Use the 4 Operations to Solve Problems

10 min

Narrative

The purpose of this True or False? is to elicit strategies and understandings students have for multiplying one-digit whole numbers by multiples of 10. The reasoning students do here helps to deepen their understanding of the associative property as they decompose multiples of 10 to make multiplying easier.

Launch

  • Display one statement.
  • “Give me a signal when you know whether the statement is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategy.
  • Repeat with each statement.

Student Task

Decide if each statement is true or false. Be prepared to explain your reasoning.

  • 2×40=2×4×102 \times 40 = 2 \times 4 \times 10
  • 2×40=8×102 \times 40 = 8 \times 10
  • 3×50=15×103 \times 50 = 15 \times 10
  • 3×40=7×103 \times 40 = 7 \times 10
Solution Steps (4)
  1. 1
    Evaluate 2×40 = 2×4×10
    →True: 40 = 4×10, so both sides equal 80
  2. 2
    Evaluate 2×40 = 8×10
    →True: 2×4=8, so 2×40 = 8×10 = 80
  3. 3
    Evaluate 3×50 = 15×10
    →True: 3×50 = 3×5×10 = 15×10 = 150
  4. 4
    Evaluate 3×40 = 7×10
    →False: 3×40 = 3×4×10 = 12×10 = 120, not 7×10=70

Sample Response

  • True: 4×104 \times 10 is 40, so the sides are the same.
  • True: 2×42 \times 4 is 8 in the first expression, so 8×108 \times 10 is also equal to 2×402 \times 40.
  • True: I can write 3×503 \times 50 as 3×5×103 \times 5 \times 10, and 3×5=153 \times 5 = 15, so 3×503 \times 50 is the same as 15×1015 \times 10.
  • False: 3×403 \times 40 is the same as 3×4×103 \times 4 \times 10 or 12×1012 \times 10, not 7×107 \times 10.
Activity Synthesis (Teacher Notes)
  • “How can you justify your answer without finding the value of both sides?”
  • Consider asking:
    • “Who can restate _____’s reasoning in a different way?”
    • “Does anyone want to add on to _____’s reasoning?”
    • “Can we make any generalizations based on the statements?”
Standards
Addressing
  • 3.NBT.3·Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
  • 3.NBT.A.3·Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., <span class="math">\(9 \times 80\)</span>, <span class="math">\(5 \times 60\)</span>) using strategies based on place value and properties of operations.
  • 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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