Represent Products as Areas

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. The arrangement of the groups of dots encourages students to see 5 groups of dots in the first image and then 6 groups of dots in the next image. When students use equal groups and a known quantity to find an unknown quantity, they are looking for and making 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?

Groups of dots.

6 groups of 3 dots.

Groups of dots.

Solution Steps (4)
  1. 1
    Identify equal groups in first image
    →5 groups of 3 dots
  2. 2
    Skip-count to find total
    →3, 6, 9, 12, 15 = 15 dots
  3. 3
    Use first result to find second image total
    →6 groups of 3: 15 + 3 = 18 dots
  4. 4
    Apply strategy to third image
    →6 groups of 4: 20 + 4 = 24 dots

Sample Response

Sample responses:

  • 15: I saw 5 groups of 3, which is 15. I counted by 3 five times like 3, 6, 9, 12, 15.
  • 18: I know that 5 groups of 3 make 15, and one more group of 3 makes 18.
  • 24: I saw 5 groups of 4 and 1 more group of 4. Five groups of 4 is 20, and one more group of 4 is 24.
Activity Synthesis (Teacher Notes)
  • “How did the first image help you find the number of dots in the second image?” (I know that 5 groups of 3 are 15, and 1 group of 3 more would be 18.)
  • “How did the first and second images help you find the number of dots in the third image?” (I figured out 5 groups of 4 pretty quickly, and then added another group of 4.)
Standards
Addressing
  • 3.OA.5·Apply properties of operations as strategies to multiply and divide.
  • 3.OA.B.5·Apply properties of operations as strategies to multiply and divide.<span>Students need not use formal terms for these properties.</span> <span>Examples: If <span class="math">\(6 \times 4 = 24\)</span> is known, then <span class="math">\(4 \times 6 = 24\)</span> is also known. (Commutative property of multiplication.) <span class="math">\(3 \times 5 \times 2\)</span> can be found by <span class="math">\(3 \times 5 = 15\)</span>, then <span class="math">\(15 \times 2 = 30\)</span>, or by <span class="math">\(5 \times 2 = 10\)</span>, then <span class="math">\(3 \times 10 = 30\)</span>. (Associative property of multiplication.) Knowing that <span class="math">\(8 \times 5 = 40\)</span> and <span class="math">\(8 \times 2 = 16\)</span>, one can find <span class="math">\(8 \times 7\)</span> as <span class="math">\(8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56\)</span>. (Distributive property.)</span>

15 min

20 min