How Crowded Is This Neighborhood?

10 min

Teacher Prep
Setup
Notice and wonder about the image. 5 minutes of quiet work time followed by whole-class discussion.

Narrative

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.

Student Task

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
Square A: 8 by 8 dot array

B
Square B: 16 by 16 dot array.

C
Square C: 4 by 4 dot array.

D
Square D: 8 by 8 dot 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?

Sample Response

  1. Completed table:
    square area of the square
    in square inches    		 number
    of dots    		 number of dots
    per square inch
    A 4 64 16
    B 4 256 64
    C 1 16 16
    D 1 64 64
    • Squares A and B have the same area. Squares C and D have the same area.
    • The number of dots in Square A is the same as the number of dots in Square D. The two other squares have different numbers of dots.
    • The number of dots per square inch is the same for Squares A and C. The number of dots per square inch is the same for Squares B and D.
Activity Synthesis (Teacher Notes)

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.

<p>Two larger squares each divided into four smaller squares. Two smaller size squares. </p>

MLR8 Discussion Supports. Invite students to repeat their reasoning using mathematical language: “Can you say that again, using the word ‘density’?”
Advances: Speaking, Conversing
Anticipated Misconceptions

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?”
Standards
Addressing
  • 7.G.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.
  • 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.
  • 7.RP.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <em>For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction <sup>1/2</sup>/<sub>1/4</sub> miles per hour, equivalently 2 miles per hour.</em>
  • 7.RP.A.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <span>For example, if a person walks <span class="math">\(1/2\)</span> mile in each <span class="math">\(1/4\)</span> hour, compute the unit rate as the complex fraction <span class="math">\(\frac{1/2}{1/4}\)</span> miles per hour, equivalently <span class="math">\(2\)</span> miles per hour.</span>

15 min

15 min