Problems with Remainders

10 min

Narrative

This Number Talk encourages students to compose or decompose multiples of 7 and to rely on properties of operations to mentally solve problems. The ability to compose and decompose numbers will be helpful when students divide multi-digit numbers. It also promotes the reasoning that is useful when finding multiples of a number, or when deciding if a number is a multiple of another number.

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

Student Task

Find the value of each expression mentally.

  • 21÷721 \div 7
  • 35÷735 \div 7
  • 140÷7140 \div 7
  • 196÷7196 \div 7

Sample Response

  • 3: Three times 7 is 21.
  • 5: Five times 7 is 35.
  • 20: Two times 7 is 14, so 20 times 7 is 140.
  • 28: Sample responses:
    • 196=140+56196 = 140 + 56, and 56=21+3556 = 21 + 35. This means 196=140+21+35196 = 140 + 21 + 35, so 196=(20×7)+(3×7)+(5×7)196 = (20 \times 7) + (3 \times 7) + (5 \times 7).
    • Three times 7 is 21, so 30 times 7 is 210. 196 is 14 less than 210, which means it is 2×72 \times 7 less than 30×730 \times 7.

Synthesis

  • “What do the expressions have in common?” (They all involve division by 7. The dividends are all multiples of 7. The results have no remainders.)
  • “How did the first three expressions help us find the value of the last expression?”
  • 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

15 min