20  Modeling with Functions

In this chapter, we’ll discuss one particular kind of modeling: approximating data with functions.

This kind of work, at the high-school level, is often not very data-sciencey. You get two columns of data, X and Y, and you’re trying to find a function so that if you make a scatter plot, the function comes close to the points. You are not awash in data.

You do, however, have a mathematical challenge involving data. It’s also the case that challenges like this “smell like” regular science-science. So it can be data and science without being data science.

That said, we can note:

• Some rich data-science problems could well have components that require some modeling of this sort.
• Techniques for modeling using functions or other structures (such as trees) often end up connecting to machine learning, which connects to AI, which is increasingly a part of data science. So being able think flexibly about modeling can be a good foundation for learning about AI.

20.1 Resource for Algebra II, Precalc, and beyond

I have another book, available on paper, called “The Model Shop.” It’s a collection of activities that connect geometry to functions through data. The activities are cool, but not within the scope of this book here.

That book has illustrations from Desmos and Fathom, but you can almost do everything in CODAP.

20.2 Basic modeling tools

If your graph has two numeric attributes—that is. it’s a scatter plot—various modeling tools are available.

20.2.1 Lines

You can find both a least-squares line and a movable line in the ruler palette. The movable line is especially cool; a line appears with handles, and as you change the line, the formula for the line updates.

If you have grouped the data (by dragging a categorical leftwards), the least-squares tool creates a separate line for each group.

20.2.2 Show squares

If you check the Show squares box in the ruler palette, CODAP constructs a square on each residual segment and displays the sum of squares.

This is perfect for explaining the meaning of a least-squares model. That said, I am personally more interested in getting a model that’s pretty good than optimizing one.

20.2.3 More general functions

The ruler palette also has a Plot function command. If you select it, a white area appears at the top of the plot; if you click it, the formula editor appears.

Two caveats:

• As with formulas for attributes, what you enter should only be the part after the equals sign. That is, if you’re entering \[A = \pi r^2\],

what you enter is pi * r^2

• As that formula suggests, the formula editor does not do implied multiplication. So enter a function like \[y = 2x(x-a)\] as 2 * x * (x - a)

20.2.4 Sliders

In the previous example, 2 * x * (x - a), that quantity a is a parameter. In CODAP, you should use a slider to represent it.

You’ll find sliders in the toolbar. Click the icon to create a slider in your workspace.

By default, the slider will have a name like V1 (that is, value 1) and a value of 0.5. Obviously, you slide the widget to change the value. Two things are not obvious:

• The slider is actually an axis just like the axis of a graph, so you can rescale the axis by dragging the numbers, just as you do in a graph axis.
• The name of the slider is editable as well. So rename it to something sensible (like a or density or whatever).

20.3 Nonlinear practice

Want to practice? The live example below show data from a video of a cotton ball falling. It’s very rough data, but the plot shows a distinct curve that we suspect of being quadratic. It may not start at time zero, however, and it might not have been released at exactly location = 0. So we might need three sliders, one for the quadratic coefficient, one for the vertical translation, and one for the horizontal translation.

20.3.1 Residual plots

One thing I miss in CODAP that was present in Fathom was an easy way to see residuals. In Fathom, we learned how powerful residuals could be in helping create and assess functional models like these.

Until CODAP learns that trick, you can make your own. They’re not automatic, but my students were (mostly) able to handle it.

I suggest that students make two new attributes, often called model and residual. In the cotton-ball data, the formula for residual is location - model. The formula for model is whatever you wrote for the dynamic function with the sliders.

Then you can make two plots:

• Your main plot, location vs frame, with the quadratic function.
• Your residual plot, residual vs frame.

As you adjust your sliders—and adjust the scale on the residual plots as the points move—you try to get the points in the residual plot to be roughly flat and roughly zero. By “flat,”, we mean that there is no systematic pattern to the residuals.

If you can’t do that, for example, if no matter what you do there’s some pattern, that’s a clue that there is something wrong with your model.

In the case of the cotton-ball data, I can get a pretty good fit except for the last point. That suggests that the quadratic model might be OK for a few frames, but that it breaks down as time goes on.

20.4 Remember chemistry?: that Glucose thing

In the NHANES data, you will sometimes find multiple columns that measure the same thing. When they report glucose, for example, there are two columns for the value of “fasting glucose.”

• One is measured in mg/dl, that is, milligrams per deciliter.
• The other is in mmol/L, that is, millimoles per liter.

If you plot them against one another, what function would you expect?

Do it, find a model, and use the result to determine the molecular weight of glucose! (Check your work by googling it.)

20.5 More complex example: BMI

Using the NHANES plugin, get data that includes BMI.

What do you think influences BMI? Get that data too, and figure out the formula for BMI.