Size of Divisor and Size of Quotient

5 min

Teacher Prep
Setup
Display one problem at a time. 30 seconds of quiet think time per problem.

Narrative

This Math Talk focuses on division of whole numbers. It encourages students to think about how the size of the divisor affects the quotient. It also prompts them to rely on what they know about properties of operations and the relationship between multiplication and division, to mentally solve problems. The reasoning elicited here will be helpful later when students further explore meanings of division and the relationship between the dividend, divisor, and quotient.

To divide large numbers mentally, students need to look for and make use of structure (MP7). In explaining their reasoning strategies, students need to be precise in their word choice and use of language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies and record and display their responses for all to see.
  • Use the questions in the Activity Synthesis to involve more students in the conversation, before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each expression mentally.

  • 5,000÷55,000 \div 5
  • 5,000÷2,5005,000 \div 2,500
  • 5,000÷10,0005,000\div 10,000
  • 5,000÷500,0005,000\div 500,000

Sample Response

  • 1,000. Sample reasoning: 5 thousands divided by 5 is 1 thousand.
  • 2. Sample reasoning: There are 2 groups of 2,500 in 5,000.
  • 12\frac12 (or 0.5). Sample reasoning: 5,000 is half of 10,000, and 5,000 divided into 10,000 groups means 0.5 in each group.
  • 1100\frac{1}{100} (or 0.01). Sample reasoning: 5,000÷1,000=55,000\div 1,000 = 5, and 500,000÷1,000=500500,000\div 1,000 = 500, so 5 ÷500=55005 \div 500 = \frac {5}{500}, which is 1100\frac {1}{100}.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”

After evaluating all four expressions, ask students:

  • “What do you notice about the value of each expression as the number we use to divide gets larger?” (It gets smaller.)
  • “Why do you think that is?”

Highlight explanations that support two ways of thinking about division, though at this point it is not important to discuss both if one of them is not mentioned.

  1. Dividing means breaking a number into a certain number of equal parts, and when there are more parts, the size of each part gets smaller.
  2. Dividing means breaking a number into parts of a particular size, and when the size of each part gets larger, the number of parts gets smaller.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Anticipated Misconceptions

If students say that they (mentally) cross out the zeros to divide, consider asking the class during discussion to explain what they believe is happening mathematically when zeros are crossed out. Clarify any confusion accordingly.

Students may say that 5,000÷10,0005,000 \div 10,000 is 2 because they automatically assign the larger number to be the dividend. Urge them to check their reasoning by referring to the preceding expression or to related division expressions with smaller numbers:

  • “Can 5,000÷2,5005,000 \div 2,500 and 5,000÷10,0005,000 \div 10,000 both have the same value of 2?”
  • “What is 10÷510 \div 5? What is 5÷105 \div 10?”
Standards
Building On
  • 5.NBT.6·Find whole-number quotients of whole numbers with up to four-digit dividends and two-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.
  • 5.NBT.B.6·Find whole-number quotients of whole numbers with up to four-digit dividends and two-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.

15 min

15 min