What Is Area?

10 min

Narrative

The purpose of this How Many Do You See? is for students to subitize or use grouping strategies to describe the images they see. Students may see equal groups in the rows or the columns of the array. Recording the equations for each way of seeing the groups is an opportunity to reinforce the commutative property.

When students use different ways to group dots within the same array to find the total number of dots, they look for and make use of structure (MP7).

Launch

  • Groups of 2
  • “How many do you see? How do you see them?”  
  • Flash the image.
  • 30 seconds: quiet think time
Teacher Instructions
  • Display the image.
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Record responses.
  • Repeat for each image.

Student Task

How many do you see? How do you see them?

A group of dots.

A group of dots with 4 rows of 3.

4 rows of 5 dots.

Solution Steps (4)
  1. 1
    Identify the array structure in each image
    →Image 1: 2 rows × 4 columns. Image 2: 4 rows × 3 columns. Image 3: 4 rows × 5 columns.
  2. 2
    Count total by viewing rows as equal groups
    →Image 1: 2 groups of 4 = 8. Image 2: 4 groups of 3 = 12. Image 3: 4 groups of 5 = 20.
  3. 3
    Count total by viewing columns as equal groups
    →Image 1: 4 groups of 2 = 8. Image 2: 3 groups of 4 = 12. Image 3: 5 groups of 4 = 20.
  4. 4
    Connect to commutative property
    →Both ways give the same total: 2×4=4×2=8, 4×3=3×4=12, 4×5=5×4=20

Sample Response

Sample responses:

  • 8: I counted 4 dots in each row and there are 2 rows, which is 8. There are 2 dots in each column and there are 4 columns.
  • 12: I saw 3 dots in each row and there are 4 rows. I saw 4 dots in each column and there are 3 columns.
  • 20: I saw two 10-frames, and 10 plus 10 is 20. I saw 4 rows of 5, which is 20. I saw 5 columns of 4, which is 20.
Activity Synthesis (Teacher Notes)
  • “How did the arrays allow us to find the number of dots in different ways?” (We were able to look at the number of dots in each row and think about how many rows there were. I counted the number in each column and multiplied it by the number of columns.)
  • As students share, record equations to match their thinking.
  • Consider asking, “How do the equations change if we think about the rows as the groups or the columns as the groups?” (For the image with 12 dots, if we think about the rows as the groups, we write 4×3=124 \times 3 = 12, but if we think about the columns as the groups, we write 3×4=123 \times 4 = 12. The order of the factors is reversed, but the product stays the same.)
Standards
Addressing
  • 3.OA.1·Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. <em>For example, describe a context in which a total number of objects can be expressed as 5 × 7.</em>
  • 3.OA.A.1·Interpret products of whole numbers, e.g., interpret <span class="math">\(5 \times 7\)</span> as the total number of objects in 5 groups of 7 objects each. <span>For example, describe a context in which a total number of objects can be expressed as <span class="math">\(5 \times 7\)</span>.</span>

15 min

20 min