Day 2 - Binary Representation of Numbers
Day 2: Binary Representation of Numbers
Learning Objectives
- DAT-1.A: Explain how data can be represented using bits.
- DAT-1.B: Explain the consequences of using bits to represent data.
Essential Questions
- How are different types of numbers represented in binary?
- What are the limitations of fixed-bit representations?
Materials Needed
- Presentation slides on number representation
- Handouts on binary number representations
- Worksheet for binary number conversions
- Calculators
- Colored markers or manipulatives for signed binary representation
Vocabulary
- Unsigned integer
- Signed integer
- Two's complement
- Fixed-bit representation
- Overflow error
- Floating point
- Precision
- Rounding error
Procedure (50 minutes)
Opening (8 minutes)
-
Review and Connection (3 minutes)
- Review binary conversion basics from previous lesson
- Connect to today's focus on representing different types of numbers
-
Warm-up Activity (5 minutes)
- Quick binary conversion practice (2-3 problems)
- Ask students to predict: "How might computers represent negative numbers?"
Main Activities (32 minutes)
-
Lecture: Integer Representation (12 minutes)
- Explain unsigned integers (positive numbers only)
- Introduce the concept of fixed-bit representation (e.g., 8-bit integers)
- Demonstrate the range of values in different bit lengths (8-bit: 0-255, etc.)
- Explain signed integers and two's complement representation
- Demonstrate converting between positive and negative numbers using two's complement
- Discuss overflow errors when values exceed the range
-
Hands-on Activity: Representing Integers (10 minutes)
- Students work in pairs with 8-bit binary representation cards
- Practice representing positive integers in 8 bits
- Practice representing negative integers using two's complement
- Identify cases where overflow would occur
-
Group Work: Exploring Limitations (10 minutes)
- Divide class into small groups (3-4 students)
- Each group explores a scenario involving binary number limitations:
- Group 1: What happens when adding two numbers causes overflow?
- Group 2: What's the smallest/largest number representable in 8-bit signed binary?
- Group 3: How does increasing bit length affect the range of representable numbers?
- Group 4: Real-world examples of overflow errors
- Groups document their findings and prepare to share
Closing (10 minutes)
-
Brief Introduction to Real Numbers (5 minutes)
- Explain the challenge of representing real numbers (infinite precision)
- Introduce the concept of floating point representation
- Briefly discuss how floating point uses bits for sign, exponent, and mantissa
- Explain rounding errors and precision limitations
- Preview that this is why some decimal calculations in computers seem "wrong"
-
Exit Ticket and Preview (5 minutes)
- Students complete a worksheet with binary number representation problems
- Preview that next class will focus on text representation
Assessment
- Formative: Participation in hands-on activities and group work
- Worksheet: Accuracy of binary number representations and conversions
Differentiation
For Advanced Students
- Explore floating point representation in more detail
- Investigate how to perform binary arithmetic operations
- Research real-world examples of binary representation issues
For Struggling Students
- Provide step-by-step guides for two's complement conversion
- Use visual aids for representing signed numbers
- Focus on smaller bit lengths (4-bit) before moving to 8-bit
Homework/Extension
- Complete worksheet on binary number representation
- Research a real-world example of an overflow error and its consequences
- Create a visual guide explaining how two's complement works
Teacher Notes
- Two's complement can be challenging; be prepared to explain multiple ways
- Make connections to programming languages and their integer limits
- Emphasize that understanding these limitations helps prevent bugs
- Consider demonstrating a simple program that shows overflow behavior