My Algebra II class recently wrapped on exponentials and logs. They toughed through some nutty equations with little protest. We verified our algebra with graphs. All old hat at this point. I figured it was worthwhile to look at some applications.

Unfortunately, most of them go like this:

College Algebra with Modeling & Visualization, 4th Edition, Rockswold

College Algebra with Modeling & Visualization, 4th Edition, Rockswold

The first problem isn't genuine. Or, it's glossing over the reality it's trying to present. A properly done retirement account isn't something you open with an initial deposit and ignore for 30 years. But books are full of these. Savings account problems citing 6% interest rates, looking at you.

Properly done, a savings account is a geometric series of exponential terms that collapses to:


More realistic, but more difficult. For the sake of simplicity, the authenticity is removed.

The second situation isn't half bad. Population growth is easy to understand and there are some fun misconceptions to discuss. And it does adhere to the simple growth model textbooks like. This word problem sucks all the questions out of it.

Guessing Games

How many people live in France? Spain? Kenya? the United States? The range of guesses will surprise you. Other than knowing that China has the most, my students were way off when it came to the rest of the world. The US being third in world population is a great "wait, really?" bomb to drop. Canada and Australia being so vacant is another.

Next, population trends. Population always goes up right? Not so fast. The World Bank maintains a lovely data table with historic population growth/decay rates.

The mere appearance of this table launched the class in a 20 minute discussion about developed vs. developing, birth rates, contraception access, AIDS, geography, the looming Japanese retirement crisis, you name it.

Task Time

Once the discussion is over, demonstrate why these growth rates are important.

  • Pick 5 countries
  • Look up their current population on your phone ("population of [blank]" does the trick)
  • Use the World Bank projections to estimate their size in 10 years, 20 years

Here's another point where you can have a 15 minute discussion. It's also a moment where you can nix a trick and "divide by 100" instead of "move the decimal" for percentages. What would you do if you were Japan? Kenya? Saudi Arabia? When would Ireland catch Singapore?

It's a natural writing task if I ever saw one.


Present (or have them create) a set of graphs like this:

What's going on here? Why are they intersecting? What's that imply? As a government official in South Africa, what are your concerns? Would they be different or similar to your French counterpart?

Social Studies and Math, together forever.

AuthorJonathan Claydon