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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=x^{2}), and observe the result (y=9).

But is that the only way to see a scenario? The setup y=x^{2} implies that y only moves because of x. But it could be that y just coincidentally equals x^{2}, 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 x^{2} pints of ice cream.

We could write the algebra relationship like this:

And it’s correct… but misleading!

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

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 = x^{2}. 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?

Update: Eddie Woo made an excellent video using this analogy.

Watch the full series (part 2, part 3), I really loved how he explained the history of the word (para=beside, i.e. you have a hidden variable beside the ones you see). Thanks Eddie!

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?”

hunter patton

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.

kalid

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.

Seth Reichard

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!

Kalid

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!

Omer Abid

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.

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?

kalid

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

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).

Anonymous

hi, great explanations!

Susan Socha

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

Eric V

@ Bill
Great question about Ohm’s law.
Here’s the dirty little secret about resistance: it doesn’t exist. Instructors would like to tell you it derives from a property (resistivity) of a substance. If you make a cylinder out of a substance and want to know the R, you just take the resistivity, multiply by length and divide by area. But resistivity isn’t really a property of a substance. It is a concise way to approximate the complex relationships between the energy levels of electrons in the various atoms. The picture is slightly easier to understand when you understand that when several atoms get together the combination of all valence shell electrons combine to form something like a single valence energy shell of the crystal as a whole. In other words, resistance and resistivity is not so much a property of a substance, but a description of behavior. It describes how a certain object responds to stimulus. As this post describes parametric analysis (remember the parameter doesn’t have to be time) consider the following. V is determined by an unstated parameter. I is determined by the same unstated parameter. R is just the ratio of V and I (R=V/I). To understand that resistance doesn’t exist (and don’t fret, many instructors and some electrical engineers I work with never break the veneer of equations to see the idea underneath) consider the way we characterize a diode, or other PN junction device, with an I-V characteristic curve. We draw a 2D graph, voltage side to side (independent variable) and current up and down (dependent variable). We apply a voltage and measure the resultant current and plot points on the graph. The slope of the graph (I/V, mathematically the inverse of R) just tells us how the device responded (in current) to our applied stimulus (voltage). The slope just tells us behavior. For a diode this I-V curve has a certain shape, for the base-emitter junction of a basic transistor, this I-V curve has a different shape (and is related to other parameters). For a device called a resistor it has a very simple shape, a line. In all of these examples resistance is just a behavior, not a property.

Ready to peak behind the parametric analysis curtain for a little better glimpse of the Great and Powerful Oz? Do you remember I said an ‘unstated parameter’ determines both voltage and current? This parameter is just a label for the cumulative effect of electrostatic field interactions of many electrons. In a similar fashion as the idea that R doesn’t exist, you could almost say that you don’t really have voltage across the terminals of a battery, almost. Voltage is not a property of a battery any more than resistivity is a property of a substance. It is a dynamic relationship. In the positive plate (of, say a lead-acid battery, like in your car) electrons jump out, and into the H2SO4-H2O mixture thanks to chemical action. They likewise build up on the negative plate. As electrons are bunched up on the neg plate there is an electric field in 3D space around the charges. From an equation standpoint electric field and electric voltage are not really two different things, just two different descriptions of the same electric phenomenon. It’s like an elevation map of hills and valleys. Rivers are in the direction of elevation change, an elevation line shows a region of equal elevation, the line and the river a perpendicular to each other. They are two ways to observe the same phenomenon. E field = river, E voltage = elevation line. One underlying principle (the ‘push’ of electrons against each other) in one way makes it appear to us observers that there is an electric field / voltage across the terminals of the battery. The same principle (the ‘push’ of electrons) describes the migration of electrons along a path that they can travel, that is, current. One unseen parameter drives both voltage and current. One behavior function (resistance) is just the relationship between those two functions. It is one large interplay of several parametric equations.

Bob

Eric V,

A few folks have asked me what “causes” voltage and current. Thanks for the explanation.

Mike

Hi Dan and Kalid
Proof of why 1 = 0.999999…

0.99999.. can be expressed as the sum (S) of 0.9 + 0.09 + 0.009 + 0.0009..infinite series
therefore 10 S = 9 + 0.9 + 0.09 + 0.009 + 0.0009….
subtracting S = 0.9 + 0.09 + 0.009 + 0.0009….
yields 9 S = 9 or S = 1
therefore 1 = 0.99999…

Prof__Lee

Beautifully explained! I particularly like the video! Thanks so much!

janis

This is fantastic! Thanks so much. Easy to follow.

Jacob

If the idea was to explain the needs for using parametric equations, I think the explanation is a little bit short. For instance, there is a different in the solutions of the following two equations 1) y = 5x + 3 and 2) x^2 + y^2 = r^2. The following explanation comes from the calculus II notes by Paul Dawkins (http://tutorial.math.lamar.edu):
In the first equation, there is a one to one relationship between the variable y and x that falls in the definition of a function. In the second equation (not a function), however, solving ‘y’ as a function of ‘x’ would result in dealing with two separate equations due to the square root operation that requires plus or minus. Therefore, the curve of the second equation would require two separate equations, one equation for points on the upper have plane (y1 = (+) sqrt(r^2 – x^2)) and another equation for the points located in the lower have plane (y2 = (-) sqrt(r^2 – x^2)).
To avoid dealing with two functions for different parts of the curve represented by the second equation, it is better to deal with two parametric equations (x = r cos(p) & y = r sin(p)) representing all points of the second equation and the curve. There should be no need to deal with parametric equations if the second equation would have accounted for all points (x,y) of the curve it represents.
On the other hand, I am not completely sure the two equations above were meant to represent a set of parametric equations or two different functions of the temperature variable. If the intent is to plot Ice cream as a function of Sunscreen, then we should have a system of parametric equations.

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