Area of a Rectangle without a Grid

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.

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?

4 groups of 5 dots.

5 groups of 6.

4 groups of 4 dots.

Solution Steps (3)
  1. 1
    Count dots in first image (4 groups of 5)
    →Count by 5s: 5, 10, 15, 20 → 20 dots
  2. 2
    Use first result to find second image
    →4 groups of 6 = 4 groups of 5 + 4 more = 20 + 4 = 24
  3. 3
    Use first result to find third image
    →4 groups of 4 = 4 groups of 5 - 4 = 20 - 4 = 16

Sample Response

Sample responses:

  • 20: I counted by 5 four times. I know that 4 times 5 is 20.
  • 24: It was like the first problem, but there was 1 dot more in each group. So I added 20 to 4 to get 24.
  • 16: It was like the first problem, but each group was missing 1 dot so I subtracted 4 from 20 to get 16.
Activity Synthesis (Teacher Notes)
  • “What numbers were easy to see in the images?” (4, 5, 6)
  • “How did the first image help you find the number of dots in the next 2 images?” (I know each group in the second image has 1 dot more than each group in the first image, so I figured out 4 groups of 5, then added 4 dots more. For the last image, I subtracted 4 from 20, since 1 dot was missing from each group.)
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>

20 min

15 min