Unlocking Repeating Decimals: A Geometric Series Approach

Algebra 2 Grades High School 2:58 Video
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Lesson Description

Convert repeating decimals to fractions using the infinite geometric series formula. This lesson explores a challenging example, separating repeating and non-repeating parts, and applying the formula for the sum of an infinite geometric series.

Video Resource

Repeating Decimal to Fraction Using Geometric Series(Challenging)

Mario's Math Tutoring

Duration: 2:58
Watch on YouTube

Key Concepts

  • Repeating decimals
  • Infinite geometric series
  • Common ratio
  • Sum of an infinite geometric series
  • Fraction conversion

Learning Objectives

  • Students will be able to identify the repeating part of a decimal.
  • Students will be able to express the repeating part of a decimal as an infinite geometric series.
  • Students will be able to apply the formula for the sum of an infinite geometric series.
  • Students will be able to convert a repeating decimal into a fraction.
  • Students will be able to verify their answer using a calculator.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing what repeating decimals are and why they are rational numbers (can be expressed as fractions). Briefly recap geometric series and the formula for the sum of an infinite geometric series (A1 / (1 - r)).
  • Video Explanation (5 mins)
    Play the Mario's Math Tutoring video on 'Repeating Decimal to Fraction Using Geometric Series (Challenging)'. Encourage students to take notes on the example problem.
  • Step-by-Step Breakdown (10 mins)
    Walk through the video's example (.08333...) step-by-step on the board. Emphasize: 1. Separating the non-repeating part (.08). 2. Expressing the repeating part (.00333...) as a geometric series (.003 + .0003 + .00003 + ...). 3. Identifying the first term (A1 = .003) and the common ratio (r = .1). 4. Applying the infinite geometric series formula (Sum = A1 / (1 - r) = .003 / (1 - .1) = .003/.9 = 1/300) 5. Adding the non-repeating part back in (.08 = 8/100 = 24/300). 6. Simplifying the resulting fraction (24/300 + 1/300 = 25/300 = 1/12).
  • Practice Problems (15 mins)
    Provide students with practice problems of similar difficulty. Encourage them to work individually or in pairs. Example problems: .1666..., .272727..., .12333...
  • Review and Q&A (5 mins)
    Review the solutions to the practice problems and address any remaining questions. Emphasize the importance of checking the answer using a calculator.

Interactive Exercises

  • Think-Pair-Share
    Present a repeating decimal. Have students individually try to convert it to a fraction. Then, have them pair up with a classmate to compare their approaches and solutions. Finally, have a few pairs share their methods with the class.
  • Error Analysis
    Provide students with an incorrectly worked-out example of converting a repeating decimal to a fraction. Have them identify the error and correct it.

Discussion Questions

  • Why can repeating decimals be expressed as fractions?
  • How does identifying the common ratio help in converting repeating decimals to fractions?
  • What are some real-world applications of converting repeating decimals to fractions?

Skills Developed

  • Problem-solving
  • Analytical thinking
  • Application of formulas
  • Fraction manipulation

Multiple Choice Questions

Question 1:

What is the first step in converting a repeating decimal like 0.2333... to a fraction using the geometric series method?

Correct Answer: Separate the repeating part from the non-repeating part

Question 2:

In the infinite geometric series formula S = A1 / (1 - r), what does 'r' represent?

Correct Answer: The common ratio

Question 3:

What is the common ratio in the geometric series representation of 0.0777...?

Correct Answer: 0.1

Question 4:

When does the formula for the sum of an infinite geometric series apply?

Correct Answer: When the common ratio is between -1 and 1

Question 5:

What is 0.666... expressed as a fraction?

Correct Answer: 2/3

Question 6:

If a repeating decimal has a non-repeating part of 0.1 and a repeating part of 0.0444..., how would you combine these after converting the repeating part to a fraction?

Correct Answer: Add the fractions

Question 7:

What is the geometric series representation of the repeating decimal 0.454545...?

Correct Answer: 0.45 + 0.045 + 0.0045 + ...

Question 8:

Which of the following is NOT a repeating decimal?

Correct Answer: 0.5

Question 9:

The repeating decimal 0.111... is equal to what fraction?

Correct Answer: 1/9

Question 10:

After converting a repeating decimal to a fraction, what is the best way to check your answer?

Correct Answer: Use a calculator to divide the numerator by the denominator

Fill in the Blank Questions

Question 1:

The repeating part of the decimal 0.12555... is just the digit ____.

Correct Answer: 5

Question 2:

The formula for the sum of an infinite geometric series is S = A1 / (1 - ____).

Correct Answer: r

Question 3:

In the repeating decimal 0.777..., the common ratio is ____.

Correct Answer: 0.1

Question 4:

Before applying the geometric series formula, you may need to ____ the repeating and non-repeating parts of the decimal.

Correct Answer: separate

Question 5:

The fraction equivalent of 0.333... is ____.

Correct Answer: 1/3

Question 6:

When converting a repeating decimal to a fraction, the repeating part forms an infinite ____ series.

Correct Answer: geometric

Question 7:

The first term in the geometric series 0.02 + 0.002 + 0.0002 + ... is ____.

Correct Answer: 0.02

Question 8:

A repeating decimal can always be written as a ____.

Correct Answer: fraction

Question 9:

If the common ratio in a geometric series is greater than or equal to 1, the series does not ____.

Correct Answer: converge

Question 10:

Checking your answer with a ____ is a good way to ensure you correctly converted the repeating decimal to a fraction.

Correct Answer: calculator

Educational Standards

A2.A.3:Represent and solve mathematical and real-world problems involving arithmetic and geometric sequences and series. A2.A.3.4:Solve problems that can be modeled using finite geometric sequences and series given the ๐‘›th terms and sum formulas. Graphing calculators or other appropriate technology may be used. A2.N.1:Extend the understanding of numbers and operations to include complex numbers, radical expressions, and expressions written with rational exponents.

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