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).
Launch
Display the image of the four squares with dots. Invite students to share what they notice and what they wonder.
Give students 5 minutes of quiet work time followed by whole-class discussion.
Activity
None
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.
A
B
C
D
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
Compare each square to the others. What is the same and what is different?
Student Response
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Building on Student Thinking
If students do not understand the purpose of the last column in the table, consider asking:
“Can you explain how you figured out the other parts of the table?”
“The word ‘per’ has appeared in many places including ‘dots per square inch,’ ‘miles per hour,’ and ‘price per gallon.’ What does ‘per’ mean?”
“How can you use the parts of the table you already filled in to figure out how many dots per square inch?”
Activity Synthesis
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.
MLR8 Discussion Supports. Invite students to repeat their reasoning using mathematical language: “Can you say that again, using the word ‘density’?” Advances: Speaking, Conversing
8.2
Activity
Standards Alignment
Building On
Addressing
7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
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 uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Activity Synthesis
The goal of this discussion is for students to make sense of the constant of proportionality in the case where the dots are not uniformly distributed. Invite students to share their interpretations of the constants of proportionality.
Consider asking questions like:
“Why is one of the constants of proportionality larger than the other? How can this be seen in the picture of the squares?” (There are more dots in the same area.)
“What are the units of the constants of proportionality?” (The number of dots per square inch).
Students will have more opportunities to think about this when working on the activities that follow.
8.3
Activity
Standards Alignment
Building On
7.RP.A.2
Recognize and represent proportional relationships between quantities.
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
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.
Launch
Give students 5 minutes of quiet work time followed by partner and whole-class discussion. Provide access to four-function calculators.
Engagement: Provide Access by Recruiting Interest. Invite students to share how their neighborhood looks compared to the two in the activity. Supports accessibility for: Conceptual Processing, Memory
Activity
None
Here are pictures of two different neighborhoods.
This image depicts an area that is 0.3 kilometers long and 0.2 kilometers wide.
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.
Student Response
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Building on Student Thinking
If students are accustomed to the area (of a rectangle) being numerically bigger than its length or width, and the areas in the activity seem incorrect to them, consider asking:
“Can you explain how to calculate the area of a rectangle?”
“What happens when 2 fractions that are less than 1 are multiplied? For example, ?”
Activity Synthesis
Invite students to share their answers for the two densities. Invite students to share their reasoning. Consider asking the following questions:
"One number is a lot bigger than the other. How can this be seen in the images?" (There are a lot more houses per the same area.)
"You only counted 48 houses in the first image but you say that there are 800 houses per square kilometer. Why is that happening?" (The area in the image is just a fraction of a square kilometer. If there were a square kilometer with the same density, there would be 800 houses in it.)
"Some of you said that there are 112.5 houses per square kilometer in the second neighborhood. How can there be half houses?" (If there were an image with an area of 2 square kilometers, there would be 225 houses in the image, or if they are uniformly distributed, the image might contain a part of a house.)
If desired, display this image to help students make sense of their answers.
The map shows a rectangle 0.3 km by 0.2 km. This means that any one of the six squares is 0.1 kilometer by 0.1 kilometer, which has an area of 0.01 square kilometer.
If 10 of these were lined up, there would be a strip 1 kilometer long. 10 of these strips would make a square of 1 kilometer by 1 kilometer; that is, 1 square kilometer. Now it’s clear that it takes 100 of these small squares to make a square kilometer, so the small square indeed has an area of a square kilometer. In fact, there are 8 houses in this square, so if the entire square kilometer were filled the same way, there would be 800 houses.
8.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.RP.A.2
Recognize and represent proportional relationships between quantities.
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.
Launch
Arrange students in groups of 2–4. Provide access to calculators.
Consider demonstrating situations that feel more crowded or less crowded by having a certain number of students stand in an area. “What would you have to do to feel more crowded with the same number of people?” (Stand in a smaller space, which would make people stand closer to each other.) “Less crowded?” (Take up more space, so that people are farther apart.) Then mark off a region on the classroom floor with tape. Ask some students to stand inside it, and then ask, “What would make the space feel more crowded?” (If more people stood in the same space.) “Less crowded?” (Fewer people.)
If desired, share some background information about New York City and Los Angeles and display satellite images:
New York City is the U.S. city with the largest population. The city has five parts (called boroughs): Manhattan, Brooklyn, The Bronx, Queens, and Staten Island. In Manhattan, most people live in apartment buildings, many in high-rises. In the other boroughs, many people also live in single-family houses. Staten Island is quite different, almost suburban.
Los Angeles is the U.S. city with the second largest population. Although there are some high-rise apartment buildings, many people live in single-family houses, and many of these houses are single-story.
Give students 5 minutes of quiet work time, followed by small-group and whole-class discussion.
Activity
None
At the time that data was collected in 2023:
New York City had a population of 8,468 thousand people and covered an area of 1,214 square kilometers.
Los Angeles had a population of 3,849 thousand people and covered an area of 1,302 square kilometers.
Each point, labeled or , corresponds 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.
Write an equation for the line that passes through and . What is the constant of proportionality?
Write an equation for the line that passes through and . What is the constant of proportionality?
What do the constants of proportionality tell about the crowdedness of these two cities?
Activity Synthesis
Invite some students to display their graphs and equations for all to see. Ask all students if they agree or disagree and why. Once students agree, focus on the meaning of the constants of proportionality and what they reveal about the crowdedness in the two cities.
Consider asking the following questions:
“Why did you choose point to represent New York City?” (New York City has more people per square kilometer than Los Angeles, so New York is more crowded than Los Angeles.)
“What does the constant of proportionality reveal about the crowdedness of the two cities?” (Even though people aren’t distributed uniformly throughout the cities, if this were the case, there would be about 7,000 people in every square kilometer in NYC and about 3,000 people in every square kilometer in LA.)
Standards Alignment
Building On
Addressing
7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks mile in each hour, compute the unit rate as the complex fraction miles per hour, equivalently miles per hour.
Arrange students in groups of 2. Introduce the context of dot density. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Display only the problem stem and related image, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as comparing dot density.
Reveal the axes and the question “On this coordinate plane, represents the area of the square and represents the number of dots enclosed by the square. For each square, plot a point that represents its values of and ” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
“Which of your questions is most similar to or different from the ones given? Why?”
“Is there a main mathematical concept that is present in both your questions and those given? If so, describe it.”
Have students answer the questions in the statement. If time allows, they can also answer mathematically interesting questions created by their classmates.
Activity
None
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.
On this coordinate plane, represents the area of the square and represents the number of dots enclosed by the square. For each square, plot a point that represents its values of and .
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".
Draw lines from to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality mean in terms of the dots and squares?
Student Response
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Building on Student Thinking
Student Response
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Building on Student Thinking
If students have difficulty understanding population being expressed in units of 1,000, consider asking:
“How many people live in New York City?”
“How does expressing large numbers in terms of thousands help one understand them?”
