Density, proportional reasoning, and misconceptions

It’s a simple question: what is the relationship between mass and volume?


Some students immeadiately said, “Density!” So, I asked them to describe density in terms of particles, and they couldn’t.

There was a whole day of collecting data. A bunch of plastic blocks, mass on a triple-beam balance and cubic centimeters. Aha! As the volume increases, the mass increases. Blocks that have more volume also have more mass! 



It wasn’t long before a misconception appeared in the group. 

“No! There is no relationship!” said one student. The other students were silent and looked at me.

Me: “What do you mean?”

Student: “If you take a block, it has an amount of stuff in it. If that stuff got bigger, it would take more volume but be the same stuff and so, the same mass, so it would be less closely packed and the ratio of mass to volume would change. Therefore, there is no relationship.”

Me: “Would the stuff be the same, then?”

Student: “Yes?”

Me: “In your experiment, is that what happened?”

Student: “No. We looked at different pieces of the same stuff, different sizes.”

Me: “None of the ‘stuff’ changed volume?”

Student: “No, they were different sizes.”

Me: “Class, anyone agree or disagree?”

Student2: “He’s right, but that wasn’t our experiment.”

Student3: “Maybe we should check gases.”

Me: “I think he’s right too, and let’s save gases for another day.”

At this point, I wonder how many students we lose during discussions about density because they think we are talking about somehow changing matter, rather than talking about different kinds of matter. I would have never known my student was thinking this if I had been lecturing and assigned homework from the text book.

Then, I prompted the students to have a discussion about what is density at the particle level (By saying, “OK, remember our particle boxes? Would the particle boxes for each kind of matter be different?”), which was slow-going. We did not reach a confident consensus, but we will circle around to density again with density of a gas. 

Proportional Reasoning

I was very interested in developing the students’ ideas of density, in terms of the way we had measured and identified it. Because they were so comfortable with the relationship between mass and volume being density, and also Density = Mass / Volume, I asked them to find it on the graph. 

Nope. This was a new idea for them.

I asked what the units of density were (since they kept insisting on density = grams / cm^3). 

And then one student saw it. Then, another. “its the slope!”

But not all the whiteboards had the same graphs.  Some were grams / cm^3, others cm^3 per gram. They had a discussion about maybe next time they should check with each other to make sure the groups set rules for the board meetings, so it will be easier to compare data.

To close the day, I reminded them to make sure the info on their whiteboards was in their notebooks. Then I asked, “Ok, so which of the blocks will float?”

More complaints – one student suggested floating was related to surface area, so multiple shapes of the same kind of stuff had to be tested. Finally, it clicked for him too – what mattered was density, that the floating thing had fewer particles in each unit of space compared to water, sinking stuff had more.


(There were also good ideas about why the cork floated so “high” – it’s in the back right corner.)

There were a few things that are standing out to me at this point in the year:

  • Three weeks down, and we still aren’t using usual chemistry words like “element”, or “atom”. We are using “stuff”, “matter”, “space” a lot.
  • We do not say “per”, as in “grams per milliliter”. We say “for every”, as in grams of matter for every cubic centimeter of that matter.”  We try to speak intentionally, with clarity.
  • I do not do most of the talking, which is the most amazing thing because I can constantly check for student misconceptions and challenge them with Socratic questioning.
  • The students are really having fun! And so am I.

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