Does Modeling Mean Using Harder Functions?

Here’s something I’m puzzled about in trying to push this picture of math education, the one where we collect data and try to model it with functions: when I come up with ideas for suitable activities, they often require “harder” functions than students may be used to seeing. Let me give an example based on a super-traditional problem such as

A 5-meter ladder is leaning against a wall. The bottom of the ladder is 2.5 meters from the foot of the wall. How high is the top of the ladder?

We know what the kid is supposed to do, traditionally. Draw the picture, recognize the right triangle, notice that the hypotenuse is 5, and write something like

$(2.5)^2+x^2=5^2$

and solve the equation to find that $x = \sqrt{25-6.25} = \sqrt{18.75} \approx 4.33$.

But in the approach I’m trying to push, we de-emphasize the specific answer and focus on the relationship. We take a 5-meter ladder (or its more practical equivalent, a chair) and set it next to a wall, at different distances (let’s call that distance base), and see how high the ladder reaches (height). We plot height against base, and try to figure out a function that is a good model for the data.

Allegedly, students can approach this two ways:

• If they can figure out the formula from the geometry, they can immediately use the formula as a model. Then the data verify their analysis.
• If they can’t figure out the formula on their own, the shape of the data will give them a clue about what function might work. They find the function, thereby using the data to illuminate the  geometry.

This gives students loads of practice plotting functions and data, and, importantly, transforming “parent” functions by incorporating parameters, often in the form of sliders.

Aside: Of course, real students and adults often look at the graph and say, “Wow! Quadratic!” and try to fit a function of the form $y = b - ax^2$. This is great because (a) it makes intuitive symbolic-mathematical sense that something will be squared; and (b) it’s wrong, so you have a chance to have the data inform you.

In this case, when we do the analysis correctly, we have a problem with the formula. Although the Pythagorean equation above is not too bad, when we write it as a function,

$h(x) = \sqrt{C^2-x^2}$,

(where C is the length of the ladder or the height of the chair,) It’s  just not one of the standard functions students study. You might be able to derive it if you recognize Pythagoras in the situation. But can you come up with that function (and get illuminated) if you start with the data but do not recognize the right triangle?

I want it to be possible, so here are some conjectures about why this possibly-damning example does not make my whole crusade worthless:

1. Some problems are better one way than the other. This one is just better if you recognize the right triangle and derive the formula.
2. The current collection of “parent” functions we teach students—linear, $y = kx^2$, exponential, logarithmic, $y = k / x$, trig functions—is just not enough. If we have technology to help with graphing, we can expand this functional toolbox to include more stuff.
3. This example is not really that hard, but students need to have seen the equation for a circle. You have them take lots of data and ask what shape the data form. With that prompt, they’re more likely to come up with a good model. (And ideally, the software will plot relations if they don’t solve for the height).

Any encouragement or correction is welcome…