Introducing the concept of density
There is a strong tradition of beginning physics with careful measurements of volumes and masses and calculations of densities, and with considerable care over arithmetic.
Generally, students find the measuring a fairly interesting routine which does not require much thought. The calculation of density appears an unnecessary interruption of interesting experiments.
Many teachers find the calculation simple and so are tempted to rush to arrive at a characteristic physical quantity. Later, when they discover their students are having difficulties, they may revisit the concept of density with greater care. By that time, it's likely that damage has been done to the picture of science which students are forming.
Solid blocks of material, which are the same size, aid the comparison of density of different materials. Rectangular shaped containers for liquids and gases help in the measurement and calculation of volume. Using spreadsheets to cope with arithmetic problems enables you to emphasize the concept of density and handle materials of different densities. Once the concept has a secure foundation, the arithmetic skill can be introduced quickly, at a later stage.
We suggest, with an average group, that you get students to compare blocks of the same dimensions. Then ask them how they can bring other blocks into comparison. If they are interested (if you've succeeded in making it an intriguing problem), you can coax the class into a discussion of ways and means. If necessary, offer the suggestion of finding out the mass of one little block of 1 cm x 1 cm x 1 cm.
Yes. If you had them, you could weigh little blocks like that little cube. But you have not got them. No, we cannot cut them up with a saw. That would take too long and spoil the big blocks. Can you count the cubes in a big block without cutting it up?
At this point, draw pictures on the blackboard/whiteboard in a progression of problems, or give problems for homework, such as the following:
Here is a block of wood 2 cm long, 1 cm wide and 1 cm high. Here is a little cube of Plasticene 1 cm by 1 cm by 1 cm. How many little cubes are there in this block?
Here is a block 2 cm long, 3 cm wide and 1 cm high. How many little cubes would fit along the 2 cm by 3 cm face? How many layers of cubes from front to back? How many cubes altogether will fit into the block?
So it is all a matter of counting cubes by multiplying length x breadth x height.
Of course, students will have learnt this in mathematics but you hope to have produced some practical meaning.
A well-packed box of sugar cubes helps this discussion. There may also be a set of Tillich's bricks, either in the science preparation room or in the mathematics department. These bricks are 1 cm3 and have a density of 1 g/cm3.
Density is just the name that scientists use for the mass of a unit cube.