Model One Change

A news story might say a city's population will pass two million within twenty years, or that an outbreak will peak in six weeks. A bank's website might say your savings will reach a certain sum by the time you're 65. Each number arrives looking like a fact. It's the output of a model of change, and the quickest way to learn to read one is to build one.

Most projections of this kind share a structure. They start from a quantity measured now and a rule for how fast that quantity is changing. The rate, multiplied by a short stretch of time, gives roughly the change over that stretch. You add the change, work out the new rate from the new value, and repeat. This is called stepping a solution forward, and its simplest version is Euler's method (Strang and Herman 2016b, sec. 4.2). You'll meet it in many familiar places:

• A savings calculator steps a balance forward month by month, adding interest on whatever the balance has become.
• An epidemic projection steps forward the numbers of people who are susceptible, infected, and recovered, with rules for how fast people move between those groups.
• A weather model steps forward equations for the temperature, pressure, and motion of the air, in short time steps, on a computer.

Every such model has to answer the same four questions. What is the rule for the rate of change? What are the starting values, and how well are they measured? What has the rule left out? And how fast do small errors grow as the steps pile up?

The last question can set a hard limit. In 1963 the meteorologist Edward Lorenz studied a small set of equations for convection, the rolling motion of a fluid heated from below. He found that two starting states differing only slightly could evolve into very different states. Since real observations always carry some error, he argued, precise forecasts of the distant future may be impossible even when the equations are right (Lorenz 1963). Not every model behaves this way. A cooling drink is far more forgiving than the atmosphere. But every model has some horizon beyond which its numbers stop meaning much.

In this project you'll write for a reader who has seen a projection in the news, of a price, a population, or an outbreak, and wants to know how such things are made. You'll show them by building one. Pick a process you know, state the quantity and its rule for change, step it forward in a table, and say plainly what it assumes and how far ahead you'd trust it.

That last part matters as much as the numbers. A write-up that hides its assumptions teaches your reader to trust projections too much. One that stays vague teaches them nothing.

This content reflects the field as of October 2026.

References

• Lorenz, Edward N. 1963. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20: 130–141.
  • Free: Edward Lorenz publications list, MIT
• Strang, Gilbert, and Edwin "Jed" Herman. 2016b. Calculus Volume 2. Houston, TX: OpenStax. CC BY-NC-SA 4.0.
  • Free: OpenStax: Calculus Volume 2

Deliverable Overview

You'll write one model write-up of 500 to 800 words, with one table. It describes a real process that changes over time and builds a simple model of it. The reader is someone who has never studied calculus, so every term you use needs a short definition the first time it appears.

The write-up has six required components:

1. The process and the quantity you track, with units
2. The rule for the quantity's rate of change, in words and in plain notation
3. A table of at least five stepped-forward values
4. Any equilibrium or long-run behavior
5. At least two assumptions the model makes
6. A judgment of how far ahead the model can be trusted, with a reason

A short heading for each component makes the write-up easy to follow. Citations are welcome but not required. If you use them, give the author, year, title, and where to find the work, and list them at the end. A reference list doesn't count toward the word range.

Detailed Requirements

1. The Process and the Quantity

Name a real process and the one quantity you'll follow through it. State the quantity's units and the unit of time you'll use, such as degrees Celsius and minutes, or dollars and months. Keep to one changing quantity. A model of a cooling cup of coffee tracks its temperature, not its temperature, its volume, and its flavor.

2. The Rate Rule

A rate of change is how fast the quantity changes per unit of time. Your rule says what that rate depends on. State it first in a sentence, then in plain notation.

For a cup of coffee, the sentence might read: the coffee cools at a rate equal to one-tenth of the gap between its temperature and the room's temperature, per minute. With T for the coffee's temperature in degrees Celsius and a room at 20 degrees, the plain notation is:

The minus sign says the temperature falls while the coffee is warmer than the room. The units of the rate are degrees per minute. The constant 0.1 here is chosen for illustration, not measured. Say so if yours is too, and say where it came from if it isn't.

3. The Table of Stepped Values

Pick a starting value and a step size, the length of time each step covers. For each step, multiply the current rate by the step size, add the result to the current value, and recompute the rate from the new value (Strang and Herman 2016b, sec. 4.2). Each row comes from the row before it.

Starting the coffee at 90 degrees with one-minute steps:

Your table needs at least five stepped values after the start. Round sensibly, but carry the unrounded value into the next step.

4. Equilibrium and Long-Run Behavior

An equilibrium is a value at which the rate of change is zero, so the quantity stays put once it gets there. For the coffee, the rate is zero when T = 20: the coffee settles at room temperature (Strang and Herman 2016b, sec. 4.2). Say whether your model has an equilibrium and what happens in the long run: growth without end, decay toward zero, settling to a level, or something else. If there's no equilibrium, explain why not.

5. Assumptions

An assumption is something the model takes to be true without checking it. List at least two. For the coffee, the room temperature stays at 20 degrees, and the cooling constant stays at 0.1 whether the cup is full or half empty, lidded or open. Say what would change in the predictions if an assumption failed.

6. How Far Ahead to Trust It

A model's trust horizon is how far into the future you'd rely on its numbers. State yours and give a reason. A practical test is to rerun the model with a slightly different constant or starting value and see how soon the results drift apart. With a cooling constant of 0.12 instead of 0.1, the coffee's temperature after five steps is about 56.9 degrees instead of 61.3, a gap of more than four degrees within five minutes.

For some systems the drift is severe. Lorenz found that in a simple set of convection equations, slightly different starting states could evolve into very different states, which limits how far ahead any forecast of such a system can reach (Lorenz 1963).

Success Criteria

Your write-up meets the requirements when:

• The quantity and its units are stated
• The rate rule appears in words and in plain notation
• The table has at least five stepped values, each computed from the one before
• Any equilibrium is identified, or its absence explained
• At least two assumptions are listed
• The trust horizon is stated with a reason
• A reader with no calculus could follow it

Step-by-Step Guidance

Step 1: Choose a process with one changing quantity. Coffee cooling, savings growing, and a rumor spreading all work. Pick something you know well enough to judge whether the model's output is sensible.

Step 2: State what the rate of change depends on. Ask what makes the quantity change faster or slower. For savings, the rate depends on the balance. For a rumor, it may depend on how many people know it and how many don't yet.

Step 3: Pick starting values and a step size, and step forward at least five times. A good step size is small enough that the rate changes only a little from one row to the next. Work the steps by hand or in a spreadsheet, and record the rate in each row so a reader can check you.

Step 4: Identify any equilibrium and the long-run behavior. Set the rate to zero and ask which value makes that true. Then ask where the table is heading if you keep stepping.

Step 5: List the assumptions and say how far ahead you'd trust the model. Rerun a few steps with a slightly different constant or starting value. If the results drift apart quickly, your horizon is short.

Step 6: Write it up for a reader with no calculus. Define each term on first use, including rate of change, step size, and equilibrium. Read the draft as that reader would, and replace any symbol you haven't explained.

Resources

• OpenStax: Calculus Volume 2, chapter 4. Section 4.2 covers stepping a solution forward and equilibrium solutions; section 4.4 covers growth that levels off at a limit. Free online.
• Edward Lorenz publications list, MIT, which links Lorenz's 1963 paper on sensitive dependence in a simple weather-like model.
• Notes or practice work of your own on rates, totals, and stepping a rule forward.

Common Pitfalls

• A rule with no stated units. "The rate is 0.1 times the gap" doesn't say per minute, per hour, or per year. The same number with different units gives a different model. State the units of the quantity, the time, and the rate.
• Steps too large for the rate. Large steps assume the rate holds steady for the whole step, and it doesn't. With 25-minute steps, the coffee model drops from 90 degrees to −85 in a single step, then swings above boiling. If the rate changes a lot from one row to the next, shrink the step.
• Claiming precision the model doesn't have. A table that reports 61.3343 degrees suggests an accuracy that a guessed constant can't support. Round to what your inputs justify, and say so.
• Forgetting that growth models eventually hit limits. A rule where the rate is proportional to the quantity predicts growth without bound. Real populations, audiences, and outbreaks run into limits of food, space, or people, which is why growth that levels off is modeled with a different rule (Strang and Herman 2016b, sec. 4.4). Say when your model would stop applying.
• Leaving out assumptions because they seem obvious. A steady room temperature or a fixed interest rate seems too plain to mention. Those are the assumptions a reader most needs to see, because they're the ones that quietly fail.

References

• Lorenz, Edward N. 1963. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20: 130–141.
  • Free: Edward Lorenz publications list, MIT
• Strang, Gilbert, and Edwin "Jed" Herman. 2016b. Calculus Volume 2. Houston, TX: OpenStax. CC BY-NC-SA 4.0.
  • Free: OpenStax: Calculus Volume 2