How Much for One?

10 min

Teacher Prep
Setup
Display the problem for all to see. 2 minutes of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on division by a two-digit number. It encourages students to think about the numbers in a computation problem and to rely on what they know about numbers in base-ten, patterns, division with remainders, and the relationship between multiplication and division to mentally find quotients. To divide larger numbers prompts students to look for and make use of structure (MP7).

Notice how students handle a remainder in a problem, which may depend on their prior experiences with division. When students begin finding unit price, they will need to be able to interpret non-whole-number quotients in either decimal or fraction form.

Launch

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 quotient mentally.

  • 24÷1224 \div 12
  • 6÷126 \div 12
  • 30÷1230 \div 12
  • 246÷12246 \div 12

Sample Response

  • 2. Sample reasoning: 2⋅12=242 \boldcdot 12 = 24
  • 12\frac{1}{2} or 0.5. Sample reasoning: 
    • 12⋅12=6\frac{1}{2} \boldcdot 12 = 6
    • Two groups of 6 make 12, so one group of 6 is half of 12.  
  • 2122 \frac{1}{2} or 2.5. Sample reasoning: There are 2 groups of 12 in 24, and 12\frac{1}{2} group of 12 in 6. Thirty is 24+624 + 6, so there are 2+122 + \frac{1}{2}, or 2122\frac{1}{2}, groups of 12 in 30.
  • 201220\frac{1}{2} or 20.5. Sample reasoning: 
    • 20⋅12=24020 \boldcdot 12 = 240 and 12⋅12=6\frac{1}{2} \boldcdot 12 = 6, so 2012⋅12=24620\frac{1}{2} \boldcdot 12 = 246.
    • 120÷12120 \div 12 is 10, so 240÷12240 \div 12 is 20. Adding 6÷126 \div 12, which is 12\frac{1}{2}, gives 201220 \frac{1}{2}.
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 have 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?”

If students express the result of the last two divisions with “2 with a remainder of 6” and “20 with a remainder of 6,” respectively, ask them if the 6 could be divided by 12, or remind them that they divided 6 by 12 in a preceding problem.

At the end of discussion, if time permits, ask a few students to share a story problem or context that 246÷12=20.5246\div 12 = 20.5 could represent.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because….” or “I noticed _____ 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
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.
  • 5.NF.3·Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. <em>For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?</em>
  • 5.NF.B.3·Interpret a fraction as division of the numerator by the denominator <span class="math">\((a/b = a \div b)\)</span>. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. <span>For example, interpret <span class="math">\(3/4\)</span> as the result of dividing <span class="math">\(3\)</span> by <span class="math">\(4\)</span>, noting that <span class="math">\(3/4\)</span> multiplied by <span class="math">\(4\)</span> equals <span class="math">\(3\)</span>, and that when <span class="math">\(3\)</span> wholes are shared equally among <span class="math">\(4\)</span> people each person has a share of size <span class="math">\(3/4\)</span>. If <span class="math">\(9\)</span> people want to share a <span class="math">\(50\)</span>-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?</span>

10 min

15 min