Three-Variable Charts

In the context of preparing my students for their IB portfolios, I realized that it is absurd/funny that in secondary math we don't often deal with three-variable relationships, for example when both volume and temperature will affect pressure. In science, I am pretty sure that at the secondary schooling level you'd discuss things like when you change only one variable to see its effects, and afterwards you hold that one as constant and change another variable, and in the end you try to aggregate their overall impact on the outcome. In secondary math, we (ironically?) shield the kids from this. So, for much of their schooling, kids only know how to make and analyze two-column t-charts involving two variables.

In IB, the curriculum certainly expects a bit more, at least in its portfolio investigations. The IB portfolio tasks go straight into the investigation of the individual and aggregate effects of two variables on a third variable. This notion is unfamiliar to the students, so I was thinking about how to introduce them to this idea of observing and representing three-variable data.

In thinking about how to broach this, I was reminded of the sizing charts I often see on the back of the commercial packaging of tights/stockings. If you have never bought a pair of tights before from a certain brand, you often need to read the sizing chart to check the correct size to purchase. And what are the TWO factors that determine your size of tights? --Height and weight, you say. (Or, at least I hope you do. One of my girls immediately replied today, "Height and color!" sigh. Is it too much to hope that they like math over fashion?)



The boys in my class giggled when I brought this up as an "everyday" example of visualizing the relationship between three variables. (One of the boys said, "These IB tasks are sexist.") Two variables go on the outside of the table (ie. weight across and height vertically down), and the contents of the cells -- ie. the size in this case -- would represent the third variable. In mathematical terms, when you organize something in a table of this form, you can more easily find the formula f(a, b), where the "key" or input to function f is a tuple of two variables, rather than just a single variable.

Again, it's funny to think that this type of relationship must be very common in "the real world", and yet we discuss so little of it in school. Can you think of other everyday examples (ones that won't make my high-school boys giggle)?