The teachers I coach are beginning their investigation of exponential functions. They are using Core Plus Math, and are working really hard to do increasingly problem-based, student-centered teaching. I’m helping the teachers with pacing, lesson-planning, and formative assessment, as well as coaching them on facilitating student-teacher and student-student interactions.
So… my project over the break is to put the “fun” back in exponential function (and maybe put a little “expo” back in there too). “Why do we have to learn this? What matters? What does this have to do with what we already know?” are valid questions I hope students ask and I can help them answer. (By the way, I just learned that exponential functions can help us predict when everyone on earth will be following Lady Gaga on Twitter (October 2025), but I don’t think we even need such fascinating applications to help make sense of exponential functions. Source: http://www.youtube.com/watch?v=51AhdTj-irQ&feature=youtu.be)
What’s the big idea behind exponential functions? And what experiences help kids ask the big idea questions themselves?
For an algebraist, I think the big idea is always about quantities (x and y, money and years, population and seconds, etc.) and the operations done on one to get the other. What are exponential functions? Just another subset of operations done to x to get y. In this case, exponentiation is the big relationship. What matters? Identities and inverses. Logarithms are the inverse operation — to find x given y, use logarithms. 1 is the identity element: input 1 into an exponentiation operation, and get back what you started with. Exponentiation builds on work with linear functions and polynomials because:
- There are quantities, related by operations
- The questions we ask are, “what’s happening to x to get y?” and “how can you undo that to get x given y?”
- The problems we solve are finding x’s given y’s (x-intercepts, specific values) and y’s given x’s (y-intercepts, specific values)
What’s hard is that when we first learn about solving equations, we talk loosely about inverses, I think. Solving y = 3x + 5 by subtracting 5 and dividing by 3 (or adding -5 and multiplying by 1/3) is really applying two inverse functions, which in this case is the same as operating by the inverse element. But with exponentiation and logarithms, the only way to make sense of “undoing” is applying an inverse function. e^x is a function, 2^x is a function, and there is no meaningful inverse under exponentiation that we could use to exponentiate both sides to find x. Instead we have to use a different function, the logarithm function. And we have to use different functions depending on the base. Thinking sensibly about the set of all real numbers under the operation exponentiation is not easy, and when we try to apply understandings of multiplication and addition, it gets tangled. At least in my head. (Also, I think that when algebraists think about exponentiation, they’re thinking about repeated operating, so exponentiation is almost like a notation rather than a well-defined operation with identities and inverses. This further befuddles me).
For an analyst (which, admittedly, I am definitely not), I think the big questions are about measurement and change. How do small changes in x affect y? What about large changes in x? What can we say about the ratio of change in y to change in x at infinitely small moments in time? What can we say about the accumulation of y over time? Exponentiation builds on work with linear functions because:
- We can use tables of values to define the patterns of change recursively, and write explicit functions
- We can notice first, second, third order differences (and how with exponential functions, they are always exponential, whereas with polynomials they are eventually constant)
- We can talk about how the rate of change (slope) seems to increase as x increases (for exponential growth functions; vice versa for exponential decay)
- We can use graphs (as well as tables) to characterize the shape and behavior of exponential functions
Why should a kid, who’s not an algebraist or analyst, care?
- Algebraists are thinking about solving equations, and ask good questions that help us do that. Thinking like an algebraist makes sense of the idea of equations and solutions to an equation, and apply those basic ideas across many kinds of equations. If you want to get good at solving equations, you want to learn to think like an algebraist — what are the quantities? the relationships? the inverse functions?
- Analysts are thinking about patterns that are pretty accessible to students. Rates of change in tables and graphs, what graphs look like, what the difference between arithmetic and geometric growth is… Those are all patterns that students see and want to be able to describe in order to make predictions and solve problems.
Okay, but why care about solving exponential equations and looking at the patterns they make?
- You can answer cool questions about money and when Lady Gaga will have more followers than there are people on Earth.
- Lots of natural phenomenon grow or decay exponentially, and it’s not too hard to see why they do, and predict what other phenomenon might grow the same way, if you think like an analyst and ask, “what are situations in which the rate of change increases or decreases proportionally to how the input increases?”
- There are some interesting puzzles and patterns to be explored. If you understand linear (arithmetic) growth, and can predict the nth value in a linear sequence, find what value of x gives a particular output in a linear function, etc. it’s pretty natural to wonder, “what if we had a repeated-multiplication situation? What’s the same? What’s different?”
So real contexts like interest, population, exponential decay, number of people receiving a chain letter, or number of Twitter users, can drive meaningful math exploration, in which the context helps students understand the concept (population is an example of exponential growth because as the population grows, more babies are born, so population grows more quickly as it gets larger).
Students’ own inquiry and natural wondering process, can drive meaningful math exploration: extending what we know about linear functions to the “next fronteir.”
The questions that I hope students carry around as they explore exponential functions, in kid-friendly-language, then, are:
- Can I do everything I could do with linear functions with exponential functions (i.e. with functions that grow by multiplying instead of adding)? Can I find inputs and outputs? Make graphs and tables? Write recursive and explicit rules? Do rates and “starting points” still matter? How?
- What kinds of situations grow by multiplying instead of adding, and how can I recognize them? What are good questions to ask about those situations? (When? How much?)
- Can I find an unknown number in a grow-by-multiplying situation, whether it’s an input or output number? Can I do it algebraically? With tools like tables, graphs, or calculators?
- What’s cool about this kind of function? (For one thing, the tables of first and second differences and accumulation are all exponential. Why?)
Those sorts of questions can be motivated through real-world problems and through good reflection and summarizing of linear functions understanding, coupled with the innocent wondering, “what if we multiplied instead of adding each time?”