When I was younger (read: before I had kids) I always thought of leaving as a point in time. Think of the graph of a line crossing the x-axis at x=2. When x is less than 2, f(x) is negative. When x<2 graph="" is="" negative="" the="" when="" x=""> is greater than 2 f(x) is positive, and the graph is only equal to zero at one point, x=2. Similarly, I thought that leaving was one time, say 12:00. Earlier than 12, you haven't left, and after 12, you've already left.
This worked pretty well for many years, but then I had kids and the strategy developed problems. I learned to try to be early to places (if you're not early, then you're late). I learned to leave time for traffic depending on the time of day. But even leaving time to be early and for traffic, I still ended up being late, even when I left at precisely the time I expected.
Then I realized that the problem was that it took between 5 and 10 minutes to leave, maybe more if I were leaving for the whole day and had a lot to remember. Even when I was just by myself, sometimes it took 5 minutes to leave (which was one reason I was late a lot before I had kids). It takes a lot of time to get 3 small people moving and to help them remember the things they need.
So leaving is more like a piece-wise defined function. The time before leaving and the time after leaving are the same, but there is a 10-15 minute stretch while we are in the process of leaving. It makes me late less often (not never). It also makes the process of leaving less stressful: if I know it's going to take 10 minutes from the time we start going out the door to the time we are driving away, I can let it take that time and not yell at the kids for being slower than I expect.
Yet another way math helps understand the world.
Original graph, then new graph.
This worked pretty well for many years, but then I had kids and the strategy developed problems. I learned to try to be early to places (if you're not early, then you're late). I learned to leave time for traffic depending on the time of day. But even leaving time to be early and for traffic, I still ended up being late, even when I left at precisely the time I expected.
Then I realized that the problem was that it took between 5 and 10 minutes to leave, maybe more if I were leaving for the whole day and had a lot to remember. Even when I was just by myself, sometimes it took 5 minutes to leave (which was one reason I was late a lot before I had kids). It takes a lot of time to get 3 small people moving and to help them remember the things they need.
So leaving is more like a piece-wise defined function. The time before leaving and the time after leaving are the same, but there is a 10-15 minute stretch while we are in the process of leaving. It makes me late less often (not never). It also makes the process of leaving less stressful: if I know it's going to take 10 minutes from the time we start going out the door to the time we are driving away, I can let it take that time and not yell at the kids for being slower than I expect.
Yet another way math helps understand the world.
Original graph, then new graph.
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