Bouncing into Infinity: Exploring Geometric Series with a Bouncing Ball

Algebra 2 Grades High School 4:20 Video
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Lesson Description

Explore the concept of infinite geometric series in a fun and engaging way using the classic bouncing ball problem. Learn how to apply the sum to infinity formula to real-world scenarios.

Video Resource

Sum To Infinity Bouncing Ball Algebra 2

Kevinmathscience

Duration: 4:20
Watch on YouTube

Key Concepts

  • Geometric Series
  • Infinite Geometric Series
  • Sum to Infinity Formula
  • Ratio of a Geometric Sequence

Learning Objectives

  • Students will be able to identify geometric sequences and series.
  • Students will be able to calculate the common ratio of a geometric sequence.
  • Students will be able to apply the sum to infinity formula to solve real-world problems.
  • Students will be able to analyze and interpret the results in the context of the problem.

Educator Instructions

  • Introduction (5 mins)
    Begin by introducing the concept of geometric series and their real-world applications. Briefly review the formula for the sum of an infinite geometric series: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio (|r| < 1).
  • Video Presentation (7 mins)
    Play the video 'Sum To Infinity Bouncing Ball Algebra 2' from Kevinmathscience. Instruct students to pay close attention to how the sum to infinity formula is applied to the bouncing ball problem. Pause at key points to emphasize the importance of identifying the first term and the common ratio.
  • Guided Practice (10 mins)
    Work through the bouncing ball example from the video step-by-step, emphasizing the need to double the distances for the bounces going up and down, and to add the initial drop distance separately. Discuss why the sum to infinity formula is applicable in this scenario (the ball eventually stops bouncing).
  • Independent Practice (10 mins)
    Provide students with similar bouncing ball problems with varying initial heights and ratios. Have them work independently to solve for the total vertical distance traveled by the ball.
  • Wrap-up & Discussion (3 mins)
    Review the key concepts and address any remaining questions. Discuss the limitations of this model (e.g., air resistance not accounted for).

Interactive Exercises

  • Bouncing Ball Simulation
    Use a graphing calculator or online tool to simulate the bouncing ball. Students can input different initial heights and ratios to observe how the total distance changes. They can compare the simulated results with the values calculated using the sum to infinity formula.

Discussion Questions

  • Why is the condition |r| < 1 necessary for the sum to infinity formula to be valid?
  • In what other real-world scenarios can the sum to infinity formula be applied?
  • How would air resistance affect the total distance traveled by the bouncing ball?

Skills Developed

  • Problem-solving
  • Mathematical Modeling
  • Analytical Thinking
  • Application of Formulas

Multiple Choice Questions

Question 1:

What is the formula for the sum to infinity of a geometric series?

Correct Answer: S = a / (1 - r)

Question 2:

In the bouncing ball problem, what does 'r' represent?

Correct Answer: The common ratio

Question 3:

Why is it important to consider the upward and downward motion of the ball separately?

Correct Answer: The bouncing up and down motion needs to be doubled.

Question 4:

A bouncing ball has an initial height of 5 meters and a ratio of 0.6. What is the value of 'a' in the sum to infinity formula?

Correct Answer: 3 meters

Question 5:

For a geometric series to have a sum to infinity, what condition must be true about the common ratio 'r'?

Correct Answer: |r| < 1

Question 6:

A ball is dropped from a height of 4 meters. After each bounce, it reaches 75% of its previous height. What is the common ratio?

Correct Answer: 0.75

Question 7:

A ball is dropped from 6 meters and bounces with a ratio of 0.8. What value do you need to remember to add to the final calculation of the sum to infinity?

Correct Answer: 6

Question 8:

If a bouncing ball eventually stops bouncing, what concept applies to the total distance traveled?

Correct Answer: Infinite Geometric Series

Question 9:

When solving a bouncing ball problem, you calculate the sum to infinity of the bounces (up and down) to be 15 meters. The initial drop was 4 meters. What is the total vertical distance traveled?

Correct Answer: 19 meters

Question 10:

Which real-world factor is NOT considered in the standard bouncing ball sum to infinity problem?

Correct Answer: Air resistance

Fill in the Blank Questions

Question 1:

The formula for the sum to infinity is S = a / (1 - ____).

Correct Answer: r

Question 2:

In the bouncing ball problem, 'a' represents the first ____ after the initial drop.

Correct Answer: bounce

Question 3:

If the absolute value of the common ratio 'r' is greater than 1, the geometric series does not have a finite ____.

Correct Answer: sum

Question 4:

To find the total distance, you must remember to ____ the distance traveled during the upward and downward bounces.

Correct Answer: double

Question 5:

In a bouncing ball problem, the initial height must be ____ to the calculated sum to find the total vertical distance.

Correct Answer: added

Question 6:

The value 'r' is also called the ____ ratio.

Correct Answer: common

Question 7:

A ball dropped from 10 meters reaches a height of 6 meters on the first bounce. The common ratio 'r' is _____.

Correct Answer: 0.6

Question 8:

The sum to ____ formula is used when the number of terms approaches infinity

Correct Answer: infinity

Question 9:

If a geometric sequence is 2, 4, 8, 16,... then the ratio is ____.

Correct Answer: 2

Question 10:

For the sum to infinity to converge, the absolute value of r must be ____ than 1.

Correct Answer: less

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.

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