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Linear Relationships in Real Life?
July 21, 2011 5:06 AM   Subscribe

I am redesigning a high-school Algebra course so as to make it more meaningful to the students. To that extent, I am trying to gather as many examples throughout our world of linear relationships. I was hoping to tap into the collective wisdom of this group to come up with as long a list as possible of linear relationships that actually exist (not contrived stuff). I once knew a teacher who had a plant that grew at a linear rate. I don't know the type of plant, however. And there are so many other linear relationships around us. I don't want everything they see be about a car's distance from home or the amount of money they saved. What do you know of that exhibits a linear relationship? I'd love to have a long list for students so that they see that studying linear relationships in Algebra really does matter. Thanks for any help in making this course more worth while to the students!
posted by mathmatters to Education (16 answers total) 5 users marked this as a favorite
Any sufficiently small perturbation to a system from an equilibrium position is linear. This can be shown by applying a Taylor series expansion to the system's energy:

E(x0 + Δx) ≈ E(x0) + E'(x0))Δx + E''(x0))Δx2/2

Measuring the energy from the bottom of the energy well, and noting that the first derivative is zero at the minimum, we have

E(x0 + Δx) ≈ E''(x0))Δx2/2

Take the derivative of this and you get Hooke's Law, F = kx. The implication is that any stable material (metal, ceramic, polymer, or foam) has a constant stiffness (i.e., deforms linearly with applied load) for sufficiently small loads.
posted by Mapes at 5:28 AM on July 21, 2011

Oh, man, now my head's spinning :)

An example from my field: foreign vocab words - I learn X a day for Y days... At least I think that's a linear relationship. I'm a teacher and writer, not an engineer!
posted by chrisinseoul at 5:44 AM on July 21, 2011

When you say linear, you mean it in the continuous sense, not the simplifies to y=mx+b sense, right?

Antibody binding and capacitor charging both do a saturation curve (as do roughly a zillion other things).
posted by Kid Charlemagne at 5:56 AM on July 21, 2011

Miles per gallon, E=MC^2, I=prt, d=rt, Tail of an alligator compared to total length,
posted by Obscure Reference at 6:09 AM on July 21, 2011

I hate to say this... I know you are trying to be pedagogical. But... nothing in this world is linear, we approximate things by straight lines because they are about the only thing humans can understand. Humans are linear, the world isn't.

that being said, supply and demand, in the linear approximation can be the basis for a discussion of where linearity fails.
posted by at 6:12 AM on July 21, 2011

In electrical engineering, Voltage is linearly related to the Current times the Resistance (V = IR)
posted by garlic at 6:35 AM on July 21, 2011

If you haven't yet, you'll want to watch this TED Talk from Dan Meyer.
posted by bah213 at 6:38 AM on July 21, 2011 [1 favorite]

I am not sure if the purpose is to awe them with real-life applications in fields they know little about, or if the point is to show them that multipling $7 by 13 is easier than adding $7 + $7 + ... + $7.

Concert tickets. You are buying tickets for the AwesomeBand concert for friends and they are $25 each. The TicketStore fee is $10, regardless of how many tickets you buy. What is the relationship between cost and number of tickets?

Pizzas. Your friends each eat 3 slices of pizza for dinner, there are 8 slices in a pie, and the delivery fee is $10. You must tip 10% on the pizza order but not on the delivery fee. Express the number of pies you need in an expression. Express the total cost in an expression.

Do you mean things like this? Or natural/social science relationships?
posted by teragram at 6:44 AM on July 21, 2011

- Library late fines (10¢/day)

- Leaky faucet

- Rate at which an adult dog covers the neighborhood in feces (causing a non-linear increase in the distance necessary to walk and find a new spot)
posted by General Tonic at 7:37 AM on July 21, 2011

- A recipe makes 4 servings, but you need to feed 3 or 6 people. How much of each ingredient do you need?
- Mileage; a car consumes gasoline a a roughly linear rate.

- Acceleration: v = v0 + a t
- Heat and temperature: The amount of heat dQ required to change the temperature of an object by an amount dT is dQ = c m dT, where c is the substance's specific heat and m is its mass. You can do a simple experiment with a hot pot or hot plate and a thermometer to plot this relationship. They apply heat at a constant rate.
- Doppler shift

- Hubble's Law: The distance to a galaxy is d = H o z where z is the redshift. (See also Doppler shift!)
- Light travel time. Okay, you said distances of cars from home is boring, but... How long does it take light to reach us from Pluto? Alpha Centauri? The Andromeda Galaxy?

posted by BrashTech at 8:48 AM on July 21, 2011

Could you check in and let us know if the answers so far have been what you had in mind? Without a little more guidance, it's a tough question to respond to, simply because there's pretty much an infinite range of answers. (For the same reason -- and with no dismissiveness intended -- it's kind of a weird question. "I'm teaching my students about vowels. So that they can see the real-world applications, help me list all the words with vowels in them.")

Maybe you specifically want rates with linear relationships to time? Because if there are no limitations on what the independent variable can be, then all patterns are linear as against something. The number of papers in my wastebasket is probably approximately linear as against time, but it's definitely linear as against the number of times I've had a bad idea and crumpled up the paper. The number of bacteria in a culture might be exponential as against time, but it's linear as against the number of bacteria in the next petrie dish over (even though there's no causal relationship).

When you say linear, you mean it in the continuous sense, not the simplifies to y=mx+b sense, right?

Since one of the tags is "correlation", I take it the OP is okay with relationships that are not exactly one-to-one.
posted by foursentences at 9:15 AM on July 21, 2011

When you say linear, you mean it in the continuous sense, not the simplifies to y=mx+b sense, right?

Given that the OP is talking about high school algebra, it's almost certainly the y=mx+b sense.

The classics are rate x time = distance (cars, boats (with/against the current), running foot races, etc.) and mixing (dissolving sugar in water or whatever) problems, I think. You can invent some econ problems--make cost and revenue linear and get students to computer profit or something along those lines.

The real masterstroke isn't in convincing the students they should want to learn algebra, though--it's getting them to realise that all these examples are fundamentally the same problem. If you can do that, they'll be excited for applications, rather than scared of them.
posted by hoyland at 9:53 AM on July 21, 2011

So if I have an object that is W x H right in front of me, how big would it appear if I moved it a distance n away from me?

W/n x H/n

This is the basis of perspective projections used in all the 3D games and special effects that your students see every day.

For the visual learners, this is an easy one to demonstrate, especially if you're inclined to set up a camera obscura.
posted by plinth at 12:01 PM on July 21, 2011

Buying (widgets) with a fixed shipping cost.
posted by anaelith at 4:49 AM on July 22, 2011

Check out Kahn Academy Linear Relationships - he covers this in a variety of ways. Foursentences has a point, it would focus us a bit if you could point out what strikes your fancy.
posted by ptm at 12:58 AM on July 24, 2011

Thank you so much for all your replies. There are some good ideas in here.
Maybe the focus is simply linear relationships that are *interesting*. Every textbook has things like concert tickets or a traveling car. These are decent enough, but I think it's nice to have a myriad of other things that aren't typically found in textbooks (or maybe are, but I don't see them often or haven't seen them at all). Hooke's Law is interesting. So is the cost of a Starbucks coffee (dependent variable: oz of coffee). Heat and temperature -- I think that's interesting. As are things like Voltage vs current and resistance. An object moving away is interesting (link to computer graphics!). The number of papers in the waste basket against number of bad ideas is probably not as interesting. Right?

I think the most interesting things are not totally linear -- real life data that might be KIND of linear. And YES -- by linear, I mean can be MODELED with Y=AX+B where Y is the predicted value dependent on X the independent (input)variable and A is a rate and B is (often though not always) an initial rate. The growth rate of Starbucks' stores used to be linear -- IF you took the log. I still don't know the name of the plant that grows at a linear rate. Bamboo maybe?

We want students to be able to MODEL REAL data. The tools that they have are the linear model, the quadratic and the exponential. They also use logs. But initially, it's all about the linear model. So as they create equations that can best fit the data (some time perfectly like with concert tickets, often not perfectly because of data measurement, etc) -- they can interpret what the model actually means and use it to make conclusions and predictions. It's just a lot more meaningful/enjoyable when I as a teach seem to have a limitless source of situations (that are inherently interesting). Kids only what to do a problem about the cost of a ticket or a cell plan so many times. :)

Anyhow -- thanks for all the answers. You've already given me many cool ideas.

If you are interested in this at all, that Dan Meyer clip is really great. He's "gets it". His video with the escalator really gets it right. A simple every day thing... made into something the average student will want to know how to figure out.
posted by mathmatters at 3:11 PM on July 29, 2011

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