Exploring Exponential Models

Students discuss an example of Simpson’s Paradox, which motivates splitting a dataset into grouped samples using filters. They then explore the Covid Spread Starter File, and discover another motivation for filtering: some scatter plots show multiple correlations, instead of just one. Finally, they learn how to filter a dataset and apply that knowledge to filtering the covid dataset into samples grouped by state.

A college is looking at enrollment and housing data for a sample of students who’ve decided what their major will be, vs. those who are undecided:

It looks like the two variables are significantly related: undecided majors are more likely to live on campus than decided ones!

But there’s a third variable hiding in the background: freshmen college students are far more likely to live on campus, and they generally have not picked a major. We need to take the year into account.

Because year at school is an important 3rd variable, we should first filter by that variable. And it turns out that for both Freshmen and Non-Freshmen, there is no correlation between between deciding on a major and living on- or off-campus.

How is this possible? When a third variable is lurking in the data, it can play tricks by obscuring relationships between two other variables - or by creating the appearance of a relationship where none exists!

Normally we think that "more data means more power!", and that the more data we include in our sample the more clearly we’ll see any potential relationships. But in certain circumstances - like in our Covid dataset - the correlations in our sub-groups cancel each other out when we put the groups together. This is called Simpson’s Paradox. (Learn more about Simpson’s Paradox at Wikipedia.)

"More data" isn’t always better. Simpson’s Paradox shows us that being able to filter the data into subsets is sometimes the only way to see what’s really going on.

Starting in 2019, Covid spread across the globe. Most of us heard terms like "flatten the curve" and "infection rate" in videos and on the news.

Even in 2020, very few people could have predicted the impact Covid would have on the world. But Data Scientists who were looking at the data knew differently. Let’s take a look at some of that data!

Note: This dataset is available for all 50 states (and Washington, D.C!), but for pedagogical purposes we’ve written the starter file to pull only data from New England.

The scatterplot you made doesn’t look much like a scatterplot all! It looks like someone took a marker and drew in five different curvy lines. While it’s clear that there are strong patterns in the data, these patterns are so distinct from one another that there isn’t really one, single model that fits them all. Each relationship appears very strong, almost as if there is more than one model here.

With all these clear, tight curves, we might think this would be a dataset with a very strong relationship. Unfortunately, that’s not what we see when we group all the data together!

Datasets like these are very difficult to model all at once, because there will always be lots of points that are far from any single function. But it’s not that there’s no relationship between the x- and y-variables. Instead, we have several sub-groups each with their own very strong relationships.

The filter function consumes a Table and a helper function! The helper function is used on every Row of the Table, producing true or false. The filter function takes all the Rows for which the helper produced true, and combines them all into a new table.

Optional: While filtering is introduced in this lesson, the primary goal is for students to explore exponential functions. If your students need more practice with filtering - or wish to filter their own datasets - we recommend checking out the Filtering and Building lesson.

It’s extremely common for students to think that filtering a table changes the original table. This is NOT how it works in Pyret! Instead, the filter function always produces a new table, containing only the Rows for which the supplied function evaluates to true.

Students explore their newly-filtered MA-table dataset, trying to fit different kinds of models to it. This section makes heavy use of interactive slider activities we’ve built in Desmos to support open-ended experimentation.

Linear models capture straight-line relationships, where one quantity varies proportionally based on another. In linear models, we expect the response variable to grow by equal amounts over equal intervals in the explanatory variable.

Have students share their resulting models. Which one fits best?

If we make the line go from the start to the peak of the curve, almost all of the points bulge out below our line of best fit. If we make the line hit the bottom of the curve, all the points fall above it. Splitting the difference (orange line) is better than both of those options, and we might even get a pretty good 𝑅²! But ultimately, straight-line, linear models just don’t behave like this curve, and we’ll never get the best-possible fit with them. It’s growing too fast to be fit with a linear model that grows at a constant rate!

Have students share their resulting models. Which one fits best?

Quadratic models capture parabolic relationships, where one quantity varies based on the square of another. In quadratic models, we expect the response variable to grow by differing amounts over equal intervals in the explanatory variable.

Quadratic models change their rate of growth over time, which definitely makes them a better fit for this data than linear ones. It’s very likely we could find a quadratic model with a pretty good 𝑅² value! But this data starts out almost flat and then suddenly takes off like a rocket - quadratic models just don’t have that kind of explosive growth, so our model will never be as good as it could be.

Having identified that the Covid scatter plot is neither linear nor quadratic, students learn about characteristics of exponential functions in tabular, graphical, and function notation form.

Let’s review what we know about the behavior of the models we’ve seen so far:

There is, however, a class of functions that grows even faster than quadratics: exponential functions.

We generally write exponential functions like this: 𝑓(𝑥) = 𝑎𝑏^𝑥 + 𝑘.
Let’s explore what each coefficient means!

The base of an exponential function (𝑏) must always be positive, because exponential functions grow and decay uniformly. A negative 𝑏 would bounce from one side of the y-axis to another. When raised to a fractional power, negative values of 𝑏 might also lead to things like √−2!

An exponential function with a growth factor will always start close to a horizontal line, then gradually shoot up to ever-increasing values. An exponential function with a decay factor will drop quickly, then level out close to a horizontal line. This horizontal line is called an asymptote, and the equation of the line will always be 𝑦 = 𝑘.

Adjusting 𝑘 shifts the asymptote up and down, along with the rest of the exponential curve that approaches it.

The y-intercept appears differently in exponential function definitions than in linear and quadratic definitions:

Exponential growth and exponential decay show up all the time!

In the following two activities, students will decide whether various scenarios and definitions represent quadratic, linear, or exponential functions. They will also have opportunities to think about and apply their knowledge of growth, decay, initial value, and growth factor.

Have students share their answers, asking them to notice and wonder about the sequences for the exponential examples. How are these sequences growing or decaying? How is that growth or decay different from what they’ve seen before?

As students discuss their answers, pay special attention to their use of vocabulary when describing the initial value and the growth factor.

Students extend their sampling techniques to exponential relationships. Students continue experimenting in Desmos, but eventually switch back to Pyret to formalize their understanding.

Now that you’re familiar with exponential functions, let’s use them to model this Covid data!

Direct students to create a scatter plot showing the change in positive Covid cases for MA-Table. Then, support them in making educated guesses about the values of 𝑎, 𝑏, and 𝑘. Have students respond to the discussion questions below in pairs or small groups.

In the next activity, students use Desmos to find promising exponential models, and then fit the model programmatically in Pyret!

★ Optional: Build models for other states. How do the coefficients differ from state to state? What differences between states could explain the different values of the coefficients?