In mathematical modeling, inverse proportionality (Links to an external site.) plays an important role. Remember that direct and inverse proportionality (Links to an external site.) are related but different. In direct proportionality, as one item increases, the corresponding item would also increase, and vice versa; and in an inverse proportionality, as one item increases, the corresponding item would decrease, and vice versa. This video (Links to an external site.) explains both direct and indirect proportionality.
Let’s begin our discussion:
Many people find that time seems to pass more quickly as you get older. So, the older you are, the shorter a year seems. Let’s model this experience mathematically. Let’s theorize that your perception of a length of time is inversely proportional to your age. We will investigate together and determine as a group if we think this is a good model of the true experience.
Let’s use ptime as the variable for your perception of a length of time, and use age as the variable for your age.
1. Our theory is that your perception of a length of time is inversely proportional to your age. Using the variables above, how would you state the theory in a mathematical equation?
2. Using this theory, how long will a year seem when you are two times older than you are now? To investigate, let ptime1 be how long a year seems now, and let ptime2 be how long a year will seem when you are twice as old. Pick a number for ptime1 (maybe a year feels like 157 days to you, for instance), and then calculate ptime2. Pick a number for ptime1 that is different from other students.
3. Compare your ptime1 and ptime2. What is the relationship between them? Would this relationship change depending on how old you are now? Take a look at other student answers. Do you see a pattern? Can you prove that pattern mathematically?
4. Let’s say that ptime1 is how long a year seemed when you were one year old. Let’s say ptime1 = 365 “time units.” Using age as our x variable and ptime as our y variable, use this online graphing tool (Links to an external site.) to graph our theoretical mathematical model of how your perception of time changes over the course of your life. Here is a short video on how to use the basics of the graphing tool.
5. Use “Settings” within the graphing tool to create a window that covers a typical human lifespan (for age) and a reasonable span of “time units” (for ptime). Attach a screen shot of your graph with your post. (You can use your Print Screen key to capture a screen shot and paste it into a Word Document).
6. Do you think our theory is an accurate model of what you experienced over the course of your lifetime so far? If so, why? If not, why not