Lesson 8How Crowded Is this Neighborhood?

Learning Goal

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

Activity 1: Dot Density

Problem 1

The figure shows four squares. Each square encloses an array of dots. Squares A and B have side length 2 inches. Squares C and D have side length 1 inch.

Four squares labeled A, B, C and D, each with an array of dots inside, as follows: Square A: 8 by 8 array. Square B: 16 by 16 array. Square C: 4 by 4 array. Square D: 8 by 8 array.
  1. Complete the table with information about each square.

    square

    area of the square
    in square inches

    number
    of dots

    number of dots
    per square inch

    A

    B

    C

    D

  2. Compare each square to the others. What is the same and what is different?

Activity 2: Dot Density with a Twist

Problem 1

The figure shows two arrays, each enclosed by a square that is 2 inches wide.

Two equal sized squares with an array of dots inside each. The left square contains an 8 by 8 array of dots. In the right square, the array of dots are as follows: Row 1: 4 blue dots, 2 yellow dots. Row 2: 4 blue dots, 2 yellow dots. Row 3: 4 blue dots, 2 yellow dots. Row 4: 4 blue dots, 2 yellow dots. Row 5: 2 blue dots. Row 6: 2 blue dots. Row 7: 2 blue dots. Row 8: 2 blue dots.
    • Let be the area of the square and be the number of dots enclosed by the square. For each square, plot a point that represents its values of and .

    • Draw lines from to each point. For each line, write an equation that represents the proportional relationship.

  1. What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?

Print Version

The figure shows two arrays, each enclosed by a square that is 2 inches wide.

Two equal sized squares with an array of dots inside each. The left square contains an 8 by 8 array of dots. In the right square, the array of dots are as follows: Row 1: 4 blue dots, 2 yellow dots. Row 2: 4 blue dots, 2 yellow dots. Row 3: 4 blue dots, 2 yellow dots. Row 4: 4 blue dots, 2 yellow dots. Row 5: 2 blue dots. Row 6: 2 blue dots. Row 7: 2 blue dots. Row 8: 2 blue dots.
  1. Let be the area of the square and be the number of dots enclosed by the square.

    • For each square, plot a point that represents its values of and .

    • Draw lines from to each point. For each line, write an equation that represents the proportional relationship.

    A blank coordinate grid with one tick mark labeled on each axes. Both tick marks are the same distance from the origin. The horizontal axis is labeled "a" and the tick mark is labeled 1. The vertical axis is labeled "d" and the tick mark is labeled "10".
  2. What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?

Activity 3: Housing Density

Problem 1

Here are pictures of two different neighborhoods.

This image depicts an area that is 0.3 kilometers long and 0.2 kilometers wide.

An aerial image of a neighborhood displaying the roofs of houses. Each 0 point1 kilometer by 0 point 1 kilometer area has a total of 8 houses.

This image depicts an area that is 0.4 kilometers long and 0.2 kilometers wide.

An aerial image of a neighborhood displaying the roofs of houses. A length of 0 point 1 kilometer indicating one fourth of the length of the neighborhood is indicated. There are a total of 9 or 10 houses displayed.

For each neighborhood, find the number of houses per square kilometer.

Activity 4: Population Density

Problem 1

  • New York City has a population of 8,406 thousand people and covers an area of 1,214 square kilometers.

  • Los Angeles has a population of 3,884 thousand people and covers an area of 1,302 square kilometers.

  1. The points labeled and each correspond to one of the two cities. Which is which? Label them on the graph.

    Two points labeled A and B plotted on a coordinate plane. The a-axis is labeled "area". The p-axis is labeled "population." Point A is to the right of the p-axis and high above the a-axis. Point B is to the right of point A and above the a-axis.
  2. Write an equation for the line that passes through and . What is the constant of proportionality?

  3. Write an equation for the line that passes through and . What is the constant of proportionality?

  4. What do the constants of proportionality tell you about the crowdedness of these two cities?

Are you ready for more?

Problem 1

Predict where these types of regions would be shown on the graph:

  1. a suburban region where houses are far apart, with big yards

  2. a neighborhood in an urban area with many high-rise apartment buildings

  3. a rural state with lots of open land and not many people

Problem 2

Next, use this data to check your predictions:

place

description

population

area
km²

Chalco

a suburb of Omaha, Nebraska

Anoka County

a county in Minnesota, near Minneapolis/St. Paul

Guttenberg

a city in New Jersey

New York

a state

Rhode Island

a state

Alaska

a state

Tok

a community in Alaska