Introduction to the Binary # system
by
Carolyn Snively @ Franklin High School
OVERVIEW
I spent my job shadowing time with the Tech Department at Otterbein Senior Lifestyle Community.
Therefore, it was my charge that week to find a way to connect math with computer technology. This activity
introduces the students to the binary number system, and its connection to computers.
TYPE/LEVEL OF INQUIRY
I would classify this activity as a #3, guided inquiry activity. The extensions could be done as a #1, open inquiry
activity.
GRADE LEVEL/COURSE
This activity would be appropriate for students from grade 7 through 12.
ANTICIPATED LENGTH OF LESSON/MODULE (MINUTES)
This activity can be accomplished in as little as one class period, or could be extended into a week-long unit with
the introduction of other number bases such as hexadecimal, etc.
PREREQUISITE KNOWLEDGE
Students will need to have an understanding of place value in base ten, and how to add, subtract, multiply, and
divide in base ten.
STATE/NATIONAL STANDARDS
As a teacher who's often introduced binary to students with good results, "binary" does not appear in the
standards at any grade level! So I looked at the ACM Model Curriculum for K-12 Computer Science (The ACM is the
major professional and academic society for Computer Science, roughly equivalent to the AMA for Mathematics).
In that curriculum, binary is introduced in Level II, which is grades 9-10. Despite all this, it does appear that binary
is taught in many state and local mathematics school curricula. As I said, I have found that students pick up the
concepts of binary nicely even at early grade levels, and it helps them gain perspective on a number of conceptual
issues in school mathematics. That said, there are a ton of crucial topics that are typically covered in school
mathematics curricula, and I can understand why binary isn’t on the list. It is too bad, though.
MATERIALS
I have developed a sheet of guiding questions that will gradually lead students to discovery of the binary
number system by connecting it to the base ten number system, that they are already familiar with. A
teacher would simply need to copy a question sheet for each group of students. The questions are meant
to be read and discussed out loud by a small group of 2-4 students.
INSTRUCTIONAL PLAN
A. INTRODUCTION
 Begin by discussing as a whole class, what they know about place value and counting
numbers in base ten.
 Have a discussion about aliens and see where it leads! Let the student’s excitement build a
little. Then, tell the students that they are going to learn how to do “alien” math!

Let the students know that this “alien” math is really how humans and computers
communicate. Let the students know that by the end of their guided question sheet, you
will expect them all to be able to talk to their computers!
B. ACTIVITY
Divide the students into small groups, and give each group a guided question sheet. (This sheet
and answer key are provided as separate documents.) Instruct them to choose a person to read
each question, quietly to the group, and then as a group discuss what they will write down on the
sheet. It is also a good idea to tell the students to use a cover sheet as they work through the
questions. The reason for this is that the questions are “guiding” questions, and sometimes the
answer to a question is used in the question below. As the groups are working, the teacher
should walk around observing the discussions, and providing clarification as needed.
C. POST ACTIVITY DISCUSSION
Once all the groups have finished the guided question sheet, the teacher should lead a whole
class discussion/sharing session regarding what they just learned about binary, and possibly sharing any
further knowledge and connections they may have realized since doing the activity.
EXTENSIONS
At this point, the teacher could be done with this particular topic, and simply move on to something else,
or they may want to spend the next day or days, working on writing and operations with numbers in other bases.
The students should be able to make the same base ten connections with the other bases as they did with binary.
The teacher would not need to make a new question sheet. They could simply use the same questions, but just
use a different base. If a teacher really wants to show them something cool and “out-there”, then the other base
that has a great connection to computers is hexadecimal, or base 16, where alphabetical letters combined with the
digits 0 thru 9 are used!
ASSESSMENT
Being that teaching binary is not included in the standards, this activity is meant to be used as something
fun! I would not give my students a formal assessment over this material, although it would provide great
opportunities for bonus questions on assessments for standards based topics. There are also plenty of online
quizzes and such for students to practice operations in binary. A teacher could even take their students to a lab to
give them an online assessment if they wanted. A good website I found with a nice online practice set is
http://www.binarymath.info/practice-exercises.php
REFERENCES
http://www.garlikov.com/soc_meth.html
http://www.ehow.com/how-does_4759517_binary-code-work.html
SAFETY CONSIDERATIONS: NONE
If a person held up all the fingers on both hands, how many would that be?
List as many ways as you can to represent the number ten:
Thinking of the WORD ten, what are written words made up of?
How many letters are there in the English alphabet?
How many words can you make out of them?
Thinking of the NUMBER "10", what is this way of writing numbers made up of?
How many numerals are there?
Starting with zero, what are they?
How many numbers can you make out of these numerals?
How come we have ten numerals? Could it be because we have 10 fingers?
What if we were aliens with only two fingers? How many numerals might we have?
How many numbers could we write out of 2 numerals?
Thinking of the numerals you wrote down, 0 to 9, for our ten numerals, if we only have two numerals and write them the same way we do for
our ten, what numerals would we probably use?
What do we do on this planet when we run out of numerals at 9?
Think of this process as using two columns. You are putting the 1 in a different column from the 0.
What do we call the column you put the 1 in?
Why do think we call it that?
What does a one followed by a zero mean, based upon the idea of columns?
What was the first number that needed a new column for you to be able to write it?
Could that be why it is called the ten's column? What is the first number that needs the next column?
And what is the name for this column?
After you write 19, what numerals do you have to change to write down 20?
“20” means 2 tens and no ones, because 2 tens are the same as ___?
What is the first number that needs a fourth column, and what is the column called?
Okay, let's go back to our two-fingered aliens arithmetic, where we only have a 0 and a 1.
What would we do to write "two" if we followed the exact same process we use with ten numerals?
What should we call this column, if we use the same process as we do with ten numerals?
What would we put in this two's column to show how many two’s we have to represent?
What should go in the one’s column to show how many ones we have to represent?
How does the alien way of writing “two” look like using columns ?
So, when an alien sees a one followed by a zero, like this: 10, they think of “two” and not “ten”, because the one is in the two’s column, not in a
ten’s column. Now this may seem difficult to you, but it’s easy to them because they learn it that way in pre-school just as you learned that
when you see a one followed by a zero, you call it "ten". If you give it a little time, you will get used to it; the alien children do. Okay, so what
number comes next after two?
How many two’s are there in three? How many would be left over?
What would need to go in the two’s column? What would need to go in the one’s column?
Show how you write the numbers we have so far in our alien counting.
Now we're out of numerals again. How do we get to four? What do you always do when you run out of numerals?
What should this column be called? Remember we name it based on the same principles we use for ten numbers.
You now have three columns. What will we put in the four’s column? The two’s column? The one’s column?
Next up, five! Remember that five will be the same as four, but with an extra one. What will we put in the four’s column? The two’s column?
The one’s column?
Next up, six! Think of six in terms of the columns we have so far. We have a column for four’s, two’s, and ones’ . So, what will we put in the
four’s column? The two’s column? The one’s column?
Next up, seven! Think of seven as just having an extra one compared to six. What will we put in the four’s column? The two’s column? The
one’s column?
Next up, Eight! Now our columns are full, so that means we will need a new one. What will we call this column?
Now we have four columns. The eight’s column, the four’s column, the two’s column, and the one’s column. So how do you think you will
write eight?
Make a list of all the numbers we have so far, and then add nine and ten to the list:
So now, how many numbers do you think you can write with a one and a zero?
Let's see what happens if we try to multiply in alien here. Let's try two times three and you multiply just like you do in tens [in the "traditional"
American style of writing out multiplication].
Ok, look on the list of numbers you wrote above, what is 110?
And how much is two times three in real life?
So alien arithmetic works just as well as your arithmetic, huh? Even easier, right, because you just have to multiply or add zeroes and ones,
which is easy. Of course, until you get used to reading numbers this way, you may need a chart, because it is hard to read something like
"10011001011" in alien, right?
So who uses this stuff?
How many positions does a light switch have?
What could you call these positions?
If you were going to give them numbers what would you call them?
What I hope you have just discovered is how to write numbers in base two, or what is called binary code! Binary code is a breakdown of
complex language into very simple zeros and ones. Using a broad perspective, the individual zeros and ones in binary code are predetermined
instructions for the processor that reads it. In essence, binary code is merely a translation of an understandable language into what has come to be
known as computer language. Binary code manipulates a series of circuits into a recognizable pattern of "off" and "on" electrical pathways.
This series of circuits might be compared to the teeth of a very intricate key, which produces a specific action from the CPU that has been
preprogrammed to understand each pattern and respond. As the code is read, zeros traditionally switch a circuit off, while ones switch the next
circuit on, until a unique pattern is produced from each code. The purpose of each binary code, representing letters, numbers and symbols, is to
display the translated result on the computer screen. In its simplest form, the binary code in computer language translates into the activation, or
lack thereof, of each individual pixel on the computer's screen, eventually determining the shape of each letter -- represented by a series of pixels
-- as well as its color, shade, and size. As each pixel becomes its designated color more rapidly than the human eye can detect, a very complex
picture is formed, showing the computer's user a single image for a predetermined amount of time. This image is then replaced with the next, and
the process continues to produce the illusion of a moving picture and the electronic signals required to transmit sound through the computer's
speakers.
Read more: How Does Binary Code Work? | eHow.com http://www.ehow.com/how-does_4759517_binary-code-work.html#ixzz23Xg726rJ
If a person held up all the fingers on both hands, how many would that be?
TEN
List as many ways as you can to represent the number ten:
10
2 x 5 TEN X
Thinking of THE WORD TEN. What are written words made up of?
LETTERS
How many letters are there in the English alphabet?
26
How many words can you make out of them?
ZILLIONS
Thinking of THE NUMBER "10" , What is this way of writing numbers made up of?
NUMERALS
How many numerals are there?
TEN
Starting with zero, what are they?
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
How many numbers can you make out of these numerals?
An infinite amount
How come we have ten numerals? Could it be because we have 10 fingers?
COULD BE
What if we were aliens with only two fingers? How many numerals might we have?
2
How many numbers could we write out of 2 numerals?
NOT MANY ?
Thinking of the numerals you wrote down, 0 to 9, for our ten numerals, If we only have two numerals and write them the same way we do for
our ten, what numerals would we have?
0, 1
What do we do on this planet when we run out of numerals at 9?
WRITE DOWN, "ONE, ZERO"
Think of this process as using two columns. You are putting the 1 in a different column from the 0.
What do we call the column you put the 1 in?
TENS
Why do think we call it that?
DON'T KNOW
What does a one followed by a zero mean, based upon the idea of columns?
1 TEN AND NO ONES
What was the first number that needed a new column for you to be able to write it?
TEN
Could that be why it is called the ten's column? What is the first number that needs the next column?
100
And what is the name for this column?
HUNDREDS
After you write 19, what numerals do you have to change to write down 20?
9 to a 0 and 1 to a 2
“20” means 2 tens and no ones, because 2 tens are the same as ___?
TWENTY
What is the first number that needs a fourth column, and what is the column called?
1000
THOUSANDS
Okay, let's go back to our two-fingered aliens arithmetic, where we only have a 0 and a 1.
What would we do to write "two" if we followed the exact same process we use with ten numerals?
START ANOTHER COLUMN
What should we call it, using the same process as with ten numerals?
THE TWO'S COLUMN
What would we put in the two's column to show how many two’s we have to represent?
What should go in the one’s column to show how many ones we have to represent?
1
0
How does the alien way of writing “two” look like using columns ?
10
So, when an alien sees a one followed by a zero, like this: 10, they think of “two” and not “ten”, because the one is in the two’s column, not in a
ten’s column. Now this may seem difficult to you, but it’s easy to them because they learn it that way in pre-school just as you learned that
when you see a one followed by a zero, you call it "ten". If you give it a little time, you will get used to it; the alien children do. Okay, so what
number comes next after two?
THREE
How many two’s are there in three? How many would be left over?
There is one "TWO" and a "ONE" left over
What would need to go in the two’s column? What would need to go in the one’s column?
PUT A 1 IN THE TWO’S COLUMN, AND A 1 IN THE ONE’S COLUMN
Show how you write the numbers we have so far in our alien counting.
zero: 0
one: 1
two: 10
three: 11
Now we're out of numerals again. How do we get to four? What do always do when you run out of numerals?
START A NEW COLUMN!
What should this column be called? Remember we name it based on the same principles.
THE FOUR'S COLUMN
You now have three columns. What will we put in the four’s column? The two’s column? The one’s column?
ONE, ZERO, ZERO
Next up, five! Remember that five will be the same as four, but with an extra one. What will we put in the four’s column? The two’s column?
The one’s column?
ONE, ZERO, ONE
Next up, six! Think of six in terms of the columns we have so far. We have a column for four’s, two’s, and ones’ . So, what will we put in the
four’s column? The two’s column? The one’s column?
ONE, ONE, ZERO
Next up, seven! Think of seven as just having an extra one compared to six. What will we put in the four’s column? The two’s column? The
one’s column?
ONE, ONE, ONE
Next up, Eight! Now our columns are full, so that means we will need a new one. What will we call this column?
THE EIGHT’S COLUMN
Now we have four columns. The eight’s column, the four’s column, the two’s column, and the one’s column. So how do you think you will
write eight?
ONE, ZERO, ZERO, ZERO
Make a list of all the numbers we have so far, and then add nine and ten to the list:
0
1
10
11
100
101
110
111
1000
1001
1010
zero
one
two
three
four
five
six
seven
eight
nine
ten
So now, how many numbers do you think you can write with a one and a zero?
MEGA-ZILLIONS ALSO/ ALL OF THEM
Let's see what happens if we try to multiply in alien here. Let's try two times three and you multiply just like you do in tens [in the "traditional"
American style of writing out multiplication].
10
x 11
10
100
110
two
times three
Ok, look on the list of numbers you wrote above, what is 110?
SIX
And how much is two times three in real life?
SIX
So alien arithmetic works just as well as your arithmetic, huh? Even easier, right, because you just have to multiply or add zeroes and ones,
which is easy. Of course, until you get used to reading numbers this way, you need your chart, because it is hard to read something like
"10011001011" in alien, right?
So who uses this stuff?
NOBODY/ ALIENS
How many positions does a light switch have?
TWO
What could you call these positions?
ON AND OFF/ UP AND DOWN
If you were going to give them numbers what would you call them?
ONE AND TWO/