# A Quick Intuition For Parametric Equations

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Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they’re completely independent?

Normal equations assume an “input to output” connection. That is, we take an input (x=3), plug it into the relationship (y=x2), and observe the result (y=9).

But is that the only way to see a scenario? The setup y=x2 implies that y only moves because of x. But it could be that y just coincidentally equals x2, and some hidden factor is changing them both (the factor changes x to 3 while also changing y to 9).

As a real world example: For every degree above 70, our convenience store sells x bottles of sunscreen and x2 pints of ice cream.

We could write the algebra relationship like this:

$\displaystyle{ice \ cream = (sunscreen)^2}$

The equation implies sunscreen directly changes the demand for ice cream, when it’s the hidden variable (temperature) that changed them both!

It’s much better to write two separate equations

$\displaystyle{sunscreen = temperature - 70}$

$\displaystyle{ice \ cream = (temperature - 70)^2}$

that directly point out the causality. The ideas “temperature impacts ice cream” and “temperature impacts sunscreen” clarify the situation, and we lose information by trying to factor away the common “temperature” portion. Parametric equations get us closer to the real-world relationship.

Don’t Think About Time. Just Look for Root Causes.

A reader pointed out that nearly every parametric equation tutorial uses time as its example parameter. We get so hammered with “parametric equations involve time” that we forget the key insight: parameters point to the cause. Why did we change? (Maybe it was time, or temperature, or perhaps sunscreen really does make you hungry for ice cream.)

Most algebraic equations lay out a connection like y = x2. Parametric equations remind us to look deeper (lost on me until recently; I’d been stuck in the “time/physics” mindset).

Sure, not every setup has a hidden parameter, but isn’t it worth a look?

## Other Posts In This Series

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1. Dan Stillit says:

this is great – many thanks. could you do one on why 1 = 0.99999… ? I know this is a very taxing one for many.

2. kalid says:

Hi Dan, glad you liked it! That’s a great topic, I have an article on it here:

Instead of laying out a definitive answer, I prefer to think “Under what assumptions about numbers does 0.999… = 1?” and also “Are there any assumptions that could mean 0.999… is different from 1?”

3. hunter patton says:

when I teach scatter plots we often talk about underlying causation. Is x really correlated to y, or is something else causing that? — One I did was ice cream sales vs bee stings. — great post.

4. kalid says:

Thanks Hunter — funny how we thought of similar examples! (I guess ice cream represents summer for everyone.) x-y plots are a good example, just putting things on the same graph doesn’t mean one causes the other.

5. Seth Reichard says:

Hey Kalid, I appreciate the shout out in the article! I wanted to let you know that I created some real awesome investigative mathematical experiences for my students with the ideas we discussed over email. Just today we were looking for how to graph these kinds of functions as a class and the equations I wrote on the board the kids still refer to as “store sales,” “ice cream sales,” and “sunscreen sales” because it was such a great, concrete example for them to fall back on. This unit on Parametrics has been absolutely incredible and as an intern for teaching, that means the world to me (and the kids). I wish you the best!

6. Kalid says:

Hi Seth, that’s so awesome to hear! Wow, I love the idea of kids learning a new analogy that sticks :). It’s really gratifying to hear when things are clicking in the actual classroom (vs. just ideas bouncing around in our heads), thanks for letting me know!

7. Omer Abid says:

Hi Kalid,

Great article!

I am an epidemiologist and we try to find how certain factors affect outcomes.

But we need to control for certain confounding factors to find out if the factor is really a causal factor or if the relationship is confounded by another variable.

I think your insight can help epidemiologists and thus doctors and thus health!

Anyhow, it would be nice if you can somehow integrate your ideas with some medical problems that epidemiologists like me try to investigate.

Cheers.

8. Bill says:

I was looking at Ohm’S law where current is directly related to resistance but voltage is inversely related to resistance so what is the relationship between current and voltage?

9. kalid says:

Hi Omer, thanks for the suggestion, medical applications would be really fun :).

Basically, Ohm’s Law can be written V = IR

In this formulation, I see see “R” as the Oomph needed to push one charge through the system, and I as how frequently you wish to push charges through.

The amount of Voltage to create this scenario is V = IR. That is, if you double the voltage, you’ll double how many charges you can pull through in a given amount of time. R is the “difficulty” required to move a single unit charge through the system. The better the conductor, the easier it is (so the same Voltage can move more charges when pulling through metal wires, vs. wood, for example).

10. Anonymous says:

hi, great explanations!

11. Susan Socha says:

I used your ice cream sunscreen idea with parametrics. Now I am wracking my brain for more ideas that have one independent and two dependent variables. It must be too late in the day…my brain is fried. Thanks for posting this…it has been integrated into my unit

12. Eric V says:

@ Bill