Lesson 5
How Crowded Is this Neighborhood?
Let’s see how proportional relationships apply to where people live.
5.1: Dot Density
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.](https://cms-im.s3.amazonaws.com/BZjTY4WGvRnpt1gqmioaaW4J?response-content-disposition=inline%3B%20filename%3D%227-7.2.G1.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.G1.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T142007Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dee32ec526ce1d0b88041380b4beda8fd7f611a2fd7baac0aba532098a1837ea)
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Complete the table with information about each square.
square area of the square
in square inchesnumber
of dotsnumber of dots
per square inchA B C D -
Compare each square to the others. What is the same and what is different?
5.2: Dot Density with a Twist
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.](https://cms-im.s3.amazonaws.com/DY6hQKZUthHvHJ2c2SSU9hPd?response-content-disposition=inline%3B%20filename%3D%227-7.2.G1.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.G1.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T142007Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ef391708cb0ee7d696b19c6397a6b93b5d79bfaec40963556bc777d32aaa9195)
- Let \(a\) be the area of the square and \(d\) be the number of dots enclosed by the square. For each square, plot a point that represents its values of \(a\) and \(d\). Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
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Draw lines from \((0, 0)\) to each point. For each line, write an equation that represents the proportional relationship.
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What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
5.3: Housing Density
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.](https://cms-im.s3.amazonaws.com/uXjWukNAKfmYSqE1gcwCSqqh?response-content-disposition=inline%3B%20filename%3D%227-7.2.G1.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.G1.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T142007Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7a813a82e9ed259f2ec7f35aa41d5fd95a5592c21b2b1c9236e31708697b6803)
This image depicts an area that is 0.4 kilometers long and 0.2 kilometers wide.
![An aerial image of a neighborhood.](https://cms-im.s3.amazonaws.com/3n8xtsNmGHtikNryFqt2X6Ky?response-content-disposition=inline%3B%20filename%3D%227-7.2.G1.Image.09.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.G1.Image.09.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T142007Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9d56e7e11e5d86df5490eb708fc8b55f2d1e30ade2a407938d1a1104b39c794f)
5.4: Population Density
- 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.
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The points labeled \(A\) and \(B\) each correspond to one of the two cities. Which is which? Label them on the graph.
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Write an equation for the line that passes through \((0, 0)\) and \(A\). What is the constant of proportionality?
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Write an equation for the line that passes through \((0, 0)\) and \(B\). What is the constant of proportionality?
- What do the constants of proportionality tell you about the crowdedness of these two cities?
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Predict where these types of regions would be shown on the graph:
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a suburban region where houses are far apart, with big yards
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a neighborhood in an urban area with many high-rise apartment buildings
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a rural state with lots of open land and not many people
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Next, use this data to check your predictions:
place description population area (km2) Chalco a suburb of Omaha, Nebraska 10,994 7.5 Anoka County a county in Minnesota, near Minneapolis/St. Paul 339,534 1,155 Guttenberg a city in New Jersey 11,176 0.49 New York a state 19,746,227 141,300 Rhode Island a state 1,055,173 3,140 Alaska a state 736,732 1,717,856 Tok a community in Alaska 1,258 342.7