Started a list for linear relationships. Not all linear across an infinite domain but that adds in discussion on piece-wise functions. Can anyone add more to the list?

**Purely mathematical:**

*1) Diameter vs Circumference of a Circle*

2) Length of Side vs Perimeter of a Square

3) Number of Sides of Polygon vs Sum of Interior Angles

**Connected to Real Life:**

1) Temperature of Coffee vs Time

2) Amount of Fuel in Tank vs Miles driven

3) Amount of Ink in Pen vs Pages Written

*4) Profit made in first weekend of Film in Cinema vs Total Profit** (yummymath.com)*

5) Altitude vs Temperature (data can be recorded on some airlines during descent)

6) Taxi Cost vs Time

7) Gym Membership Cost vs Time (Initial Payment to begin member)

8) Internet Cost vs Time (Initial Payment for Installation)

9) Distance-Time Graphs and Velocity-Time Graphs

10) Hooke’s Law and Newton’s Second Law ** **

**– Physically Measure:**

*1) Length of Foot vs Height of Person*

2) Pull back distance of Model Car vs Total Distance travelled of Model Car

3) Number of Bites of Liquorice vs Length of Liquorice (Discussion on the type of bites you would need to keep linear)

*4) Queueing time vs Number of Items per Customer (Dan Meyer)*

5) Number of Sips vs Amount of Water in Glass (Discussion of the types of sips you would need to keep linear)

6) Time Jogging on Treadmill vs Calories Burned ** **

**– Service Related:**

1) Number of Cigarettes Smoked per Day vs Life Expectancy

2) Number of Questions Practiced vs Grade in Test (in class data taken)

3) Percentage of Population in Extreme Poverty vs Time (ATM: MT245)

4) Income per Person vs CO2 Emissions (www.gapminder.org)

*5) Math Achievement 8th Grade vs Income per Person (www.gapminder.org)*

**Things I’d like to know how strong the correlation would be:**

1) Weight of Person vs 1 Rep Max on Benchpress in First Gym Session

2) Salary of Striker vs Number of Goals Scored

I did recently ask Grade 10 to look into Number of Tweets vs Number of Followers on Twitter (restricting the domain so as not to include anyone famous). Whilst I knew it wouldn’t be linear, I think it’s important to discuss examples of why relationships are not linear. **Often there is a correlation between how many variables are involved and how linear the relationship is. **For example, the start up capital of a business vs profit in first five years. There are simply too many variables at play in this, just as there are for number of followers on twitter. I’d love to add more to this list…

1. Charging an iPhone (Michael Fenton): Linear at first and then tails off.

2. UK shoe size vs Europe/US shoe size (*mathematical beginnings*)

Great examples! I have a comment on this one, though:

1) Temperature of Coffee vs Time

From teaching calculus, I’m aware of “Newton’s Law of Cooling,” which indicates that the rate at which an object cools is proportional to the *difference* between the temperature of the object and the temperature of the ambient air. So the temp of a very hot cup of coffee in a very cold room will drop more quickly than that of a lukewarm cup in a warmish room. But my point is that the temperature changes exponentially, not linearly!

Hi James, thanks for the detail in your comment – I wasn’t aware that it was Newton that developed this law. Of course exponential based on general solution to differential equation (dT/dt = k(T-a)) but a few examples on the list aren’t “officially linear” and so with the coffee example I love that some students think it’s going to be linear and then realise that it won’t be.