Binary Numbers - NCRG



Introduction to Computer Logic

RET Lesson Plan I, University of South Florida

Teacher: Michaela Westlake

Professor: Sanjukta Bhanja

Subject Area: Mathematics

Grade Level: 7th/8th

Lesson Purpose:

This lesson is designed as an enrichment activity to introduce students to mathematical applications in computer technology. The lesson will cover basic operations with binary numbers, Boolean gates, and truth tables.

Student Objectives:

1. Students will convert binary numbers into decimal numbers.

2. Students will complete truth tables for majority gates.

3. Students will construct and explain a variety of Boolean gates, such as AND-, OR-, NOT-, and Majority gates in a cooperative learning activity.

Student Assessment (formal):

• Successful completion of worksheets

• Group project

Student Assessment (informal):

• Discussion Log

Sunshine State Standards:

MA.A.1.3.1; MA.A.2.3.2; MA.D.1.3.1; MA.D.1.3.2; MA.E.1.3.1

Approximate Time Needed:

Three 50-minute class periods (three days):

Day 1: Binary Information

Day 2: Boolean Gates and Introduction to Poster Group Project

Day 3: Continuation of Poster Group Project and Discussion Log

Prerequisite skills:

1. Students should be able to find place values of digits in numbers.

2. Students should be able to evaluate powers.

3. Students should have a basic understanding of theoretical probability.

Procedures:

Day One:

• Hand out copies of Binary Numbers (attached to this lesson plan).

• Use a transparency of the Binary Numbers handout to introduce students to the binary number system, and to demonstrate how binary numbers are converted into decimal numbers.

• Have students independently solve Worksheet 1 (attached).

• Use a transparency of Worksheet 1 to review the answers.

Day Two:

• Hand out copies of Boolean Gates and Worksheet 2 (both attached).

• Use a transparency of the Boolean Gates handout to give students the background information necessary to design basic AND-, OR-, NOT-, and Majority gates, with the proper labeling.

• Demonstrate to students how to find the information for and complete a truth table.

• Have students independently work out the truth table on Worksheet 2.

• Use a transparency of Worksheet 2 to review the inputs and outcomes of the truth table.

• Hand out copies of the Cooperative Learning Rules, and read it with the students.

Day Three:

• Divide each class into groups of three or four students (depending on class size, student compatibility, and special needs).

• Hand out Poster Group Activity instructions (attached).

• Explain assignment, choices, responsibilities, and grading procedures to students.

• Have each individual group come forward to receive the required material.

• When finished with the assignment, have each student complete a Discussion Log (attached).

Material and Resources Required:

Technology:

Overhead projector

Printed Materials:

Transparencies with lesson content and examples

• Worksheets with directions, examples, and problems

• Discussion logs

• Cooperative Learning Rules

Others:

Poster board, yarn, pipe cleaners, tooth picks, scissors, glue, rulers, coloring utensils

Additional Resources:

• Stencils with appropriate geometric shapes

Modification for Differentiated Instruction:

For the Special Needs Student:

• Teacher will speak slowly and clearly when giving students directions (ESOL students).

• ESOL students should be placed in non-ESOL groups for help from their peers and should try to use English to participate in group activities (ESOL goal 1).

• Co-teacher will be able to assist SLD students during problem-solving activities.

• Students will be able to complete pre-printed worksheet.

• SLD students will get extra time on their individual assignments, if required.

Extensions:

1. Take and additional class period, and have one student from each group explain the details of one of the Boolean gates and the associated truth table to the class.

2. Have an open discussion with the students about this enrichment activity.

3. Or, take two additional days, and continue with the lesson Introduction to Nanotechnology, also posted on this web site.

Teaching Tips and Resources:

It is important to be familiar with the covered topics. Review information can be found in

• Givone, D. D., Digital Principles and Design, New York: McGraw-Hill, 2003.

Binary Numbers

Information stored in a computer may be thought of as a series of switches, which are either on or off and are indicated by either the number 0 (off) or the number 1 (on). Computers use the binary number system. In the binary, or base-2, system of numbers, numbers are formed using the digits 0 and 1. Each place in a binary number is associated with a power of 2. Binary numbers are written with the subscript two so that they are not confused with numbers in the decimal system. A digit in the binary system is called a bit, an acronym for binary digit.

Example:

|Decimal |Binary |

|0 |0 |

|1 |1 |

|2 |10 |

|3 |11 |

|4 |100 |

|5 |101 |

|6 |110 |

|7 |111 |

|8 |1000 |

|9 |1001 |

|10 |1010 |

|11 |1011 |

|12 |1100 |

|13 |1101 |

|14 |1110 |

|15 |1111 |

Binary Place Value

|Eights (2³) |Fours (2²) |Twos (2¹) |Ones (2º) |

|1 |1 |0 |1 |

You can use the expanded form of 1101two to find the value of the number as a decimal, or base-10, number.

(1·2³)+(1·2²)+(0·2¹)+(1·2º) = (1·8)+(1·4)+(0·2)+(1·1)

= 8+4+0+1

= 13

So, 1101two = 13ten.

Worksheet 1

Example:

1001two = (1·2³)+(0·2²)+(0·2¹)+(1·2º)

= 8+0+0+1

= 9

Write each binary number as a decimal number. [Show all steps of your conversion!]

1. 1011two

2. 1100two

3. 10001two

4. 11011two

5. 11111two

6. 100000two

7. 111101two

8. 100011two

Boolean Gates

George Boole (1815-1864), an English mathematician, proposed a mathematical system to represent logic. Computer logic systems consist of elements such as gates, which are electronic circuits. Therefore, we also talk about switching circuit theory, when Boolean mathematics is applied to computer logic design. In a logic system, the outputs are a direct result of the inputs.

The following diagrams show different gate symbols and their behavior:

1. AND-gate (Output is “1,” if all inputs are “1” else “0”)

“1”

INPUT

“1”

2. OR-gate (Output is “1,” if at least one input is “1” else “0”)

“1”

INPUT

“0”

3. NOT-gate or Inverter (Output is always opposite of input)

INPUT “1”

4. Majority gate (Output reflects majority of inputs)

A

INPUT B

C

Worksheet 2

For a majority gate, a truth table can be used to show the possible

input combinations, and the resulting output.

Truth table for the majority gate in the example:

|A |B |C |M |

|0 |0 |0 |0 |

|0 |0 |1 |0 |

|0 |1 |0 |0 |

|0 |1 |1 |1 |

|1 |0 |0 |0 |

|1 |0 |1 |1 |

|1 |1 |0 |1 |

|1 |1 |1 |1 |

PRACTICE:

Complete the truth table for the diagram below.

A

INPUT B

C

D

|A |B |C |D |M |

|0 |0 |0 |0 |0 |

|0 |0 |0 |1 | |

|0 |0 |1 |0 | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Poster - Group Activity

1. Each group will be responsible for creating a poster with four diagrams of logic gates and a truth table.

2. The group members will share all responsibilities for the project, meaning that each student in the group will do a specific task, and will be able to explain how he/she did it.

3. Each group will be given a poster board, yarn, pipe cleaners, toothpicks, scissors, glue, rulers, and coloring material. Decide as a group, which of these materials you want to use for your poster.

4. You have one class period to finish the project.

5. Your poster should show a diagram of an AND-gate, an OR-gate, a NOT-gate, and a majority gate. All of the diagrams need to be labeled completely. The truth table for the majority gate should show all the possible combinations for the number of inputs in your diagram, as well as the correct outputs.

6. Write the names of all group members, as well as your class period on the back of the poster.

7. After finished with the group project, students will individually complete a discussion log regarding their experiences with this project.

COOPERATIVE - LEARNING GROUND RULES

1. Cooperate with everyone in your group.

2. Give praise liberally.

3. Use no put-downs.

4. Listen carefully to others.

5. Get all ideas out in the open. (There’s no such thing as a bad idea!)

6. Achieve a group solution for each problem.

7. Make sure that everyone understands the solution before continuing.

8. Share the leadership of the group.

9. Make sure that everyone participates and no one dominates.

Discussion Log

Name:_____________________ Period:_____ Date:_________

1. What I already knew about math and computers was: _________________________________________________________________________________________________________________________________________________________________________________

2. What I would like to know more about math and computers is: _________________________________________________________________________________________________________________________________________________________________________________

3. What I have learned from this lesson is: _________________________________________________________________________________________________________________________________________________________________________________

4. What I liked / did not like about the group work was: _________________________________________________________________________________________________________________________________________________________________________________

5. Additional comments or suggestions I would like to make are: _________________________________________________________________________________________________________________________________________________________________________________

-----------------------

M

“1” OUTPUT

“1” OUTPUT

“0” OUTPUT

M [A, B, C] OUTPUT

M

M [A, B, C, D] Output

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