Objective: 

Students will approximate irrational numbers using rational values, place them on a number line, and discuss how computers represent numbers. They will explore the challenges computers face with irrational numbers due to limitations in representing data as sequences of 0s and 1s.

Materials Needed: 

• Number line posters

• Sticky notes

• Calculators

• Rulers

• Graph paper

Steps:

• Introduction: 

  • Students review irrational numbers like √2 and π, recalling that they cannot be expressed as exact fractions or finite decimals. 

  • Transition to a discussion on how computers store numbers. 

  • Explain that computers use binary (0s and 1s) to represent all data, including numbers, and this poses challenges with irrational numbers since they require infinite precision. 

  • Because of this, computers use rational approximations to handle them, much like what students will do today.

• Group Activity: 

  • Students work in pairs to approximate irrational numbers such as √2. 

  • First, they use calculators to estimate that √2 is between 1 and 2, then refine their approximation to between 1.4 and 1.5, and so on. 

  • As students calculate better approximations, they place sticky notes on a large number line, visually representing the irrational number using rational values. 

  • While working, students will reflect on how their approximations, like computers, have limitations and cannot represent the true irrational number.

• Discussion: 

  • After completing the activity, guide a discussion on how computers translate numbers into binary and how they approximate irrational numbers due to their inability to represent infinite decimals. 

  • Ask, "What challenges do you think computers face when calculating with irrational numbers like √2 or π?" 

  • Lead into how this relates to data representation, where the same number might be displayed differently in various contexts, such as rounded values in a calculator or different visual representations (graphs, charts, etc.).

Equity and Access: 

Provide pre-drawn number lines for students needing extra support and pair students with varying strengths in math for peer collaboration.

Real-World Application: 

Connect this activity to real-world uses, such as how computers round irrational numbers when calculating distances or using GPS systems, emphasizing the limitations of representing data precisely.

CS Practice(s):

• Developing and Using Abstractions: Students represent irrational numbers as rational approximations on a number line, mirroring how computers approximate and display numbers.

• Communicating about Computing: Students explain how their approximations are similar to the way computers approximate numbers, and the limitations of both.

Standard(s):

• CA CCSS Mathematics 8.NS.1

• CA CCSS Mathematics 8.NS.2

• CA CS 6-8.DA.8

• CS CS 6-8.IC.20

Objective: 

Students will approximate irrational numbers using Python, exploring how computers handle these approximations. They will understand the challenges computers face when representing irrational numbers using binary and examine how the same data can be represented in different ways.

Materials Needed: 

• Computers with Python installed (or access to an online Python IDE)

Steps:

• Introduction: 

  • Students review the distinction between rational and irrational numbers, focusing on the fact that irrational numbers, like √2 and π, cannot be exactly expressed as fractions or finite decimals. 

  • Transition to discussing how computers represent numbers in binary and the challenges they face when working with irrational numbers. 

  • Explain that students will use Python to approximate irrational numbers today, similar to how computers use approximations when dealing with them.

• Coding Activity: 

  • In pairs, students write Python code to approximate irrational numbers like √2 by using the math.sqrt() function. 

  • Have students compute and display these values in multiple ways: as truncated decimal approximations (e.g., 1.414) and by comparing the binary representation using Python’s built-in tools (like bin() for integers). 

  • Students will reflect on how the computer stores and manipulates these approximations due to limitations in precision.

• Discussion: 

  • After running their code, students will compare the rational approximations generated by Python with the results of their calculations. 

  • Facilitate a class discussion on the limitations of computing irrational numbers and how these values can be represented in different formats. 

  • Highlight how computers approximate irrational numbers using rational values and store them as finite sequences of 0s and 1s. 

  • Encourage students to think about how this connects to real-world problems, like GPS systems that rely on approximate values for distances.

Equity and Access: 

Provide pre-written code templates for students needing extra support, ensuring all students can participate. Group students with different coding experience levels to encourage peer learning.

Real-World Application: 

Discuss how computers use approximations to display data in various formats, such as using binary, decimals, or graphs. Relate this to real-world scenarios like rounding distances in navigation systems or using different visual representations in data charts.

CS Practice(s):

• Developing and Using Abstractions: Students use Python to approximate irrational numbers and display them in various forms, such as decimal, binary, and visual outputs.

• Communicating about Computing: Students explain how Python handles the approximations of irrational numbers and the challenges involved in storing such numbers.

Standard(s):

• CA CCSS Mathematics 8.NS.1

• CA CCSS Mathematics 8.NS.2

• CA CS 6-8.DA.8

• CS CS 6-8.IC.20