Generate Equivalent Fractions

10 min

Narrative

This Number Talk encourages students to look for structure in multiplication expressions and to rely on properties of operations to mentally solve problems. Reasoning about products of whole numbers helps to develop students’ fluency.

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time
Teacher Instructions
  • Record answers and strategies. 
  • Keep expressions and work displayed. 
  • Repeat with each expression.

Student Task

Find the value of each expression mentally.

  • 2×82 \times 8
  • 6×86 \times 8
  • 10×810 \times 8
  • 12×812 \times 8
Solution Steps (4)
  1. 1
    Solve 2×8 mentally
    →16 (known fact or 8+8)
  2. 2
    Solve 6×8 using prior results
    →48 (3×16 or double 3×8=24)
  3. 3
    Solve 10×8 using place value
    →80 (known pattern: ×10 adds zero)
  4. 4
    Solve 12×8 using decomposition
    →96 (12=10+2, so 80+16=96)

Sample Response

  • 16: I just knew it. It’s 8+88+8, which is 16.
  • 48: It’s 3 groups of 2×82 \times 8, so it's two more groups of 16, or 16+16+1616 +16+16, which is 48. I knew 3×83 \times 8 is 24, and 6×86 \times 8 is twice that number, which is 48.
  • 80: I know 5×85 \times 8 is 40 and 10 is twice 5, so 10×810 \times 8 is 80. I just knew it.
  • 96: 12 is 10+210 + 2, so 12×812 \times 8 is (10×8)+(2×8)(10 \times 8) + (2 \times 8) or 80+1680 + 16, which is 96. 
Activity Synthesis (Teacher Notes)
  • “How did the earlier expressions help you find the value of the last expression?”
  • Consider asking:
    • “Did anyone have the same strategy but would explain it differently?”
    • “Did anyone approach the problem in a different way?”
Standards
Addressing
  • 3.OA.5·Apply properties of operations as strategies to multiply and divide.
  • 3.OA.7·Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • 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>
  • 3.OA.C.7·Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that <span class="math">\(8 \times 5 = 40\)</span>, one knows <span class="math">\(40 \div 5 = 8\)</span>) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

20 min

15 min