Density Curriculum

Section 1—Lesson 2: What Are Some Models That Help Us Think About Density?

Background Information

Introducing Additional Models of Density

The purpose of this lesson is to help students develop mental models of what density is and to conceptualize what "more dense" and "less dense" looks like. This will help students begin to realize that density explains why objects of different materials have different mass-to-volume relations.

Density is an intensive quantity. It is called intensive because it cannot be directly measured. It is determined by the relationship between mass and volume, a quantity because it can be calculated mathematically. This means that you can't measure it directly. It must be inferred. For older students, this can involve calculating it mathematically, but for younger ones, the concept can be introduced by controlling for either volume or felt weight.

This lesson builds on the previous lesson by offering additional models in an attempt to form a solid anchor for students. It asks students to consider possible models in terms of what works and does not work about them. In Section 3, students return to these models and critique them from a different stance—how well they capture the micro-causes of density.

The Issue of Air Versus Space

When thinking about density, students often think that either the object must be hollow or the spaces between the dots in a dot-per-box model must be air. In this module, objects that include air are discussed as having mixed density (the matter of the object plus the matter of the air). For instance, think about eating a brownie. It has a certain density. If there is an air bubble in it, that bubble is not actually brownie (and not a part of the density of the brownie), instead it is air. What you really have is brownie plus air. This is the case with hollow objects. Other mixed density materials have air and another material mixed together such that it is hard to find where the air is. For instance, ice cream is a mixture of cream particles and air particles that are so small it all looks like one material.

However, some materials do not have any air in them. There are spaces between the particles, but these spaces are not filled with air (and in most cases, are not large enough to allow an "air molecule" to fit inside). For instance, the aluminum and copper cylinders in Lesson 1 are made of particles and space, but not air.

This distinction between air and space is tricky for students, but is important for a deep understanding of density and density-related phenomena. For this reason, the module comes back to these distinctions numerous times and in numerous ways.

Models for Visualizing Differences in Density

No model is a perfect representation of a concept. They all include various trade-offs—ways that they fit the real world phenomenon and ways that they don't. Therefore, the focus in discussing models is not about which model is right, but rather what works well about a model (ways that it fits and ways that it doesn't). Ultimately, the model that has the most explanatory power in a given situation is likely to be the most useful. Here is a brief explanation of each of the models:

Wooden Balls and Marbles Model: This model is different from the ones that most students draw. Most students end up drawing crowdedness models. This model introduces the idea that differences in density are due to differences in material. It also introduces a fairly sophisticated concept that we will return to in a different lesson—the concept that different particles have different masses, or the concept of atomic mass. This is a nice model to share because it is so different from what the students generate. It invites them to be more flexible in thinking about models. It controls for the volume, but it lets students feel and measure differences in mass. It uses different materials to depict different materials. Students can think of the balls as particles. It invites those students who may be ready to think about the mass of different particles being different. At the same time, it does not address the idea of crowdedness, and therefore doesn't help students to see what accounts for the differences in mass. It just shows that those differences can exist.

Dots-Per-Box Model: Dots-Per-Box models use crowdedness as a basic analogy to density. Crowdedness models, as the students will see later, work on a number of levels. They work on an abstract level by describing stuff per unit space. They also work as an analogy to the crowdedness of protons and neutrons around an atom. (The amount of space that atoms take up doesn't vary very much, but the crowdedness of the neutrons and protons in the nucleus does. Crowdedness of electrons also varies, but they have little mass.) The Dots-Per-Box model also works as an analogy for how crowded atoms are as they squeeze together into a molecule. There is no need to go into these details with students now. One of the advantages of these models is that they give a sense of particles and the spacing between those particles. The models are similar to those used in a social studies setting to think about population density.

Dots-Per-Box models are less useful for capturing a sense of how mass is a part of the density equation. On the other hand, you could think of each little black box as weighing a certain amount, or guess that boxes of the same size with different numbers of dots will have different masses.

Bread Model: The bread model¹ is a nice model for helping students to think about the amount of stuff in a given amount of space. If you listen carefully to students' responses, you can get a fair amount of information about how they are reasoning. Some students may reveal that they do not understand conservation of matter. Others may not think of air as matter. The model controls for mass (allowing for a small amount of mass due to the air inside the bread to be released when squished), but lets students see visual changes in volume. It illustrates how something that is more compact is more dense because the same amount of "stuff" has been packed into a smaller amount of space. It also introduces the concept of mixed density, in that students must consider air matter and bread matter.

However, it is important to talk explicitly about air and space when talking about the bread model. Otherwise the model may inadvertently reinforce the notion that what makes something dense is whether or not it has air in it. When they learn about the micro-causes of density, they will see that there are some spaces that are too small for "air molecules" (or more correctly, the molecules that make up the air) to fit. While the bread model provides a good visual picture of "compact-ness," it doesn't necessarily help students develop a more sophisticated understanding of what's happening on a molecular level. However, it offers a good starting point for thinking about the relationship between volume and mass.