Solving Problems Involving Decimals

10 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time, followed by a whole-class discussion for each problem.

Narrative

This Math Talk focuses on division of a decimal. It encourages students to think about the reasonableness of a quotient by looking closely at the values of the dividend and divisor and to rely on what they know about base-ten numbers to mentally solve problems. The reasoning elicited here will be helpful later in the lesson when students solve contextual problems involving division.

To find the value of the last two expressions, students need to look for and make use of structure (MP7). In explaining their reasoning, 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

Decide mentally which value is the best estimate for each expression.

  • 124.3÷2124.3 \div 2
    • 6
    • 60
    • 600
  • 124.3÷24124.3 \div 24
    • 0.5
    • 5
    • 50
  • 12.43÷8012.43 \div 80
    • 0.15
    • 1.5
    • 15
  • 1.243÷1.61.243 \div 1.6
    • 0.075
    • 0.75
    • 7.5
       

Sample Response

  • 60. Sample reasoning:
    • 124.3 is close to 120, and 120÷2=60120 \div 2 = 60.
    • 2⋅62 \boldcdot 6 is only 12, which is much less than 124.3, and 2⋅6002 \boldcdot 600 is 1,200, which is much greater than 124.3. 2⋅602 \boldcdot 60 is 120, which is very close to 124.3.
  • 5. Sample reasoning:
    • 124.3 is close to 125 and 24 is close to 25. 125÷25125 \div 25 is 5. 
    • There are more than 1 group of 24 and fewer than 10 groups of 24 in 124, so the quotient is between 1 and 10.
    • The divisor 24 is 12 times the divisor in the first problem, so the quotient is one-twelfth of the first quotient. One-twelfth of 60 is 5.
  • 0.15. Sample reasoning:
    • The quotient is less than 1 because there is less than 1 group of 80 in 12.43.
    • 12 is about 1.5 times 8, so 1.2 is about 0.15 times 8.
  • 0.75. Sample reasoning:
    • The divisor is slightly greater than the dividend, so the quotient is close to 1.
    • 1.243 is close to 1.2, and 1.2÷1.61.2 \div 1.6 is equivalent to 12÷1612 \div 16, which is 1216\frac{12}{16}, or 34\frac{3}{4}.
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?”

If time permits, ask students if the actual value of each expression would be greater than or less than their estimate, and ask them to explain how they know.

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
Standards
Building On
  • 5.OA.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <em>For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.</em>
  • 5.OA.A.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <span>For example, express the calculation “add <span class="math">\(8\)</span> and <span class="math">\(7\)</span>, then multiply by <span class="math">\(2\)</span>” as <span class="math">\(2 \times (8 + 7)\)</span>. Recognize that <span class="math">\(3 \times (18932 + 921)\)</span> is three times as large as <span class="math">\(18932 + 921\)</span>, without having to calculate the indicated sum or product.</span>

25 min