In this activity, students compare dot densities when dots are uniformly distributed. The squares are sized so that students can compare dot density in large and small squares by drawing a partition of the larger square into four smaller squares and comparing the number of dots in squares of the same size. Students reason abstractly and quantitatively as they compare the density of dots and make decisions about how to partition the squares (MP2).

Invite students to share what is similar and what is different about the arrays.

Define density as “things per square inch,” in this case dots per square inch. Demonstrate the correct use of “dense” and “density” by saying things like:

• “The green dots in Square B are more densely packed than the red dots in Square A and the blue dots in Square C.”
• “The density of the red dots in Square A and the blue dots in Square C is the same.”

If students haven’t noted it already, point out that Square A can be partitioned into four smaller squares. Each has an array of red dots that is spaced the same as the array of blue dots in Square C. The same is true for Squares B and D.

In this activity, the dots are distributed uniformly in the first square but not in the second square:

However, these dots are drawn so it is not too hard to see that if they were redistributed, each square inch would have 8 dots:

Two squares. Each square is divided into four equal sized smaller squares. In the first square, each smaller square has four rows of four dots, equally spaced. In the second square, each smaller square has four rows of two dots. 

The fact that there are 8 dots per square inch means that if the dots were distributed uniformly throughout the square in the right way, there would be 8 dots in each square inch. This prepares students to be able to interpret the constant of proportionality in the next two activities when working with actual houses per square mile or people per square kilometer. Students reason abstractly and quantitatively as they interpret the distributions of dots (MP2). The phrase “500 houses per square kilometer” can be thought of as “If all of the houses in the region were spread out uniformly, then there would be 500 houses in every square kilometer.”

In the digital version of the activity, students use an applet to graph lines representing the relationships between the squares’ areas and the number of dots. The applet allows students to adjust the scale on each axis. The digital version may be helpful for thinking through how to scale each axis to best represent the data.

This activity starts to transition students from arrays of dots to real-world objects distributed over the surface of Earth. This task concerns housing density, which is very similar to dot density. The two images in the activity have the following properties:

• It is fairly easy to distinguish the houses and not too tedious to count them all.
• The scale of the images is similar, but the size of each image and the number of houses in each image is not the same.
• In the first image, the houses look fairly uniformly distributed, and in the other, they look less uniformly distributed but not to the point that it is hard to interpret the image.

This task can be customized to any location, for example, different neighborhoods in the school’s city. Care should be taken in selecting the images to include noticeably different housing densities and easy-to-count houses.

The purpose of this activity is to introduce the concept of population density. One added step going from houses in a neighborhood to people in a location is the fact that people do not stay at a fixed spot but rather move around. In this activity, students make sense of what it means to say there are 42.3 people per square kilometer in some location (MP1).

This activity gives some information about New York City and Los Angeles and ultimately asks students to decide which city is more crowded. Students may benefit from a demonstration of situations that feel more crowded vs. less crowded.

In the data for this task, people are given in “blocks” of 1,000 people. It’s perfectly fine to make up a new unit customized to the situation, even if it doesn’t have an official name. This is a more sophisticated use of nonstandard units. In earlier grades, nonstandard units tend to be the length of a paper clip, or the length of your shoe. When dealing with any units, it’s important to list the units: in the heading of a table, in labels for the axes of a graph, or in writing numbers.