Is Rate Of Change Really The Same As Slope?

  Over the past week or so, I've introduced my Algebra I class to slope and interpreting slope in real life situations.  This actually lead to a question from one of my students. He asked if slope and rate of change are the same?  The two are used interchangeably but are they really the same?

I told them I see rate of change as something that is more real world such as miles per hour, or typing per minute, or any rate whose change is constant while slope is the mathematical representation of the real world rate of change.


I answered it the way I did because we usually teach the topic with slope always being associated with the line on a graph but if you think about it, most slope in real life is already calculated such as the pitch of a roof, or the grade of a mountain road. 

In the past, I've usually just taught the slope mathematically with a visual graph but I've not extended it into the real world.  I've thrown in a few graphs for students to interpret but nothing more.  This year, I've teaching slope hand in hand with real rates of change and with interpretation so students get a better background in the topic.

This week is short week, so I've planned a few activities such as
1.  Calculating slope from a topographic map because the rate of change determines the grade for a road, the rate of erosion and other things.  It helps students see that rise over run in this case has real meaning.  The rise is a change in elevation, while the run is a change in distance.  This site has a nice explanation, instructions, and practice problems.  It covers math for the student who is studying the geosciences.

2. Calculating the slope of a roof after its built.  WikiHow has three ways to do this and emphasizes the idea that rise and run are both distances. The information has good pictures and instructions.

3. If you've ever watched the news, they always report on the NASDEQ or Dow Jones, both of which has slope but how is that calculated?  It turns out, the lines are actually a line of best fit using regression as explained here.  To simplify this, why not get the daily prices for one or two stocks such as Coca-cola or Apple to see how they move up or down and calculate the rate of change for those?

4.  Prices of gas go up and down in a cyclic manner.  Monitor local prices and calculate the slope of the increase or decrease in price over say 30 days of even a year.  Where I live, the price stays quite stable and only changes once maybe twice a year when the barge arrives with the fuel or the store put it on sale to get rid of it before the new barge load arrives.

5. Of course there is always having students calculate the slope of the ramp at school or any stairs as they are both rates of change.  These two are easy to find and use without much trouble.  You could even have students look up the law to see if the school's ramp meets ADA requirements. We have a ramp but there is a nasty turn in it so I'm not sure if it is fully legal.

So over the week, my students are going to measure the ramp, the stairs, and do a change in elevation off a topo map.  I'll let you know how it goes.  Let me know if you have any other suggestions for students to experience rate of change.