Fractions and Whole Numbers

10 min

Narrative

This Number Talk encourages students to rely on their knowledge of multiplication, place value, and properties of operations to mentally solve division problems. The reasoning elicited here helps to develop students' fluency with multiplication and division within 100.

To find the quotients of greater numbers, students need to look for and make use of structure in quotients that have less value or are more familiar, or rely on the relationship between multiplication and division (MP7).

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.
  • 12÷412 \div 4
  • 24÷424 \div 4
  • 60÷460 \div 4
  • 72÷472 \div 4
Solution Steps (4)
  1. 1
    Solve 12÷4 using multiplication fact
    →3×4=12, so 12÷4=3
  2. 2
    Solve 24÷4 using doubling relationship
    →24 is twice 12, so 24÷4 is twice 3 = 6
  3. 3
    Solve 60÷4 using decomposition
    →60=40+20, so 40÷4=10, 20÷4=5, total=15
  4. 4
    Solve 72÷4 using prior results
    →72=12+60, so 3+15=18

Sample Response

  • 3: I just knew it. 3×4=123 \times 4 = 12
  • 6: 24 is twice 12, so the value of 24÷424 \div 4 is twice that of 12÷412 \div 4 or twice 3, which is 6.
  • 15:
    • I knew that 4×104 \times 10 is 40 and 4×54 \times 5 is 20, so 4×154 \times 15 is 60.
    • I knew that 20÷420 \div 4 is 5 and 60 is 3 times 20, so 60÷460 \div 4 is 3 times 5.
    18:
    • 72=12+6072=12+60, so I added the values of 12÷412 \div 4 and 60÷460 \div 4, which is 3+153 + 15.
    • 4×10=404 \times 10 = 40, 4×8=324 \times 8 =32, 10+8=1810+8=18.
Activity Synthesis (Teacher Notes)
  • “How did the earlier expressions help you find the values of the later expressions?”
  • 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.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.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.

15 min

20 min