How to Guess an Equation Without Math: Exploring Cat vs. Bird Populations

Have you ever wondered how scientists figure out how one thing affects another—like how the number of cats in an area might impact the number of birds?

You might think they always start with a complicated equation, but in real science, it often works the other way around: they start with observation and intuition, and only later guess or derive an equation.

In this post, we’ll walk through how to guess a relationship (an equation) between two populations: cats and birds. No advanced math needed—just logic and curiosity.

🧠 Step 1: What Are We Curious About?

We want to explore the question:

"How does the number of cats affect the number of birds in an area?"

We don’t know the answer. That’s the fun part.

📋 Step 2: List the Variables

Let’s define what we can measure:

C = Number of cats
B = Number of birds

We assume more cats might mean fewer birds. Now we try to guess how they might be related.

🔮 Step 3: Make an Educated Guess

We’re going to guess simple equations—just like scientists do before they collect data.

🤏 A. Linear Relationship

B = a - bC

This says that each new cat removes a fixed number of birds. It’s simple, but might work!

📉 B. Exponential Decay

B = a \cdot e^{-kC}

Now, each additional cat reduces the bird population by a percentage, not a fixed amount. This better fits situations where the bird population drops quickly at first.

🪝 C. Inverse Relationship

B = \frac{a}{C}

This suggests a very steep drop when cat numbers increase—more dramatic than the others.

🧪 Step 4: Plan a Simple Experiment

Suppose you observe different neighborhoods and count cats and birds:

Area	Cats (C)	Birds (B)
1	0	120
2	2	100
3	5	65
4	8	25
5	10	15

Now plot these points on a graph.

If the points form a straight line → use the linear model
If they curve downward → exponential decay might be better
If bird numbers fall fast → maybe the inverse model fits

📈 Step 5: Refine Your Equation

After plotting, you can estimate which guessed equation fits best. You can even fit the data using tools like Excel, Python, or just graph paper!

Congratulations—you just did what real physicists, ecologists, and engineers do:
Guess a relationship, test it, and refine.

🌱 Final Thoughts

This process—starting from observation and variables, then guessing and testing relationships—is the core of the scientific method.
You don’t need to start with math. You start with questions.

Next time you’re curious about something in nature—cats and birds, plants and light, cars and fuel—remember: