Math
Practicum in Information Technology
Lesson Plan
Performance Objective
Upon completion of this lesson, each student will be able to convert between different
numbering systems and correctly write mathematical formulas for use in computer programs.
Specific Objectives
• Students will explain the concept of binary.
• Students will convert numbers between binary and decimal.
• Students will explain the concept of hexadecimal.
• Students will convert numbers between hexadecimal, binary, and decimal.
• Students will solve and write equations as utilized in computer programming.
This lesson should take 10-12 class days to complete.
Preparation
TEKS Correlations
This lesson, as published, correlates to the following TEKS. Any changes/alterations to the
activities may result in the elimination of any or all of the TEKS listed.
Practicum in Information Technology
130.280 (c)
(3) The student applies academic knowledge and skills to research and develop projects. The
student is expected to:
(B) demonstrate proper use of mathematics concepts in the development of products or
services; and
(C) demonstrate proper use of science principles in the development of products or
services.
Interdisciplinary Correlations
Reading I, II, III
110.47(b)
(1) The student uses the following word recognition strategies. The student is expected
to:
(A) Apply knowledge of letter-sound correspondences, language structure, and
context to recognize words;
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1
(B) Use reference guides such as dictionaries, glossaries, and available
technology to determine pronunciations of unfamiliar words;
(2) The student acquires an extensive vocabulary through reading and systemic word
study. The student is expected to:
(A) Expand vocabulary by reading, viewing, listening, and discussing;
(B) Determine word meanings through the study of their relationships to other
words and concepts such as content, synonyms, antonyms, and analogies;
(4) The student comprehends text using effective strategies. The student is expected to:
(A) Use prior knowledge and experience to comprehend;
(B) Determine and adjust purpose for reading; and
(D) Summarize texts by identifying main ideas and relevant details.
Public Speaking I, II, III
110.57 (b)
(4) Organization. The student organizes speeches. The student is expected to:
(B) Organize speeches effectively for specific topics, purposes, audiences, and
occasions.
(7) Delivery. The student uses appropriate strategies for rehearsing and presenting
speeches. The student is expected to:
(C) Develop verbal, vocal, and physical skills to enhance presentations.
Tasks
• Students will utilize provided notes and slide presentation to complete notes pages.
• Students will complete worksheets.
Accommodations for Learning Differences
It is important that lessons accommodate the needs of every learner. These lessons may be
modified to accommodate your students with learning differences by referring to the files
found on the Special Populations page of this website (cte.unt.edu).
Preparation
• Copy the handouts.
• Have multimedia presentations ready to show the class.
Instructional Aids
• Student Notes sheet
• Student worksheets
Materials Needed
• Copies
• Pencils
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2
Equipment Needed
• Teacher computer
• Projector (for slide presentation)
Introduction
Learner Preparation
• Review slides 1 - 6 of the multimedia presentation with the class (positional numbering
systems - specifically decimal).
Lesson Introduction
• Ask the class:
o How is data represented within a computer system?
o How would information be more readable or user friendly for programmers and
technicians?
• Tell the class that every character and command given to a computer by a user is
translated into binary before it can be “sent” to the CPU for processing.
• Explain that in-depth computer troubleshooting and programming requires a firm
understanding of numbering systems and how to convert between them.
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3
Outline
OUTLINE
NOTES TO TEACHER
Slides 1 – 6 Decimal Review
a. Natural numbers
b. Integers
c. Positional numbering
system
d. Number base
e. Review of exponent laws
regarding 0 and 1st power
f. Review positional
numbering system
a. Hundreds
b. Tens
c. Ones
Slides 8 – 11
Converting from binary to decimal
Extension notes are provided in the
multimedia presentation. Timeline of
delivery is at discretion of instructor.
Guidelines regarding division of
information are suggestions only.
MI
Slides 9 – 16
Converting from decimal to binary
Slides 17 – 22
Introduction to hexadecimal
Conversion from hexadecimal to
binary
Slides 23 – 26
Conversion from decimal to
hexadecimal
Slides 27 – 31
Rewriting math in programming
Slides 32 – 36
Using math in programming
Upon completion of Slide 11, give
students the binary to decimal
conversion worksheet.
Upon completion of Slide 16, give
students the decimal to binary
worksheet.
Upon completion of Slide 22, give
students the hexadecimal to binary
worksheet.
Upon completion of Slide 26, give
students the decimal to hexadecimal
worksheet.
Upon completion of Slide 31, give
students the evaluation of
expressions worksheet.
Upon completion of Slide 36, give
students the worksheet on using
programs to write and solve
mathematical formulas worksheet.
Multiple Intelligences Guide
Existentialist
Interpersonal
Intrapersonal
Kinesthetic/
Bodily
Logical/
Mathematical
Musical/Rhythmic
Naturalist
Verbal/Linguistic
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Visual/Spatial
4
Application
Guided Practice
Students will use the provided slide presentation to complete the notes handouts pages.
Independent Practice
• Students will complete all provided worksheets:
o Binary to decimal
o Decimal to binary
o Hexadecimal to binary
o Decimal to hexadecimal
o Rewriting math for computer programs
o Using math in computer programs
Summary
Review
• Why is it important to know how to use math properly?
• What is the fundamental language of computers?
• Why was binary chosen as machine language?
Evaluation
Informal Evaluation
• The teacher will check frequently for understanding
• Very small work group option (peer tutoring)
Formal Evaluation
• Worksheets and test over the material
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5
Binary, Hex, and Decimal, “Oh My”!
Student Notes Handout
a.
numbers
a. Zero and any number obtained by repeatedly adding one to it
b. Ex. 0, 1, 3, 100
b. Integers
a. Any
number
b. Includes negative numbers and zero
c.
Numbering System
a. The number depends on the position of the digits to represent the number.
d. Number base
a. The
(or radix) is the number of
digits used to represent
numbers in a positional numerical system.
Any number to the
power is equal to 1.
0
Example: 2 =
Any number to the
power is equal to that
1
Example: 2 = 2
.
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6
Converting from Decimal to Binary
• Base
• The base of a number is indicated by a
value at the “end” of the number. If
there is no subscript the base is assumed to be 10 (or decimal).
• Uses
and
• Binary uses two as its base so each
is equal to two to an exponent.
• Since we are using two single characters we use
and
as our two digits.
• Consider the binary number 000011102.
Binary Table
Do you see how each
in value? This makes binary one of
the easiest numbering systems to convert to decimal. Using basic addition, we can take any
binary number and convert it to a decimal equivalent. As long as you memorize the pattern all
you have to do is
the value of the placeholders where a 1 appears.
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7
Look at the table. In the top row you notice there are only zeros and ones. This one is a little
easier for you by adding the table and colors. This is the number we are going to convert to
decimal,
.
By looking down the chart you can see that we are going to use the following values:
,
, and
because there are 1s in those placeholders. Note that red indicates which
numbers we will use to convert to a decimal value. Using the formal method, we would transfer
the digits to the blue box you see on the right of the slide. I like to use the vertical alignment to
make things easier to read. We start with
; determine that there is a zero in that
placeholder.
What is the answer when you multiply any number by zero?
Answer:
Moving down the line we have zeros until we get to the place holder
right? This placeholder has a ________ instead of a zero.
, which is equal to
What is the product of any number multiplied by one?
Answer:
number
Once you have calculated the
appears, we add the answers together.
What is the sum of 8+4+2?
Answer:
The decimal equivalent of
of all the placeholders where a 1
is
, which is shown in the green box.
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8
Converting from Decimal to Binary
• Can be done two different ways
–
–
Long Division
Convert
from decimal to binary.
Sort by
the original number by our base, which is . Notice that we do
not directly divide 35 by 2 and get 16. You must divide the numbers
so that
you have a
of
or
at the end of the process.
Once you reach a 0 or 1 as the remainder you move to the next step.
Using the
we got in the first step (35 divided by 2), we move to the second
step, which is dividing
by 2. We
the process until we have
in the
,
, and
. Now
here comes the tricky part.
Example:
1
2 3
2
1
5
8
2 1
1
4
2 8
8
2
2 4
4
0
0
1
2 2 2 1 2 0
0
0
2
Binary
Each of the zeros and ones in the remainders represent individual
. These bits are in
order so we have to do what we call “
,“ meaning we
rewrite them putting the
bit in the
position when
we write the number in binary format.
is flipped to
.
Once we rewrite the number, we see that 35 in binary is 01000112 . You can always check your
answer by using the binary to decimal conversion method. Go ahead, check our math.
=
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9
Subtraction
Convert 3710 to binary using subtraction
Looking at our binary table what is the
?
Answer:
number that is still less than
To indicate that 32 will be used, we put a
in the
from 37 we have
left, right? What is the
lower than 3?
Answer:
placeholder. Once we subtract 32
number in our table that is still
Skip down to the
placeholder and put in a 1. What is the only number left that we can use
that is lower than or equal to 1?
Answer:
Now we put a
in the
placeholder.
Continue the subtraction process until we get zero as an answer.
What do you think we do with all those blank spaces in the table?
Answer:
Unlike using the division method, you
the binary representation of
=
have to “flip the bits” in this case, so
.
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10
Hexadecimal
Characters
–
–
–
–
You can convert directly from hex to decimal but not from decimal to hex.
In order to represent the numbers
–
A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
(to give us our 16 characters) we use letters.
Each
is equal to
(combination of zeros and ones),
and like binary, can be combined to create even larger numbers.
Hexadecimal
(Hex)
Binary
Decimal
0001
1
2
3
2
0011
0100
5
6
5
0110
0111
8
9
1001
B
1100
E
As with binary and decimal,
system. Each placeholder uses
10
11
1101
F
7
8
1010
C
4
13
14
1111
is a
numbering
as its base and is equal to 16 to an exponent.
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11
Look at the table.
is the hexadecimal number we are going to convert to decimal.
By looking at the chart, I can see that we are going to use the following values: and
because there are characters in those placeholders. Note that the color red indicates which
numbers we will use to convert to a
value.
Using the formal method again and for easier reading, we will
we use to the blue box on the right of the slide.
Just like converting binary, once you have calculated the
where a character appears, we
the answers together.
What is the sum of 240 + 15?
Answer:
The decimal equivalent of
is
the
of all the placeholders
,which is shown in the green box.
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12
Decimal to Hex Conversion
Convert 5410 to Hexadecimal
The easiest way to convert a decimal number to hex is to go to
at it using the subtraction method to go from decimal to binary.
first. Let’s look
Convert from
to binary
Convert from
binary to
Now we know that
is equal to
in binary. Here is where we
take the next step. Remember that when we first discussed hexadecimal and looked at the
hex/binary/decimal table you learned that each hex character represents
binary bits.
Note that we have
of the four bits are then
the binary bits into
into a
Looking first at the left-hand group of four:
hex?
Answer:
Look at the second group of four:
Answer:
of
. Each
character.
. How would is that number represented in
. How would you represent that number in hex?
While it looks like 5410 in hex is thirty-six ,it is actually read as
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.
13
Convert 12810 to Hexadecimal
Decimal to binary
Binary to
hex
Rewriting math in Programming
•
– Parentheses
–
– Multiplication and/or division in the order they appear
– Addition and/or subtraction in the
Writing Formulas for Computer Programs
What is the formula in the green box?
Answer: Area of a circle
Computer programs do not recognize the symbol for Pi and there are no options for
superscripts to indicate exponents in the majority of compilers. This very common formula must
be rewritten in a way a computer program can decipher it.
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14
In order to represent exponents, we use what’s called a “carrot top,” which you see in the red
box and you get by pressing
. To represent
you must use the
value
. For most applications 3.14 is adequate; however, for more
precise calculations, you can use as many decimal places as you require.
You can use virtually any variable to represent the A; however, you must remember that certain
words are “
“ and can vary from language to language, so be cautious and
document your code so that other programmers will be able to understand your code easily.
Looking at the blue box, we see how the same formula would be written in a computer
program. Notice we have used the “
top” to indicate the
and the number 3.14
to represent . r has been left alone with the assumption that the radius is either
in or
by the user at some earlier point in the program.
Rewrite the formula for the slope of a line
Using Math in Computer Programming
Math is universal. The
of programs that are written involve some type of
. The formulas can range from very simple to highly complex.
• Used to
– Solve
and or complex calculations
•
for a company
• Track
data sets in
calculations
– Create
files (programs)
– Create
systems
• Machine Language
Everything that happens within a computer all the way from the GUI we see to the files we save
is
into
language. Each
entered into
the machine; each
the user issues is translated into
codes.
Boolean Operators
Because of some of the programs you will write, we also need to discuss the
or
Boolean operators in computer programming. These operators have their own
, fit into
, and have a specific
of operation.
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Boolean/Logic (or bool) Operators
a. Also called
Operators or just
b. Logical operators that
or
a.
= false
b.
= true
c. Three basic bool operators
a.
=
–
arguments must be true for the statement to return True.
b.
=
–
argument may be
for the statement to return
True.
c.
=
–
the statement to
if it returns
and
if it returns
.
d. Order of operations – as with PEMDAS
, groups are evaluated
and operators are evaluated in the following order:
a.
b.
c.
d.
e.
f.
g.
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Truth Tables
These are truth tables.
tables help us to visualize all possible results of
or
comparisons. The results of AND and NOTAND are in the blue
tables while the results of OR and NOTOR are in the green tables.
Truth tables help us step through our program code and determine if our
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is sound.
17
Binary, Hex, and Decimal Oh My
Student Notes Handout
a. Natural numbers
a. Zero and any number obtained by repeatedly adding one to it
b. Ex. 0, 1, 3, 100
b. Integers
a. Any whole number
b. Includes negative numbers and zero
c. Positional Numbering System
a. The number depends on the position of the digits to represent the number.
d. Number base
a. The base (or radix) is the number of single digits used to represent numbers in a
positional numerical system.
Any number to the 0 power is equal to 1.
Example: 20 = 1
Any number to the 1st power is equal to that number.
Example: 21 = 2
2
7 * 10 = 7 * 100 = 700
1
2 * 10 = 4 * 10 = + 20
0
5 * 10 = 5 * 1 = + 5
725
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18
Converting from Decimal to Binary
•
•
•
•
•
•
Base 2
The base of a number is indicated by a subscript value at the “end” of the number. If
there is no subscript the base is assumed to be 10 (or decimal).
Uses 0 and 1
Binary uses two as its base so each placeholder is equal to two to an exponent.
Since we are using two single characters we use zero and one as our two digits.
Consider the binary number 000011102.
This is where binary digits (0s and 1s) go
Do you see how each placeholder doubles in value? This makes binary one of the easiest
numbering systems to convert to decimal. Using basic addition, we can take any binary number
and convert it to a decimal equivalent. As long as you memorize the pattern all you have to do
is add the value of the placeholders where a 1 appears.
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19
Convert 00011102 from binary to decimal
Look at the table. In the top row you notice there are only have zeros and ones. This one is a
little easier for you by adding the table and colors. This is the binary number we are going to
convert to decimal, 00001110.
By looking down the chart you can see that we are going to use the following values: 23, 22, and
21 because there are 1s in those placeholders. Note that red indicates which numbers we will
use to convert to a decimal value.
Using the formal method, we would transfer the digits to the blue box you see on the right of
the slide. The vertical alignment makes things easier to read. I like to use the vertical alignment
to make things easier to read. We start with 128; determine that there is a zero in that
placeholder.
What is the answer when you multiply any number by zero?
Answer: zero
3
Moving down the line we have zeros until we get to the place holder 2 , which is equal to 8
right? This placeholder has a one instead of a zero. What is the product of any number
multiplied by one?
Answer: That number
Once you have calculated the products of all the placeholders where a 1 appears, we add the
answers together. What is the sum of 8+4+2?
Answer: 14
The decimal equivalent of 000011102 is 14, which is shown in the green box.
Converting from Decimal to Binary
• Can be done two different ways
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20
– Long division
– Subtraction
Long Division
Convert 3510 from decimal to binary.
Start by dividing the original number by our base, which is 2. Notice that we do not directly
divide 25 by 2 and get 16. You must divide the numbers individually so that you have a
remainder of zero or one at the end of the process. Once you reach a 0 or 1 as the remainder
you move to the next step.
Using the quotient we got in the first step (35 divided by 2), we move to the second step, which
is dividing 17 by 2. We repeat the process until we have zero in the quotient, dividend, and
remainder. Now here comes the tricky part.
Example:
1
2 3
2
1
5
1
8
2 1
1
4
2 8
8
2
2 4
4
1
0
0
0
0
1
2 2 2 1 2 0
0
0
2
0
1
Binary
0
Each of the zeros and ones in the remainders represent individual bits. These bits are in reverse
order so we have to do what we call “flipping the bits,” meaning we rewrite them putting the
rightmost bit in the leftmost position when we write the number in binary format.
1100010 is flipped to 0100011.
Once we rewrite the number we see that 35 in binary is 01000112 . You can always check your
answer by using the binary to decimal conversion method. Go ahead, check our math.
0100011 = 35
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21
Subtraction
Convert 3710 to binary using subtraction
Looking at our binary table what is the highest number that is still less than 37?
Answer: 32
To indicate that 32 will be used, we put a 1 in the 25 placeholder. Once we subtract 32 from 37
we have 3 left, right? What is the highest number in our table that is still lower than 3?
Answer: 2
1
Skip down to the 2 placeholder and put in a 1. What is the only number left that we can use
that is lower than or equal to 1?
Answer: 1
0
Now we put a 1 in the 2 placeholder.
Continue the subtraction process until we get zero as an answer.
What do you think we do with all those blank spaces in the table?
Answer: Fill them with zeros because they are not used.
Unlike using the division method, you do not have to “flip the bits” in this case, so the binary
representation of 3710 = 001000112.
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22
Hexadecimal
Characters
– 0–9
– A–F
You can convert directly from hex to decimal but not from decimal to hex.
In order to represent the numbers 10 – 15 (to give us our 16 characters) we use letters. A = 10, B
= 11, C = 12, D = 13, E = 14, and F = 15.
Each character is equal to four bits (combination of zeros and ones), and like binary, can be
combined to create even larger numbers.
Hexadecimal
(Hex)
Binary
Decimal
1
0001
1
2
0010
2
3
0011
3
4
0100
4
5
0101
5
6
0110
6
7
0111
7
8
1000
8
9
1001
9
A
1010
10
B
1011
11
C
1100
12
D
1101
13
E
1110
14
F
1111
15
As with binary and decimal, hexadecimal is a positional numbering system. Each placeholder
uses 16 as its base and is equal to 16 to an exponent.
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23
Look at the table. FF is the hexadecimal number we are going to convert to decimal.
1
0
By looking down the chart, I can see that we are going to use the following values: 16 and 16
because there are characters in those placeholders. Note that I have used the color red to
indicate which numbers we will use to convert to a decimal value.
Using the formal method again and for easier reading, we will transfer the digits we use to the
blue box on the right of the slide.
Just like converting binary, once you have calculated the products of all the placeholders where
a character appears, we add the answers together.
What is the sum of 240 + 15?
Answer: 255
The decimal equivalent of FF16 is 255, which is shown in the green box.
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24
Decimal to Hex Conversion
Convert 5410 to Hexadecimal
The easiest way to convert a decimal number to hex is to go to binary first. Let’s look at it using
the subtraction method to go from decimal to binary.
Convert from decimal to
Convert from
binary to hex
Now we know that 5410 is equal to 00110110 in binary. Here is where we take the next step.
Remember that when we first discussed hexadecimal and when we looked at the hex/binary/
decimal table you learned that each hex character represents four binary bits.
Note that we have divided the binary bits into groups of four. Each of the four bits are then
translated into a hex character.
Looking first at the left-hand group of four: 0011. How would you represent that number in
hex?
Answer: 3
Look at the second group of four: 0110. How would you represent that number in hex?
Answer: 6
While it looks like 5410 in hex is thirty-six, it is actually read as “three six.”
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25
Convert 12810 to Hexadecimal
Rewriting math in Programming
• PEMDAS
– Parentheses
– Exponents
– Multiplication and/or division in the order they appear
– Addition and/or subtraction in the order they appear
Writing Formulas for Computer Programs
What is the formula in the green box?
Answer: Area of a circle
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26
Computer programs do not recognize the symbol for Pi and there are no options for
superscripts to indicate exponents in the majority of compilers. This very common formula must
be rewritten in a way a computer program can decipher it.
In order to represent exponents, we use what’s called a “carrot top,” which you see in the red
box and you get by pressing shift+6. To represent Pi you must use the numerical value 3.14. For
most applications 3.14 is adequate; however, for more precise calculations, you can use as many
decimal places as you require.
You can use virtually any variable to represent the A; however, you must remember that certain
words are “reserved words” and can vary from language to language, so be cautious and
document your code so that other programmers will be able to understand your code easily.
Looking at the blue box we see the way we would write the same formula in a computer
program. Notice we have used the “carrot top” for the exponent and the number 3.14 to
represent Pi. r has been left alone with the assumption that the radius is either read in or
entered by the user at some earlier point in the program.
Rewrite the formula for the slope of a line
Using Math in Computer Programming
Math is universal. The majority of programs that are written involve some type of calculation.
The formulas can range from very simple to highly complex.
• Used to
– Solve repetitive and or complex calculations
• Payroll for a company
• Track complex data sets in scientific calculations
– Create executable files (programs)
– Create automated systems
• Machine Language
Everything that happens within a computer all the way from the GUI we see to the files we save
is translated into machine language. Each character entered into the machine; each instruction
the user issues is translated into binary codes.
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27
Boolean Operators
Because of some of the programs you will write, we also need to discuss the logical or Boolean
operators in computer programming. These operators have their own symbols, fit into
PEMDAS, and have a specific order of operations.
Boolean/Logic (or bool) Operators
a. Also called Logical Operators or just bool
b. Logical operators that return true or false
a. 0 = false
b. 1 = true
c. Three basic bool operators
a. and = and - AND: both arguments must be true for the statement to return True.
b. || - OR: either argument may be true for the statement to return True.
c. ! – NOT: toggles the statement to False if it returns True and True if it returns False.
d. Order of operations – As with PEMDAS, parenthetical groups are evaluated first and
operators are evaluated in the following order:
a. NOT (!)
b. Multiplication/division (*, /)
c. Addition/subtraction (+, -)
d. Relational operators (<, <=, >, >=)
e. NOT Equal (!=)
f. AND (andand)
g. OR (||)
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28
Truth Tables
These are truth tables. Truth tables help us to visualize all possible results of Boolean or logical
comparisons. The results of AND and NOTAND are in the blue tables while the results of OR and
NOTOR are in the green tables.
Truth tables help us step through our program code and determine if our logic is sound and
rational.
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29
Binary to Decimal Worksheet
Convert the following numbers from binary to decimal. Please show your work. Do not use
electronic devices.
a. 10101001
b. 00110010
c. 00111100
d. 11101100
e. 00001000
f. 00101000
g. 00011000
h. 10011001
i.
11111111
j.
01100000
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30
Binary to Decimal Worksheet KEY
Convert the following numbers from binary to decimal. Please show your work. Do not use
electronic devices.
1. 10101001 - 169
2. 00110010 - 50
3. 00100100 - 36
4. 11101100 - 236
5. 00001000 - 8
6. 01001000 - 72
7. 00011000 - 24
8. 10011001 - 153
9. 11111111 - 255
10. 01100000 - 69
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31
Decimal to Binary Worksheet
Convert the following decimal numbers to binary. For the first five you may use the subtraction
method. For the second five you must use the long division method. You must show all work.
No electronic devices are allowed. Please use a separate sheet of paper.
1. 192
2. 168
3. 253
4. 169
5. 17
6. 25
7. 173
8. 127
9. 5
10. 39
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32
Decimal to Binary Worksheet KEY
Convert the following decimal numbers to binary. For the first five you may use the subtraction
method. For the second five you must use the long division method. You must show all work.
No electronic devices are allowed. Please use a separate sheet of paper.
1. 192 = 11000000
2. 168 = 10100100
3. 253 = 11111101
4. 169 = 10101001
5. 17 = 00010001
6. 25 = 00011001
7. 173 = 10101101
8. 127 = 01111111
9. 5 = 00001001
10. 39 = 00100111
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33
Hexadecimal to Decimal Conversion
You will need to convert the numbers below from hexadecimal to decimal format. Please do
not use a calculator (or any other electronic device) and show all work.
1. 0216
2. 3016
3. 1016
4. 0F16
5. F016
6. E116
7. 8016
8. AB16
9. C116
10. 0D16
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Hexadecimal to Decimal Conversion KEY
You will need to convert the numbers below from hexadecimal to decimal format. Please do
not use a calculator (or any other electronic device) and show all work.
1. 0216 = 2
2. 3016 = 48
3. 1016 = 16
4. 0F16 = 15
5. F016 = 240
6. E116 = 225
7. 8016 = 128
8. AB16 = 171
9. C116 = 193
10. 0D16 = 13
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35
Binary to Hexadecimal Conversions
Convert the following numbers from binary to hexadecimal. Please show your work. Do not use
electronic devices.
1. 10101001
2. 00110010
3. 00111100
4. 11101100
5. 00001000
6. 00101000
7. 00011000
8. 10011001
9. 11111111
10. 01100000
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36
Binary to Hexadecimal Conversions KEY
Convert the following numbers from binary to hexadecimal. Please show your work. Do not use
electronic devices.
1. 10101001 = A916
2. 00110010 = 3216
3. 00111100 = 3C16
4. 11101100 = EC16
5. 00001000 = 0816
6. 00101000 = 2916
7. 00011000 = 1816
8. 10011001 = 9916
9. 11111111 = FF16
10. 01100000 = 6016
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37
Decimal to Hexadecimal Conversions
Convert the numbers below from decimal to hexadecimal. Please show all your work and do
not use electronic devices.
1. 255
2. 33
3. 26
4. 51
5. 169
6. 172
7. 89
8. 64
9. 22
10. 47
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38
Decimal to Hexadecimal Conversions KEY
Convert the numbers below from decimal to hexadecimal. Please show all your work and do
not use electronic devices.
1. 255 = FF16
2. 33 = 2116
3. 26 = 1A16
4. 51 = 3316
5. 169 = A916
6. 172 = AC16
7. 89 = 5916
8. 64 = 4016
9. 22 = 1616
10. 47 = 2F16
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39
Evaluations of Expressions and Formulas
Evaluate the following expressions as the computer does. Show your work NEATLY and include all
calculations. Round your final answer to ONE decimal place.
1. 28 – 2 * 6 + 4 – 2 * 12 / 4 + 3^2
3. 8 * (4 – 6 / 3) – (5 – 6 * 4 / 3)
5. 9 * 6 / 5 * 2 + 4 – 4 ^2 + 9 / 4 – (-6) / 2
2. 7.3 – 4.2 + 1.2^ 2 / (-0.6)
4. (4.3 – 2.5) / (7 + 2.1 / 0.7 – 1) -2 * (4.5 / 2)
6. 35 / (6 / 3 + 2^3) / (6 – 2 – 5)
Write the following formulas in the correct form to be used in a BASIC expression. Alphabetic characters
represent variables. Special symbols should be written numerically.
7. V = πh (r2 + rA + A2)
3
8. V = P 1 – (1 + i) –n
i
9. A = P 1 + r
m
10. C = (x – h)2 + (y – k)2
3a2
4b
mt
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40
Evaluations of Expressions and Formulas KEY
Evaluate the following expressions as the computer does. Show your work NEATLY and include
all calculations. Round your final answer to ONE decimal place.
1. 28 – 2 * 6 + 4 – 2 * 12 / 4 + 3^2
2. 7.3 – 4.2 + 1.2^ 2 / (-0.6)
28 – 2 * 6 + 4 – 2 * 12 / 4 + 9
7.3 – 4.2 + 1.44 / (-0.6)
28 – 12 + 4 – 24 / 4 + 9
7.3 – 4.2 + (-2.4)
28 – 12 + 4 – 6 + 9
0.7
23
3. 8 * (4 – 6 / 3) – (5 – 6 * 4 / 3)
8 * (4 – 2) – (5 – 24 / 3)
8 * (4 – 2) – (5 – 8)
8 * 2 – (-3)
16 – (-3)
19
4. (4.3 – 2.5) / (7 + 2.1 / 0.7 – 1) -2 * (4.5 / 2)
(1.5) / (7 + 3 – 1) – 2 * 2.25
1.5 / 9 – 4.5
.166 – 4.5
-4.3
5. 9 * 6 / 5 * 2 + 4 – 4 ^2 + 9 / 4 – (-6) / 2
9 * 6 / 5 * 2 + 4 – 16 + 9 / 4 – (-6) / 2
54 / 5 * 2 + 4 – 16 + 9 / 4 – (-6) / 2
21.6 + 4 – 16 + 2.25 – (-3)
16.9
6. 35 / (6 / 3 + 2^3) / (6 – 2 – 5)
35 / (6 / 3 + 8) / (6 – 2 – 5)
35 / (2 + 8) / (-1)
35 / 10 / (-1)
-3.5
Write the following formulas in the correct form to be used in a BASIC expression. Alphabetic
characters represent variables. Special symbols should be written numerically.
7. V = πh (r2 + rA + A2)
8. V = P 1 – (1 + i) –n
3
i
V = (3.14 * h * (r^2 + r * A + A^2)) / 3
V = P * (1 – ((1 + i) ^ -n) / i)
9. A = P 1 + r mt
m
A = P * ((1 + r / m) ^ (m * t))
10. C = (x – h)2 + (y – k)2
3a2
4b
C = ((x – h)^2) / (3 * a^2) + ((y – k) ^2) / (4 * b)
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41
Numbering Conversions Test
Convert the following from binary to decimal:
1. 11000011
2. 11110010
3. 10010010
Convert the following from decimal to binary:
4. 254
5. 128
6. 33
Convert the following from hexadecimal to decimal:
7. F2
8. 1D
9. 50
Convert the following from decimal to hexadecimal:
10. 256
11. 89
12. 22
Convert the following from binary to hexadecimal:
13. 11001100
14. 11110000
15. 00111100
Evaluate the following equations:
16. 2 + 16 (33 + 22) / 6
17. 6 – 12 (2 + 12)
Rewrite the following equations as you would in a computer program:
18. 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2
19.
20.
𝑥𝑥 =
2
−𝑏𝑏±√𝑏𝑏 2 −4𝑎𝑎𝑎𝑎
2
2𝑎𝑎
𝑎𝑎 + 𝑏𝑏 = 𝑐𝑐 2
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42
Numbering Conversions Test KEY
Convert the following from binary to decimal:
7. 110000112 = 19510
8. 111100102 = 24210
9. 100100102 = 14610
Convert the following from decimal to binary:
10. 25410 = 111111102
11. 12810 = 100000002
12. 3310 = 001000012
Convert the following from hexadecimal to decimal:
21. F216 = 24210
22. 1D16 = 2910
23. 5016 = 8010
Convert the following from decimal to hexadecimal:
24. 25610 = 10016
25. 8910 = 5916
26. 2210 = 1616
Convert the following from binary to hexadecimal:
27. 110011002 = CC16
28. 111100002 = F016
29. 001111002 = 3C16
Evaluate the following equations:
30. 2 + 16 (23 + 22) / 6 = 8210
31. 6 – 12 (2 + 12) = -3010
Rewrite the following equations as you would in a computer program:
32. 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2
A = 3.14 * r^2
33.
34.
𝑥𝑥 =
−𝑏𝑏+𝐴𝐴𝐴𝐴
2𝑎𝑎
2
𝑎𝑎2 + 𝑏𝑏 = 𝑐𝑐 2
x = (-b + A *C) / (2 * a)
c^2 = a^2 + b^2
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43