Students investigate exponential relationships in data about Covid spread, using an inquirybased model involving hypothesizing, experimental and computational modeling, and sensemaking. They are introduced to table transformations and inverse functions, which are used to fit exponential models onto nonlinear data.
Lesson Goals 
Students will be able to…

Studentfacing Lesson Goals 

Materials 

Preparation 

🔗Looking for Patterns 45 minutes
Overview
Students explore the covid dataset, focusing on the growth in positive test cases for the state of Massachusetts. They try to fit different kinds of models to it, ultimately discovering the need for models beyond linear and quadratic functions. This section makes heavy use of interactive slider activities we’ve built in Desmos to support openended experimentation.
Launch
Starting in 2019, COVID19 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!

We’re going to look at the daily total of confirmed, positive cases for Rhode Island, Maine, Vermont, New Hampshire, Massachusetts, and Connecticut  all the states formed out of what the English called "New England".

Open the Covid Spread Starter File, which imports data from a spreadsheet that looks at the Covid infection rates for New England states from summer 2020 until the end of the year.

From the File menu, select "Save a Copy", and click "Run."

Working in pairs or small groups, complete Exploring the Covid Dataset.
Why just New England, starting from June 9th?!?
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.
And even within New England, we’ve artificially constrained this dataset, showing only the data from June 9th to December 26th, 2020. We’ve made this choice in order to showcase the most purelyexponential behavior of the infection curve, for the sake of this lessons' math learning goals.
For students who are farther along in mathematics, we recommend showing them all the data through 2020, starting in January rather than June. The first portion of the infection curve shows a gradual, linear growth pattern before exploding in the Fall of 2020. A purely exponential function will underpredict the growth during this time period by that it adds enormous friction to the modeling goal of this unit!
(The functions necessary to model this kind of growth have multiple terms showing different kinds of growth, and are just out of reach for students right now. Students can return to this unit once they’ve learned about hybrid models in later lessons.)
Based on the strength of your students, we encourage you to choose the data that best fits your learning goals.
To use all available data, open the Covid Spread Starter File and change the source sheet on line 7 from "New England"
to "All"
Discuss in groups or pairs, and prepare to share out to the class:  Based on the look of the scatter plot you just made, do you think there’s a strong relationship here?  If we fit a curve or straight line to this data, do you think it would fit the scatter plot well?
Review student answers to confirm that students have made a number of observations:

There appears to be more than one relationship in this dataset

Every relationship appears to be extremely strong

Most/all of these relationships appear nonlinear
Investigate
Let’s start out by looking at just one state. We’ll use Massachusetts for this investigation, but you can do your own investigation about any state you like after finishing this one!
The definition MAtable = filter(covidtable, isMA)
filters our dataset, keeping only the rows for which state = "MA"
.
Complete Linear Models for MAtable.

Did you see a correlation between date and the total number of confirmed, positive cases in this dataset?

Yes


Describe its form, direction, and strength.

Sample response: It’s definitely strong and appears to be nonlinear. Nonlinear functions don’t usually have direction, so not sure about that one.

Linear models capture straightline 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.

Are linear models a good fit for this data?

Why or why not?
If we make the line go from the start to the peak of the curve (top line), 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 (bottom line). Splitting the difference (orange line) is better than both of those options, and we might even get a halfway decent S! But ultimately, straightline, linear models just don’t behave like this curve, and we’ll never get the bestpossible fit with them. The number of positive cases is growing too fast to be fit with a linear model that grows at a constant rate!
Maybe linear isn’t the way to go, here!
Make sure you’ve:

Clicked on "pacing" and set your teacher dashboard of Modeling Covid Spread (Desmos) to the first slide so that students are looking at the "Quadratic Models" screen

Generated your own link in Desmos for sharing the file with your students

Open the Desmos link I shared with you to the Modeling Covid Spread file.

You should be on Slide 1 (Quadratic Models).

Using the file, complete Quadratic Models for MAtable
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.

Are quadratic models a good fit for this data?

Why or why not?
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 lower Svalue than our linear model! 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.
Synthesize

This data grows very slowly in the beginning and then grows very quickly. Can you think of any other situations in real life that act like this?

Can you think of any math that acts like this?
🔗Exponential Functions 55 minutes
Overview
Having identified that the Covid scatter plot appears neither linear nor quadratic, students learn about characteristics of exponential functions in tabular, graphical, and function notation form.
Launch
Let’s review what we know about the behavior of the models we’ve seen so far:
Remember that linear functions grow by fixed intervals, so the rate of change is constant. In the table shown here, each time the xvalue increases by 1, we see that the yvalue increases by 2. This is true for any set of equalsized intervals: a line needs to slope up or down at a constant rate in order to be a straight line!
If the "growth" is constant, the relationship is linear.
Quadratic functions grow by intervals that increase by fixed amounts! In the table to the right, the blue arrows show a differentlysized jump between identical intervals, meaning the function is definitely not linear! However, if we take a look at the difference between those differences(shown in red), we’re back to constant growth!
If the "growth of the growth" is constant, the relationship is quadratic.
Exponential functions grow even faster than quadratics.
When we calculate the growth between the yvalues, we can immediately tell it’s not linear. When we try to calculate the "growth of the growth", we see that it’s not quadratic either.
If we calculate the "growth of the growth of the growth" (shown in green)… we still haven’t found a constant… but we should notice that each of these "growths" just repeats the original pattern of yvalues! We could calculate the third difference, fourth difference, and so on  and never reach a stopping point.
Exponential functions grow so rapidly that looking for "what is added to y?" isn’t helpful at all.
In order to talk about the growth of an exponential function we need to identify the growth factor by asking ourselves "what is y being multiplied by?"
In this case, we can see that the yvalues are doubling each time, so the growth factor is 2!

Turn to What Kind of Model? (Tables), and decide what kind of pattern is represented in each table.

For each table that is growing exponentially, identify the growth factor.

Turn to What Kind of Model? (Graphs & Plots), and whether the shape of the scatter plot suggests a linear, quadratic, or exponential relationship.
Investigate
Exponential Functions can be written in the form: f(x) = ab^{( }x  h) + k
Most textbooks only present exponential functions with a horizontal shift of zero. When h=0, we can safely remove it and use this simplified form of the equation: f(x) = ab^{x } + k.
Advance your teacher dashboard of Modeling Covid Spread (Desmos) to the second slide ("How does horizontal shift transform exponential functions?") and give students a chance to reinforce their understanding of hvalues from other models by connecting it to exponential functions.
Let’s explore what each coefficient of f(x) = ab^{x } + k means!
Make sure you’ve advanced your teacher dashboard of Modeling Covid Spread (Desmos) to the third slide ("Exploring Exponential Functions") so that students are looking at the correct screen.
Decide whether you want to debrief this activity with your class after each section or at the end.

Let’s return to the Modeling Covid Spread Desmos file.

You should now be on the third slide ("Exploring Exponential Functions").

Use it to complete Graphing Exponential Models: f(x) = ab^{x} + k.
Invite students to consider what new information they have gained by looking at graphical representations rather than tables.
Base b
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 yaxis to another.
And, when raised to a fractional power, negative values of b might also lead to things like √!
Exponential Growth  Flat  Exponential Decay 




b > 1

b = 1

0 < b < 1

Vertical Shift…and Horizontal Asymptote k
The equation of the horizontal line that an exponential function approaches will always be y = k. This horizontal line is called an asymptote.
Adjusting k shifts the asymptote up and down, along with the rest of the exponential curve that approaches it.
Initial Value a
In exponential function definitions the yintercept is not represented by a single value (as it would be for linear and quadratic functions).
Since any value raised to the power of zero is 1, when x = 0 in exponential equations, part of the exponential term remains, so we can’t just cross out the other terms and look at the constant term.
For example, the yintercept for the function below will not be 3 (as you might have expected it to be).
 f(x) = 4(2^{x }) + 3
 f(0) = 4(2^{0 }) + 3
= 4(1) + 3
= 7

So what is the yintercept of an exponential function?

Give students time to discuss…

The yintercept of an exponential function is a + k!

If k is "missing", the coefficient a is the initial value where x = 0.

If a is "missing", the value of the coefficient is 1.

If we don’t see a or k in an exponential equation, the yintercept of the function is 1.
Exponential growth and exponential decay show up all the time!
Suppose you deposit $100 in a savings account, earning 3% interest each year.

How much money would be there after 1 year?

$103, because 3% of $100 is $3


How much is there after 2 years?

$106.90, because 3% of $103 is $3.90


After 3 years?

$109.27

Every year there’s a little more money to grow, and the total grows faster than the year before. If you start saving early, that $100 will grow into a lot more money down the road!

Most cells (e.g. bacteria, the cells in a growing fetus, etc.) divide every few hours, doubling the number of cells each time. A single cell will split into 2, those 2 cells will split to become 4, which will become 8, then 16, and so on.

Unstable atomic nucleus decay into stable nuclei over time, emitting ionizing radiation as a byproduct. We use the term halflife to refer to the length of time it takes for 50% of the atomic nuclei in a radioactive sample to decay.
If you’ve ever heard of something called "interest rate", then you know that sometimes we want to think in terms of percentages instead of factors. When your savings account has a 3% interest rate, it means your money is growing by 3%  a growth factor of 1.03.
Converting between growth rate and growth factor is easy:
b = 1 + r
If a $50,000 car loses 20% of its value each year, the growth rate is  20%. Modeling this with an exponential function would mean a growth rate b of 1  .20 = .80, for a function value(years) = $50,000 * (1 +  .20)^{years } = $50,000(.80)^{years }.
In the following activities, students will:

identify whether various plots, scenarios, and definitions represent linear, quadratic, or exponential functions

think about and apply their knowledge of growth, decay, initial value, and growth factor
Decide whether you’d like to pull your class back together to discuss after each activity or once they’ve completed all three.

Let’s practice identifying linear, quadratic, and exponential growth. With your partner, complete:

For more practice, complete What Kind of Model? (Descriptions 2)

What strategies did you use to decide if a function was linear, quadratic, or exponential?

When a function was exponential, how did you recognize whether it was growing or decaying?

What new insights did you gain about exponential functions by thinking about them in realworld scenarios?
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?
Synthesize

You looked at several different representations of exponential functions: tables, graphs, descriptions, and equations.

Which representation was the most useful for you? Why?

Which representation was the least useful for you? Why?
🔗Fitting Exponential Models 30 minutes
Overview
Students extend their sampling techniques to exponential relationships. Students continue experimenting in Desmos, but eventually switch back to Pyret to formalize their understanding.
Launch
Now that you’re familiar with exponential functions, let’s use them to model this Covid data!

Make a scatter plot showing the change in positive Covid cases for
MAtable
. 
What can you tell about the base b from this plot?

What about the initial value a?

What about k?
Have students respond to the discussion questions below in pairs or small groups.

Does your scatter plot show exponential growth or exponential decay?

The scatter plot shows growth. The "hockey stick" is pointing up, meaning that positive cases are increasing.


Can we make any conclusions about the value of b? Explain.

Because we see exponential growth, we know that b must be greater than one.


Can we make any conclusions about the value of k?

Can we make any conclusions about the value of a? Explain.

a must be positive, because the curve is consistently above k.

Investigate
Make sure you’ve advanced your teacher dashboard of Modeling Covid Spread (Desmos) to the fourth slide ("Exponential Model for MA") so that students are looking at the correct screen. In the next activity, students use Desmos to find promising exponential models, and then fit the model programmatically in Pyret!

Let’s return to the Modeling Covid Spread Desmos file.

You should now be on the fourth slide ("Exponential Model for MA").

Use it to complete the first section of Exponential Models: f(x) = ab^{x} + k.

Then use Covid Spread Starter File to complete the rest of the page.

Is an exponential model a good fit for this data? Why or why not?
On Exponential Models: f(x) = ab^{x} + k you’ll see a note about the use of ~1
to tell Pyret to prioritize speed over precision. Unlike most calculators, Pyret usually prioritizes precision.
In a math classroom, this is the difference between ^{2 }/_{3} rendering as 0. or being rounded to 0.666666667.
In data processing, choosing speed over precision can have ethical or technical consequences!
For example:
1) When calculating a path over an extremely long distance, being off by just a billionth of a degree could result in a Marsbound rocket missing its destination.
2) For an extremely large population like China, rounding to 10 decimal places might result in discounting an entire group of people!
Optional Activity: Guess the Model!

Divide students into teams of 24, and have each team come up with an exponential, realworld scenario, then have them write down an exponential function that fits this scenario on a sticky note. Make sure no one else can see the function!

On the board or some flipchart paper, have each team draw a scatter plot for which their exponential function is best fit. They should only draw the point cloud  not the function itself! Finally, students title display to describe their realworld scenario (e.g.  "money in a savings account vs. years").

Have teams switch places or rotate, so that each team is in front of another team’s scatter plot. Have them figure out the original function, write their best guess on a sticky note, and stick it next to the plot.

Have teams return to their original scatter plot, and look at the model their colleagues guessed. How close were they? What strategies did the class use to figure out the model?

The coefficients can be constrained to make the activity easier or harder. For example, limiting these coefficients to whole numbers, positive numbers, etc.

To extend the activity, have the teams continue rotating so that each group adds their sticky note for the bestguess model. Then do a gallery walk so that students can reflect: were the models all pretty close? All over the place? Were the guesses for one coefficient grouped more tightly than the guesses for another?

Synthesize

What makes exponential models different from the linear and quadratic models you’ve seen before?

Is it always okay for Data Scientists to round off their numbers to speed up computation? Why or why not?
Have students share their predictions for each of the timespans in question 5.

How accurate were your "guesstimates" for your models' predictions after 50 days? (Very accurate? Not accurate at all?)

How accurate were your "guesstimates" after 250 days?

How accurate were your "guesstimates" after 450 days?

How accurate were your "guesstimates" after 550 days?
Chances are, your guesses got less accurate as the number of days increased!
Why was it so much guesstimate the farther out we got, when the number of days was always increasing by a fixed amount?
We are creatures of nature, so our brains are designed to be really good at working with things we see all the time. It’s normal to see groups of 2, 5, or even 10 or 100, and we have a pretty good intuition for comparing group sizes as long as they’re small.
But when numbers grow really, really, really fast…we get lost! Our brains lose track of differences when two numbers get really enormous.
Mathematically, the number line is composed of equal intervals forever. But we don’t actually process it that way at all.
Exponential growth poses a problem for those of us with human brains, because the numbers get so big, so fast that it can be difficult to wrap our heads around it!
This may have played a role in the sluggish response of many countries, and the tragic loss of life and decrease in public health that followed.
Fortunately, there’s another mathematical tool that can help us get control of these wildly gigantic numbers. (Stay tuned!)
🔗(Optional) Why Just One State at a Time?
Overview
Students discuss an example of Simpson’s Paradox, which motivates splitting a dataset into grouped samples using filters. They then discover another motivation for filtering: scatter plots like our covid dataset 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.
Launch
A college is looking at housing data for a sample of students and comparing choices among those students who’ve decided what their major will be to choices made by students who are undecided about their major:
# On Campus  # Off Campus  % On Campus  

Undecided 
120 
80 
120/200 = 60% 
Decided 
80 
100 
80/180 = 44% 

According to the table, how many Undecided Majors live offcampus?

80


How many Decided Majors live oncampus?

80


Who is more likely to live on campus: Decided or Undecided Majors?

(Give students time to talk about this, and explain their thinking! )

If you’d like to distribute printed copies of this table and the accompanying questions, they are available here.
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 less likely to have picked a major than seniors, and they are much more likely to live on campus.
When we filter by this important third variable, it turns out that for both Freshmen and NonFreshmen, there is no correlation between between deciding on a major and living on or offcampus.


What looks like a correlation between havingamajor and livingoncampus is actually a correlation between age and livingoncampus.
A third variable lurking in the data 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 the more data we include in our sample the more clearly we’ll see any potential relationships. But in certain circumstances the correlations in our subgroups cancel each other out when we put the groups together. This is called Simpson’s Paradox.
Simpson’s Paradox: visible trends in subgroups disappear or even reverse when the groups are combined.
Sometimes filtering the data into subsets is the only way to see what’s really going on. That’s exactly what this starter file does, by filtering the data for Massachusetts only.
Investigate
Datasets like the one used in our Covid Spread Starter File 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 yvariables. Instead, we have several subgroups each with their own very strong relationships, and another variable lurking in the background.
In fact, the scatter plot for all our New England states didn’t look much like a scatter plot all! It looks like someone took a marker and drew in five different curvy lines.

What do you think might be the variable lurking in the background, which accounts for these separate lines?

Give students time to discuss!

Diseases spread more rapidly in denselypopulated areas, since it’s easier for the infection to jump from one person to another. Unfortunately, we can’t see the density data in our table, so that dimension is missing from our dataset! This is exactly what happened in our college example: we couldn’t see the age of the students, which skewed our interpretation of the scatter plot.
These patterns are so distinct from one another that we’re going to need to make more than one model.

We needed to break the Covid data up into grouped samples, so that all of the data for Massachusetts is in one table. We would then do the same for Maine, Rhode Island, etc.

How is a grouped sample different from a random sample?

A grouped sample is a nonrandom subset chosen from a larger set. Grouped samples are nonrandom by design!

Make sure you’ve advanced your teacher dashboard of Modeling Covid Spread (Desmos) to the fifth slide ("Exponential Model for VT") so that students will be looking at the correct screen when they are directed to return to Desmos part way through Modeling Other States

Working in pairs or small groups, complete Modeling Other States.

You will be working with both Covid Spread Starter File and the fifth slide of Modeling Covid Spread Desmos file.
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.
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.
Common Misconceptions
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
.
Synthesize

In what other situations would it be useful to filter a dataset?

Can you think of other examples where Simpson’s Paradox might arise?

When comparing one country’s schools to another’s, a researcher finds that students living in poverty in country A outperform students living in poverty in country B. They also find that the wealthy students in A outperform their wealthy peers in B. In fact, for every income level, country A outperforms country B! But if country B has less child poverty overall, it will still outperform A.

Another, thoroughlyexplained example involving soft drinks can be found on this web page.

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