Multiplication Strategies on Ungridded Rectangles

10 min

Narrative

This Warm-up prompts students to compare four representations of multiplication. It gives students a reason to use language precisely as they talk about characteristics of the items being compared (MP6). During the discussion, ask students to explain the meaning of any terminology they use, such as “strategies,” “area,” and “parts.”

Launch

  • Groups of 2
  • Display the image.
  • “Pick 3 that go together. Be ready to share why they go together.”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 2–3 minutes: partner discussion
  • Share and record responses.

Student Task

Which 3 go together?

A
Diagram. Rectangle split into 2 parts. One part partitioned into 3 rows of 2 of the same size squares, the other partitioned into 3 rows of 5 of the same size squares.

B
<p><span>Rectangle split into two parts. One part labeled 6 with horizontal side 2, the other labeled 12 with horizontal side 4.</span></p>

C
Addition. Three times 2 plus three times 4.

D
<span>Array. 3 rows of 6 dots.</span>

Solution Steps (5)
  1. 1
    Analyze representation A
    →Rectangle with 3×2 + 3×5 = 6 + 15 = 21 squares
  2. 2
    Analyze representation B
    →Rectangle with areas 6 and 12, so 6+12=18, which is 3×(2+4)=3×6
  3. 3
    Analyze representation C
    →Expression 3×2 + 3×4 = 6 + 12 = 18
  4. 4
    Analyze representation D
    →Array with 3 rows of 6 dots = 3×6 = 18
  5. 5
    Identify which 3 go together
    →B, C, D all represent 3×6 = 18 (distributive property)

Sample Response

Sample responses:

A, B, and C go together because:
  • They show 3×23 \times 2 and another part (3×43\times 4 or 3×53 \times 5) separately.
  • They show a whole split into two parts.
A, B, and D go together because:
  • They are diagrams.
A, C, and D go together because:
  • They have 3 represented with a number or with countable objects.
B, C, and D go together because:
  • They represent 3×63 \times 6.
  • They show a total of 18.
Activity Synthesis (Teacher Notes)
  • “What numbers do the diagrams and the expression in C represent? How do you know?” (B, C, and D represent 18. There are 18 dots in the array. If I add the values of the two parts of the expression or the areas of the two parts of the rectangle, I get 18. A shows 21 because there's 6 squares in the first part and 15 squares in the second part.)
  • “What might be the length of the unlabeled side of the rectangle in B? How do you know?” (3 units. I know because the rectangle is the same one as in A, just not showing a grid. I know because the 6 and 12 show the area of each part and 3×2=63 \times 2 = 6 and 3×4=123 \times 4 = 12).
Standards
Addressing
  • 3.MD.7·Relate area to the operations of multiplication and addition.
  • 3.MD.C.7·Relate area to the operations of multiplication and addition.
Building Toward
  • 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>

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