Ten years after I took Calculus, I signed up to teach it. Limits, Derivatives and Integrals; I recalled the basics. However, not wanting to look stupid in front of my class I took home the book over the summer and did most of the problems. Sometimes I needed the examples and other times I didn’t. Even if I was feeling great about my abilities on a set I continued to work the problems. (This by the way is a trait that I would love to see in my students). My ability to solve Calculus problems returned quickly and my understanding of Calculus followed. I jumped into teaching AP Calculus BC in the Fall of my 7^{th} year teaching and 3^{rd} at the high school level.(There’s another story here) It was a great year that I really enjoyed. I found that most of the concepts were just like I had learned them in high school and college, but I was surprised to find some things that I didn’t remember doing in my Calculus classes. Some of these ideas made complete sense and I admit some took me awhile (years if I’m being honest) to fully digest.

Accumulating a rate is something that I didn’t recall from my Calculus experience, but I fell in love with almost immediately. I remember the first time that I came across the problem about the amusement park with equations for the rate that people were entering and the rate that people were leaving throughout the day. The questions asked were about how many people came to the park that day, what time was the population of the park at it’s highest, and how much money would the park make based on the equations and a price. I searched my book for good problems to help prepare my students for the AP test to no avail. I did later find a problem about recording your speed while you drive to find the total distance traveled. When I came across it I immediately saw the connection and realized the potential.

I then created a project where students would go on a drive somewhere together with a driver, a time tracker, a recorder, and a speedometer reader. This is the only project that I have done every single year and my students always enjoy it. I have them plot their speeds on a graph every 10 or 15 sec and then draw rectangles using a Left or Right Riemann Sum. They would then find the area under all the rectangles and do some conversions to come up with the total distance traveled. Then they would compare their number with the trip meter and Google maps. Then when we visited problems like the amusement park problem I connect it to their understanding of accumulating the speeds to get a distance traveled. This really helped my students to grasp the idea of accumulating a rate, but it didn’t help me with having problems for them to practice so they could demonstrate their understanding.

I wasn’t able to find many problems with accumulating rates out there, so I started to write my own. I was happy with many of them, but some I felt just barely got the job done. Regardless, I now had a collection of problems that students could use to practice demonstrating their understanding of Accumulating a Rate.

As we go over each problem in class I would say, “To accumulate a rate – Integrate.” I’ve now said it so much that it’s stuck in my head and so for me felt like a very natural name for my Blog.

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I love the rhyme! The driving project seems like a nice way to work in multiple methods of solving a problem. I’m making my classes ‘go google’ this year, but I hadn’t included google maps yet. Clearly I’ll need to work on that.

I would love it if you could share your project and your rate accumulation questions… I name our whole unit on integration “Accumulating Rates” so I am equally enamored by the idea. I also tried to do a project with a speedometer that didn’t go that well, so I’d love to see yours… (http://bowmandickson.com/2012/04/06/the-time-i-chickened-out-dealing-with-fears-of-lessons-not-working/ –> don’t know why the pictures aren’t working…)

So, I was also thinking I should share the stuff too, and I put it in the next post ( https://accumulatearate.wordpress.com/2012/08/25/finding-your-distance-traveled-with-speeds/ ) I included my video too, but wish I would have covered up the odometer and somehow secured the camera to the dash so it wouldn’t bounce all over the place. I guess if it gets the job done

Hi Kaleb,

I enjoy your blog. Do you have a collection of “accumulate a rate” problems you can share? Thanks!

Found them! on the follow-up post. Keep up the good work!