An Algorithm with Partial Quotients

10 min

Narrative

This Number Talk encourages students to look for and make use of the structure of numbers in base-ten to mentally solve division problems (MP7). The reasoning elicited here will be helpful later in the lesson when students divide large numbers using increasingly more abstract strategies. In explaining their reasoning, students practice being precise in their use of language (MP6).

Launch

  • Groups of 2
  • 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 strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Task

Find the value of each expression mentally.

  • 90÷390 \div 3
  • 96÷396 \div 3
  • 960÷3960 \div 3
  • 954÷3954 \div 3

Sample Response

  • 30: I know that 90 is 10×910 \times 9 and 9÷39 \div 3 is 3, so 90÷390 \div 3 is 10×9÷310 \times 9 \div 3, which is 30.
  • 32: Ninety is 30×330 \times 3 and 6 is 2×32 \times 3, so 96 is (30×3)+(2×3)(30 \times 3) + (2 \times 3), which is (30+2)×3(30 + 2) \times 3 or 32×332 \times 3.
  • 320: I know that 96÷396 \div 3 is 32, so 960÷3960 \div 3 is 10×96÷310 \times 96 \div 3 or 10×3210 \times 32, which is 320.
  • 318: I know that 954 is 960−6960 -6, so it is (320×3)−(2×3)(320 \times 3) - (2 \times 3), which is (320−2)×3(320 - 2) \times 3 or 318×3318 \times 3.
Activity Synthesis (Teacher Notes)
  • “How did each expression help you find the next one?”
  • Consider asking:
    • “Who can restate _______ 's reasoning in a different way?”
    • “Did anyone have the same strategy but would explain it differently?”
    • “Did anyone approach the expression in a different way?”
    • “Does anyone want to add on to____’s strategy?”
Standards
Addressing
  • 4.NBT.6·Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 4.NBT.B.6·Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

20 min

15 min