As par of changing my language this year, I have moved to using the phrase “For every ______, we have _____.” We started with describing the relationship of milliliters and cubic centimeters by establishing the slope of 1.0 ml/ 1cm3. We then described this saying, “for every one milliliter, we have one cubic centimeter.” Now students did not yet see this as a conversion factor since it was a ratio and they did not feel the need to put it mathematically.
Before mentioning the word density, we looked at graphs of mass vs volume and saw that they created lines with constant slopes. We described the relationship for aluminum using “for every 2.7 grams of aluminum, we have 1.0 ml.” We then used this relationship to solve questions that required converting between mass and volume. In the process we linked the algebraic, graphical, diagrammatic and verbal definitions of density.
It is only when we get to discussing the size of particles that we begin to see that this relationship between slope, conversion factors and “for every” statements are applicable everything. For instance we can say, “for every 100 cm, we have one meter.” Graphing cm v m would give us a line with slope of 100cm/m and we can create a conversion factor of the same.
So why is this important? It means students can visualize how units scale. For instance, they can envision a massive slope when converting m to cm. So if they they try to convert 2.75 m to cm they will envision a large number and can tell wether they calculated an answer with the appropriate scale.
Now you may say this isn’t that important for the metric scale, but it will give a solid foundation for the stoichiometry to come.