INTRODUCTION TO THE AESTHETIC
COiPUTlING METHOD FOR TEACHING
ALGEBRA IN MIDDLE AND HIGH SCHOOL
Pgui Fishwick, PhD
httffp//www.cisve. ofldv/ fishwick/ escompuvtfi
Draft lDte: 9/28/200o
 1 UNIVERSITY OF
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TABLE OF CONTENTS
Topic
Page
Introduction to Aesthetic Computing for Algebra 3
Middle School Lesson Plans, Handouts, Teacher Resources 10
High School Lesson Plans, Handouts, Teacher Resources 24
Software Tools (2D and 3D) 41
Motivation
Modem media shape the way we view the world. With a natural language such as English,
we have seen a gradual evolution of representation from voice to handwriting, print,
photography, radio, television, cinema, and video game consoles. This only briefly covers
the possibilities of media, since there is active research in holography, 3D displays, and
new areas on the horizon such as pervasive and tangible computing. The same semantic
"content" is represented in different ways. This evolution is not based on replacement.
Television does not eliminate the radio, nor do video games eliminate the book. Instead
of replacement, the world of media create an augmentationnew ways of seeing and
hearing similar content. We see the same content through multiple representations.
Considering this history of English, and contrasting this with formal languages such as
mathematics, we find a different kind of evolutionone based on improving the richness
of mathematics without significantly affecting representation. Why is it that we cannot
also have an evolution, or perhaps a revolution, in the ways in which we might view and
hear mathematics? To some extent, this evolution is taking place in what we might call
the solution space of mathematics: plotting graphs, solving equations, running programs.
In this solution space, new technologies help us to visualize geometric proofs and
equation plots in new ways. The problem is that there has been little emphasis on the
problem space. When we express and manipulate mathematical expressions, we are
dealing with the problem space. Can the problem space also be a place where we might
create new representations using our latest technologies? Can we create gamelike, highly
visually appealing, worlds to attract students to mathematics? Can mathematical
expressions look like a video game, feel like a physical sculpture, or be read like a story?
This is the emphasis on aeI/theic computing, and the corresponding method described in
this document.
The method provides the following definite benefits:
Order of operations: students learn the order of operations when constructing the
expression trees. This provides the student with a clearer understanding of the
structure of algebraic expressions.
Interdisciplinary activity: students when making expression trees, and the 2D, 3D,
or physical objects that they represent are forced to think in terms of design, art,
and English sentence structure (i.e., for the story type of target object). Thus,
mathematics is linked closer to the literary and fine arts. This provides a way for
math teachers and other teachers to relate their work to each other.
Introduction to computing: the exercise involving expression trees has a close link
with learning how to use/program a calculator or computer since mathematical
expressions need to be made explicit, and the expression tree can be used as a
means of computation or calculation to obtain results.
The method has these perceived, untested, benefits which may eventually become
hypotheses for testing: (a) improved memory retention for complex expressionsthe
mnemonic effect of using icons; (b) improved motivation for doing algebra; and (c)
improved grades in math class.
This document is divided into four parts:
1. Introduction what you are reading now, to create an overview of the aesthetic
computing method
2. Middle School Track lessons and handouts for Middle School: prealgebra or
algebra (courses: Math 6,7,8)
3. High School Track lessons and handouts for High School: algebra and
integrated math (course: Integrated Math 3)
4. Tools 2D and 3D tools (virtual algebra machine) to assist students and teachers
If you see a small icon that looks like Rodin's The Thinker (the figure on the left),
you can probably skip this extra material unless you want more detail. When we
consider problem space versus solution space, these two phrases relate in ways
similar to syntax versus semantics (the structure of a thing versus its meaning, which is
often the interpretation, or behavior, of the syntax). You can also think of it from a
computing perspective: an equation is fed into a machine, which then operates on it,
producing output in the form of a plot, a rearranged symbolic expression, or something
else. Thus, mathematics has focused on novel representations for what comes out of the
machine, but not what goes into it.
1. Goal
The goal of these lessons, handouts, and implementation should be to provide a creative
exercise in teaching the subject of algebra that instills student interest in the subject. The
purpose is to help motivate the student in algebra by allowing them to be creative. The
most basic assessment strategy is a videotaping of the class, surveying of students after
the lessons are complete, and teacher interview as to the success or failure of the method.
The documentation is composed of this introduction to the aesthetic computing method,
followed by lesson plans, handouts, and assessment mechanisms. There are two tracks:
MIDDLE SCHOOL and HIGH SCHOOL. We recommend that teachers look at both
tracks and take examples that are relevant to their classes.
The overall strategy is for the student to begin with a source algebraic expression in text
and for this source expression to be represented by a target product. The target may be
something dynamic such as music, animation, or a story or something physical and
concrete such as a sculpture, architectural layout, or form of theatre.
2. Method
The aesthetic computing method is composed of the following steps. Some of these steps
may take different amounts of time to teach, depending on the grade, and some might be
merged with other step to save time.
mnemonic effect of using icons; (b) improved motivation for doing algebra; and (c)
improved grades in math class.
This document is divided into four parts:
1. Introduction what you are reading now, to create an overview of the aesthetic
computing method
2. Middle School Track lessons and handouts for Middle School: prealgebra or
algebra (courses: Math 6,7,8)
3. High School Track lessons and handouts for High School: algebra and
integrated math (course: Integrated Math 3)
4. Tools 2D and 3D tools (virtual algebra machine) to assist students and teachers
If you see a small icon that looks like Rodin's The Thinker (the figure on the left),
you can probably skip this extra material unless you want more detail. When we
consider problem space versus solution space, these two phrases relate in ways
similar to syntax versus semantics (the structure of a thing versus its meaning, which is
often the interpretation, or behavior, of the syntax). You can also think of it from a
computing perspective: an equation is fed into a machine, which then operates on it,
producing output in the form of a plot, a rearranged symbolic expression, or something
else. Thus, mathematics has focused on novel representations for what comes out of the
machine, but not what goes into it.
1. Goal
The goal of these lessons, handouts, and implementation should be to provide a creative
exercise in teaching the subject of algebra that instills student interest in the subject. The
purpose is to help motivate the student in algebra by allowing them to be creative. The
most basic assessment strategy is a videotaping of the class, surveying of students after
the lessons are complete, and teacher interview as to the success or failure of the method.
The documentation is composed of this introduction to the aesthetic computing method,
followed by lesson plans, handouts, and assessment mechanisms. There are two tracks:
MIDDLE SCHOOL and HIGH SCHOOL. We recommend that teachers look at both
tracks and take examples that are relevant to their classes.
The overall strategy is for the student to begin with a source algebraic expression in text
and for this source expression to be represented by a target product. The target may be
something dynamic such as music, animation, or a story or something physical and
concrete such as a sculpture, architectural layout, or form of theatre.
2. Method
The aesthetic computing method is composed of the following steps. Some of these steps
may take different amounts of time to teach, depending on the grade, and some might be
merged with other step to save time.
STEP 1: Standard Notation this is where we begin, with traditional (standard) textual
algebraic notation for one or a set of algebraic expressions. The "grammar" of traditional
notation is discussed along with the implicit operations by virtue of their relative position
in the expression. For example multiplication are sometimes expressed implicitly as "ab"
2,, A ..
instead of"a*b", and powers as "a2 instead of "a 2" or "a*a".
STEP 2: Prefix Notation the standard algebraic expression is modified into a prefix
expression. This is still in text form, but the operators are located before the operands.
For example, a(b+c) becomes a (b + c) to first make all operations explicit, and then the
prefix notation is ( a, + (b,c)). In "computer parlance", this is also known as "functional
notation" since all operators (i.e., functions) are specified first and grouped with their
arguments (i.e., operands). The transformation from standard to prefix notation is easiest
in gradual steps: (1) a(b+c), (2) *(a,b+c), (3) *(a,+(b,c))
STEP 3: Expression Tree we take the textual prefix expression and convert it into an
expression tree with the root of the tree at the top and the leaves at the bottom. Thus, this
tree is inverted, or flipped, from what a tree normally looks like. A tree is composed of
nodes and branches, with external nodes (i.e., "leaves") and internal nodes. The
expression tree is created from the prefix notation expression by reading it from left to
right, and building the expression tree in a topdown manner from root to the leaves.
Operators appear as internal nodes, while variables and numbers are leaves. For example,
reading the prefix expression "*(a,+(b,c))" from left to right gives us a root tree node of
"*" with a left branch ending with a leaf "a" and a right branch ending with an
expression "+(b,c)". This expression is then expanded further to be a node of"+" with
two leaves "b" and "c".
The creation of the expression tree reinforces the order of operations. The order of
operations proceeds from the bottomup, or from the leaves of the tree up to its root.
STEP 4: Concept Mapping Key the first stage in mapping is to develop a key by
taking each element of the expression tree and mapping it to a corresponding English
word or phrase. The key is like the key of a chart or map with two columns: the left
column containing the element (branch and node) of the tree and the right column
containing the target English representation.
STEP 5: Concept Map the mapping associations (left and right columns) inside the
key created in the previous step are applied to yield a modified expression tree which
now has English labels for both the tree nodes and branches. Each branch of the tree
should be represented with an arrow rather than a line if there is the possibility of
ambiguity. The arrow is useful in capturing the node relationship as with "a isplugged
into b", rather than "b ispluggedinto a", which has a different meaning. We call this
resulting tree, with arrows for branches, a concept map. This map is a visually
represented tree with node and branches labeled with English words or phrases.
STEP 6: Design Mapping Key the second mapping stage involves creating a set of
mappings, in the form of another key that maps the English word/phrase concept to a 2D,
3D, or physical element.
STEP 7: Design Map The design mapping key is used to create a new tree that is
labeled with visual icons present from the previous step.
STEP 8: Representation The design map is "realized" into a 2D, 3D, or physical
representation using the design elements in the prior step. This realization may take
several forms: story, structure, or space for example.
This description of the aesthetic computing method is one that is made specific
to teaching algebra. More generally, the method is slightly altered:
Step 1 involves textual notation for a mathematical structure which may be a
data or program structure, in addition to an expression. Step 3 is generalized to a
graph, rather than a tree.
3. Mapping Strategies
Steps 4 through 8 in Sec. 2 require creative mapping approaches. There are some
guideline templates that will help this process. They are based on whether we consider
sample domains, tree mapping as a whole, or whether we are simply looking at potential
mapping possibilities for individual tree nodes and branches.
3.1 Domain, Style and Case Study
The first strategy might be to consider a specific application theme or domain of interest.
For example, a domain might be theatre, cinema, architecture, or sculpture. Within each
domain, there are styles and case studies. For cinema, we might have fantasy as a style,
and then Harry Potter and Lord of the Rings as specific case studies for the fantasy style.
For architecture, we might have a modern style with Bauhaus or Gaudi serving as the
targeted case studies. In some cases, the teacher may wish to constrain the assignment to
a given domain, style or case. Or, students could be free to choose.
3.2 Category
The categories story, structure, and space serve as useful archetypes to help constrain and
guide mapping.
Story: operations will be internal nodes in the tree diagram, and will usually map to verb
phrases with other nodes being nouns. The branches would be adverbs or left unlabeled.
Structure: all tree nodes will be physical objects, and branches will either be connecting
objects (wires, tubes, cables) spatial relations (i.e. adjacentto, below, nextto, ontopof
etc), or containment relations (i.e. insideof, outsideof).
Space: all tree nodes will be spaces, and branches will be connecting or containment
relations as in Structure.
The story is the only type that involves the flow of time and it must be made clear as to
which nodes occur before other nodes (i.e., node order) to "tell the story." For example,
one might say that the story begins at the leaf nodes of the tree and moves upward with
the passage of time. Explicitly labeling nodes or branches with numbers (i.e., 1,2,3 ...)
indicates the temporal order of events in the story. Separate branches, in such an
approach, may designate parallel story lines. For the story, the student should number
each node to tell in what order the tree is traversed.
Semantics that are used to label the tree can ultimately result in representations
that span a wide variety of human interfaces. For example, it is possible to map
music or sound instead of, or in addition to, existing relationsthus obtaining
hybrid audiovisual representations. Moreover, different media can be used in the
representation physical or computer generated.
3.3 Relation
SIf we look at the tree structure itself with nodes and branches, we are led to two
sorts of relations between two nodes: temporal and spatial. Nodes generally map
tophysical objects. A branch connects two tree nodes, and the semantics for this
branch depends on one of two cases:
If the branch represents a temporal relation between two nodes then we will use a
verb phrase for the branch semantics, and the resulting tree will be something
dynamic such as a story or animation with the flow of time being explicit.
If the branch represents a spatial relation between two objects then this relation
can be one of these types: proximity, connection, containment, which tend to
indicate prepositional phrases. Proximity relations suggest relative position and
include phrases such as nextto, above, below, infrontof. Connections are
physical objects that connect two objects such as a wire, tube, or pathway.
Connections are phrases with object arguments such as connectedto(wire3)
meaning connected to the object "wire3". Containment refers to encapsulation of
one object in another as in a Venn diagram.
4. Guidelines
4.1 Completeness: The mapping from source to target should be complete: all elements
in the source should be represented in the target.
There are two parts to this: completeness in syntax and in semantics. The syntax of an
expression is its denotational quality referring to its grammatical structure. The source is
an expression that contains many elements such as numbers, relations, variables, and
operators. In any representation of the source expression, we should try to ensure that
each of these elements has a counterpart in the target representation. This is so that we do
not miss out on mapping an element. This syntactical aspect of completeness is the most
important. The other part is semantic mapping we'd like to ensure that we map any
semantics in the source to the target. For example, if"x" represents a variable in the
source, then we need a corresponding thing in the target that represents a variable even if
it looks very different than "x".
Semantic mapping can become complicated. For example, there are many
expressions that have "implicit semantics". For example, representing Y = aX
+ bZ has no additional semanticsit is an abstract expression and will likely
not remind the student of anything in particular other than being an expression
containing variables, a "+" operator and a relation "=". However Y = aX + b is a "taught
template" for the equation for a line. If the student has been taught this previously, then
the preexisting knowledge of "line" represents information that may need to be carried
forth to the target in some way.
4.2 Consistency: The mapping from source to target should be consistent: similar
elements in the source should map to similar elements in the target.
There are two types of consistency, based on syntax and semantics. For syntax, we must
ensure that if we represent an "x" as a grey stone in the target, we must ensure to
represent all instances of "x" with grey stones. This can be thought of as consistency with
respect to instances.
Regarding consistency in categories (or types), in a source expression "x" and "y" are
both types of "Variable", which is the category. One would then expect that target
representations of "x" and "y" should also be related (i.e., fall within the same category).
It is not yet clear how stringent we should be on these categories. For example, if"x" and
"y" map to a red stone and a green stone, clearly the category of "stone" is at a fairly low
level, consistent with "variable". However, if we decided to represent "x" and "y" as a
pink bear and a blue lion, the common category of "animal" is at a fairly high level in
comparison.
" Another aspect of this rule to consider deals with the visual or structural
complexity of the objects used in the target representation. A stone is fairly simple
in structure, but a lion has a lot of structure and attributes with preconceived
student notions of what a lion looks like, or the way it behaves. The potential trap is that a
student may become confused with the complexity of a target, imagining that certain
object attributes (i.e., the size or color of the lion) has some meaning when reflecting, say,
the magnitude of the variable "X". The question is: what target object attributes do we
ignore during sign formation? The mapping chosen may be singular to one specific
algebraic expression, or it may provide scalability. This suggests a rule in mapping that
hints that target representations that are scalable (when scaling the source), may represent
a more powerful means for representation. Scaling is difficult, however, for a story
mapping, since stories will tend to work well for that one algebraic expression, and then
fail if other variables or operators are added. Structures and spaces appear to have greater
scalability potential.
5. Assessment Methods
5.1 Teacher to student
In assessing the student's representation work, you may wish to consider the following:
The completeness of the representation (Sec. 4.1)
o Are the keys onetoone in terms of mapping?
o Can you go "in reverse" from the target to the source, given a key?
The consistency of the representation (Sec. 4.2)
o Take similar or identical elements in the target are they similar or
identical in the source?
The aesthetic quality of the representation:
o Are the target elements clearly perceived (visually or aurally)?
o Does it look like the student spent considerable time on the product?
o Does the target product have an appeal?
o Does the target product represent a strong degree of creativity?
5.2 Student to student
It might be useful for students to grade, or otherwise judge, others' work in a group
assessment.
5.3 Teacher to Curriculum
What are the teacher's views of the approach strengths and weaknesses?
5.4 Student to Curriculum
What are the student's views of the approach strengths and weaknesses?
6. Acknowledgments
Micah Branum created the figure on page 1. The field of aesthetic computing has been
forged through the activities of numerous University of Florida students. During the
Summer of 2005, two teachers, Jodee Alice Rose and Katrina (Katie) Indarawis of
Gainesville, Florida have been instrumental in putting together the attached two tracks,
with Jodee being chiefly responsible for the Middle School track and Katie for the High
School track. Both Jodee and Katie are recipients of the National Science Foundation
Research Experiences for Teachers (RET) award given for the Summer 2005 session.
The work performed by the author (Paul Fishwick) and teachers were funded in part by
the National Science Foundation under grant EIA0119532 entitled "An Investigation
into Aesthetic Computing within the Digital Arts and Sciences Curricula" with coPIs
Timothy Davis and Jane Douglas. There are many students who have in the past
contributed to the area of aesthetic computing. The students actively involved in aesthetic
computing at this time are Salam Daher, Hyunju Shim, Zachary Ezzell, Yuna Park, and
ChenYuan Huang. Minho Park finished the PhD and recently accepted a position as
Assistant Professor at Stephen F. Austin University in Texas. Zachary Ezzell is an
Undergraduate Engineering Scholar, 6 month appointment.
midJie school
The purpose of this level is to encourage creativity in representation, and to spend
additional time on the steps from the original expression to the final product.
Aesthetic Computing: Teacher Resources
Lesson One: Changing Mathematic Notation to Prefix Notation (2
days)
Objectives:
* Students will realize that math problems often use a shorthand notation that needs to be
expanded to gain full understanding of what is being asked of the student.
* Students will learn to group algebraic expressions by a main operator
Activities:
Objective One
* As a class, lead by teacher, students will identify different types of operators.
Questions to ask:
What is an operator? (Tells us what to do in a math problem)
What is another word we could use for that? (Signs)
List the different operators. (+, , *, /, powers, square root, division bar)
What are other ways we are told to do these operations? (2a, 2(a), 2/a, etc.)
Explain "implicit" versus "explicit". (Use examples from above to solicit responses
from students as to which are implicit and explicit)
* Teacher will present several problems with varying complication and show the students
where implicit operations are changed to explicit operations. (5 examples)
Work out the following problems by changing implicit operations to explicit:
6g = 6*g
2(a + b)= 2 *(a + b)
x(9c) = x (9*c)
14t 3/u =14*t3/u
2c + 4j 2/y = 2*c + 4*j 2/y
* Teacher will walk students through several examples (implicit to explicit) picking
volunteers and random students to answer aloud. (10 examples)
9(5y)= 9 *(5y) 4 + 7x= 4 + 7*x
2z(5g)= 2* z (5*g) x 3= x2 3
8z + 3= 8*z + 3 8(4x +9)= 8 (4 *x + 9)
6 (9+y)= 6* (9+y) 4y + 5y + 8y= 4*y + 5*y + 8*y
7y (4/h)= 7* y *(4 /h) (7z + 5)(2g + 3)= (7*z +5) (2 g +3)
* Students work on problem set. (WS or Overhead Set 20)
See attached problem set titled Changing Implicit Operations to Explicit Operations
Objective Two
* As a class, lead by teacher, students will discuss the order of operations.
Questions to ask:
What is the order of operations? (PEMDAS)
Why is the order of operations important? (It tells us what order to do a problem in
when there are several different ways. Without it, answers would be different
throughout a class)
Show an example of a problem using order of operation incorrectly and then correctly
4 + 10 *2 (wrong =14 *2= 28) correctt 4+20= 24). The answers are different!
What was the first thing we did in the last example? (adding)
What two parts were we adding? ((10*2) and 4)
Show students an example of Prefix Notation in three steps:
4+10*2
+ (4, 10*2,4)
S+ (4,* (10,2))
Does that make sense to anyone?
This is a thing called "Prefix Notation" That means what you are supposed to do (the
operation) is placed at the front of the problem. Pretty confusing, huh? Well, it
should help you to understand what is going on in a problem a little better.
* Teacher will present several problems with varying complication and show the students how
to group operations and operators in prefix notation. (5 examples)
As with most algebraic work, it will help the students to go through the three steps for each
problem to avoid confusion. The answers below only represent the final answer. Notice that the
students may write the words or use the symbols.
6g = *(6,g)
2(a + b) = *(2,+ (a, b))
x (9c) = *(x, *(9,c))
14t 3/u = (*(14,t) (3,u))
2c + 4j 2/y = (+ (* (2,c), (4,j)) / (2,y)) wow! That was hard work!
* Teacher will walk students through several examples (implicit to explicit) picking volunteers and random
students to answer aloud. (10 examples)
9(5y)= *(9,(5,y)) 4 + 7x= +(4,*(7,x))
2z (5g)= (*(2,z), (5,g)) x2 3= (^ (x,2), 3) or (* (x, x), 3)
8z + 3= + (*(8,z),3) 8(4x + 9)= (8, + (*(4,x) 9)
6(9+y)= *(6,+(9,y)) 4y + 5y + 8y= + (*(4,y), *(5,y), *(8,y))
7y (4/h)= *(*(7,y) ,/(4,h)) (7z + 5)(2g + 3)= *(+ (*(7,z),5) ,+(* (2,g) 3)
* Students work on problem set independently. (WS or Overhead Set 20 examples)
Use worksheet attached Titled Implicit Operations to Explicit Operations. Answers attached.
Wrap Up Questions:
How many of you felt that this method was interesting?
...Helpful?
...Really, really hard?
Aesthetic Computing: Teacher Resources
Lesson Two: Expression Trees
Objectives:
Identify operators in given math equation
Demonstrate knowledge of how the order of operations will affect the order of
branches on a tree
Using previous lesson (prefix notation), students will arrange various
algebraic structures into expression trees
New Vocabulary:
Define expression tree (branch, leaf, and root)
Activities:
Objective One was covered in the last lesson. Teacher should give students the following quick
quiz to test that objective has been met.
Identify the Operators in the following problems:
(7z + 5)(2g + 3) = (7*z +5) (2 *g + 3) x3 +6k = (x*x*x) + 6 *k
9(5y) = 9*(5y) y/3(5) = y/3 5
14t3/u =14*t3/u 4+9c =4+9*c
5 3y 2 =53*y*2 q(uiz) =q*u*i*z
Review answers with students.
Objective Two and Three:
Review lesson one's results briefly (Remember how we grouped the problems by
operators? It was easy for some and more difficult for others. We will be doing
something like that today, but a little different, and more fun)
Questions:
What was an operator? (+,, *, /, etc.)1
How did we figure out which operator to put out front in the sets we did?
(Whichever operation was last in PEMDAS.)
How many of you have heard of trees? (All will) and in math? (Tree diagrams are
used for prime factorization and branch and leaf diagrams are somewhat like
trees)
Introduction: "Today, we are going to be making math problems into to 'trees'. What are the
parts of a tree? (Wait time) Well, our trees will have branches, leaves, and roots. I will show you
two examples and then we will do several together and then you will work on some on your own.
We are going to use the problems we have already worked out so that we have even less work to
do! Please take out your Lesson One worksheet."
Objective Three:
Using the handouts for the previous two assignments, student should put their prefix
notation problems into tree diagrams. Teacher will lead in two examples and then solicit student
1 Teachers may wish to choose whether to use "/" or "+" for "division", or both.
help for 4 of the previously worked out examples (see lesson one) and the teacher will assign
student to do 18 on the lesson one worksheet.
Assignment:
Work out the following problems FOR the students.
yV +by mx+b(y+r i, 4264*z 6  ((4,2),6)
b S
m 4 z
1. 2.
As you work, explain that the circles are the nodes, the lines are the branches and the
leaves come off of the branches. It will help the students to see the correlation if the teacher talks
through the prefix notation as a guide of what to do next.
Example of "out loud thinking" for example one: Y equals MX plus B. Okay, so if we put that into
expanded form, we have "Y equals M TIMES X plus B" okay! Now if we put that into "prefix
notation" we have this: Equals at the front, because that is the last thing we have in a problem,
right? So, Equals, what equals, hmm, well, Y equals something so, Y. And then what else? Hmm.
Oh, What kind of problem is on the other side of the equals sign? Well, it is an addition problem.
We are adding a multiplication problem and the variable B. Okay, so what should go out front? It's
a fancy addition problem so, + should go out front, okay so now we have "=(y, (+..." what now...?
Oh okay, the multiplication of m and x. that looks like *(m, x) and then we are adding b to it, so it
is + (*(m, x), b) okay, whew, got that, now how do we put it into an expression tree?
Equal came first so we put that in the top ROOT (It's just a circle, don't get nervous). Then we
will have two branches coming from the root. Why two? (There were two sets in the prefix
notation). On the left one, I put a Y underneath. Notice I didn't put a circle, you don't have to.
Now on the right, at the bottom of that branch I am going to put another root and in it I am going
to put a plus. How many things was I adding in the problem? Two really; a multiplication problem
and B. So I have two stems coming off my plus root. Got it, almost there! On the left branch I am
going to do ANOTHER root. Remember, that is a circle. Inside it, I will put a multiplication sign.
Then on the right branch I will put a B. Are we done? Nope, not quite yet. We need to put our last
branches and leaves on the tree. How many things were being multiplied? Two? So two stems
and then leaves of m and x. Now we are done.
Work out the following problems WITH the students
3. 6 (7+) 4. y2 + 7 6 4x+2 bh
(6,+ (7,x)) + y,2),7) + ((4,x),2) 2
S,/ (* (bh),2)
r xy 2 b
After working put problems with students, assign #18 on lesson one worksheet as class
work/homework. Check in class on a board/overhead the following day.
Wrap up questions:
Did anyone think that making expression trees was really easy once you had the Prefix Notation
in front of you?
Do you think you could solve these problems more easily now that you have seen their trees?
Aesthetic Computing: Teacher Resources
Lesson Three: Concept Mapping
Objective
Using previous two lessons, the students will learn to associate trees and their
connections with English meanings to make the trees more interesting to the students.
Students will utilize familiar expression trees and attach English semantics, as discussed
in class to make logical connections on their trees.
Words to know:
Proximity
Containment
Connectivity
Introduction: "Today we are going to do something that won't seem like math at all. We are
going to use all of the lessons we have done so far with aesthetic computing to accomplish it
though. Please take out your expression tree homework."
Have completed expression trees available for students who were absent or don't have their
homework.
Activities:
While students are taking out materials, pass Concept Mapping Rule and Guideline
Sheet to each student. Read over instructions for concept mappings as a class.
Present the students with an example of a concept map.
Using a(b c)
asugar
Into cup
L on
b blueop c redcup
Try a concept map with students. You may choose to use the map given as a guide and may
prompt students with the following theme.
Structure Candy Mobile
Using aA2 + bA2=A2
2
a 2b
What roots must be the same? Which leaves?
What candy could we use as connectors? As roots?
Do you think you could build something like this?
Label each root, leaf and branch as the class comes up with
consistent and follows all of the rules on the sheet.
it. Make sure that it is categorically
Class work:
Have students choose and copy three of their completed expression trees and try one of
each type of concept map (story, structure, space). Remind the students that they must actually
write the English words on the expression tree. Like in an English class, the students should be
encouraged to brainstorm before they start their final concept maps.
Inform the students that they will be choosing one of these concept maps for a final
project during the next lesson.
Wrap up Questions:
How many of you knew that you could connect English or Art or Architecture to math?
How many of you enjoyed changing trees into concept maps?
What other rules or tips could you add to the rule page to make concept maps easier to do?
Ask:
Concept Mapping Rule and Guideline Sheet
RULE 1: The mapping from source to target should be complete that is all elements in
the source (numbers, relations, variables, operations) should be represented in the target.
This means everything in the problem needs to be represented by SOMETHING
in the final product.
RULE 2: The mapping from source to target should be categorically consistent.
Categorically consistent? It means that operators should all be in the same
category (or family) and the same with numbers and variables and relations. For example,
saying x is cheese and y is a gopher is not categorically consistent (no matter how much
you like cheese and gophers!).
Guidelines:
1. First you must create an expression tree from your source (source here means
problem). It is best to put your source into prefix notation as this makes it MUCH
easier, and your teacher will probably require it.
2. Then take the basic expression tree and perform a semantic tagging procedure,
which means that each root, leaf and branch in the tree must be clearly labeled with
an English phrase.
3. Next, make sure to make a Mapping Key. It is important to make a key so anyone
who looks at your work can know which word relates to what part of the expression
tree. This also helps because sometimes when creating a concept map, your words
may get all scrunched up. This way you can keep your thoughts clear and neat.
4. Choose from three different types of representations:
a. Story Here you will label the roots with verbs and the leaves with nouns.
The branches will be adverbs or indicate the passage of time (next, after,
then, etc.). Time will usually go from the leaves up. You should number the
story sequence on your concept map. Remember, two things CAN happen at
the same time (if the leaves are on the same level this might apply to you!).
b. Structure Here all the roots and leaves will be physical items. The
branches will connect the objects, contain them, or tell how close they are to
one another (ex., tube, wire, inside, around, on top of, next to, etc.). Your
final product will look like a sculpture with this choice.
c. Space  Here the roots and leaves will be rooms or areas and the branches
will either connect the areas or contain them. The final product will look like
a place (ex. Dorm, garden, spaceship, etc.).
Aesthetic Computing: Teacher Resources
Lesson Four: Representation
Objective:
The students will be asked to make a final representation of one of their previously
tagged concept maps.
The actual representation maps the tagged tree diagram to a target object. That object may be in
various forms: (a) a drawing or painting, (b) a 2D or 3D computer model, or (c) a physical model
made of a material such as paper, cardboard, clay, or wood.
Introduction: "Remember, last lesson I told you that we would be using those concept maps for a
special project. Well this is it. Today is the day. Today, we will begin the final stage of this unit.
I will be asking you to actually create the ideas that you started last time. What do I mean?
Remember the candy mobile we talked about last time. Imagine that we actually had to make
that for a grade. Do you think you could do it still? Well that is similar to what I am asking of you.
What were the three types of representations we talked about last time? (Wait time) Right,
story, structure and space. So here is the assignment. You will be asked to either draw, or build
your concept map. What do I mean? Let's say I had to make that candy mobile, I could either
draw (or paint) it, model (or draw) it on a computer, or go out and buy the candy and stick it all
together and bring it in. That is what you will have to do. Of course, since we did a candy mobile
in class, I am asking that you not copy that idea, which is why you did your own maps. I will show
you some finished examples. If you would please take out your completed concept maps, we can
get started."
Activities:
Show students examples provided of different types of representations.
Pass out guideline sheet and set due date on the project.
Have students write up a proposal of what type of project they would like to create.
Posting the following questions for the students will make it easier for them to focus their
ideas which the students should submit for your review.
o What is the expression you are using?
o What does its expression tree look like? Copy your concept map.
o What kind of metaphor have you chosen (story, structure, or space)?
o What is your theme (ex. skateboarding, nature, manga, food)?
o Do all of your roots, leaves and branches match in theme? Explain how they
match.
o Have you made both a mapping key and a design key? Attach both.
o Draw a labeled sketch of your final product.
Have students meet with the teacher to finalize their plans.
Collect project on due date and grade using attached rubric.
Aesthetic Computing: Teacher Resources
Lesson Four: Grading Rubric
Requirements:
Students will turn in a written proposal for their project (6 pts)
o Points based upon six questions given to the students to answer.
Project is worked on during given class time (3 pts)
o Three days maximum should be given in class, and points can be deducted for
offtask behavior.
Project is turned in on time (5 pts)
o Points may be deducted for each day the project is late.
Student uses good craftsmanship (5 pts)
o Points can be deducted per major construction/ cleanliness issue.
All nodes are categorically consistent (8 pts)
o Teacher may choose to give multiple points per correct node, or deduct for
inconsistencies.
Theme is apparent and consistent (5 pts)
o Teacher may deduct per inconsistency.
Student turns in both the mapping and design key (2 pts)
o Teacher should deduct points for incomplete keys.
Project conforms to proposal idea (6 pts)
o Teacher may choose to deduct points per unnecessary change.
Total points possible = 40
Grading Scale
"A" = 40 to 36 points
"B" = 35 to 32 points
"C" = 31 to 28 points
"D" = 27 to 24 points
"F" = below 24 points
Name
Class Period
Representation Guidelines and Instruction Sheet
Due Date
Using a completed concept map from our aesthetic computing lessons,
you are being asked to create an art project. Your project may be 2dimensional
(drawing, painting, collage, etc.) or 3dimensional (sculpture, mobile, model, etc.)
or created on a computer in either 2 or 3 dimensions.
Use the following guidelines to achieve the best grade possible.
Have a strong theme in mind, something that means something to you. It
is always easier (and more fun) to do something if you are interested in it.
Always make sure your categories are consistent. Even if you think they
match, it is always a good idea to ask a friend, family member or other
teacher to see what they think.
Before you turn in your work, make sure that you have completely
represented ALL of the roots, branches and leaves on your expression
tree. Having a design key will keep you on track.
o To make a design key simply draw or write what symbol stands for
what. That is, if you have a peppermint that stands for c, draw
"peppermint=c". Do this for each part of your concept map.
Use good craftsmanship. That means no glopping glue or paint. In other
words, make it look good, like you put TONS of effort into the assignment.
Keep all documentation for this project. All of your trees, maps, keys and
questions will be due with the project on the due date. Points will be
deducted from the final total for missing information.
With that said, your project is due on the date above. At that time, you must turn
in a packet containing the 7 questions from lesson four WITH all the
documentation the questions request (attach your key, mapping, etc.). You must
also turn in your completed project at the same time. Points are deducted for
late assignments. For more information, you may ask me anytime before the
due date (Please make it before the last day!). You will be given a maximum of
three days to work on this project in class. Beyond that, the final project is your
responsibility. This should be a fairly easy and fun project so don't let it get you!
Aesthetic Computing: Teacher Resources
Dorm Life Example of Structure
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Aesthetic Computing: Teacher Resources
Dorm Example of Space
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Aesthetic Computing: Teacher Resources
Burger Shop Example of Story
I (,
*s N
Joe is a chef at Burgeroo Fast Food Palace. He absolutely loves his job. Everyday he goes to
work first thing in the morning and mixes the spices and meat together. Then after he had made
all of the patties, he puts them in a frying pan and cooks them until they are well done. The next
step is to prep the buns. Once Joe has cooked up all the burgers, he adds them to the buns and
puts them in their special yellow wrapping. Everyone loves Joe's burgers. In fact, you can
always find a line of people outside Burgeroo Fast Food Palace who are just waiting for a bite!
blaim r BERM
y=mx + b y=m*x+b> =(y,+(*(r x),b)) ,...
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high school
The purpose of this level is to encourage creativity in representation, and to alleviate
student frustration and lack of motivation for the trigonometric identities.
Aesthetic Computing: Teacher Resources
Lesson One: Expression Tree
Objectives:
Students will be able to construct 'trees' given simple algebraic equations.
Students will be able to match the correct tree to trigonometric identities.
New Vocabulary:
Define expression tree
Activities:
Objective One
Intro: "Today we are going to try something new. We will start with some
simple equations and turn them in to what we call 'expression trees.' The way this
is going to work is that I will do two of them on the board, then we will do two
together, and then you will do four of them on your own. Any questions?"
On the board, teacher writes: y = mx + b
Questions:
o "What is the primary operation? Answer: "addition" "mx + b"
o "What are we adding together?" Answer: "mx and b"
"So, we can start our tree like this:
y +
x (b
Question: "Are there any other operations in this that we haven't dealt with yet?"
(wait time) Answer: "mx is multiplication." "Great! So mx is multiplication
and we need to expand that out in our tree."
On the board, teacher adds...
b
x
Teacher will continue in this manner for the following examples:
Teacher does: a) y = mx+b b) x2 + y2 = 1
Together do: c) xy = 1 d) y2 = 1 + x2
Students do: e) 2y = xy2 f) y/2 = x/(l+x)
g) y/2 = (1x)/z h) y = x(90z)
As students are working, the teacher will be circling the room and helping
students. When they are finished, teacher will go over e)h) above (that was
assigned to students).
If the teacher decides that the students need a little more practice with these, have
the students do the following:
1. y= b + c 2. y = a/b 3. y = 1/x2 4. y = c (a/b) 5. y = a (b + c)
Objective Two
If the above five problems were assigned, the teacher will go through the answers
and answer any questions as necessary. Otherwise,
The teacher then puts the transparency of the trigonometric identities on the
overhead. The students are asked to copy these down in their notes and then to
observe the forms of the equations.
"Let's compare these identities to the forms of the equations we just did. For
example, let's look at x2 + y2= 1. Which trig identity has the same form as this
equation? (wait time) Answer: "cos20 + sin2 = 1"
"Now, categorize the rest of these equations by writing the correct letter beside
each equation that matches the form. Don't make this harder than what it is, it's
simply a matching game."
Teacher will walk around and help students, giving them approximately 8 minutes
to match all the equations with the corresponding trig identity. Then go through
the correct forms (approx. 5 min).
Questions: "Does everyone see how these identities can take the form of some
simple algebraic equations? Anyone having a problem with this or have
questions?" (wait time)
Tomorrow we will start getting creative with these expression trees.
X2 + y2 = 1 (Here
examples of expression trees)
XY = 1
Y2 = 1 + X2
2Y = XY2
Y/2 = X/(1+X)
Y/2 = (1 X)/Z
Y=B+C
Y = A/B
y=1/X2
Y=C
Y=A
Pythagorean Identities:
Cos2 0 + sin2 e = 1
1 + cot2 e = csc2
Reciprocal Identities:
1 + tan2 e = sec2
sin e = 1/cos e
csc e = 1/sin e
cos e sec e = 1
tan e = l/cot e
cot e = 1/tan e
sin e csc e = 1
cos e = 1/sec e
sec e = 1/cos e
tan e cot e = 1
tan e = sin e/cos e
cot e = cos e/sin e
The cofunction Identities:
sine = cos (90e)
tane = cot (90e)
sece = csc (90e)
cose = sin (90e)
cote = tan (90e)
csce = sec (90e)
Aesthetic Computing: Teacher Resources
Lesson Two: Semantic Tagging and Visualization
Objective
Students will be able to add English meaning to the links of their tree diagrams.
Students will be able to understand proximity, containment, and connectivity in
the context of aesthetic computing.
Words to know: Proximity, Containment, Connectivity
Activities:
Objective One
Intro: "Today we are going to learn how to turn these expression trees into
works of art. I want to show you an example of what I will be asking you to do. I
will give you a handout with more specifics at the end of class. This example
uses the y = mx + b equation. Each of you will be given a different equation for
your project."
"Now let me explain a little about each of these examples. First, the dorm life
example. This is an example of a 'structure'. Each part of the equation is
represented by something in the structure. Notice the expression tree in its basic
form and then another one to the right of it with the words actually attached to the
parts of the tree that mean something."
representation is using the key to the right of the second expression tree.>
Teacher goes through the other two examples, explaining the 'space' (dorm)
example and the 'story' (burger) example. Teacher should emphasize the words
being placed on the expression tree, and the key that explains what each part of
the equation is represented by.
"So these are some examples of what will be expected of you for your project.
Obviously, you may not have the resources to do this on the computer and that is
okay. This project will be hand written for most of you. If you choose to do
something like a story, you may not need pictures at all...the story would describe
everything that is going on. You may use pictures if they pertain to your idea, but
don't box yourself in to one idea based on one particular picture you are dying to
use."
"So let's go through a few things before I give you the formal writeup of the
project. If you notice in all these examples, there are categories of items in your
equations. For example, in y = mx+b,
We have 'm' and 'b' as constants.
'x' and 'y' are variables.
'+' and '*' are operations.
And '=' is a relation."
Teacher then explicitly shows the students how each of the parts of the tree
diagram have English semantics added to it. This is done by literally writing the
English words for these representations next to the corresponding parts on the tree
diagram itself.
Objective Two
"Now, there are three styles of metaphor that can help you in deciding what each
part of your tree should be represented by. That is,
o Containment An example of containment would be something like
addition being represented by two objects inside a container of some sort.
o Proximity An example of proximity is two objects next to each other
representing multiplication. It's something that is "next to," "above,"
"beside," etc.
o Connectivity Connectivity two objects being connected by something to
represent addition. The connection could be anything from a vine to a
path or even just a line or string."
"For your project, you will choose one of the trig identities to change. You will
make it your structure, space, story, or whatever art form you choose to express
this equation. The first step we did together in class."
"Please take those expression trees out for just a minute. (wait time for the
students to get out their last assignments). Everyone should have his/her
'expression trees' out. When you change your equation from a regular equation to
a trigonometric identity, there is just one adjustment you need to make to your
expression tree. In the same way that a '+' or '' is treated as a function, the sin,
cos, tan, sec, csc, and cot, needs to be treated as a function. For example, the
expression tree for sin 0 = 1/cos 0, is as follows:
sin
0 1 co
"This should not be too big of an adjustment from the expression trees you have
already completed, therefore, this should be the easiest part of your project.
Teacher hands out the "Steps to get started" and the write up of the project.
"The first page I gave you outlines some steps to get you started. The last step
describes what you will be physically turning in to me at the end of this project. It
will be different depending on what you decide to do for your project. You will
then fill out the last page and turn it into me by tomorrow. Any questions?"
Aesthetic Computing: Teacher Resources
Dorm Life Example of Structure
dwrm 1~P
Y= g+ b Ymfrx+b (y,+(*&hx),b))
= wdM
remlt, h
y Y bill
.9 2lPhw
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plug 21.0
prtata
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ius .6tuFIor
b is Lws Is
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7 i nI Nil
Wmo~e m M~t
In a iait
of MOIM
Aesthetic Computing: Teacher Resources
Dorm Example of Space
dprm (mx)b))
cbrm
ymx + b> y=m*x+b > (y,+(*(mx),b))
yviwo
Don
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y is livij rom
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Iroahe te o grpiw
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Aesthetic Computing: Teacher Resources
Burger Shop Example of Story
S,::'..':'..":...
49 .. .....
Joe is a chef at Burgeroo Fast Food Palace. He absolutely loves his job. Everyday he goes to
work first thing in the morning and mixes the spices and meat together. Then after he had made
all of the patties, he puts them in a frying pan and cooks them until they are well done. The next
step is to prep the buns. Once Joe has cooked up all the burgers, he adds them to the buns and
puts them in their special yellow wrapping. Everyone loves Joe's burgers. In fact, you can
always find a line of people outside Burgeroo Fast Food Palace who are just waiting for a bite!
blonrer Ba
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Aesthetic Computing: Additional Examples
sin 9 coo (900)
sn 0
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Alien Invasion
mother ship
in
saucer GSaucer
sin co
pilot tco~pe
in skhttle
n
90
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invaders
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sin is saucer
cos is saucer
 Is escape shuttle
theta Is pilot
90 Is alien
invaders
branches represent
inside
note: "sin" and "cos" are assumed to be written on each saucer, differentiating them
sin e = 1/cos 8
light shade
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note: the two lightbulbs are the same but each accepts different wattage
Steps to get started:
Step 1:
Create your tree diagram and make sure it is correct.
Step 2:
Write out a list of all the variables, constants, and operations.
Step 3:
Write out what you enjoy doing or some things you are interested in that you could use
for this project. Use the following list of antonyms to help you or come up with some
more of your own:
shrinkgrow spendget paid lose gasrefuel
leak outpour in givereceive drinkfill up
accumulationdecretion expandreduce add colorblack/white
deepenshallow developmentdestruction bloatedshriveled
weakenstrengthen downgradeupgrade elevatedemote
extendcontract accumulatedispense groupscatter
adjoinseparate connectunfasten balloonshrivel
expand uponabbreviate transparentopaque clearunclear
Step 4:
Decide what your theme will be (based on your interests in step 3) and how you will
express that theme (through a story, a picture, a sculpture, etc).
Step 5:
Decide what will represent your variables, constants, and operations. Make sure that you
choose the same category for similar types of operators. Think about the three types of
metaphors: containment, proximity, or connectivity. Start making connections with
these metaphors. Remember you can always use shapes and colors to represent these
things as well.
Step 6:
Add English semantics to the tree diagram. In other words, label your tree diagram with
the words you used to describe each part of the equation in step 5.
Step 7:
Write (or sketch) everything out. If it follows more of a story, describe the story. If it is
represented more by a picture, sketch it out.
Step 8:
Type everything up, making sure to include:
1. Project title
2. Your name
3. Describe what each of the operations, constants, and variables are represented by.
4. Include a picture, sculpture, a 300 word description of your story, or any other form of
art you managed to represent your equation with.
Name:
The equation I chose to do my project on is
The correct tree diagram for this equation is:
Some things I enjoy are:
A possible theme for my project is:
Aesthetic Computing: Teacher Resources
Lesson Four: Grading Rubric
Requirements:
Students will turn in a written proposal for their project (6 pts)
o Points based upon six questions given to the students to answer.
Project is worked on during given class time (3 pts)
o Three days maximum should be given in class, and points can be
deducted for offtask behavior.
Project is turned in on time (5 pts)
o Points may be deducted for each day the project is late.
Student uses good craftsmanship (5 pts)
o Points can be deducted per major construction/ cleanliness issue.
All nodes are categorically consistent (8 pts)
o Teacher may choose to give multiple points per correct node, or deduct
for inconsistencies.
Theme is apparent and consistent (5 pts)
o Teacher may deduct per inconsistency.
Student turns in both the mapping and design key (2 pts)
o Teacher should deduct points for incomplete keys.
Project conforms to proposal idea (6 pts)
o Teacher may choose to deduct points per unnecessary change.
Total points possible = 40
Grading Scale
"A" = 40 to 36 points
"B" = 35 to 32 points
"C" = 31 to 28 points
"D" = 27 to 24 points
"F" = below 24 points
software
tools
These tools allow the student and teacher the potential to interact with algebraic
expressions. The 2D tool is specifically targeted toward moving numbers and variables
for basic arithmetic, and illustrating the distributive property. The 3D tool may be used in
one of two directions: given an algebraic expression, derive a 3D model, and vice versa.
2D Virtual Algebra Machine
This program is interactive and allows the student to view the distributive property of
algebra. Parenthetic grouping is replaced by a circle, Numbers may be moved on top of
one another to add or multiply. Dragging the "4" circle through the circle containing
"2+3":
distributes the "4". The student may, for example, drag the 4 closer
to the 2, which yields an "8".
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00 UMH
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3D Virtual Algebra Machine
The 3D Virtual Algebra Machine is a piece of software that demonstrates some of the
fundamental concepts behind Aesthetic Computing. The software takes traditional
algebraic syntax and represents it with a 3D expression tree. The 3D tree is composed of
nodes that are connected by tubelike edges. The resulting expression tree is a full
binary tree (each node has zero or two children). Variables and constants (x, y, 4, 12)
are children, and operators (+, , *, /) are roots. Since all operators are roots that have
two children, they are referred to as connectors. The set of connectors is defined in
figure 1.
ADDITION SUBTRACTION MULTIPLICATION DIVISION
Figure 1
Connectors
A sample output can be seen in figure 2. This particular expression is 20(x + y).
Figure 2
20(x + y)
Figure 3 showcases the Distributive law of algebra, A(B + C) = AB + AC. The software
is still in its development phase so the variable names have been superimposed on the
individual roots.
Figure 3
A(B + C) AB +AC
The engine behind the 3D virtual algebra machine will allow for custom renders of a
given expression tree. This means high quality still images can be rendered with custom
textures and camera angles. This is demonstrated in figure 4.
Figure 4
Custom Renders
To produce an expression tree the user will interface with a calculatorlike control system.
Figure 6 provides a digital sketch of the 3D Virtual Algebra Machine's userinterface.
Figure 6
UserInterface Sketch
