Teaching Exponential Decay: A Lesson from "The Incredible Shrinking Man"
Recently, I watched The Incredible Shrinking Man, a classic 1957 science fiction film directed by Jack Arnold and based on Richard Matheson's novel The Shrinking Man. As a math teacher, I couldn’t help but smile at the story's amusing parallel to a mathematical concept we teach—exponential decay. In the film, the main character faces a bizarre fate, shrinking smaller and smaller but never quite disappearing, much like how an object undergoing exponential decay never fully reaches zero. Spoiler alert: just as in exponential decay, the man continues to get smaller and smaller, facing life’s challenges as his size diminishes, but he does not cease to exist.
This idea of continuously reducing size, but never fully vanishing, offers a perfect way to introduce the concept of exponential decay to students. Not only does it make the math more relatable, but it also brings a sense of intrigue, especially if your students enjoy thinking about real-world applications or even science fiction.
What is Exponential Decay?
In mathematical terms, exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. Over time, the quantity becomes smaller and smaller, but it never truly reaches zero. This is why exponential decay is often illustrated using graphs that asymptotically approach zero but never actually touch the x-axis.
Connecting Exponential Decay to the Movie
In The Incredible Shrinking Man, the main character’s shrinking process follows an intriguing path, much like an exponential decay function. He faces new challenges as his size diminishes rapidly at first, then more slowly over time—just as exponential decay initially decreases at a rapid rate but gradually slows as it continues. However, much like an exponential decay function that never reaches zero, the character doesn’t simply disappear. He continues to exist, facing challenges with each new reduction in size.
This storyline can serve as a helpful analogy for students learning exponential decay. It highlights how, no matter how small a quantity becomes, it never truly ceases to exist. This can also help clarify common misconceptions, such as the assumption that an object undergoing exponential decay will eventually "run out" or hit zero after a certain time period. The truth is, it approaches zero but never quite gets there.
Teaching Exponential Decay with Real-World Applications
To make exponential decay more engaging for students, we can draw from real-life examples that they might be familiar with:
Radioactive Decay: Perhaps the most well-known example of exponential decay (half-life) in the real world, radioactive substances decay over time, becoming less and less radioactive but never fully disappearing. This is a great example to introduce in science or math classes.
Depreciation of Assets: The value of a car or piece of technology decreases exponentially after it is purchased. While it loses value quickly in the first few years, the depreciation slows down over time.
Population Decline in Ecology: In environmental science, populations of species undergoing decline often follow exponential decay patterns, with their numbers shrinking over time but never fully disappearing.
Medicine: The concentration of a drug in the bloodstream often follows an exponential decay pattern, decreasing over time as the body metabolizes it.
Graphing Exponential Decay
Visualizing exponential decay through graphs is a powerful way to reinforce the concept. By plotting exponential decay functions, students can see how the line approaches zero but never quite reaches it. This helps them grasp the abstract idea that, although the quantity decreases continuously, there is always something left, no matter how small.
One fun activity could involve having students graph different exponential decay functions and compare their rates of decay. For example, comparing radioactive decay to the depreciation of a car can help students see how different factors affect the rate at which something decays.
Exponential Decay in Word Problems
Another effective strategy for teaching exponential decay is through word problems. You could even weave in a bit of narrative from The Incredible Shrinking Man to keep students engaged. For example:
Imagine a shrinking man whose height decreases by 20% each day. If he starts at 6 feet tall, how tall will he be after 7 days?
Such problems not only strengthen students' algebraic manipulation skills but also help them apply exponential decay in real-world scenarios. The storytelling aspect makes it more memorable and relatable.
Why Exponential Decay Matters
Understanding exponential decay is crucial not just for math classes, but for real-life decision-making and critical thinking. Whether it's understanding the half-life of a substance, the long-term value of an investment, or the effects of population decline, the concept has far-reaching implications.
By teaching exponential decay in a relatable and engaging way, you can give your students a solid grasp of this important mathematical concept while also encouraging them to think critically about how it applies to the world around them.
Conclusion
Exponential decay can be a tricky concept for students to grasp at first, but by connecting it to fun and unexpected places—like the plot of The Incredible Shrinking Man—we can make it more approachable and engaging. Whether through graphs, word problems, or real-life examples, exponential decay teaches students that even as things shrink, they never truly disappear. By building a strong understanding of this concept, students are better prepared for more advanced math and science topics in the future.
So, just like the shrinking man in the movie, let's help our students understand that while things may get smaller, they never quite vanish!
