Sir Isaac Newton was a physicist who wanted to know why things move the way they do and fall the way they do. To tackle those questions, he started thinking about speed. If a person walked from one place to another and kept the same pace along the way, then her speed at any time would always be the same. And her speed would be the total distance she traveled divided by the time it took her to arrive, like this:

Instead of keeping a steady pace, if she varied her speed during the trip, slowing down sometimes, stopping, and speeding up other times, then you could still describe her AVERAGE speed as the total distance she traveled divided by the time it took, like this

But Newton wasn't satisfied with knowing the average speed. He wanted a way to know her speed at any instant along her journey. This created a mathematical problem since you would have to divide the distance traveled during an instant (which would be zero) by the amount of time it took to go that distance (again, zero). And this didn't make any sense at all since surely at each instant along the journey, she had to have some speed, or she would never arrive in the first place!

So Newton thought that he might be able to sneak up on the speed at one instant by starting with the average speed, and making the time interval shorter and shorter until it almost became one instant of time. Like this

The study of slopes is part of the mathematics of calculus. Newton had invented calculus in order for him to think more clearly about speed and acceleration. Of course he found plenty of other uses for it, too.

Amazingly, at the same time in Germany, Leibnitz, a mathematician, was thinking about the slopes of lines in general (as mathematicians are wont to do), and invented calculus independently of Newton.

Here are some discussion questions you might want to think about ==> ?

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