I was working with 5 middle school students on math through a Waldorf-Inspired Homeschool Co-op. We met once a week and studied Math for an hour. I would always start the hour with a 5 min lecture on some aspect of math that related to the world around us. One week I took plant growth as the topic for the mini lecture. If you look at the height or mass of a plant over it's life span it is quite an amazing graph. First you have the seed which sends out it's root and then it cotyledons (baby leaves). The plant growth begins to accelerate with an exponential curve. Then it tops out at it's max height. The dying process of the plant is rather quick in which it withers away so you have a very interesting graph. You have an exponential growth curve followed by a logorithmic-like deceleration curve, and then a rapid decline to zero at some point after the max height is reached.
This curve can be understood at a middle school level to post grad and beyond. It involves calculus and many math concepts. It resembles the bell curve but rapidly declines to zero on the right side. There is probably a name for this curve. My goal with presenting this lesson about the math way of understanding the growth of a plant, was not to diminish the complex nature of growth and plant life, but rather to add to our appreciation of plant growth through our understanding of math. It was not a gimick to push some math concept. It was five minutes of exploration using tools we were learning. We did plenty of written math that day and skills were honed, but what was memorable? I have noticed that enthusiasm is infectious, especially when it is about something inherently interesting.