# Lesson 5

How Crowded Is this Neighborhood?

### Lesson Narrative

This lesson is optional. This lesson involves a sequence of four activities that prepare and introduce students to the concept of population density. The lesson can be adjusted depending on available time and teacher-identified goals from 1 to 2 class days.

Contexts involving population density are useful for helping students understand how derived units arise from a proportional relationship (MP2). Population density arises from two familiar quantities, number of people and area. The way the lesson develops helps students make sense of the somewhat abstract idea of density in very concrete terms: They start by comparing the number of dots distributed in squares and move on to houses in different neighborhoods. Finally they compare the number of people living in different cities. Unlike speed or unit pricing, density is not likely to be familiar to students, so it provides an opportunity to make sense of an unfamiliar situation by thinking about familiar quantities in a new way.

This lesson engages students in important aspects of modeling (MP4). In particular, the rates are used to model rather than represent. For example, houses may not be uniformly distributed in any given area, but rates for houses per square mile characterize differences between rural and urban areas. This lesson begins students’ transition from contexts that involve constant rates to contexts that involve average rates of change. Average rate of change is a high school topic, but before high school students begin to investigate situations modeled by proportional relationships, for example, bivariate data with measurement error or quantities that change at rates that are close to constant.

As with all lessons in this unit, all related standards have been addressed in prior units. This lesson provides an optional opportunity to go deeper and make connections between domains.

### Learning Goals

Teacher Facing

• Compare and contrast the density of uniformly distributed dots in squares.
• Create an equation and a graph that represent the proportional relationship between the area of a square and the number of dots enclosed by the square.
• Interpret the constant of proportionality in models of housing per square kilometer or population of people per square kilometer.

### Student Facing

Let’s see how proportional relationships apply to where people live.

### Required Preparation

If desired, prepare to display satellite images that show the housing density in different neighborhoods of your city, New York City and Los Angeles.

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