Geometric sequences and series
• Analyze situations that involve geometric sequences and series (A-SSE.A.1$^\star$).
• Derive the formula for the sum of a finite geometric series (A-SSE.B.4$^\star$).
Analyze situations that involve summing a exponential sequence (which is generally called a geometric series) These arise naturally in some saving and banking problems as well as in some interesting geometric contexts.
WHAT: The A Lifetime of Savings task gets at the heart of how the total length of time makes a big impact in an investment, and how a small amount of savings can grow very large over a given time. It also ties in what students already know about exponential functions and finances, and ties these topics to geometric series.
WHY: The purpose of this instructional task is to give students an opportunity to construct and find the value of a geometric series in a financial literacy context. The task assumes that students have already developed the formula for a geometric series themselves; having them recognize the need for this formula (and look up if necessary) allows them to engage in MP.5 (use appropriate tools strategically). The task also provides students with an opportunity to look for and express regularity in repeated reasoning (MP.8). This task also asks students to interpret the variables in the future value formula in the context of the problem.
WHAT: The Course of Antibiotics task has students use data about a dosage of medicine, including dosage times and how much medicine remains in the body, and develop a geometric series to determine how much medicine is in the body at a given time. It is a good problem for basic practice of using data to generate a series. and how to sum the series.
WHY: The purpose of this task is to present a real-world application of finite geometric sequences in a tangible context—the amount of medicine that remains in a body over time. It is also an opportunity for students to use repeated reasoning as they try to find a pattern that will result in a formula (MP.8).
WHAT: The Triangle Series task applies the same concepts of a geometric series and sums, but in a geometric context.
WHY: This task provides an opportunity for students to apply geometric series in geometric contexts. It is an opportunity to reason abstractly (MP.2).
WHAT: The Cantor Set task is more practice with geometric series and sums, but might be conceptually more abstract for students.
WHY: This task provide an opportunity for students to apply geometric series in geometric contexts. It is an opportunity to reason abstractly (MP.2).
WHAT: From the website: “Everyone knows that vampires are hungry, immortal monsters. What if they turned a new victim into a vampire every week? Because of exponential growth, they would quickly run out of victims, not just in the local area, but on the face of the Earth! This must mean that vampires do not exist. Of course, there are real contagious, fatal diseases in the world. Unlike with vampirism, some people are immune, some people can be cured, and some people die from these diseases. Students will analyze the consequences of a more realistic contagion and discuss factors that keep a contagion from spreading among a population.”
WHY: This lesson modeling the spread of disease is very engaging. It is placed in this section because it provides an opportunity for students to use their newfound skill in summing a geometric series (A-SSE.B.4$^\star$). (A few questions use a logarithm to solve an equation. These could be handled, instead, with a graphing solution using technology.)