# Category Archives: Concavity

## Concavity and Population – A Calculus Essay

*(got a little burned out last month, but feeling back on track… so I’m trying to get on a blog schedule again because I really enjoy reflecting and have missed keeping up with the blog)*

For the past two years, I have done a really awesome project with my students in my non-AP Calculus class using UN Population data. The kids all cite it as one of the most interesting things that we do, and it really helps me see who truly understands the concepts of concavity and inflection points and who can just set a second derivative equal to zero.

GOAL:Using data from the UN World Population Prospects, graph the population data for three different countries from the year 1950-2010. Then, describe how the population has changed over the past 60 years and will change for the next 90 using the Calculus concepts of increasing/decreasing, concave up/concave down, relative extrema and inflection points. If inclined, try to connect some of the trends that you see to history.

Of the so-called “four representations of Mathematics” (verbal, numerical, analytic and graphical), I think we tend to emphasize the analytic, and rightfully so. It’s the most sophisticated and the cleanest to use in a classroom. So I wanted to do a project that focused on the other three representations. Basically, I wanted students to tell me everything they could about concavity without getting caught up in the calculations.

We spent a couple days in class just making the graphs, which was a real test of my students’ limited Excel skills, and analyzing the data with Excel by calculating manual first and second derivatives (just calculating the difference between the population between successive years, and the difference between the differences, just to get an idea of how the function is changing and how the change is changing). This was really fun because students would be calling me over excitedly to show me things on their graph (like huge spikes in migration in a country going through a genocide, or the insane and sudden shift in the population of the former Soviet countries when the Soviet Union broke up). Then we spent another class period writing outlines for the analysis, mostly just picking out inflection points and where the population was concave up and concave down. Then the students wrote the analyses for homework. Most students wrote around 2-3 pages not including the graphs… about Calculus… and I really enjoyed reading them.

### Things I liked about the project:

**Students picked their own countries**, which meant that many became pretty invested in the topic. Talking about real countries that students really care about made things like “switching from increasing at an increasing rate to increasing at a decreasing rate” actually interesting. And every student’s project was different which made it awesome to grade.- I think having students describe verbally what was going on with the population
**forced some to really think hard about the huge idea of concavity**and brought out a lot of misconceptions, especially the perennial confusion between inflection points and relative extrema. The wonderful thing about using Standards Based Grading is that I gave them really detailed feedback and about half of them went back and edited their paper and rewrote parts to make the Calculus better – I never edited something that had already been graded in high school! - I think
**students really learned a lot about academic technology**. I purposefully didn’t give them guidelines on how I wanted them to present everything, and I think this struggle is something that is going to help students in the future, in a more independent environment of college or work.

### Things I didn’t like about the project:

- Some students felt limited by population and
**wanted to examine data for other things.**But I didn’t have other reliable data ready and didn’t want to waste time having them search for data and find bad data. So I kind of nixed some creativity and student independence for the sake of efficiency and logistics… which is something I do a lot. And something I want to get better at. But I don’t know how. **Having them do three different countries was a bit overkill,**as they kind of did the same thing for each one.- Last year we came back to this project and added the fourth representation by fitting logistic curves using Geogebra to the populations and comparing our predictions with that of the UN, but
**I didn’t have time this year for the wrap up this year**, which was sad.

The addition we made this year is that we are going to send our work over to the AP World History class and they are going to try to explain the graphs historically. I’m not sure how that is going to work out, but I’m all about trying interdisciplinary things.

Overall, great project and a great change of pace!

## PART OF ONE STUDENT’S WORK:

DRC’s population has increased in the past century and is projected to significantly increase within the next century as well (

Graph 1). The overall population rate (Graph 2)for DRC significantly increased in 1990-1995. This sudden population increase was caused by the Rwandan Genocide which reached a fever pitch in 1994. In 1995, the population jumps from 36.5 million in 1990 to 44 million. The migration rate in 1990-1995, mainly consistent of refugees, was the rate that contributed most to the overall population number fluctuation. Many Rwandans fled to DRC as refugees from the massacre that plagued their country, which was a destructive conflict that erupted between Rwanda’s two main tribes: the Hutus and the Tutsis. As aforementioned, this conflict was initiated by the colonizers, who set the stage for the genocide by favoring one tribe over the other. Also, when looking atGraph 1,I noticed there is an inflection point in the population of year 2040. Before 2040, the population of DRC had been increasing at an increasing rate. After 2040, it is projected to begin increasing at a decreasing rate. The occurrence of this inflection point is probably due to DRC reaching its carrying capacity, or reaching the point where it cannot support any more people due to lack of sufficient resources to support a consistently growing population.

(not perfect, but I love it!)

## PROJECT DESCRIPTION AND INSTRUCTIONS:

## GRADING RUBRIC:

## STUDENT FEEDBACK:

- It was a nice project for math as it gave us the freedom to relate concepts we learn in class to historical measurements, keeping my interest. However, I didn’t feel as comfortable with the analysis as the graphs as it was exactly clear what I should be writing. I gave it a shot and got a 3 but I feel like the instructions on that part couldve been a bit more clear.
- I just find it interesting and important to teach students that what they learn in calculus isn’t just for the sake of being able to do math but because its essential for many people to come up with things we never really consider, like predicitions about the population of countries. The chanign world population is a big deal and without knowledge on critical points, derivatives and so on the world would not know what to expect in the future or how to react based on the change.
- I found the project interesting as I researched and got to know a little bit more about a certain historical conflict that interested me. I learned skills in Excel that I didn’t really know before, like doing certain types of graphs. I think I could have done better if I had more time to work on it.
- Of course the population project was very helpful as i stated above. It was interesting to see how some regions or countries differed in population growth and calculus terms helped us relate why each population graph looks the way it does.
- I loved the population project!!! I even sent you an anonymous feed back about it. I think it was one of the most interesting assignments I’ve ever done in my high school career at King’s, and I really enjoyed getting it done, even if we didn’t have much time to do it. I was genuinely interested in what the population trends on my graph meant, and I was amazed at how much historical and political events have an impact on the population. Super interesting. Thanks for assigning it!
- the population project was a good activity. it was helpful in terms of understading the material because we were able to explain the population/time graphs and why they curved up and down. It just took me a lot of time to do because there was so much to talk about for every country, so I ended u writing an essay-like analysis of each country. but I totally recommend doing this project next year.
- this project was very interesting and helped me learn new skills using excel but did not really help me learn anything new using calculus it just helped me cement the ideas we learnt in my head. I thought that the time we used doing the project could have been used to help us understand other topics better.
- I learned things I never knew I never knew. if that makes sense.
- I really enjoyed doing the population project!! As you can tell it took me a while to do. I like to use skills that I learn in class and apply them to situations like these. makes me feel that I did learn something. It was also a fun way to get us used to claiming what an inflection point is and what critical points is and how they look like on a graph.
- Although I want to recommend this for the next year I would not make the students do three different countries because I feel liek it was tedious and repetative talking about concavity, derivaties, and inflection points. Also I feel like that would solve your 16 page issue when grading 😛
- I liked the project and learned a lot but I thought it was a lot of work.
- Although it was a bit hard at first because i wasn’t sure what I was supposed to write in the analysis, (if it was supposed to be explaining math concepts or history) I found it very interesting and I enjoyed doing it especially since I’m really interested in demographics.
- I learned so much about Excel! I think it was a very fun way of reasserting our knowledge to you and to ourselves. I didn’t know that what I wrote would come out of me, turns out I’m smarter than I think. The only problem was that I felt it was due to soon. That might have been because I had another project and a test due on the same day and the pressure was on.
- I feel like I did a good job, though after looking at the feedback I got I didn’t really understand exactly what I did wrong. I have to look over it again and put more work into it. But overall I feel it was a large project but It wasn’t that bad.
- It helped show how calculus us used in real life rather than on paper. plus it wasn’t that much work, you gave us two lessons to work on in class which was helpful because you were there to answer all the question at the beginning of the project where it’s usually most difficult.
- Even though i thought it was more about History and less about Math, I thought it was intresting
- I learned a lot of interesting things and tricks on the Excel that I didn’t know, which helped me a lot in other subjects.
- it was a bit hard and i didnt understand the concepts and the method of making it and describing the project
- i would not recommend it because I personally don’t know how to write math papers, and analyse graphs because i don’t know what answers you’re looking for.
- The project was not that hard, but what i found difficulties in was the analysis part of it.
- I think that the population project was interesting, as well as, I learned skills in Excel that I didnt know before. It really helped in raising our grades, and I recommend doing it next year.
- The population project definitley helped to improve my skills in Excel since the graphs we were required to create were for the most part new to me. Even though this project took a good amount of time and effort, it definitley helped me learn several important skills and I regret not submitting it on time because had I worked about an hour maximum each night and used my time in class effectively I would have been able to complete it on time.
- I just think that the project could be improved for the class next year so they get more into it and the due date was too soon which is why so many people didn’t hand it in on time and got a reduction in their grade
- It was really good, I think time was sort of an issue
- No comment. It was an interesting and novel way of teaching. Typical Mr. Bowman (That’s a good thing btw):-)
- The population project was very interesting. I learned a lot from it, but it was too much work, and it was due right before midterm week. I wouldn’t recommend doing the project next year. I didn’t feel I doing taking math. It felt like physics.
- I checked the last two boxes for a reason, because i think that students will definatly benefit from the project by learning more about certain countries and why and how population changes. Helps them determine how every change would look like on a graph and why it looks like that. At the same time i think its too much work and i feel 7arram for them to go through all of that when they can simply right a quiz on that topic :S
- We could have had a simple explanation of the subject in class and not do a really big project. To me, it was kind of really pointless, and I think that you should not do it again for the sake of the students.