Rational vs. Irrational Numbers: Cracking the Code

Algebra 1 Grades High School 5:54 Video
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Lesson Description

Explore the difference between rational and irrational numbers, learn how to identify them, and discover why some numbers are considered 'irrational'.

Video Resource

Introduction to rational and irrational numbers | Algebra I | Khan Academy

Khan Academy

Duration: 5:54
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Key Concepts

  • Rational Numbers: Numbers expressible as a ratio of two integers (a/b, where b ≠ 0).
  • Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers; they are non-repeating, non-terminating decimals.
  • Integers: Whole numbers (positive, negative, and zero).

Learning Objectives

  • Students will be able to define and identify rational and irrational numbers.
  • Students will be able to convert terminating and repeating decimals into fractions (ratio of integers).
  • Students will be able to recognize common irrational numbers like π, e, and square roots of non-perfect squares.

Educator Instructions

  • Introduction (5 mins)
    Begin by asking students what they already know about different types of numbers (integers, fractions, decimals). Briefly review the concept of integers and fractions as a foundation for understanding rational numbers.
  • Video Viewing (10 mins)
    Play the Khan Academy video "Introduction to rational and irrational numbers". Instruct students to take notes on the key definitions and examples provided in the video.
  • Discussion and Examples (15 mins)
    Lead a class discussion about the video. Clarify any confusing points. Work through additional examples of rational and irrational numbers, emphasizing the process of converting repeating decimals to fractions. For irrational numbers, discuss why they cannot be written as a fraction.
  • Practice Exercises (15 mins)
    Provide students with a worksheet or online activity with a variety of numbers (integers, fractions, decimals, square roots). Ask them to classify each number as either rational or irrational and explain their reasoning.
  • Wrap-up (5 mins)
    Summarize the key differences between rational and irrational numbers. Preview the next steps, which might include proofs about irrational numbers or applications in geometry.

Interactive Exercises

  • Number Sort
    Create a set of cards with different numbers written on them (integers, fractions, decimals, square roots, π, e). Have students work in pairs or small groups to sort the cards into two categories: Rational and Irrational. They must justify their choices for each card.
  • Repeating Decimal Challenge
    Give students a list of repeating decimals and challenge them to convert them into fractions. Provide guidance or resources for converting repeating decimals, as needed.

Discussion Questions

  • Can you give an example of a rational number that is not an integer?
  • Why is the square root of 4 a rational number, but the square root of 5 is irrational?
  • How can you tell if a decimal is rational or irrational just by looking at it?
  • Are all fractions rational numbers? Explain.

Skills Developed

  • Critical Thinking
  • Classification
  • Mathematical Reasoning
  • Problem Solving

Multiple Choice Questions

Question 1:

Which of the following is a rational number?

Correct Answer: 1/3

Question 2:

Which of the following is an irrational number?

Correct Answer: √5

Question 3:

Which of the following decimals represents a rational number?

Correct Answer: 0.3333...

Question 4:

Which set of numbers does 7 belong to?

Correct Answer: Rational Numbers

Question 5:

Which of the following is NOT an integer?

Correct Answer: 3/4

Question 6:

Which of the following numbers can be expressed as a ratio of two integers?

Correct Answer: 0.75

Question 7:

If a number is a non-repeating, non-terminating decimal, then it is a(n) _______ number.

Correct Answer: Irrational

Question 8:

Which of the following is equivalent to 2/3?

Correct Answer: 0.666...

Question 9:

Which of the following is the sum of two rational numbers?

Correct Answer: Always Rational

Question 10:

Which statement is always true?

Correct Answer: Rational + Rational = Rational

Fill in the Blank Questions

Question 1:

A number that can be written as a fraction of two integers is called a _______ number.

Correct Answer: rational

Question 2:

A decimal that neither terminates nor repeats is called a _______ number.

Correct Answer: irrational

Question 3:

The number π is an example of an _______ number.

Correct Answer: irrational

Question 4:

The set of whole numbers and their opposites are called _______.

Correct Answer: integers

Question 5:

The number 0.333... is a repeating decimal that can be expressed as the fraction _______.

Correct Answer: 1/3

Question 6:

The square root of any non-perfect square is a(n) _______ number.

Correct Answer: irrational

Question 7:

A _______ decimal can be written as a ratio of two integers

Correct Answer: repeating

Question 8:

A _______ number has an infinite, non-repeating decimal representation.

Correct Answer: irrational

Question 9:

Integers are part of a set of _______ numbers.

Correct Answer: rational

Question 10:

The sum of an irrational number and a rational number will always be _______.

Correct Answer: irrational

Educational Standards

A1.N.1:Extend the understanding of exponents to include square roots and cube roots. A1.A.3.4:Evaluate linear, absolute value, rational, and radical expressions. Include applying a nonstandard operation such as x ⨀ y=2x+y.

Teaching Materials

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