Measure to Find the Area

10 min

Narrative

In this Warm-up, students observe two dot images that show the same number of dots in groups of 3 but arranged in different ways. The purpose is to elicit observations about the similarities in their structure (5 groups of 3 dots plus 1 group of 3 dots more) and to prepare students to make sense of expressions with a similar structure in an upcoming lesson.

When students notice that there is an additional group of 3 in both the diagram of equal groups and the array, they are looking for and making use of structure (MP7).

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Task

What do you notice? What do you wonder?

<p>6 groups of 3 dots. 15 black dots, 3 blue dots.</p>

Rectangular array of dots. 6 groups of 3 dots. 15 black dots, 3 blue dots. 

Solution Steps (4)
  1. 1
    Count groups and dots in first image
    →6 groups of 3 dots (15 black + 3 blue = 18 total)
  2. 2
    Count rows and columns in array image
    →6 rows of 3 dots, same 18 total
  3. 3
    Notice both have extra group highlighted in blue
    →5 groups of 3 black + 1 group of 3 blue
  4. 4
    Connect equal groups to array structure
    →Both represent 6 × 3 = 18 in different visual arrangements

Sample Response

Students may notice:

  • There are 6 groups of 3 in both diagrams.
  • There are 15 black dots and 3 blue dots in each diagram.
  • There are 18 dots in both diagrams.
  • Each image has a group of dots that's a different color.

Students may wonder:

  • Why are some dots a different color?
  • What situations could these diagrams represent?
  • Why are the dots arranged differently in the 2 images?
Activity Synthesis (Teacher Notes)
  • “How do you see the extra group of 3 in each diagram?” (In the diagram of equal groups, it’s an extra group that is separate from the 5 groups of 3. In the array, the extra group of 3 is the column of dots that is a different color.)
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