Seven Must-Have Graph Grooming Tips

A style guide for our number rockets as they learn what a well-groomed graph wears before it ever even thinks of stepping out of the house.

Known and Unknown Data Devils

Independence and Dependence: Examples of two kinds of stories that show up as 2-variable data analysis models in math instruction.

I thought I understood independent and dependent variables. Then I tried to explain it to myself so that I could help explain it to the 7th grade RTI3 students that I support. Then I fell into a philosophy hole of cause and effect. The kinds of basic data sets (e.g., size of stamp collection per year, inches grown over several years) that they were working with seemed to get pretty slippery when attempts were made to get them (the data) to hold still long enough to be able to determine the independent and dependent variables. What follows is my attempt to crawl out of that hole and back into the well-reasoned light of day, hopefully carrying with me a sackful of language that I can use with our students.

1. A Clear, Intuitively Available Case of Cause and Effect

The closer you get to a campfire*, the hotter it feels.
Suppose you have a tape measure and a thermometer.
You are some distance from a campfire, and as you move closer to it, the thermometer shows higher temperatures.
You record your distance from the fire at regular intervals.
At each interval, you record the temperature.
Question
Does being closer to the fire cause the temperature to increase, or
Does an increase in temperature cause you to be closer to the fire?
Question (alternate)
Does your distance to the fire get shorter because the height of the thermometer column gets taller, or
Does the height of the thermometer column get taller because your distance to the fire gets shorter.
What are the two variables?
distance and temperature
Which variable changes because the other variable changes?
temperature changes because distance changes
Which variable depends on the other?
temperature depends on distance
Which is the dependent variable?
temperature
Temperature is the dependent variable because it changes depending on how close you are to the fire.
Which is the independent variable?
distance
Distance is the independent variable because WE determine how it changes, rather than the temperature of the fire. How close you are to the fire depends on where you decide to stand, NOT on how hot it is where you stand.
*The campfire is stable and burns at about the same intensity while the data is being collected.

2. A Not Very Clear Case of Cause and Effect, or Maybe Even No Case At All

The temperature outside changes over a 24 hour period.*
Suppose you have a clock and a thermometer.
You go to the same spot outside and you notice that the temperature is different at different times of the day.
You record the time you go outside at regular intervals.
At each interval, you record the temperature.
Question
Does the time of day you go outside with the thermometer cause the temperature to be what you measure, or
Does the temperature you measure cause you to go outside when you do.
Ummm, neither? There is no cause here. Temperature changes, but not because of when we measure it or the length of time between measurements. On one day, the temperature might drop 3 degrees in the hour between 2:00 PM and 3:00 PM; on another day, during the same interval at the same time of day, it might rise or it might stay the same. If there is no cause and effect, and if we want to talk about dependent and independent variables, then we need a different perspective on the variables in this story.

What are the two variables?
time and temperature

Do these variables change in predictable ways or in ways that WE control?
TIME
Time changes in a completely predictable way. If we record the first measurement of temperature at 8:00 AM, and then again at regular (evenly spaced) intervals of 15 minutes, we can predict with 100% certainty what time it will be when we record the fortieth measurement, but we can NOT be certain what the temperature will be at that time (at 9:45 PM).
We have chosen to measure temperature at regular intervals of time. We have NOT chosen to measure time at regular intervals of temperature (for example: what time is it at every temperature change of 2º?). We have freely selected and determined time intervals as the feature to compare unknown changes in temperature with.
TEMPERATURE
We may have a hypothesis about how temperature will change over time on a given day, but the exact manner of change is unpredictable. We can not be certain of the exact temperature at 8:00 AM or 9:45 PM or at any other time until we actually measure it.
This story looks at how temperature changes compared with just one feature, or variable: time. We could have compared temperature with humidity, or proximity to the ocean, or the angle at which sunlight falls, or weather, or altitude, longitude, or latitude, or many other variables (probably along with time). We have chosen change in time to be the (sole) basis for comparison with change in temperature.

Question
Which variable changes in ways that we understand and can predict with absolute certainty?
Question (alternate)
Which variable changes in some certain way regardless of how the other variable changes?

These new questions replace cause and effect with insulation. A variable is independent when its way of changing is fully known and when it is protected from any influence that might cause it to change differently. The independent variable will only change in some way that we know exactly, a way that we have chosen and which we control, regardless of what the dependent variable does. In other words, the independent variable only changes the way we say it will, or the way we know it will, whereas the dependent variable can change however it damn well pleases.

Does this mean that the dependent variable is a rebel, a lawless action, an event wholly random and uncontrolled? Maybe. It could mean that, unless we see a pattern in its behavior. We need the controlled, predictable, wholly known and independent variable as a background against which we can look at and compare the dependent variable. That is the only way we can ever notice pattern.

Question
Which is the independent variable?
time, because we are certain about how it will change (its rate of change)
Question
Which is the dependent variable?
temperature, because we are uncertain about how it will change (or if it even has a rate of change…that we can see)

Here, I  think, are criteria that I can ask our number rockets to evaluate: certainty, the independence of the variable that is above it all, and uncertainty, the whatever-it-is that the other variable depends on, the stuff we don’t know about yet, the mystery.

*This could just as easily be: number of stamps collected per year, number of socks lost in the laundry per month, number of watermelon seeds accidentally swallowed per summer, number of fish caught per day, number of home runs hit by bearded, left-handed hitters against right-handed pitchers in US stadiums located south of the 40th. parallel per season, etc. A dependent variable is caused to happen by the independent variable only when a pattern can be found. Until then, it’s the independent devil we know versus the dependent devil we don’t.

Then, when I finally went poking around, I found these:

Math Forum
http://mathforum.org/library/drmath/view/61593.html

Mathwords
http://www.mathwords.com/i/independent_variable.htm
http://www.mathwords.com/d/dependent_variable.htm

Physical Science
http://mathxscience.com/scientific_method_variables.html

Social Science
http://www.apexdissertations.com/articles/independent_dependent_variables.html
https://www.mtholyoke.edu/courses/etownsle/qr/Independent%20and%20dependent%20variables.htm