Teaching quadratic expressions
Quadratic expressions are the first category of non-linear expressions that algebra students typically study in detail. How do you motivate quadratics as an interesting object of study?
There are a few answers to this question. For example:
- Khan Academy starts with parabolas. After a hint that the shape of a parabola has to do with ballistics and the path of a thrown object, they go on to describe the shape of the graph in detail, including the vertex, the axis of symmetry, and the x and y intercepts.
- EngageNY starts with algebraic techniques. Specifically, it is concerned with distributing and factoring. Engage goes out of its way to avoid giving any early significance to quadratics, instead focusing on the more general category of polynomials, and solving problems with polynomials using factoring. Quadratics are presented as a special case, and associated set of tricks, for that general problem.
I have always liked a third option, which I’m writing about here. Quadratics can be taught as a natural generalization of linear expressions, keeping the focus squarely on what these expressions mean.
Reviewing linear expressions
To follow this line of reasoning, students will need a previous understanding of linear expressions. I’ll keep this brief, but the building blocks they need most are here.
First, they must understand that an expression represents a number. The specific number that it represents, though, may change depending on the values of the variables used in that expression.
I find students often get confused about the meaning of what they are doing as soon as x and y coordinates get involved. Suddenly they think the meaning of a linear expression is a line. (It’s the other way around: a line is one of several ways to represent a linear relationship!) This confusion is reinforced by terms like slope and especially y-intercept, which talk about the graph instead of the relationship between quantities. This has consequences: students who learn this way can answer questions about graphs, but don’t transfer that understanding to other changing values.
For this reason, I prefer to leave x and y out of it, and talk in terms of t, which can represent either time or just a generic parameter, instead. An expression like mt + b represents a number, m represents the rate of change of that number (relative to change in t), and b represents the starting value of that number (when t = 0). That m is also the slope a a graph, and b the y-intercept of the graph, is a secondary meaning.
Students should also understand that the defining characteristic of a linear expression is that it changes at a constant rate. (Specifically, that rate is m.)
Generalizing to non-linear functions
To reach non-linear functions, one simply changes the assumption of the constant rate of change, instead using another linear expression for the rate of change.
The resulting expression looks like: (m t + v) t + b. The rate of change is now the entire linear expression: m t + b. Now students can dig into the rate of change, and they will see that it has its own initial value and its own rate of change. (There’s one important caveat, though, discussed in the next paragraph.) There’s also just one quick application of the distributive property between this form and the more popular m t² + v t + b. But this time, the meanings of the coefficients are front and center.
Here’s the caveat: m does not represent the acceleration, or change in the instantaneous rate. Instead, it represents the change in the average rate so far. A bit of guided exploration can clarify that this must be the case: to decide how far something has traveled, you need to know its average speed, not its speed right this moment. The starting rate is v. The rate at time t is a t + v (where a is the acceleration). That means the average rate so far is the sum of these, divided by 2, or 0.5 a t + v, so m is only half of the instantaneous acceleration.
The upshot of this is that if students accept that a linear expression is the simplest kind of smooth change, then a quadratic expression is the simplest kind of smooth non-linear change!
What next?
Of course, once the importance of quadratic expressions is established, it’s still important to talk about the parabolas that appear in their graphs. It’s still important to talk about techniques for solving them. It’s important to talk about situations, such as trajectories of thrown objects, where they come up a lot. But as you do this, students will hopefully understand this as talking about an idea that has a fundamentally important meaning. They aren’t studying quadratics because parabolas are an interesting shape or because the quadratic formula is so cool; they are studying them because once you need to talk about non-linear change, quadratics are the simplest model for doing that.
If you are the whimsical sort, though, you might notice one more connection. The choice of inserting a linear expression for rate of change was the simplest option, but ultimately arbitrary. In fact, any continuous non-linear function can be written as f(t) = a(t) t + b, if a(t) is a function giving the average rate of change of f(t) over the range from t = 0 to its current value. How could one find such a value for a(t)?
In calculus, students will learn that the derivative of a function gives the instantaneous rate of change of a function at any point in time. We want a(t), then to represent the average value of that derivative. That integrals are so closely related to average values of a function over a range of input is less well-understood by early calculus students. Again, it’s more popular to present these ideas in terminology about areas, that confuse the graph representation for the fundamental meaning. But in fact, a(t) t is precisely the integral (specifically: the definite integral evaluated from 0 to t) of the derivative. A constant factor is lost by taking the derivative, so b recovers that detail. Everything fits nicely together.
You wouldn’t get into this with an algebra class, but it’s an interesting follow-on.
