Linear vs. Non-linear relationships

I think that a lot of mistakes that happen are related to relationships that are linear, whereas others are non-linear. Linear is easy to understand because the increments follow intuition. To think that a relationship is linear when it's not is the source of one of the most common mistakes.

One bottle of water weights x units. Therefore 10 bottles of water weight 10x units. A factory with 100 employees manufactures 1000 products per day. Therefore two factories with 200 employees manufacture 2000 products per day. If turning on the air-conditioner for 10 hours a day adds up to a cost of x $ by the end of the month, then running it for 5 hours a day should cut the costs in half.

10 people - 30 units of something
42 people - x units of something

Do the cross product and solve for x. We learn that before leaning about functions and that leads to a misconception that is to think that whenever some quantity relates to another, with one being proportional to the other, then we can (we can't!!) assume that the same proportion can be extrapolated for other quantities. The only case in which we can do it is when the rate of change is constant. Else, we can't solve it like that because we can't assume that there is a linear function underneath.

Ten people working for 8 hours a day, during 27 days, produce something. Now reduce the number of people from ten to eight and increase the number of hours from 8 to 9. The question is: how many days does 8 people, working for 9 hours per day, are going to take to produce the same result or quantity of something as 10 people, working for 8 hours for 72 days?

10 people - 8 hours per day - 27 days
8 people - 9 hours per day - x days

This problem is still a linear relationship, except that there are two variables now. The first reasoning that we can correctly have is that the workload per person has increased, by how much we don't know yet.

Note: this type of problem is a bit misleading because it's doing an oversimplification about life. When we have people as variables it's not always true that more people produce more and fewer produce less. There are tons of examples from real life where relationships aren't, can't and shouldn't be treated as linear relationships.

Linear graph

To say that we have a constant rate of change can mean two things: the rate of change is positive or negative; or the rate of change is zero. That's a very common source of mistakes! Without more information we don't know whether the function is increasing, decreasing or neither.

Let's say we have a free fall problem. Gravity for most daily life purposes is a constant force. It doesn't matter if we are at ground level or flying at 10 km of altitude. For distances between planets or stars we can't treat gravity as a constant force because with large distances the force decreases. Suppose a person jumps from a plane flying at 2 Km of altitude, parallel to the ground. The initial vertical velocity is zero. Over time the person accelerates. Without numbers, what we can already assume is that the acceleration is positive because the person is falling and gaining velocity.

Which graph better represents acceleration x time?

The first one could be the result of reversing the orientation of the vertical axis, but it's also a misconception. With the person falling down, one could naturally think that the graph that best describes such motion is a straight line going down. The question is about acceleration, not a trajectory!

The third one is a confusion between velocity and acceleration. We often use both interchangeably in daily life without much trouble. But we often also forget that acceleration due to gravity is a constant. Velocity is increasing during the free fall, but the acceleration is not.

The correct graph for acceleration x time for free fall is graph 2.

Non-linear graph

Now which graph better represents distance x time during a free fall?

A common mistake is to make the association between a trajectory that is a straight line and the graph. The parabola means that the rate of change of the distance from the starting position is not linear. For each unit of time, the distance increases with the square of it. I think that this is where some confusion happens. Distance is measured in meters or any multiple of it. The fact that the variation of distance over time is non-linear is not related to the unit having or not having a square in it. Over time the velocity increases during the free fall. As such, for each second that passes, the distance travelled keeps increasing faster and faster.