Decoding Repeating Decimals: An Exploration into Recurring Patterns

PreAlgebra Grades High School 7:02 Video
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Lesson Description

This lesson explores repeating decimals, their origins in division, and how to represent them using bar notation. Students will learn to identify repeating decimals and understand the concept of recurring patterns.

Video Resource

What are Repeating Decimals? | What Causes Repeating Decimals? | Math with Mr. J

Math with Mr. J

Duration: 7:02
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Key Concepts

  • Repeating Decimals: Decimals with a digit or group of digits that repeat infinitely.
  • Bar Notation: A method to represent repeating decimals using a bar over the repeating digits.
  • Division and Repeating Decimals: Repeating decimals often arise from division problems where the remainder never reaches zero, leading to a recurring pattern.

Learning Objectives

  • Students will be able to define repeating decimals and identify them in a given set of numbers.
  • Students will be able to convert fractions to repeating decimals using long division.
  • Students will be able to represent repeating decimals using bar notation.
  • Students will be able to explain why some fractions result in repeating decimals when converted.

Educator Instructions

  • Introduction (5 mins)
    Begin by asking students what they know about decimals. Introduce the concept of repeating decimals through examples like 1/3 = 0.333... and ask if they've encountered such numbers before. Briefly explain that this lesson will delve deeper into understanding these numbers.
  • Video Viewing (10 mins)
    Play the 'What are Repeating Decimals? | What Causes Repeating Decimals? | Math with Mr. J' video. Instruct students to take notes on the definition of repeating decimals, how they arise from division, and the use of bar notation.
  • Guided Practice: Division to Decimals (15 mins)
    Work through example problems converting fractions to decimals using long division on the board. Start with fractions like 2/9 and 5/11 which result in repeating decimals. Emphasize the process of identifying the repeating pattern and expressing the result using bar notation. Encourage students to participate by suggesting the next steps in the long division process.
  • Independent Practice (15 mins)
    Provide students with a set of fractions (e.g., 1/7, 4/15, 7/9) to convert to decimals using long division. Have them identify whether the resulting decimal is repeating or terminating and represent repeating decimals using bar notation. Circulate to provide assistance and answer questions.
  • Discussion and Wrap-up (5 mins)
    Facilitate a class discussion on the challenges they faced while working on the independent practice. Review the key concepts of repeating decimals and bar notation. Assign the quizzes for homework.

Interactive Exercises

  • Decimal Identifier Game
    Present a series of decimals, some repeating and some terminating. Students identify which are repeating and represent them using bar notation.
  • Fraction-to-Decimal Challenge
    Divide students into teams and give each team a set of fractions to convert to decimals. The first team to correctly convert all fractions and represent repeating decimals using bar notation wins.

Discussion Questions

  • Can all fractions be written as terminating decimals? Why or why not?
  • What are some real-world situations where repeating decimals might occur?
  • How does bar notation simplify the representation of repeating decimals?

Skills Developed

  • Long Division
  • Pattern Recognition
  • Mathematical Representation

Multiple Choice Questions

Question 1:

Which of the following is a repeating decimal?

Correct Answer: 0.333...

Question 2:

What does the bar in bar notation indicate?

Correct Answer: The digit(s) under the bar repeat infinitely.

Question 3:

Which fraction, when converted to a decimal, results in a repeating decimal?

Correct Answer: 1/3

Question 4:

How is 0.1666... written in bar notation?

Correct Answer: 0.1(6bar)

Question 5:

Repeating decimals always come from what operation?

Correct Answer: Division

Question 6:

What is the first step to converting a fraction to a decimal?

Correct Answer: Divide the numerator by the denominator

Question 7:

Which of the following decimals terminates?

Correct Answer: 0.125

Question 8:

Which of the following is the correct bar notation for 0.232323...?

Correct Answer: 0.(23bar)

Question 9:

What digit repeats in the repeating decimal 0.777...?

Correct Answer: 7

Question 10:

Which of the following fractions converts into a repeating decimal that is NOT 0.333...?

Correct Answer: 4/10

Fill in the Blank Questions

Question 1:

A decimal that repeats without ending is called a _____ decimal.

Correct Answer: repeating

Question 2:

The notation used to show repeating digits in a decimal is called _____ notation.

Correct Answer: bar

Question 3:

Repeating decimals often result from the operation of ____.

Correct Answer: division

Question 4:

In the repeating decimal 0.666..., the digit that repeats is ____.

Correct Answer: 6

Question 5:

The fraction 1/9, when converted to a decimal, is 0._____ repeating.

Correct Answer: 1

Question 6:

A decimal that ends after a certain number of digits is called a _____ decimal.

Correct Answer: terminating

Question 7:

In bar notation, a bar is placed over the _____ that repeat.

Correct Answer: digits

Question 8:

0.142857142857... is a _____ decimal.

Correct Answer: repeating

Question 9:

To convert a fraction to a decimal, you _____ the numerator by the denominator.

Correct Answer: divide

Question 10:

3/11 converted into a decimal is 0.272727..., so what numbers are repeating? _____ and _____.

Correct Answer: 2 and 7

Educational Standards

PA.N.1:Read, write, compare, classify, and represent real numbers, and use them to solve problems in various contexts. PA.N.1.4:Compare and order real numbers; locate real numbers on a number line. Identify the square roots of perfect squares to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers.

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