Build Fractions from Unit 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 with multiplication within 100.

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.

  • 3×33 \times 3
  • 7×37 \times 3
  • 10×310 \times 3
  • 3×173 \times 17
Solution Steps (4)
  1. 1
    Compute 3×3
    →9 (known fact)
  2. 2
    Compute 7×3
    →21 (6×3=18, plus 3 more)
  3. 3
    Compute 10×3
    →30 (known fact)
  4. 4
    Compute 3×17 using decomposition
    →3×17 = 3×10 + 3×7 = 30 + 21 = 51

Sample Response

  • 9: I just knew it. I know 2×32 \times 3 is 6, and 3 more is 9.
  • 21: Six times 3 is twice 3 times 3, which is 2×92 \times 9 or 18. Adding 3 more gives 21.
  • 30: I know that 3×103 \times 10 is 30, and 10×310 \times 3 is also 30. I just knew it.
  • 51: It’s the sum of 10×310 \times 3 and 7×37 \times 3, which I know is 30 and 21. I know 3×203 \times 20 is 60, and 3×173 \times 17 is 3×33 \times 3 less than 60 or 60−960 - 9, which is 51. 
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