# Real life examples to motivate the study of linear functions

Some years ago I used mobile phone or internet rates (for example, with basic fees and a given charge per minute or by data volume) to introduce and motivate the study of linear functions.

However, today (at least in Germany) most people have flat rates (or some semi- flat rates where the data speed is throttled after a given GB volume is reached). So mobile phone and internet rates are not ideal examples for students anymore.

Question: Are there other examples that students (in my case, they are ~15 years old, most of them boys who are also interested in technology and science) may find motivating and that also have the didactic clarity and variability which some years ago the mobile phone rate examples had?

(I thought about something like electricity tariffs, but I have the feeling that they would find it boring and not related to their real lives.)

In his blog post, "An Argument Against the Real World", Mr. K argues that fictional (even mythical) problems can work better than 'real-world' problems. I think he has a great idea.

• That's a good point; finding "real" and not "pseudo" real world examples which motivate your students is often a non trival task. Do you have suggestions for interesting or funny fictional or historical problems for my case? – Julia Sep 12 '14 at 18:56
• Great point. I helped run a calculus mooc, and the problem that generated the most "hype" was definitely a "grey goo" problem: under exponential growth, how long until the grey goo uses all available matter on earth? – Steven Gubkin Sep 12 '14 at 19:06
• Mr. K's post has some good examples. If those don't work for you, I don't know anything better off the top of my head. September 25 is Math Storytelling Day, so if you tell me what sorts of stories your students would like, maybe I can write something up. – Sue VanHattum Sep 12 '14 at 21:34

Maybe encourage them to come up with some examples? Affine maps are so ubiquitous that it seems hard to motivate them: it is like trying to motivate the air. Any time some quantity is changing at a constant rate with respect to some other quantity, you have a linear function. It is probably more fun for the students to try to come up with examples than listen to your examples. For a sequence of progressively more demanding examples, maybe the following will do:

1. Assuming 1 pound of bodyweight is 3500 kcal, model effect on bodyweight of various daily caloric deficits and surpluses compared to daily energy requirements. Applications to bodybuilding. This can be coupled with "real world" diet planning. How many grams of which foods would you need to include...etc

2. Starting amount of money and weekly paycheck? What is the relationship between the functions when you take day, week, or month as unit of time? When you take dollar, euro, or pound as unit of money? What happens to the graph of these functions? Try to get them to think of this in terms of composition of linear functions.

3. Linear interpolation of data. Have a bunch of discrete points, maybe stock market data, and approximate the continuous data by connecting the dots with straight lines. How can you "automate" this process? (This will motivate the "point slope" form of a line).

You could even throw some lightweight "calculus" at them, by given them discrete data for velocity of some object, have them linearly interpolate, and then find area under these piecewise linear curves using geometry to recover an approximation of distance traveled.

Patterning tasks are a great way to introduce linear equations. Something like the Pool Border Task (here's a sample lesson plan; I use a stripped-down version) builds a lot of student buy-in for small group work, and naturally leads students to discover a linear relationship for themselves. I've had a good deal of success with this task and the discussion afterward, even for students who typically have a low output. You can easily invent your own version of such a task, but I find this one is simple enough and yet rich enough to make for a good day in the classroom.

Real world situation: Weekly (or monthly) pocket money.
Constant term: agreed upon disbursement from parents.
Variable/function argument: number of nagging episodes per week (or total length thereof, for a truly continuous argument).
Argument coefficient: extra money per unit of variable.

They can even arrive at the concept of non-linear functions this way.

And if you want to turn it statistical, you can include an "error term" describing the "visiting grandparents" disturbance.

I personally tend to avoid examples that make use of money, but not for the reasons outlined by Mr. K (my reasons are more about the kids who don't have money).

How about some simple things like:

• how much your glass of water goes up each time you add an ice cube
• assuming that the public transportation system in your area has stops that are the same time apart (say 5 minutes) and that you start at stop 10, what stop are you on 20 minutes later? x minutes later?