
C
Class
Rotate
The next move we will learn is a rotate. The four students holding the ropes will
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:
1.
2.
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) is not straight forward. Experiment 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
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.
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?
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
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:
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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 : Quarter Circle Left
2. R2 : Half Circle Left
3. R3 : Three Quarters Circle Left
4. R0 : Whole Circle Left
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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:
R1
L1
F2
F1
L1
L2
F2
=
F1
L2
Caller
Caller
6) Now try the function for Chain:
Ch
L1
F2
=
F1
L2
Caller
Caller
7) ...and for Three Quarters Circle Left:
R3
L1
F2
=
F1
L2
Caller
Caller
8) What if we started in a di↵erent position and then danced a Half Circle Left?
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R2
F2
F1
=
L2
L1
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.
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F2
Ch R2
F1
=
L2
L1
Caller
F2
R2 R2
Caller
F1
=
L2
L1
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 .
R1
L1
F2
F1
L1
L2
F2
=
F1
L2
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 . We notate this as:
R1 1 = 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
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. From the 0 position, the student in the
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
B
D
Twist
B
C
Class
Two twists in a row:
A
D
A
D
B
C
Twist, Twist
B
C
Class
Rotate
The next move we will learn is a rotate. The four students holding the ropes will
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:
1.
2.
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) is not straight forward. Experiment 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.
11) Prove that t 1 (x) 6= t r. Why might it seem like t 1 (x) = t r when using the
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
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.
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
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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:
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.
137
3) What functional operation can be used to notate the action of dancing multiple
figures in a row?
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
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 noncommutative 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
Thus, quarter circle left (r) and chain (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, by using only r 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
e
r
r2
r3
c
rc
r2 c
r3 c
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c
rc
r2 c
r3 c
D.4
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}. We can create words on the set A by concatenating elements of
A and their inverses, denoted by a
1
and b 1 , in any order. For example,
aaabbb 1 b 1 b 1 ba
1
is a word.
1) Create five new words on A.
1.
2.
3.
4.
5.
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,
aaabbb 1 b 1 b 1 ba 1 a = aaabb 1 a
= aaaa
= aa
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1
1
is a reduced word.
2) Reduce your words from Exercise 1.
1.
2.
3.
4.
5.
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] be the free group on a set A and i : A ! F [A] the inclusion function
such that i(a) = a. If G is a group and f : A ! G a map of sets, then there
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:
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.
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
ababa 1 b
bbbbbb
a 1a 1b
bab 1 a
1
abbbba
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Bibliography
[1] Adams, C. C. (1994). The knot book: An elementary introduction to the mathematical theory of knots. New York: W.H. Freeman.
[2] Carter, N. C. (2009). Visual group theory. Washington, D.C.: Mathematical
Association of America.
[3] Clark, A. (1984). Elements of abstract algebra. New York: Dover Publications,
Inc.
[4] Conrad, K. (1998). SL2 (Z). [Electronic Version].
http://www.math.uconn.edu/ kconrad/blurbs/grouptheory/SL(2,Z).pdf
[5] Conway, J. (1998). The power of mathematics. [Electronic Version].
www.inference.phy.cam.ac.uk/mackay/conway.ps.gz
[6] Crowell, R. H., & Fox, R. H. (1977; 1963). Introduction to knot theory. New York:
Springer-Verlag.
[7] Davis, T. (2013). Conway’s rational tangles.
http://www.geometer.org/mathcircles/tangle.pdf
[8] Department of Higher Education. Retrieved from https://www.ohiohighered.org
[9] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ:
Wiley.
[10] Fraleigh, J. B. (1982). A first course in abstract algebra (2nd ed.). Reading,
Mass.: Addison-Wesley Pub. Co.
[11] Hungerford, T. W. (1974) Algebra. New York: Springer-Verlag.
[12] National Governors Association Center for Best Practices & Council of Chief
State School Officers. (2010). Common core state standards. Washington, DC:
National Governors Association Center for Best Practices & Council of Chief
State School Officers.
145
[13] Olds, C. D. (1963). Continued fractions. New York: Random House.
[14] Ronquillo, J. (2015). Contradance, groups and dihedral groups.
[15] von Renesse C, et al. (2013). Discovering the art of mathematics: Dance (Draft)
[electronic version]. https://www.artofmathematics.org/books/dance.
146