Area and the Multiplication Table

10 min

Narrative

The purpose of this Warm-up is for students to notice that figures composed of multiple arrays can be decomposed into smaller arrays, and that this is a strategy to determine the total number of dots. This will be helpful in later lessons when students decompose figures into rectangles to find the total area.

When students find ways to decompose the given arrangements of dots to find the number of dots, they practice 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?

Array. 2 rows of 4 dots.

A group of dots.

Groups of dots.

Solution Steps (3)
  1. 1
    Count first image: 2 rows of 4 dots
    →2 × 4 = 8 dots
  2. 2
    Count second image: first array plus one more row
    →8 + 4 = 12 dots (or 3 × 4)
  3. 3
    Count third image: second figure plus one more row
    →12 + 4 = 16 dots (or 4 × 4)

Sample Response

Sample responses:
  • 8: There are 2 rows of 4, and I know 2 times 4 is 8. There are 4 groups of 2, which is 8.
  • 12: It’s like the first one, but there’s another group of 4, and 8 plus 4 is 12.
  • 16: It’s like the second one, but there’s another group of 4. 12 plus 4 equals 16.
Activity Synthesis (Teacher Notes)
  • “What’s the same in how you found the number of dots in each image?” (I looked for groups of 4. I decomposed each arrangement of dots into smaller arrays.)
  • Consider asking:
    • “Who can restate the way ___ saw the dots in different words?”
    • “Did anyone see the dots the same way but would explain it differently?”
    • “Does anyone want to add an observation to the way ____ saw the dots?”
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>
Building Toward
  • 3.MD.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
  • 3.MD.C.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

15 min

20 min