Lesson Plan. Preparation

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1 Math Computer Technician Practicum 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 10-1 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 Computer Technician Practicum (c) Knowledge and skills. (4) The student applies communication, mathematics, English, and science knowledge and skills to research and develop projects. The student is expected to: (B) demonstrate proper use of mathematics concepts as they apply to the development of products or services; and (C) demonstrate proper use of science principles to the development of products or services. (5) The student knows the concepts and skills that form the basis of computer technologies. The student is expected to: (B) define the use of Boolean logic in computer technologies; 1

2 Interdisciplinary Correlations English 110.4(b) Knowledge and skills. (6) Reading/word identification/vocabulary development. The student uses a variety of strategies to read unfamiliar words and to build vocabulary. The student is expected to: (A) expand vocabulary through wide reading, listening, and discussing; and (B) rely on context to determine meanings of words and phrases such as figurative language, idioms, multiple meaning words, and technical vocabulary. (7) Reading/comprehension. The student comprehends selections using a variety of strategies. The student is expected to: (F) identify main ideas and their supporting details; (G) summarize texts; and (J) read silently with comprehension for a sustained period of time. Speech (b) Knowledge and skills. (1)(A) explain the importance of communication in daily interaction; ()(E) participate appropriately in conversations for a variety of purposes; (3)(A) The student uses appropriate communication in group settings; (E) use appropriate verbal, non-verbal, and listening strategies to communicate effectively in groups; (5)(B) use language clearly and appropriately; Tasks Students will utilize provided notes and slide presentation to complete notes pages. Students will complete worksheets. Accommodations for Learning Differences Lessons must 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 ( Preparation Copy the handouts. Have multimedia presentations ready to show the class. Instructional Aids Student Notes sheet Student worksheets

3 Materials Needed Copies pencils 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 requires a firm understanding of numbering systems and how to convert between the various systems. 3

4 Outline MI OUTLINE 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 1 st power f. Review positional numbering system a. Hundreds b. Tens c. Ones Slides 8 11 Converting from binary to decimal Slides 9 16 Converting from decimal to binary Slides 17 Introduction to hexadecimal Conversion from hexadecimal to binary Slides 3 6 Conversion from decimal to hexadecimal Slides 7 37 IP Addressing and logical operators NOTES TO TEACHER Extension notes are provided in the multimedia presentation. Timeline of delivery is at discretion of instructor. Guidelines regarding division of information are suggestions only. 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, give students the hexadecimal to binary worksheet. Upon completion of slide 6, give students the decimal to hexadecimal worksheet. Upon completion of slide 37, give students worksheets pertain to IP addressing. Multiple Intelligences Guide Existentialist Interpersonal Intrapersonal Kinesthetic/ Bodily Logical/ Mathematical Musical/Rhythmic Naturalist Verbal/Linguistic Visual/Spatial 4

5 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 utilize 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. 5

6 Binary, Hex, and Decimal, Oh My! Student Notes Handout 1. numbers a. Zero and any number obtained by repeatedly adding one to it i. Ex. 0, 1, 3, 100. Integers a. Any number i. Includes negative numbers and zero 3. Numbering System a. The number depends on the position of the digits to represent the number 4. Number base a. The (or radix) is the number of digits used to represent numbers in a positional numerical system. 5. Any number to the power is equal to 1. a. Example: 0 = 6. Any number to the power is equal to that. a. Example: 1 = 6

7 Converting from Decimal to Binary Base o 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 o Binary uses two as its base so each is equal to two to an exponent. o Since we are using two single characters we use and as our two digits. Consider the binary number 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. 7

8 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 which is equal to right? This place holder has a instead of a zero. What is the product of any number multiplied by one? Answer: number Once you have calculated the we add the answers together. of all the placeholders where a one appears What is the sum of 8+4+? Answer: The decimal equivalent of is which is shown in the green box. Converting from Decimal to Binary Can be done two different ways Long Division Convert from decimal to binary Sart by the original number by our base which is. Notice that we do not directly divide 35 by 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 zero or one as the remainder you move to the next step. 8

9 Using the we got in the first step (35 divided by ) we move to the second step which is dividing by. We the process until we have in the,, and. Now here comes the tricky part. Example: 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 You can always check your answer by using the binary to decimal conversion method. Go ahead, check our math. = Subtraction Convert to binary using subtraction Looking at our binary table what is the number that is still less than? Answer: 9

10 To indicate that 3 will be used we put a in the placeholder. Once we subtract 3 from 37 we have left right? What is the number in our table that is still lower than 3? Answer: 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 have to flip the bits in this case so the binary representation of =. Hexadecimal Characters You can convert directly from hex to decimal but not from decimal to hex. In order to represent the numbers (to give us our 16 characters) we use letters. A = 10, B = 11, C = 1, D = 13, E = 14, and F = 15. Each is equal to (combination of zeros and ones) and like binary can be combined to create ever larger numbers. Hexadecimal (Hex) Binary Decimal B 11 C E 14 F 1111 As with binary and decimal, is a numbering system. Each placeholder uses as its base and is equal to 16 to an exponent. 10

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 character appears we the answers together. the of all the placeholders where a What is the sum of ? Answer: The decimal equivalent of is which is shown in the green box. Decimal to Hex Conversion Convert to Hexadecimal 11

12 The easiest way to convert a decimal number to hex is to go to using the subtraction method to go from decimal to binary. first. Let s look at it 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 the binary bits into of. Each of the four bits are then into a character. Looking first at the left-hand group of four; Answer:, how would is that number represented in hex? Look at the second group of four; Answer:, how would you represent that number in hex? While it looks like in hex is thirty-six it is actually read as. 1

13 Convert to Hexadecimal Decimal to binary Binary to hex IP Addressing Every node on a network have a unique IP address Protocol version Unique -bit number Divided into divided by decimal points» EX: Separated into Commonly Used IPv4 Classes Class First Octet Shared Octets Number of Networks Maximum Addressable Hosts ,777,14 B 65,534 C 19 3 >,000,000 Highest decimal number an octet may be is 13

14 Each octet is equal to = 10 Running out of IPv4 addresses due to 3-bit limitation Combined with a to increase number of addressable nodes on a network Subnet Mask Class First Octet Default Subnet Mask B C 19 3 Ipv6 Composed of bits Eight fields Represented by hexadecimal numbers Divided into groups of characters» EX: FEE3:00FF:003D:0000:0000:0000:3015:AABC Multiple fields with zero values can be abbreviated» EX:» EX: Maximum number of IPv4 addresses is: 3 or roughly provides us with 3.4 x or 340 decillion addresses addresses whereas Assigning IP Addresses Can be done manually or by DHCP (Dynamic Host Configuration Protocol) IP address assigned Does not change Human error in duplicating addresses can cause issues IP addressing Assigned by a server Most common and simplest method 14

15 Boolean Operators Because of some of the programs you write we also need to discuss the in computer programming. These operators have their own, and have a specific of operation. or Boolean operators, fit into 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. 15

16 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 is sound. ANDING Every IP address has a default o Class A o Class B o Class C In order to locate a specific node on the network a computer must the IP address with the appropriate in Based on our truth tables we should know the following o 1 AND 1 = o 1 AND 0 = o 0 AND 1 = o 0 AND 0 = 16

17 EXAMPLE: IP address: Subnet mask: First we convert both to : IP address: ( ) Subnet Mask: ( ) AND: ( ) The result of ANDing is the address. 17

18 Student Notes Handout Binary, Hex, and Decimal Oh My a. Natural numbers a. Zero and any number obtained by repeatedly adding one to it a. Ex. 0, 1, 3, 100 b. Integers a. Any whole number a. 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: 0 = 1 Any number to the 1 st power is equal to that number. Example: 1 = 7 * 10 = 7 * 100 = * 10 = 4 * 10 = * 10 = 5 * 1 =

19 Converting from Decimal to Binary Base 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. 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. Convert from binary to decimal 19

20 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, By looking down the chart you can see that we are going to use the following values; 3,, and 1 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 on the right. The vertical alignment makes things easier to read. We start with 18, determine that there is a zero in that placeholder. What is the answer when you multiply any number by zero? Answer: zero Moving down the line we have zeros until we get to the place holder 3 which is equal to 8 right? This place holder 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 one appears we add the answers together. What is the sum of 8+4+? Answer: 14 The decimal equivalent of is 14 which is shown in the green box. Converting from Decimal to Binary Can be done two different ways Long division Subtraction Long Division Convert from decimal to binary Start by dividing the original number by our base which is. Notice that we do not directly divide 5 by 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 zero or one as the remainder you move to the next step. Using the quotient we got in the first step (35 divided by ) we move to the second step which is dividing 17 by. We repeat the process until we have zero in the quotient, dividend, and remainder. Now here comes the tricky part. 0

21 Example: Binary 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 is flipped to Once we rewrite the number we see that 35 in binary is You can always check your answer by using the binary to decimal conversion method. Go ahead, check our math = 35 Subtraction Convert to binary using subtraction Looking at our binary table what is the highest number that is still less than 37? Answer: 3 To indicate that 3 will be used we put a 1 in the 5 placeholder. Once we subtract 3 from 37 we have 3 left right? What is the highest number in our table that is still lower than 3? Answer: 1

22 Skip down to the 1 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 Now we put a 1 in the 0 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 = Hexadecimal Characters 0 9 A F You can convert directly from hex to decimal but not from decimal to hex. Hexadecimal (Hex) Binary Decimal In order to represent the numbers (to give us our 16 characters) we use letters. A = 10, B = 11, C = 1, 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 ever larger numbers A B C D E F

23 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. Look at the table. FF is the hexadecimal number we are going to convert to decimal. By looking down the chart, I can see that we are going to use the following values; 16 1, and 16 0 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 ? Answer: 55 The decimal equivalent of FF 16 is 55 which is shown in the green box. Decimal to Hex Conversion 3

24 Convert 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 is equal to 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 in hex is thirty-six it is actually read as three six. 4

25 Convert to Hexadecimal IP Addressing Every node on a network must have a unique IP address Internet Protocol version 4 Unique 3-bit number Divided into four octets divided by decimal points» EX: Separated into classes Commonly Used IPv4 Classes Class First Octet Shared Octets Number of Networks Maximum Addressable Hosts A ,777,14 B > 16,000 65,534 C >,000, Highest decimal number an octet may be is 55 5

26 Each octet is equal to eight binary bits = Running out of IPv4 addresses due to 3-bit limitation Combined with a subnet mask to increase number of addressable nodes on a network Subnet Mask Class First Octet Default Subnet Mask A B C Ipv6 Composed of 18 bits Eight 16-bit fields Represented by hexadecimal numbers Divided into groups of four hexadecimal characters» EX: FEE3:00FF:003D:0000:0000:0000:3015:AABC Multiple fields with zero values can be abbreviated» EX: 00EE = EE» EX: 0000 = 0 Maximum number of IPv4 addresses is: 3 or roughly 4 billion addresses whereas Ipv6 provides us with 3.4 x or 340 decillion addresses Assigning IP Addresses Can be done manually or by DHCP (Dynamic Host Configuration Protocol) Static IP address Manually assigned Does not change Human error in duplicating addresses can cause connection issues Dynamic IP addressing Assigned automatically by a DHCP server Most common and simplest method Boolean Operators 6

27 Because of some of the programs you will write we 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. 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. andand - 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 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 ( ) 7

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. ANDING Every IP address has a default subnet mask o Class A o Class B o Class C In order to locate a specific node on the network a computer must AND the IP address with the appropriate subnet mask in binary Based on our truth tables we should know the following o 1 AND 1 = 1 o 1 AND 0 = 0 o 0 AND 1 = 0 o 0 AND 0 = 0 8

29 EXAMPLE: IP address: Subnet mask: First we convert both to binary: IP address: ( ) Subnet Mask: ( ) AND: ( ) The result of ANDing is the network address. 9

30 Binary to Decimal Worksheet Convert the following numbers from binary to decimal. Please show your work. Do not use electronic devices. a f b g c h d i e j

31 Binary to Decimal Worksheet KEY Convert the following numbers from binary to decimal. Please show your work. Do not use electronic devices

32 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

33 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 = = = = = = = = = =

34 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 E AB F C F D 16 34

35 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 = = = F 16 = F0 16 = E1 16 = = AB 16 = C1 16 = D 16 =

36 Binary to Hexadecimal Conversions Convert the following numbers from binary to hexadecimal. Please show your work. Do not use electronic devices

37 Binary to Hexadecimal Conversions KEY Convert the following numbers from binary to hexadecimal. Please show your work. Do not use electronic devices = A = = 3C = EC = = = = = FF =

38 Decimal to Hexadecimal Conversions Convert the numbers below from decimal to hexadecimal. Please show all your work and do not use electronic devices

39 Decimal to Hexadecimal Conversions KEY Convert the numbers below from decimal to hexadecimal. Please show all your work and do not use electronic devices = FF = = 1A = = A = AC = = = = F 16 39

40 Converting IP Addresses Convert the IP addresses below from decimal to binary or binary to decimal as appropriate. Each octet in an IP address is considered its own eight-bit number

41 Converting IP Addresses KEY Convert the IP addresses below from decimal to binary or binary to decimal as appropriate. Each octet in an IP address is considered its own eight-bit number

42 Identify the address class for each IP address. Address Class Identification Address Class

43 Address Address Class Identification KEY Class A B B C C A A A C A A A C C A A A C A C Identify the address class for each IP address. 43

44 Numbering Conversions Test 1. Convert the following from binary to decimal: a = b = c =. Convert the following from decimal to binary: a = b = c = 3. Convert the following from hexadecimal to decimal: a. F 16 = b. 1D 16 = c = 4. Convert the following from decimal to hexadecimal: a = b = c. 10 = 5. Convert the following from binary to hexadecimal: a = b = c = 6. Convert the following IP addresses as indicated: a = b = 7. AND the IP addresses below with their default subnet mask to determine the network address: a b c. AND d. Network address 44

45 Numbering Conversions Test KEY 1. Convert the following from binary to decimal: a = b = 4 10 c = Convert the following from decimal to binary: a = b = c = Convert the following from hexadecimal to decimal: a. F 16 = 4 10 b. 1D 16 = 9 10 c = Convert the following from decimal to hexadecimal: a = b = c. 10 = Convert the following from binary to hexadecimal: a = CC 16 b = F0 16 c = 3C Convert the following IP addresses as indicated: a = b = AND the IP addresses below with their default subnet mask to determine the network address: a b c. AND d. Network address: