Students discuss the components of their favorite videogames, and discover that they can be reduced to a series of coordinates. They then explore coordinates in Cartesian space, and identify the coordinates for the characters in a game at various points in time. Once they are comfortable with coordinates, they brainstorm their own games and create sample coordinate lists for different points in time in their own game.
In workbooks, students create a data model that describes a simple videogame
Students identify the coordinates of characters in a picture
Students complete the Videogame Design Worksheet to design their own game
Students convert several arithmetic expressions between multiple representations
Standards and Evidence Statements:
Standards with prefix BS are specific to Bootstrap; others are from the Common Core. Mouse over each standard to see its corresponding evidence statements. Our Standards Document shows which units cover each standard.
5.G.1-2: Graph points on the coordinate plane to solve real-world and mathematical problems
Use a pair of perpendicular number lines, called axes, to define a coordinate system
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.OA.1-2: Write and interpret numerical expressions
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
6.NS.5-8: The student performs operations with negative numbers, works with the number line and coordinate plane, order and absolute value of numbers, and solves real-world problems with rational numbers.
graphs of negative numbers on the number line.
A-SSE.1-2: The student interprets the structure of expressions to solve problems in context
interpretation of complicated expressions by viewing one or more of their parts as a single entity
N-Q: The student reasons quantitatively in using units to solve problems
selection of (Identification of) appropriate quantities for the purpose of descriptive modeling
BS-CE: The student translates between structured expressions as arithmetic, code, and Circles of Evaluation
translating a simple (1-operation) equation into a Circle of Evaluation
translating a nested (multi-operation) equation into a Circle of Evaluation
translating a Circle of Evaluation into its equivalent arithmetic expression
translating a Circle of Evaluation into its equivalent programming syntax
BS-IDE: The student is familiar with using a REPL, entering expressions properly, and interpreting error messages
enter and evaluate expressions on the computer
look to error messages as a way of diagnosing syntax errors
BS-M: The student models a problem in context and determines the data needed to describe the problem
identifying which quantities are fixed and which are variable
BS-PL.1: The student is familiar with declaring values and applying built-in functions using the programming language
interpreting a function application and identifying its arguments
Length: 85 Minutes
Glossary:
apply: use a given function on some inputs
arguments: the inputs to a function; expressions for arguments follow the name of a function
circle of evaluation: a diagram of the structure of an expression (arithmetic or code)
definitions area: the text box in the Editor, where definitions for values and functions are written
editor: software in which you can write and evaluate code
error message: information from the computer about errors in code
evaluate: perform the computation in an expression, producing an answer
evaluates to: the result of computing an expression
expression: a computation written in the rules of some language (such as arithmetic, code, or a Circle of Evaluation)
function: a mathematical object that consumes inputs and produces an output
interactions area: the text box in the Editor, where we enter expressions to evaluate
legal expression: code that follows the Code Rules for a language
mathematical expression: a computation in arithmetic
programming language: a set of rules for writing code that a computer can evaluate
value: a specific piece of data, like 5 or "hello"
Materials:
Computer for each student (or pair), running WeScheme or DrRacket with the
bootstrap-teachpack installed
Welcome to Bootstrap!Welcome to Bootstrap! In this course, you’ll be learning a new programming language - a way to tell computers exactly what you want them to do. Just like English, Spanish or French, a programming language has its own vocabulary and grammar that you’ll have to learn. Fortunately, the language you’ll be using here has a lot in common with simple math that you already know!
Set expectations and rules for the class.
Connect the material to come with things students already know:
What makes a language? Do some students already speak more than one language?
Programming is very much a language, with its own syntax, semantics, etc
Ask students about their favorite videogames. Be open about the scale of a modern game: they cost millions of dollars, are built by huge teams of programmers and artists, and are developed over years.
Set expectations about what can be accomplished in a single, introductory programming class.
Dissecting a Demo
Overview
Play a simple game, then take it apart and figure out what’s going on in the computer.
Learning Objectives
Model a simple videogame
Evidence Statementes
Students will be able to identify the elements of a game and how each one changes during gameplay
Product Outcomes
In workbooks, students create a data model that describes a simple videogame
Preparation
Dissecting a Demo(Time 10 minutes)
Dissecting a DemoLet’s begin by exploring a simple videogame, and then figuring out how it works. Open this link
to play the game, and spend a minute or two exploring it. You can use the arrow keys to move
the cat left, right and jump(up and down) - try to catch the ruby and avoid the dog!
[Video] Show the kids NinjaCat, either letting them play or by demoing it in front of the class. You can move the cat up, down, left and right by using the arrow keys.
Play until the students see everything in the game (the clouds, the ruby, the dog and the cat), and be
sure to point out the title and score as well. Give them at least 3 minutes to play with the game and listen to their observations.
This game is made up of objects, each of which has its own behavior. The Ruby moves from the right to the left, as does the Dog. NinjaCat only moves when you hit the arrow keys, and she can move left, right, up and down. We can figure out how the game works by first understanding how each object works.
Turn to Page 2 in your workbook. Here we have a table, which we’ll use to reverse-engineer NinjaCat and see how it works. Fill out the first column with a complete list of all things in the game. (Remember to fill out the first column only! Place a paper or notecard over the 2nd and 3rd columns, if needed.)
This can be a team activity. Solicit a few examples from students, to make sure they understand what’s being listed here: nouns like "the cat", not verbs like "jumping". During this activity, walk around and see how groups are doing.
This is the time to encourage expectations of community and respect - call out good teamwork when you see it! When time is up, give them a countdown: "30...10... 5... 4... 3... 2... 1... PENCILS DOWN, EYES UP HERE!" Wait for total silence and complete attention.
Did you list all four moving characters? What else is in the game? Is the background a part of the game? The score?
Have students come up to the board, and fill in the table with what they’ve found. If time is short, call on students and have them tell you what to write. Emphasize "Attending to Precision" with language when giving feedback.
We also need to think about what is changing when we play the game. What about the Ruby? Does it get bigger? Does it change color? Does it spin around? The only thing that changes about the Ruby is its position! Everything else about it is the same. What about the Dog? The Cat? Fill in the rest of the second column, listing everything that changes about each character.
Did you notice that the Dog, Ruby, Cloud and Cat only change position, and nothing else? What about the Background - it doesn’t change at all, so we can leave that cell in table blank! Did you notice the Score changes value? You’ll want to add that to your table, too!
Quickly fill in the second column on the board. Assign scores.
Coordinate Planes
Overview
Students must use the Cartesian plane to describe the location of characters in a picture.
Learning Objectives
Understand the use of number lines in defining locations via coordinates
Evidence Statementes
Students apply the number line as a tool for objectively defining location, first identifying how to apply it to
describe the position of a character onscreen in one dimension. This understanding is extended to two dimensions,
ultimately motivating the use of a coordinate grid.
Product Outcomes
Students identify the coordinates of characters in a picture
Coordinate Planes(Time 15 minutes)
Coordinate PlanesComputers use numbers to represent a character’s position onscreen, using number lines as rulers to measure
the distance from the bottom-left corner of the screen. For our videogame, we will place the numberline so that the screen runs from 0
(on the left) to 640 (on the right).
The key point for students here is precision and objectivity. There are many possible correct answers, but students
should understand why any solution should be accurate and unambiguous. This requires students to propose solutions that share a common
"zero" (the starting point of their number line) and direction (literally, the direction from which a character’s position is measured).
Have students discuss this. If time allows, have a student leave the room and let other students come up with ways to write
down where the dog is on the board. Then remove the dog, call in the other student, and have the class direct them using
only what they’ve written down.Try positioning the characters at different places on the line, and have students tell you
what their coordinates are. Try giving them verbal descriptions (e.g. "The center of the screen", "Off the righthand
side of the screen", etc.) and have them translate those into coordinates.
We can take the image of the Dog, stick it anywhere on the line, and measure the distance back to
the lefthand edge. Anyone else who knows about our number line will be able to duplicate the exact position of the Dog, knowing only
the number. What is the coordinate of the Dog on the righthand side of the screen? The center? What coordinate would place the Dog beyond
the lefthand edge of the screen?
By adding a second number line, we can locate a character anywhere on the screen in either
dimension. The first line is called the x-axis, which runs from left to right. The second line, which runs up and down, is
called the y-axis. A 2-dimensional coordinate consists of both the x- and y-locations on the axes.
A coordinate represents a single point, and an image is (by definition) many points. Some students will ask whether a
character’s coordinate refers to the center of the image, or one of the corners. In this particular program, the center
serves as the coordinate - but other programs may use another location. The important point in discussion with students
is that there is flexibility here, as long as the convention is used consistently.
Suppose we wanted to locate NinjaCat’s position on the screen. We can find the x-coordinate by dropping a line down from NinjaCat and read
the position on the number line. The y-coordinate is found by running a line to the y-axis.
(If time allows, invite volunteers up to the board to try placing NinjaCat and the Ruby at different locations. Have students identify the
coordinates at the corners of the screen. For additional practice, have students bisect each side of the screen and label
the midpoints.)
A coordinate pair is always written in the form of (x, y). When we write down these coordinates, we always put the x before the y (just like in the alphabet!). Most of the time,
you’ll see coordinates written like this: (200, 50) meaning that the x-coordinate is 200 and the y-coordinate is 50.
Here are some sample teacher notes to accompany this point
Depending on how a character moves, their position might change only along the x-axis, only along the y-axis, or both. Look
back to the table you wrote in the workbook (Page 2).
Can NinjaCat move up and down in the game? Can she move left and right? So what’s changing: her x-coordinate, her y-coordinate,
or both? What about the clouds? Do they move up and down? Left and right? Both?
Fill in the rest of the table, identifying what is changing for each of your characters.
Turn to Page 3 in your game-planning workbook, and look at the
project sheet that has a picture of the NinjaCat game. What are the coordinates at the bottom-left corner
of the screen? Top-right? Center? For practice, label the coordinates at the midpoint of each side
of the screen, then label the coordinates of each of the characters.
Brainstorming
Overview
Students select the theme and characters for their videogame
Learning Objectives
Evidence Statementes
Product Outcomes
Students complete the Videogame Design Worksheet to design their own game
Preparation
Brainstorming(Time 15 minutes)
BrainstormingOn Page 4, you’ll find a planning template for you to make your own game.
Just like we made a list of everything in the NinjaCat game, we’re going to start with a list of everything in your game.
Have students put their names at the top of this page.
To start, your game will have with four things in it:
A Background, such as a forest, a city, space, etc.
A Player, who can move when the user hits a key.
A Target, which flies from the right to the left, and gives the player points for hitting it.
A Danger, which flies from the right to the left, which the player must avoid.
Have students walk through some sample themes, to make sure they understand the format of the game. For example: A football
game might have a quarterback for the player, a rival player for the danger, and a football as the target. A jungle game
might have a monkey as the player, a snake as the danger, and bananas as the target.
Now it’s time to get creative! Fill out Page 4 in your
workbook for your game, using your own player, target and danger.
Be sure to consult with every team. A lot of students will have trouble fitting their ideas into this
format, or they’ll struggle with coordinates. Be clear about what can and cannot be done! (e.g. - no
3d games, joysticks, multiplayer games, etc. Hint: students who have a well-worded description of their
images will be happier with any images you could find them. Try searching for "Person" versus "Jogger",
for example.
Order of Operations
Overview
Students identify the order of operations for complicated arithmetic expressions through diagramming (circles) and
evaluating (the numerical answer). This lesson introduces students to the Circle of Evaluation, which is a sentence
diagramming tool for arithmetic expressions. This is a powerful way to understand Order of Operations, as it forces
students to focus on the structure of expressions over their computed result.
Learning Objectives
Understand the structure of arithmetic expressions
Understand order of operations in an expression
Write expressions as Circles of Evaluation
Translate between Circle of Evaluation and arithmetic expressions
Evidence Statementes
Students will be able to explain why arithmetic expressions require structure
Students will be able to identify the order of operations in an expression that includes
adding, subtracting, multiplying and dividing
Given a Circle of Evaluation, students will be able translate the expression into the
arithmetic expression it represents
Given an arithmetic expression, students will be able to complete a partially-finished Circle
of Evaluation for the same expression
Given a bank of Circles of Evaluation and arithmetic expressions, students will be able to match them
Students will be able to translate the structure of an arithmetic expression into a Circle of
Evaluation
Product Outcomes
Students convert several arithmetic expressions between multiple representations
Order of Operations(Time 20 minutes)
Order of OperationsMath is a language, just like English, Spanish, or any other language. We use nouns, like "bread", "tomato",
"mustard" and "cheese" to describe physical objects. Math has values, like the numbers ,
or , to describe quantities.
Students should begin to look at math as a language. To help them with this, you can discuss syntax
and grammar of math, and its potential for ambiguity in the absence of a fixed order of operations. Help
students see that math is a way to communicate calculations among people.
Humans also use verbs like "throw", "run", "build" and "jump" to describe operations on these nouns. Mathematics has
functions like addition and subtraction, which are operations performed on numbers. Just as you can "slice a piece
of bread", a person can also "add four and five".
A mathematical expression is like a sentence: it’s an instruction for doing something. The
expression tells us to add 4 and 5. To evaluate an expression, we follow the instructions
in the expression. The expression evaluates to .
What does evaluate to? What does evaluate to? What does evaluate to?
Sometimes, we need multiple expressions to accomplish a task, and it will matter in which order they come. For exmple, if you were to write instructions for making a sandwich, it would matter very much which instruction came first: melting the cheese, slicing the bread, spreading the
mustard, etc. The order of functions matters in mathematics, too. If someone says "four plus two minus one",
they could mean several things:
Add four and two, then subtract one:
Add four to the result of subtracting one from two:
Write an expression of your own that can mean several things.
Depending on which way you read the expression, you might have very different results! This is a problem,
because we often use math to share calculations between people. For example, you and your cell phone company
should agree upfront on how much you will pay for sending text messages and making calls. Different results
might mean that your bill looks wrong.
We avoid problems by agreeing on the order in which to use the
different operations in an expression. There are two ways to do this:
We can all agree on an order to use
We can add detail to expressions that indicate the order
Write down one reason why it is important to have rules about the order of operations.
Mathematicians didn’t always agree on the order of operations, but now
we have a common set of rules for how to evaluate expressions. The pyramid on the right summarizes the
order. When evaluating an expression, we begin by applying the operations written at the top of the
pyramid (multiplication and division). Only after we have completed all of those operations can we
move down to the lower level. If both operations are present (as in ), we read the expression
from left to right, applying the operations in the order in which they appear.
This item covers PEMDAS (order of operations in ambiguous expressions). You can skip this item if you
do not need to cover PEMDAS.
In what order should we apply the functions in the following example?
One way to indicate the order of operations in an expression is to first
draw the expression as a diagram. This diagram is called a Circle of Evaluation. Here you can
see an example of a Circle of Evaluation, for the math expression .
[Video] This section benefits enormously from visual aids, diagrams, etc. Make sure you have plenty of board space to
draw examples!
Circles of Evaluation show the structure that’s going on inside an expression. All Circles of Evaluation have two rules:
Circle Rule 1: Each circle must have one function, which goes at the top of the circle.
Circle Rule 2: The inputs (in this case, numbers) are written below, in order from left to right.
Which rule does this Circle of Evaluation break?
Try drawing the Circle of Evaluation for each of the following
expressions:
Every Circle of Evaluation evaluates to the result of its
corresponding expression. For example, the circle on the right evaluates to -1, because the circle says to use subtraction (the function at the top) on the numbers in order (4 - 5).
It’s important for students to view the Circles of Evaluation as "just another way of writing arithmetic".
Have students discuss whether associativity and commutativity still matter here (they do).
To use multiple functions in the same expression, we can combine Circles
of Evaluation. Look at the Circle of Evaluation you’ve written for . We already
know that the Circle for will evaluate to , so we can replace the
number with the expression. Does this change what the expression evaluates to?
From this point forward, the Circles of Evaluation are your assessment tool for Order of Operations.
Quizzing students by asking them to correctly evaluate an expression is fraught with false negatives,
as many student might get the order right but may still have problems with basic calculations. This
method is also vulnerable to Commutativity, since a student will correctly evaluate
even if they get the order of operations wrong! Circles of Evaluation have neither of these flaws,
as they put the emphasis where it should be: exercising a student’s ability to see the structure
inside the arithmetic.
Finish writing the Circle of
Evaluation shown here, so that it represents
What does this Circle of Evaluation evaluate to? Let’s review how we evaluate a Circle:
We know we are multiplying because that’s the function at the top of the Circle (represented by an asterisk * ).
The Number is the first number in the multiplication, because it’s on
the left-hand side.
The second number in the multiplication is on the right-hand side. The right-hand
side has a separate circle, so we need to evaluate the number for that
circle. The second number is therefore the result of adding 4 and 5.
(the inner circle) evaluates to , and (the outer
circle) evaluates to . This circle evaluates to .
Convert this Circle of Evaluation into an arithmetic expression.
Work through several of these examples with students, asking them to come up with arithmetic
expressions and then convert them into Circles, or giving them Circles and having them
translate them back into arithmetic. When you talk about Circles, be sure to consistently use
the term function for what’s on top, rather than similar terms like "operation", "symbol",
"procedure", "name", etc. The Circles of Evaluation will help students see the similarity between
arithmetic functions (like ) and algebraic functions (like ) if you use
terminology carefully.
Match the following Circles of Evaluation with the corresponding arithmetic expressions:
(/(+24)(-63))
(-(*94)3)
(*-52)
(*9(-43))
(*(-93)4)
Intro to Programming
Overview
Students are introduced to the programming environment
Learning Objectives
Convert Circles of Evaluation into code
Enter and evaluate code expressions for arithmetic on the computer
Understand the goal of error messages from the computer
Evidence Statementes
Students will be able to identify the Interactions and Definitions areas
Students will be able to enter and evaluate simple arithmetic expressions in the Interactions area
Students will be able to convert Circles of Evaluation into correctly-formed programs
Students will be able to explain an ’unbounded identifier’ error message
Students will be able to explain the purpose of error messages
Given a Circle of Evaluation, students will be able to complete a partially-written program
Given a bank of Circles of Evaluation and programs, students will be able to match them
Product Outcomes
Preparation
Intro to Programming(Time 15 minutes)
Intro to ProgrammingOpen the programming tool of your choice: If you’ve installed, DrRacket, double-click the
application to launch it. If you are using the online-tool, click here to log in.
Enter your username and password to log in, then click "Start a New Program".
Have the students look at the editor on their computers. Refer to overhead
projector as necessary.
This screen is called the editor, and it looks something
like the diagram you see here. There are a few buttons at the top, but most of the screen is
taken up by two large boxes: the Definitions area on the left and the
Interactions area on the right.
The Definitions area is where programmers define values and functions in their program,
while the Interactions area allows them to experiment with those values and functions.
This is analogous to writing a series of function definitions on a blackboard, and having
student evaluate expressions using those function on scrap paper. As students are not yet
defining values of their own, it is not important that students understand this distinction
right now. For now, we will work only with the Interactions area.
A program is a legal expression that, when evaluated, produces a value. You’ve been
writing programs using mathematical expressions since you first learned how
to add! Just as in English, there are rules that determine whether a sentence makes sense. Programs have rules too! A program can be very complicated, but it doesn’t have to be: the simplest programs of all are just Numbers.
Draw students’ attention to the Language Table (see Lesson Overview for Unit 1),
which currently sits empty. Add "Numbers" to the Types section of the language
table.
Enter 4 in the Interactions area and hit "Return". You
will see the value 4 appear on the next line in
the Interactions area.
Type 10 in the Interactions area and hit "Return". Now the
value 10 appears in the Interactions area.
Try evaluating numbers, like 10345017, or negative
numbers, like -2. Is there a limit to how big a number can be?
What happens if you write a decimal? What happens when you click on
a decimal, like 0.75? You get a new type of number, a
fraction, like 3/4.
The editing environment evaluates all fractions and returns them as decimals by default. This can be
surprising to students at first, so you may want to take a moment to explain what’s going on, and
show them that these decimals can be converted back to fractions just by clicking on them. The
environment uses standard annotations for repeating, non-terminating decimal expressions and
properly handles expressions like . If you want to work with those kinds of
numbers in your class, enter them to get familiar with how they appear in the Interactions area.
The computer obviously knows about Numbers, but what happens
if you type in something that it doesn’t know about?
Will it complain? Crash? Guess?
Try asking the computer
to evaluate dog in the Interactions area.
These error messages are really useful for programmers. Rather than saying "this program doesn’t work", the computer does the best it can to tell you what went wrong, and to give you as much information as possible to help you fix the problem. Make sure you always read these messages carefully!
The error message uses the term "variable". Don’t worry if your students don’t already know this
term; we will teach it to them shortly. For now, students just need to get used to error messages
and the kinds of problems that they catch in programs.
We can also write more complex computer programs than just evaluating numbers, and using the Circles of Evaluation makes this simple. To translate a Circle of
Evaluation into a program, begin with an open parenthesis (, and then the function written at
the top of the circle. Then translate the inputs from left to right in the same way, adding a closing
parenthesis ) when you’re done. This process gives us the second rule for expressions:
Code Rule 2: Each open parenthesis is followed by one function, then by one or more legal
expressions, and finally by a closing parenthesis.
Here is the code for this Circle of Evaluation: (- 4 5)
Press the Return key to evaluate the program.
You should see -1 as an answer.
Have students practice converting simple Circles of Evaluation into code. If you want to help students
understand when to use the parentheses, here are two explanations that we find useful. First, the
parens look like the Circle, and the Circle encloses the function name and its inputs. Second, we
use a more visual description of an ant eating its way through the expression. The ant eats into the
Circle (an open paren), then goes to the function at the top, then to the arguments from left to right,
then finally out of the Circle (a close paren). If the ant encounters another Circle while writing
down the arguments, it needs another open paren, etc.
When a Circle of Evaluation has other circles inside of it, the
translation still follows the same rules: each Circle requires a new set of parentheses:
(* 6 (+ 4 5))
Try entering this code into the Interactions
area. What should the program evaluate to
when you hit Return?
Practice converting other Circles of Evaluation you’ve drawn into code.
[Video on Roles in Pair Programming.]
Scaffolding for Pair Programming: Talk to students about the roles of Driver
and Navigator. The Driver is the student with their hands on the keyboard - they’re
in charge of typing, using the mouse, etc. The Navigator should be telling the Driver what to
type, and pointing out mistakes or suggesting things. It’s important to clarify what these
roles are (perhaps reinforcing them with a visual), to ensure that both partners are active
and talking to one another about the task at hand. The linked video does a great job discussing these roles.
All of the expressions that follow the function name are called arguments to the
function. The following diagram summarizes the shape of an expression that uses a function.
For practice, turn to Page 5
in your workbook. For each mathematical expression, draw the Circle of Evaluation, then convert
that Circle into Racket code.
When expressions don’t follow the code rules, the computer will tell you that it found a problem. The
computer also gives you information to help you fix the problem. This information is called an
error message. We’ll talk more about error messages later. For now, we just want you to see a couple of error messages so that you’ll know what they are if you run into one while programming.
Enter each of the following illegal expressions in the Interactions area
and look at the error message or behavior that the computer gives you.
(5 * 6) [puts the function in the middle, instead of at the front]
(*5 6) [missing a space after the function]
* 5 6) [forgets the open parenthesis]
(* 5 6 [forgets the close parenthesis]
At this point, the goal is simply to have students see error messages. Students are not
expected to be experts at reading error messages this early
(you will drill this later). NOTE: When a close parenthesis is missing, hitting return will do nothing,
because the computer is waiting for the parenthesis. When an open parenthesis is missing, WeScheme
does nothing and waits for the programmer
to add the parens. If students hit enter and "nothing happens", they should check their parentheses.
Also note that if you do this sequence of exercises with + instead of *, the error
appears differently, because +5 is a perfectly valid number (just like -5).
Closing
Overview
Learning Objectives
Evidence Statementes
Product Outcomes
Materials
Preparation
Closing(Time 5 minutes)
ClosingYou’ve done a lot in this first unit!
You took a game apart to see how the parts move
You designed your own game, that you’ll make during this course
You practiced using coordinates to put characters into a screen
You learned about order of operations, Circles of Evaluation, how to turn Circles of
Evaluation into programs, and how to run those programs to get answers.
In the next unit, we’ll begin writing programs for more interesting things than just arithmetic.
Make sure student names are on the Game Design page. Take this page, or take photos of it, to prep game images for a later Unit.
Bootstrap:Algebra by Emmanuel Schanzer, Emma Youndtsmith, Kathi Fisler, and Shriram Krishnamurthi was developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598), and is licensed under a Creative Commons 4.0 Unported License. Based on a work at www.BootstrapWorld.org. Permissions beyond the scope of this license may be available by contacting schanzer@BootstrapWorld.org.
Turn to 
We also need to think about what is 
Computers use numbers to represent a character’s position onscreen, using number lines as rulers to measure
the distance from the bottom-left corner of the screen. For our videogame, we will place the numberline so that the screen runs from 0
(on the left) to 640 (on the right). 
By adding a second number line, we can locate a character anywhere on the screen in either
dimension. The first line is called the x-axis, which runs from left to right. The second line, which runs up and down, is
called the y-axis. A 2-dimensional 
Fill in the rest of the table, identifying what is changing for each of your characters.
Mathematicians didn’t always agree on the order of operations, but now
we have a common set of rules for how to evaluate expressions. The pyramid on the right summarizes the
order. When evaluating an expression, we begin by applying the operations written at the top of the
pyramid (multiplication and division). Only after we have completed all of those operations can we
move down to the lower level. If both operations are present (as in ), we read the expression
from left to right, 
One way to indicate the order of operations in an expression is to first
draw the expression as a diagram. This diagram is called a 
Which rule does this Circle of Evaluation break?
Every Circle of Evaluation 
To use multiple functions in the same expression, we can combine Circles
of Evaluation. Look at the Circle of Evaluation you’ve written for . We already
know that the Circle for will evaluate to , so we can 
Finish writing the Circle of
Evaluation shown here, so that it represents 
What does this Circle of Evaluation evaluate to? Let’s review how we evaluate a Circle:
Convert this Circle of Evaluation into an arithmetic expression.
This screen is called the 
Here is the code for this Circle of Evaluation: (- 4 5)
When a Circle of Evaluation has other circles inside of it, the
translation still follows the same rules: each Circle requires a new set of parentheses:
(* 6 (+ 4 5))
