I gave the following question on my last physical science (physical science for non-science majors) test:
Block A has a mass of 10 kg and volume of 100 cm3. Block B has a mass of 1 kg and a volume of 1 cm3. Which block has the greater density? Which block has the greater volume?
The results of this question are interesting. Of the respondents, 33 answered both parts in a meaningful way. 8 of these 33 answered that the one with the greatest density had the greatest volume (that is 24%). Most of these 24% said that 1 cm3 was greater than 100 cm3. However, there were some that said the density of the 1 cm3 block had a smaller density because the volume was smaller.
Can around 20% of the class really think that 1 is greater than 100? I don’t think so. Rather, they are falling into the classic trap of thinking that says:
If two things are compared, the one with the greater attribute will be greater in everything.
You can see this with density, mass and volume quite easily. Which has a greater mass, a tiny rock or a huge pile of pillows? Some students and people will say the rock. Really, they will. Here is another: A bug collides with a bus. Which exerts a greater force on the other? Well, the bus is bigger so it should exert a greater force. You get the idea.
I think this is similar, but different than the confusion between a quantity and the rate that quantity changes (like velocity and acceleration).